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NOAA Technical Report EDS 1 9

Separation of Mixed
Data Sets into
Homogeneous Sets

Washington D.C.
January 1977

U.S. DEPARTMENT OF COMMERCE

National Oceanic and Atmospheric Administration

Environmental Data Service

NOAA TECHNICAL REPORTS

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relations into proper historical and statistical perspective and to provide a basis for assessing
changes in the natural environment brought about by man's activities.

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of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22151. Price: $3.00 paper copy;
$1.45 microfiche. When available, order by accession number shown in parentheses.

ESSA Technical Reports

EDS 1 Upper Wind Statistics of the Northern Western Hemisphere. Harold L. Crutcher and Don K. Halli-
gan, April 1967. (PB-174-921)

Direct and Inverse Tables of the Gamma Distribution. H. C. S. Thom, April 1968. (PB-178-32b)

Standard Deviation of Monthly Averane Temperature. H. C. S. Thom, April 1968. (PB-178-309)

Prediction of Movement and Intensity of Tropical Storms Over the Indian Seas During the October
to December Season. P. Jagannathan and H. L. Crutcher, May 1968. (PB-178-497)

An Application of the Ramma Distribution Function to Indian Rainfall. D. A. Mooley and H. L.
Crutcher, August 1968. (PB-180-056)

Quantiles of Monthly Precioitation for Selected Stations in the Contiguous United States. H. C.
S. Thom and Ida B. Vestal, August 1968. (PB-180-057)

A Comparison of Radiosonde Temperatures at the 100- , 80-, 50-, and 30-mb Levels. Harold L.
Crutcher and Frank T. Quinlan, August 1968. (PB-180-058)

EDS 8 Characteristics and Probabilities of Precioitation in China. Augustine Y. M. Yao, September
1969. (PB-188-420)

EDS 9 Markov Chain Models for Probabilities of Hot and Cool Days Seguences and Hot Spells in Nevada.
Clarence M. Sakamoto, March 1970. (PB-193-221)

NOAA Technical Renorts

EDS 10 BOMEX Temporary Archive Description of Available Data. Terry de la Moriniere, January 1972.
(COM- 72-50289)

EDS 11 A Note on a Gamma Distribution Computer Proaram and Graph Paper. Harold L. Crutcher, Gerald L.
Barger, and Grady F. McKay, April 1973. (COM-73-11401)

EDS 12 BOMEX Permanent Archive: Description of Data. Center for Experiment Design and Data Analysis,
May 1975.

EDS 13 Precipitation Analysis for BOMEX Period III. M. D. Hudlow and W. D. Scherer, September 1975.
(PB-246-870)

EDS 14 IFYGL Rawinsonde System: Description of Archived Data. Sandra M. Hoexter, May 1976.
(PB-258-057)

EDS 15 IFYGL Physical Data Collection System: Descriotion of Archived Data. Jack Foreman, September
1976.

(Continued on inside back cover)

EDS

2

EDS

3

EDS

4

EDS

5

EDS

6

EDS

7

■'WfMT Of *

"^'^^^J^^co^

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NOAA Technical Report EDS 1 9

Separation of Mixed
Data Sets into
■Homogeneous Sets

Harold L. Crutcher and Raymond L. Joiner
National Climatic Center
Asheville N.C.

January 1977

>. U.S. DEPARTMENT OF COMMERCE

^ Elliot L. Richardson, Secretary

o

^^ National Oceanic and Atmospheric Administration

Robert M. White, Administrator

Environmental Data Service

Thomas S. Austin, Director
Stock Number 003-019-00036-5 Price $2.45

ACKNOWLEDGMENTS

Appreciation is expressed to the many who have helped us in the prepara-
tion of this paper. Among these are the personnel of the National Climatic
Center's Science Advisory Staff, ADP Services Division, Audio-Visual Services
Section, and the Library Group. Specific acknowledgment is made to Miss Lisa
Green for preparation of the original typescript and to Mrs. Margaret Larabee
for the careful preparation of the final typescript.

Acknowledgment is made to Prof. E. S. Pearson and the Trustees of Biometrika
to use the data and format displayed in figures 2a and 2b. Acknowledgment is
made to University Microfilms, Ann Arbor, Michigan, for permission to repro-
duce or to modify figures as these appear on figures 4, 5, and 6.

Appreciation is expressed to the U. S. Navy and to Dr. John H. Wolfe of the
U. S. Navy Personnel Research and Development Center, San Diego, California,
to adapt and use his NORMIX and NORMAP computer programs and for his ever
ready response to written, telephonic, and personal visit requests.

Acknowledgment is made to the National Oceanic and Atmospheric Administra-
tion for permission to quote material from the Monthly Weather Review.

Acknowledgment is made also to Prof. A, Clifford Cohen, Jr. and to Mr. Lee
Falls for permission to use their computer program which they furnished and
which was adapted to separate two mixed univariate normal distributions.

Appreciation also is expressed to the staff at EDS's Environmental Science
Information Center, in particular to Mr. Patrick McHugh, for the care given
to the editing of this paper.

Mention of a commercial company or product does not constitute an endorse-
ment by the NOAA Environmental Data Service. Use for publicity or advertising
purposes of information from this publication concerning proprietary products
or the tests of such products is not authorized.

n

CONTENTS

Acknowledgments ii

Symbols used in this report xiii

Abstract 1

1. Introduction 1

2. Regression techniques 6

3. Discriminant techniques 10

3.1 Discriminant functions 10

3.2 Factor analysis 15

3.3 Principal component analysis 16

3.4 Multivariate statistical methods 17

3.5 Multivariate logit 18

4. Clustering 19

5. Transformations 21

6. Separation of mixtures 23

6.1 Univariate mixtures 23

6.2 Bivariate mixtures 23

6.3 Multivariate mixtures 24

7. Wolfe - NORMIX 360 computer program 26

7.1 Maximum likelihood estimation 26

7.2 Initial estimation 27

7.3 Significance tests for number of clusters 28

7 . 4 Strategy of use 28

7.5 Usage 28

7.5.1 Storage requirements 28

7.5.2 Restrictions 28

7.5.3 Error messages 28

7.5.4 Input deck 29

7.5.5 Validation examples 29

8. Examples 31

8.1 Introduction 31

8.2 Brief descriptions of data and locations 31

8.2.1 Land-sea breeze data 31

8.2.2 Tropical stratospheric wind data 32

8.2.3 Mid-latitude troDospheric wind data 33

8.2.4 Mountain pass wind data 34

8.2.5 Marine surface data 34

8.2.6 Radiosonde and rawinsonde data 35

m

8.3 Selected data 35

8.3.1 Land-sea breeze data set 35

8.3.1.1 Input information 35

8.3.1.2 Tables - output information 36

8.3.1.3 Figures and discussion '. 36

8.3.2 Tropical stratospheric wind data set 41

8.3.2.1 Input information 41

8.3.2.2 Tables for wind configurations (1954-1964) ... 42

8.3.2.3 Figures and discussion for wind configurations
(1954-1964) 57

8.3.2.4 Tables and discussions for height, temperature

and wind configuration (1957-1967) 76

8.3.3 Mid-latitude tropospheric wind data set 82

8.3.3.1 Input information 82

8.3.3.2 Tables 82

8.3.3.3 Figures and discussion 104

8.3.4 Mountain pass wind data set Ill

8.3.4.1 Input information Ill

8.3.4.2 Tables Ill

8.3.4.3 Figures and discussion 116

8.3.5 Marine surface data set 121

8.3.5.1 Input information 121

8.3.5.2 Tables - output information 121

8.3.5.3 Figures and discussion 129

8.3.6 Radiosonde and rawinsonde data set 134

8.3.6.1 Input information 134

8.3.6.2 Tables and discussion 134

9. Multivariate quality assurance and control 137

10. Prediction 155

Summary 1 56

References 1 57

Author i ndex 1 66

IV

FIGURES

Figure 1. A mixed set of trivariate distributions showing clusters

or modes of varying sizes and shapes 4

Figure 2a. Regression of sons' statures on the fathers' stature 7

Figure 2b. Regression of sister's span for given forearm of brother.. 7

Figure 3. General schematic of points and clusters with implied

correlations, r 8

Figure 4. Schematic illustrations of discrimination for two

cl usters 11

Figure 5. Schematic illustration of discrimination in two-
dimensional form with projection onto plane and onto
one axis for linear discrimination 12

Figure 6. Schematic representation of the discriminate function
when the two bivariate populations, Pi and P2> have
unequal means but equal variances and covariances 13

Figure 7. Schematic "D" illustration of constellations of annual

temperature climates for North American stations 14

Figure 8. San Juan, Puerto Rico, surface wind distributions; period
of record, October 1-31, 1955, October 1-3, 1956, hours
0600 and 0800 and 1200-1400 local standard time; n = 200,
100 from each period; separation shown for two and three
types with covariances assumed equal and then unequal 40

Figure 9. Canton Island, U.S.A. and U.K., upper wind distribution
plots; period of record, the months of July 1954-1964;
pressure levels (a) 50-, (b) 30-, (c) 20-, and (d) 10-mb.. 59

Figure 10. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; wind plot shown in figure 9; separation
shown for two types, assumption of unequal covariances
matrices 61

Figure 11. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; wind plot shown in figure 9; separation
shown for three types, assumption of equal covariance
matrices 62

Figure 12. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure level, 30-mb;
n = 244; wind plot shown in figure 9; separation shown
for four types, assumption of equal covariance matrices... 63

V

Figure 13. Plot of Discriminant Functions 1 versus 2 for Canton
Island, U.S.A. and U.K.; July winds shown in figure 9
for winds shown in figure 11, based on assumption of
equal covariance matrices; period of record 1954-1964;
functions 1 versus 2 are plotted for (a) three types
at 50-mb, (b) three types at 30-mb, (c) four types at
30-mb , and (d ) three types at 20-mb 64

Figure 14. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; n = 263, 244, and 162, respectively;
assumption of unequal covariance matrices: distributions
are shown for (a) total for the three levels, (b) type 1
for the three levels, and (c) type 2 for the three levels. 68

Figure 15. Canton Island, U,S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; n = 263, 244, and 162, respectively,
separation shows two types, assumption of unequal co-
variance matrices; distributions are shown for (a) total
and two 2 types at 50-mb, (b) total and 2 types at 30-mb,
and (c) total and 2 types at 20-mb 69

Figure 16. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; n = 263, 244, and 162, respectively;
separation shows 3 types, assumption of unequal covari-
ance matrices; distributions are shown for (a) three types
at 50-mb, (b) three types at 30-mb, and (c) three types
at 20-mb 70

Figure 17. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-
and 30-mb; n = 263 and 244, respectively; separation
shows 4 types, assumption of unequal covariance matrices;
distributions are shown for (a) four types at 50-mb and
(b) four types at 30-mb 71

Figure 18. Canton Island, U.S.A. and U.K., upper wind distribution
plots; period of record, January 1954-1964; pressure
levels (a) 50-, (b) 30-, (c) 20-, and (d) 10-mb 72

Figure 19. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, January 1954-1964; pressure level,
50-mb; wind plot shown in figure 18; separation shows for
two and three types with assumption of equal then unequal
covariance matrices • 74

Figure 20. Bivariate distributions of winds at Rantoul , Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels.
Two cluster types (1 and 2) are assumed in the total mixed
observed distribution (0.171 + 0.829 = 1.000) 105

vi

Figure 21. Bivariate distributions of winds at Rantoul , Illinois,
October 1950-1955 at the 700-, 500-, and 300-mb levels.
Three cluster types (1, 2, and 3) are assumed in the
total mixed observed distribution (0.182 + 0.183 +
0.635 = 1 .000) 107

Figure 22. Bivariate distributions of winds at Rantoul, Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels.
Four cluster types (1, 2, 3, and 4) are assumed in the
total mixed observed distribution (0.094 + 0.193 +
0.293 + 0.421-1.000) 109

Figure 23. Stampede Pass, Easton, Washington, U.S.A.; winds and

temperatures, December 1966-1970, showing breakdown of

winds only into groups 2 and 3 from group 1 (a) and

breakdown of wind- temperature combination into 2, 3,

and 4 groups (b, c, and d) 119

Figure 24. Selected area of North American chart, 1200Z, Monday,
December 3, 1968, NMC analysis. The star represents
the approximate position of Stampede Pass, Easton,
Washington 120

Figure 25. OSV "C" surface distribution of pressure, temperature,
dew point, and wind components, February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed
to be unequal. The 0.25 probability ellipses are
shown for the wind distribution. The total distribu-
tion and the breakout into two clusters are shown 131

Figure 26. OSV "C" surface distribution of pressure, temperature,
dew point, and wind components, February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed
to be unequal. The 0.25 probability ellipses are shown
for the wind distribution. The total distribution and
the breakout into three clusters are shown 132

Figure 27. OSV "C" surface distribution of pressure, temperature,
dew point, and wind components, February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed
to be unequal. The 0.25 probability ellipses are shown
for the wind distribution. The total distribution and
the breakout into four clusters are shown 133

Figure 28. Example of a two-tailed Gaussian filter operating on a
set of heterogeneous data to isolate, set aside, and
eliminate outlying data 141

Figure 29. Distribution of wind standardized components along the

two principal axes of the Canton Island, U.S.A. and U.K.,
July, 30 mb 143

vn

Figure 30a. Schematic illustration of a sample drawn from a homo-
geneous bivariate distribution contaminated by a lone
outlier and two groups of data. The result is a hetero-
geneous distribution. The ellipse shown is a theoretical
0.95 probability ellipse. The lone outlier will be
rejected as not being part of the homogeneous distri-
bution 144

Figure 30b. Schematic illustration of a sample drawn from a homo-
geneous bivariate distribution contaminated by two small
sets. The result is a heterogeneous sample. The lone
outlier of figure 30a has been eliminated as it did not
appear within the 0.95 probability ellipse. Here, the
two contaminating sets exist outside the 0.95 probability
ellipse of this figure and will be eliminated in figure
30c '. 145

Figure 30c. Schematic illustration of a sample drawn from a homo-
geneous bivariate distribution. It exists as a result
of the filtering action of the 0.95 probability ellipses
illustrated in figures 30a and 30b. Here, the 0.95
probability ellipse contains all sample data points of
the remaining group 146

Figure 31. Distribution of wind and temperature standardized com-
ponents along the three principal axis of the Stampede
Pass, Easton, Washington, December 1968-1970 data 147

Figure 32. Distribution of wind, temperature, height, and dew
point standardized components along the 20 principal
axes of the Balboa, C.Z., July data. Four levels are
involved: surface, 850-, 700-, and 500-mb 149

VI n

TABLES

Table 1. Surface wind statistics for San Juan, Puerto Rico,

October 1-31, 1955, and October 1-3, 1956, 0600-0800

and 1200-1400 l.s.t. The assumption is that the

covariance matrices are the same in any breakdown 37

Table 2. Surface wind statistics for San Juan, Puerto Rico,

October 1-31, 1955, and October 1-3, 1956, 0600-0800

and 1200-1400 l.s.t. The assumption is that the

covariance matrices are not equal 38

Table 3. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 50-mb.
Sample size is 263. The assumption is that the
covariance matrices are the same 43

Table 4. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 30-mb.
Sample size is 244. The assumption is that the
covariance matrices are the same 44

Table 5. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 20-mb.
Sample size is 162. The assumption is that the
covariance matrices are the same 45

Table 6. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 50-mb.
Sample size is 263. The assumption is that the
covariance matrices are not the same.... 46

Table 7. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 30-mb.
Sample size is 244. The assumption is that the
covariance matrices are not the same 49

Table 8. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 20-mb.
Sample size is 162. The assumption is that the
covariance matrices are not the same. 52

Table 9. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
January 1954-1964. The pressure level is 50-mb.
Sample size is 168. The assumption is that the
covariance matrices are the same 54

ix

Table 10. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
January 1954-1964. The pressure level is 50-mb.
Sample size is 168. The assumption is that the
covariance matrices are unequal , 55

Table 11. January upper air statistics for Canton Island,

U.S.A. and U.K. The period of record is the months

of January during 1957-1967. The pressure level is

30-mb. The sample size is 244. The assumption is

that the covariance matrices are not the same 78

Table 12. January correlation coefficients for data shown in

tabl e 11 79

Table 13. July upper air statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1957-1967. The pressure level is 30-mb.
Sample size is 244. The assumption is that the
covariance matrices are not the same 80

Table 14. July correlation coefficients for data shown in

table 13 81

Table 15. A multivariate (6) set of Rantoul , Illinois,

October 1950-55, upper wind components, zonal and

meridional, at the 700-, 500-, and 300-mb levels 84

Table 16. Separation of a multivariate (6) set of Rantoul,
Illinois, October 1950-55, upper wind components,
zonal and meridional mixed distribution, at the
700- , 500- , and 300-mb levels into two separate
distributions 90

Table 17. Separation of a multivariate (6) set of Rantoul,
Illinois, October 1950-55, upper wind components,
zonal and meridional mixed distribution, at the
700- , 500- , and 300-mb levels into three distinct
distributions 94

Table 18. Separation of a multivariate (6) set of Rantoul,
Illinois, October 1950-55, upper wind components,
zonal and meridional mixed distribution, at the
700-, 500-, and 300-mb levels into four distributions 98

Table 19. Surface wind statistics for Stampede Pass, Easton, WA,
U.S.A. The period of record is the month of December
1966-1970. The sample size is 310 taken 155 from each
of the local standard time hours 0700 and 1300. The
assumption is that the covariance matrices are not the
same 112

Table 20.

Table 21

Table 22,

Table 23,

Table 24.

Surface temperature and wind statistics for Stampede

Pass, Easton, WA, U.S.A. The period of record is

the month of December 1966-1970. The sample size is

310 taken 155 from each of the local standard time

hours 0700 and 1300. The assumption is that the

covariance matrices are not the same 113

Marine observations from Ocean Station C. Februaries

12Z 1964 through 1972. Sample size 1s 251. Number

of variables is 5. Number of types is 2 122

Marine observations from Ocean Station C. Februaries

12Z 1964 through 1972. Sample size is 251. Number

of variables is 5. Number of types is 3 124

Marine observations from Ocean Station C. Februaries

12Z 1964 through 1972. Sample size is 251. Number

of variables is 5. Number of types is 4 126

Means and standard deviations for the total set and

clusters 1 and 2 of the data for Balboa, C.Z. These

data are the pressure (or height), temperature, dew

point, and the u and v components of the wind at the

surface, 850-, 700-, and 500-mb levels. The dimensions

are 20. Equal covariance matrices are assumed for types

1 and 2 136

Table 25,

Separation of standardized transformed components along
the major axis of the distribution of the Canton Island:
U.S.A. and U.K., winds at the 30-mb level during the
Julys 1954-1964. The sample size is 244. There are no
dimensions in terms of units. The assumption is that
the variances are not the same ,

153

Table 26.

Separation of standardized transformed components along
the major and minor axis of the distribution of the
Canton Island, U.S.A. and U.K., winds at the 30-mb level
during the Julys 1954-1964, The sample size is 244.
There are no dimensions in terms of units. The assumption
is that the variances are not the same.. 154

XI

Symbols used in this report

d difference; deviation

d.f. degrees of freedom

f function

ft. feet

gdkm geodynamic kilometer

i subscript or superscript

j subscript or superscript

k kth point in a sample; number of clusters; kth cluster

km kilometer

m mth item; meter

mb pressure in millibars

mi. miles

n nth item; number in a sample

r sample correlation coefficient

s sample standard deviation; second; cluster

s^ sample variance

t Student's "t"

X variate

X' variate transpose

y variate

C covariance matrix; Celsius

F function

G.C.T. Greenwich Civil Time

P probability

P(S|X. ) probability of membership of X. in the cluster s

R correlation matrix

|R| determinant of the correlation matrix R

S type; cluster type

X observed value of variate

X mean of variate X

X|^ vector of observations for the kth point in the sample

Y observed value of variate

Y mean of variate Y

xm

a alpha; proportionality factor (^5); probability level of rejection
for the null hypothesis

A lambda hat; mixing proportion for type cluster s

A lambda; the diagonal matrix of eigenvectors

y mu; population mean

y mu hat; mean vector for cluster s

IT pi

p rho; population correlation

a sigma; population standard deviation

a^ population variance

a sigma hat; covariance matrix for cluster s

z sigma; summation

ij; psi

hat (caret)

equal to
^ approximately

overbar; averaging process

transpose

XIV

SEPARATION OF MIXED DATA SETS
INTO HOMOGENEOUS SETS

Harold L. Crutcher and Raymond L. Joiner

National Climatic Center

Environmental Data Service, NOAA

Asheville, N.C.

ABSTRACT. In any study, the collection, processing, and
storage of data are important. Whether the data are clean,
biased or contaminated is also important. Pollution or
adulteration of data confuse the investigator.

Data do not necessarily fall into neatly packaged boxes or
groups. Usually the data sets are mixtures of several
types of phenomena. Some of these are basically determin-
istic in nature while others are not.

This paper illustrates the use of a clustering technique
to separate mixed data sets into subsets which exhibit
group characteristics. The investigator then assesses the
relative importance of the subsets, the nature of the sub-
sets, and perhaps makes an assumption as to whether a
particular subset is biased, contaminated, or adulterated.
That is, an assessment of the quality of the data may be
made.

The techniques are applicable to any data set which is
multivariate normal. Here, they are applied to weather
data subsets, (1) land-sea breeze, (2) tropical strato-
spheric winds, (3) mid-latitude tropospheric winds, (4)
mountain pass winds and temperatures, (5) surface marine
weather temperatures, dew points and winds, and (6) radio-
sonde observation of heights, winds, temperatures, and dew
points.

1. INTRODUCTION

Prehistoric man differentiated between the good and the harmful, between
winter and summer, between drought and floods, and among many variable
factors affecting him.

In the real world, the experienced hunter knew the differences between the
bear, boar, deer, and turkey. His senses of sight, smell, and hearing aided
him when he could not see the animal, but he could sense its presence. When
he had some models in mind, he drew symbols or wrote words for the benefit of
the inexperienced. He could even provide measurements of a sort. At this
stage of the game, he began to deal more with the abstract.

Later, man attempted to record experiences, his thoughts, and his aspira-
tions in pictographs and monuments. (Even today, artists present forms in

1

symbolic representation or as configurative concepts.) Undoubtedly, many
numbering systems had also been developed and were then lost in antiquity,
for there are some which remain indecipherable today.

The above examples may be generalized to any field. Clear-cut numerical
descriptions and configurations remain important whether the field be sociol-
ogy, psychology, geophysics or medicine, four of an infinite number of fields.

In this paper we illustrate some of the techniques used to differentiate
groups within sets of given measurements. Hopefully, the measurements are
both accurate and precise. Also, hopefully, the measurements obtained and
used are those which will provide good differentiation bases.

As Friedman and Rubin (1967) quote from Bose and Roy (1938): "The problems
of discrimination and classification are insistent in sciences."

Once we enter the realm of measurements and numbers in multidimensional
space, assumptions and decisions are made as to the better or best character-
istics. The metrics that can be used are many. Those chosen ought to provide
the most useful differentiation possible.

There are many techniques and procedures used to differentiate groups.
These usually involve some measures of central tendencies within the groups,
some measures of differences of these central tendencies, variability within
and among the groups, group shapes, and scales, etc. As more is known about
the normal distribution than other distributions, it is wise, wherever possi-
ble, to transform non-normally distributed data sets to data sets which may
be approximated by the normal distribution. Transformations have been of
interest for many years. In fact, these are represented in the change of
base in many counting systems. For example, the logarithm transformation
changes a zero bounded positively skewed distribution to an unbounded distri-
bution at both ends where the values more distant from one are scaled down-
ward faster than those nearer one.

If the various features of a data set are each transformed to normal or
near normal distributions, then the combined features may be multivariate
normal. A multivariate normal distribution has normal marginal distributions.
However, the fact that the marginal distributions are normal does not assure
multivariate normality. If multivariate normality is assumed, then proba-
bilistic statements can be made. Transformation will be discussed in greater
detail later in the paper.

If multivariate normality is assumed and the distribution has one centroid,
i.e., it is unimodal , then multivariate regression techniques can be used to
produce forecast equations. If the distributions are normally distributed
in the multivariate sense but the total set is multimodal, then ordinary
linear regression techniques will not serve. Regression equations for clus-
ters must be developed and the better sets of regression equations clustered
and examined. The unique or best equation or set of equations can be used.
If there is a best equation, it is unique.

Figure 1 illustrates a multimodal trivariate distribution. The various
clusters take various ellipsoidal forms such as spheres, ellipsoids, and
disks. Each one of these is trivariate normal. Each one represents a
centroid or grouping of characteristics.

Some may be equal in all directions, such as in the spheres. Some may be
equal in two directions but not the third, such as in the football- and disk-
shaped ellipsoids. In others the distribution may be unequal in all direc-
tions. In figure 1, no scales are indicated as the illustration is for
concept only. The illustration also could be considered as a higher dimen-
sional ensemble projected onto three dimensions. When multimodal features
are evident, and if one wishes to study the modes or clusters, then tech-
niques other than regression are required to separate the data into appro-
priate subsets.

There are many techniques to separate distributions into parts which can be
studied individually. Some of these are:

(a) discriminant function analysis

(b) factor analysis

(c) principal cluster analysis

(d) dendritic (tree) analysis

(e) cluster analysis

(f) clumping

(g) numerical taxonomy

(h) unsupervised pattern recognition
(i) typology

Some of these are essentially the same.

Mixtures always present problems when they must be separated. Characteris-
tics may or may not be so noticeable that classification and discrimination
can be made. A mixed herd of cattle, sheep, goats, and horses may be easily
separated though the sheep and goats may sometimes present a few problems.
The herdsman, the separator, or the investigator must have a clear picture in
his mind, i.e., a model which includes the necessary characterization(s) of
the populations in which he is interested. If he doesn't have his senses of
sight, smell, hearing and touch to guide him (for these provide him explicit
models) and he has only some measurements of the mixture provided to him, he
is in a quandry. He no longer has an explicit model. He has to start some-
where. Good (1965) discusses the philosophical problem of deciding what can
be the best beginning in the problem of classification and discrimination.
First of all, t>'e investigator decides to accept the characterization of each
object by a set of measurements. He believes that there should be some cate-
gories or sub-categories which will be helpful in distinguishing group char-
acteristics. Explicitness is lost and there is no external criterion with
which to define the categories. An internal criterion (Good, 1965) would be
acceptable. That is, the data themselves may suggest "natural categories."
The word "suggest" is necessary, for a different beginning in the treatment
of the data may lead to slightly different categories. This item will be
treated in more detail later.

Figure 1 A mixed set of trivariate distributions showing clusters or modes
of varying sizes and shapes. These are viewed from four different
points in 3-space. In each view, the three axes and a suggested
line of best fit are indicated.

The next few sections take the reader through a short discussion of regres-
sion techniques and through some of the various techniques used to provide
classification and discrimination within heterogeneous mixtures.

2. REGRESSION TECHNIQUES

Inevitably, due to inherent laziness or an inherent desire to get to a goal
with the least expenditure of energy, physical or mental, one attempts to read
relationships into sets of experiences or of data. This. is a canon of science.
For example, if one knows that a certain thing will happen provided that some-
thing else specific is done, then there is a clear-cut and obvious relation-
ship between the reaction and the prior action. Though the converse may not
be true, it is disregarded here.

Galton (1889) and Snedecor and Cochran (1967) noted the height of sons as
related to the heights of the fathers. With the heights of the fathers as
one set of data and the heights of the sons as a second set of data, Galton
plotted the heights of the fathers against the heights of the sons, respec-
tively. He noted that tall fathers did not always produce as tall or taller
sons but that the height of a son seemed to be oetv/een the height of the
father and the mean height of the group. That is, the height of a son re-
gressed towards the group norm. The line of best fit for the data set was
then and since then labeled the line of regression for any line relationship
between data sets. Pearson and Lee (1903), Galton's associates, collected a
set of data (more than a thousand) of stature, cubit, and span in family
groups.

Figure 2a shows the regression of sons' statures on the fathers' statures.
Please note that the heights of the sons of short fathers also tend (or re-
gress) toward the mean. This figure is taken from Pearson's and Lee's data
as illustrated, but is also shown by Snedecor and Cochran (1967). The data
are scaled in the metric system here. Also, note that the data scatter uni-
formly along the line and do not cluster.

Figure 2b illustrates the relationship between a sister's span for a given
forearm length of her brother. Both figures are adapted from Pearson and Lee
with the kind permission of the Trustees of Biometrika.

Figure 3a illustrates schematically the correlation at one point. The
correlation may be said to be perfect on the one hand, but also it can be
said to be indeterminant.

Figure 3b illustrates the correlation between two points. The correlation
may be said to be perfect on the one hand but for the space in between the
points, on the line connecting the points, it may be said to be indeterminant
or even zero.

Figure 3c illustrates a correlation of one with three col linear data.

Figure 3d illustrates the case of two clusters each with zero correlation,
one superposed on the other.

Figure 3e illustrates the case of two clusters shown in figure 3d where the
clusters are slightly separated with cluster 2 moving away on a line of 45
degrees. The cc'^relation coefficient is something greater than zero but much
less than one; .e., 0<r<<l .

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Figure 3f illustrates the case of two clusters shown in figure 3d where the
clusters are separated still further with cluster 2 moving farther away on
the 45 degree line.

Figure 3g illustrates the case of two superposed clusters. One with a
correlation of plus one, the other with a correlation of minus one. The
total cluster has a correlation of zero.

Figure 3h illustrates the case of two clusters one with a correlation of
plus one, the other with a correlation of minus one. The second cluster is
moving on a 45-degree line from the first. The distance between cluster
centroids is the same as in figure 3c. The group correlation is greater than
zero but less than that shown in figure 3e.

Figure 3i illustrates the case of the two clusters shown in figure 3g
moving along the 45-degree line farther than shown in figure 3h. The corre-
lation is increasing and the coefficient approaches one but not as rapidly as
in figure 3f as the distance between the centroids increases.

The above discussion implies that there may be two or more clusters in any
data ensemble. The scatter may be more in one cluster than in another. Also,
the internal correlation (or dispersion) within each cluster may be the same
or may be different from the other clusters. The total correlation attained
thus may be more of the relationship among the clusters than points. As the
distance between the centroids increases, the clusters are more like singleton
points in the regression analyses. If there is a clustering, i.e., effec-
tively a dearth of observations between groups either real or simply unob-
served, then regression analysis whether linear or non-linear will fail.
Please refer again to figure 1 which represents clusters in 3-space or clus-
ters in n-space projected into 3-space. The representation here is multi-
normal. This may not be the case sometimes and some clusters or all clusters
may be eccentric in shape such as eggs, starch grains, or bivalve shells.

If the variates are not all normally distributed, it is assumed that the
user will transform the variates to normal variates. It is helpful to know
by means of the Central Limit Theorem that, though individual data character-
istics may not be normally distributed, linear functions of these tend to be
normally distributed. Also, it is assumed that the user will extract the
deterministic part of the variables wherever possible. Therefore, the prob-
lems of nonlinear regression are not discussed here.

The point of the entire discussion above is simply that linear regression
analysis ought to be used only with a unimodal univariate or multivariate
distribution. Linear predictor equations developed in linear regression
models then become more useful and accurate.

Gupta and Sobel -(1962) consider this problem. Aversen and McCabe (1975)
present some subset selection problems for variances associated with applica-
tions to regression analysis.

3. DISCRIMINANT TECHNIQUES

3.1 Discriminant Functions

Many workers realized the problems induced by heterogeneous or mixed dis-
tributions. Among the first to attack the problem in a systematic mathemati-
cal treatment was Pearson (1894, 1901) in work on univariate distributions.
Many others also have worked on this problem.

Barnard (1935) and Fisher (1936) may be considered to have first attacked
the problem of classification with discrimination techniques, though the
problems of classification had involved many other workers up to that time.

Let us look at a two cluster mixture from the viewpoint of separation or
discrimination with subsequent rules for classification. Figure 4 shows an
assemblage composed of two clusters. The clusters are shown first within a
dashed circle with no axes chosen (a). Suppose that the measurements are
made in terms of the x-axis drawn horizontally (b).

Projections of the cluster points are on the x-axis and in (d) on the y-
axis. Visually there is separation in the (x, y) space or two space. This
separation can also be shown in one space, i.e., linearly. In one space or
one dimension chosen first on the x-axis, there is mixture of the projections.
In (c) where the x-axis is rotated through an angle to x", there is some
separation but two points indicate some mixing. In (d) where the rotation
has been carried through 90 degrees so that the x' axis is equivalent to the
former y-axis, separation is complete though perhaps not the optimum. There
is some angle of rotation which will produce a major separation between the
clusters and a minimum variance within the clusters. The computed linear
function which describes the above line after rotation is called the linear
discriminant function. Brown (1947) applies these techniques to establish
the discriminating procedures for azotobacter, the nitrogen-fixing bacteria.
Smith (1947) provides some discrimination examples. Crutcher (1960) applies
the techniques developed by Rao (1950) to the annual march of temperature and
rainfall climates in the United States. Figure 5 (adapted from Crutcher,
1960) shows a two cluster (bimodal) bivariate distribution with the two di-
mensions shown in three-dimensional form. These three-dimensional forms are
projected into two dimensions on the xz plane. Linear discrimination is
effected along the single axis pointing to the lower right (xy plane). Figure
6 illustrates the same idea with all projections being made onto the xy plane.
Tatsuoka (1971) presents similar ideas to illustrate geometrically how the
discriminant function operates.

Figure 7 (Crutcher, 1960) shows three constellations for monthly average
temperature galaxies where the basic variability of the constellations is
different. The covariance within a galaxy is the same where the circles pro-
vide a measure of individual cluster variances and the distances between
clusters is a measure of cluster variance. A point indicates one station
only. Miller (1962) applies the technique to weather prediction. Applica-
tions as indicated by Fix and Hodges (1952) ran into the hundreds.

10

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14

The works of Hotel ling (1931) and Mahalanobis (1936) are related closely
to Fisher's (1936) discriminant criterion. Study of these papers provides
excellent background. Hotelling describes these relationships in 1954.
Mahalanobis' work in a sense deals with a set of multi-dimensional data for
which the ensemble variance is reduced to one, i.e., the total ensemble is
standardized. Euclidean measurements then are used.

There are other techniques to categorize and classify data. All of these
require some transformation of data, the selection of those measurements
which provide the greatest amount of information, and the elimination of
those that provide the least if these may be termed unimportant. Usually,
their importance is assessed in how much they contribute to the overall
variability.

3.2 Factor Analysis

The basic mean structure and variance-covariance structure, i.e., the
matrices of means and covariances, are the bases for factor analysis and
principal component analysis. It is in the details and interpretation, some
complete and some incomplete, of the internal structures that the techniques
differ.

The term "factor analysis" comes from factoring procedures and techniques.
There are many variants each of which is a special case of the general method
of independent dimensional analysis (Tryon, 1959, Tryon and Bailey, 1970).
Briefly, as with the discriminant function analysis described previously, the
correlation (or covariance) matrix is the initial starting point. The methods
used to extract information from this matrix are many and varied. Some are
more complex than others. Some use weighting schemes either based on "a
priori" knowledge and experience or physical constraints and bases. Some
simply attempt to let the matrix itself determine the factors. All this leads
to some confusion and some competition among the proponents of the various
systems.

Multiple factor analysis with rotation to simple structure is the usual
procedure in factor analysis.

The purpose of factor analysis is to explain the matrix of covariances of a
multidimensional set by the least number of hypothetical factors. The corre-
lation matrix is used. First of all, the matrix is examined to determine
whether it is significantly different from zero (the identity matrix). If so,
the technique then identifies, extracts, and weights proportional amounts of
the correlation until the residual matrix is not significantly different from
zero. Factor analysis stems mainly from the initial work of Spearman (1904,
1926). Essentially, the technique serves to study the similarities in a set
of data.

Other pertinent references on factor analysis techniques are Thurstone

(1947), Kendall and Smith (1950), Cattell (1952), Bartlett (1953), Fruchter

(1954), Harman (1967), Sokal and Sneath (1963), Lawley and Maxwell (1963),
Mulaik (1972), and Anderson and Rubin (1956).

15

3.3 Principal Component Analysis

Karl Pearson (1901) first proposed an empirical method for the reduction
of a large body of data so that a maximum of variance could be extracted.
Hotel ling (1933) developed this method fully as the principle component
method. This under some conditions is identical to the discriminant function
[Kullback (1968), Anderson (1958), and Girshick (1936)].

For example, principal components according to Anderson (1958) are linear
combinations of random or statistical variables which have specified proper-
ties in terms of variances. The first principal component is the normalized
linear combination with maximum variance. In a swarm of points, each point
representing an n-component vector, the distribution may be ellipsoidal.
Unless the distribution is spherical, there will be an axis which is as long
or longer than any other axis. This axis is called the major or principal
axis. A plot of these points will reveal the ellipsoid. It is difficult to
represent such an ellipsoid in more than three dimensions. Measures of the
components may not directly reveal the ellipsoidal nature of the swarm. How-
ever, the variance-covariance or correlation matrix may be rotated in space
such that new axes are obtained where the components, projections, or linear
functions along these new axes are not correlated. The directions of these
axes are known variously as characteristic vectors, latent vectors, direction
cosines, or eigenvectors. If there are observations in n-dimensions, each
observation as an n-dimensional vector may be projected onto each of the n-
orthogonal (mutually uncorrelated) axes obtained above. The variability of
these components along each axis may be determined. These variances are known
respectively as characteristic roots, latent roots or eigenvalues. The sum
of these is the total variance or trace of the matrix. Their product is the
determinant. Each one divided by the total provides the proportional amount
of variance contributed by the components along each axis. The largest axis
is the major axis. The principal axes are those axes whose sum, from the
largest in sequence to the next largest, accounts for all or some preselected
or specified proportion, say 95 percent. It is the components along each axis
that are used. Thus, it is possible to reduce a large dimensional problem to
a small dimensional problem. It is also possible to select from the same data
a dimensional subset (n-1 or less) to eliminate that (those) dimension(s)
which contribute(s) little to the total overall variability.

There are tests to determine which principal components may be considered
to be significantly different from the others if indeed they are. Hotel ling
(1933) and Bartlett (1950, 1951) provide subtests. Mulaik (1972) presents a
good discussion of this problem. However, with all this, one must heed the
advice of Hotelling (1957) that there may be difficulties involved in neglect-
ing one principal axis even though it is the least in variance. Such an
omission may change radically the multiple correlation used in regression.
Only adequate testing will reveal whether an axis may be neglected.

In addition, the components along any one axis may be checked for evidence
of heterogeneity or mixtures. If multimodal ity is present on any principal
axis, then separation or clustering can be done for these data.

The principal component analysis is distinguished from the factor analysis

16

in that the principal component analysis studies the data structure from the
viewpoint of differences rather than similarities.

Any degeneracy which exists in a multivariate distribution is revealed in a
principal component analysis [Tatsuoka (1971)]. Therefore, principal compo-
nent analysis perhaps ought to be considered to be a first stage in factor
analysis, though this does not meet the views of the proponents and the oppo-
nents of the two techniques.

3.4 Multivariate Statistical Methods

As mentioned previously, the ways in which factor analyses are developed
and used lead to newer systems with their respective proponents and opponents.
This is so because the data sets must be discussed on the basis of each set.
From set to set the bases may be different. Hotel ling (1936a & b, 1957)
discusses the relations between two sets of variates, simplified calculation
of principal components, and then the newer multivariate statistical methods
to factor analysis.

Up to the time of Hotelling's (1957) paper, factor analyses of the usual
kinds were often inferior to other procedures. Unless the research worker
determines and uses (an) invariant statistic(s), the results always will be
difficult to assess. However, as in all investigative work of this nature,
these analyses may have only heuristic or suggestive value. Hypotheses may
be exposed which may be better tested by other methods.

In examination of the use of statistics in the various scientific fields,
it is readily apparent that each field develops its own names and "jargon"
for statistical terms within their specialized fields. Also, workers trying
to reach a common goal will independently develop similar techniques. Most
such techniques are beset with "nuisance" parameters induced by an arbitrarily
chosen statistic. As Hotelling (1957) points out, it is the invariance of the
multiple correlation coefficient as deduced by Fisher (1928) that illustrates
the possibility of eliminating the "nuisance" parameters which can only be-
cloud the issue. This puts a premium on the use of invariant statistics.

Student's (1925) "t" distribution is well known for its use to test the sig-
nificance of means and difference of means and to establish confidence inter-
vals with known probabilities. Its wery usefulness in the one dimensional case
led to much work towards generalizing this procedure to the multivariate case.

Hotelling (1933) in his work on principal components led to such a generali-
zation in the "T^" test. The value of "T^" is invariant under all non-
singular linear transformations among the variates. The trace or the sum of
the diagonal variances of a covariance matrix is invariant. The "T^" distri-
bution is a beta distribution or a variance ratio distribution.

The "D2" stability of Mahalanobis (1936) is closely allied to the "T^"
stability of Hotelling (1931). This also was an attempt to arrive at the use
of invariant statistics. Wilks (1932) considered generalization in multi-
variate analysis. Roy (1939, 1942a & b), Hsu (1938), Bartlett (1950), and
Bose and Roy (1938) did considerable work in the field of multivariate analysis

17

In the application of principal components in factor analysis, all compo-
nents must be used. The exclusion of even the smallest component may lead to
a complete change in the obtained functions to such an extent that interpreta-
tions or decisions may be quite erroneous.

Ridge regression is mentioned as an evolving technique which should receive
the attention of some readers [Hoerl (1962), Hoerl and Kennard (1970a, b), and
Bannerjee and Carr (1971)]. From the viewpoint of response surfaces, Davies
(1956), Draper (1963), and Myers (1971) may be consulted. Though these have
their impact on the problems of clustering and classification, they are not
discussed further.

3.5 Multivariate Logit

Distribution of quantal responses to drugs or poisons may be better de-
scribed by distributions other than the normal. The logistic curve is one of
these. As Kendall and Stuart (1968) state, "The Probit and Logit transforma-
tions of percentages, respectively to normal and logistic distribution devi-
ates, arise mainly in biological contexts and are discussed by Finney (1952).'
It has not found wide application in the geophysical field. Anderson (1958)
and Dempster (1973) discuss the Logit model. This would be most appropriate
if the mixture is composed of a set of variables one or more of which would
be considered fixed while the others are normal. This is beyond the scope of
the present paper as we here utilize only those distributions which from
experience are known to be approximated by the normal distribution. The use
of the multivariate Logit distribution mixture separation must be deferred to
later research. Hopefully, this will be within the next five years.

18

4. CLUSTERING

Grouping of individuals with similarities and separation of individuals with
dissimilarities is a continual process. This process is evident from the
smallest to the largest features in any ensemble whether it be in the universe
in the development and dissolution of galaxies, in the earth in its geologic
processes, or in life on earth in whatever form it may be. It is evident in
the abstract world, too. Readers of Aristotle and the followers of Hippo-
crates and Linnaeus will recognize attempts to order chaos in the sense of
clustering and differentiation, whether it be the assignment of some attribute
or the recognition of some measurable quality or quantity. Much has been done
and much has been written. The techniques of clustering and differentiation
are as varied as those who attempt to perform those functions. The logic and
the metrics may vary but the insistent theme is to group those that are alike
and separate those that are unlike. Each individual is, of course, an entity
in itself. A set of measurements on that individual undoubtedly constitutes
a group in itself. This much is acknowledged by any worker. The job is to
group those individuals together whose sets of measurements are not too far
different and to put aside those whose measurements are different at some
level of perception and thinking to another group or groups. Much sometimes
depends on the individual whose measurements are used as the starting point.
In a sense, those measurements of the first individual are in an "a priori"
sense weighed most heavily in the clustering process. Coalescence begins on
this individual.

Clustering is simply another term among the many in the general field of
taxonomy. Taxonomy is the scientific ordering and classification of informa-
tion. Taxonomic systems must be simplified representations of group char-
acteristics and all their interrelationships. Because these are most general
and therefore quite usable, perfection cannot be attained except in the case
of an individual. The main function of any such system is to reduce the
complex problem to a simpler problem, thus reducing the requirements of memory.
References, some of which have been given previously, are Sokal and Sneath
(1963), Tryon and Bailey (1970), Duda and Hart (1973), Fisher (1936), and
Anderson (1958). Hartigan (1975) offers a considerable number of clustering
algorithms.

Many terms are used in the grouping concept. Some are more suggestive of
the process than others. These terms depend on the field of endeavor and on
the basic background of the investigators. Some of the techniques are the
same though the words are different. Let us look at a few of them: clusters,
clumps, coalescence, condensations, aggregations, agglomerations, divisions,
similarities, cohesions, linkages, hierarchies, communal i ties, types, and
affinities. The classes obtained may be given names in the specialized fields.
In the biological sciences these are the families, genera, species, subspecies,
etc. In the field of the natural sciences, though not restricted to these
fields, these may be clusters, universes, constellations, galaxies, etc. In
the latter example, the order may be interchanged depending on the viewpoint
of the problem.

In any taxonomic problem dealing with measurements, there must be some way
to collect the like individuals together and to isolate, reject, and form new

19

groups with those which are most alike among the unlike. Some measures of
likeness or similarity must be developed and accepted. As individuals are
collected into small groups and as the smaller groups are collected into
larger and larger ensembles, a tracing of the procedures may be kept. As
diagrams or the concepts of the trace resemble some feature of the known
world, these may be called tree-diagrams (branch diagrams), root diagrams
(dendrodiagrams or dendritic diagrams), link diagrams, or hierarchical dia-
grams. These may be shown in two or three dimensions or left in abstract form,

Baker and Hubert (1975) consider procedures to measure the power of hier-
archical cluster analysis. This important feature in studying the various
techniques and their alternatives is not examined here in further detail.

The clustering procedures are allied closely to the previously discussed
techniques of factor analysis, principal component analysis, and discriminant
function analysis. Canonical analysis also may be used. The last is simply
the techniques which maximize the correlations among linear functions of the
data.

The specific clustering techniques used here are discussed in more detail
in Wolfe's (1971b) NORMIX program, section 7.

20

5. TRANSFORMATIONS

The multivariate normal distribution is only one of an infinite number of
useful multivariate distributions. The univariate normal distribution has
been exploited more than any other because the necessary statistical tools
have been developed. It is nearly so with the multivariate normal. If a
distribution is multivariate normal, then any of the subspace marginal dis-
tributions are normal. Though the normality of all marginal distributions
does not guarantee multivariate normality, such normality is a necessary
condition for multivariate normality.

If prior experience indicates that a certain measure is usually normally
distributed and tests imply non-normality, then one inference that can be
made is that the data set is mixed. It is precisely the thrust of this re-
port to separate such mixed distributions into their homogeneous parts. If
any marginal distribution is mixed, then certainly any higher order is mixed
or any lower order variate distribution is or may be mixed. If the set of
projections onto any principal axis is mixed (multimodal) and the set of
projections is a linear function of the multivariate set, then this set can
be used to establish estimates of the parameters of the various homogeneous
collectives.

With normality of distribution, the usual tests of significance or tests of
hypothesis may be made. Otherwise, decisions based on such tests may be
invalid or questionable. The robustness encountered, though, usually is such
that a lack of normality does not impair the decisions yery much. Far more
important perhaps is the practical significance of the decision.

If it is known that an unmixed marginal distribution is not normal, then
some transformation towards normality of that distribution should be sought.
A marginal distribution is a subset of the higher order distribution. If the
marginal distribution itself is not normal in the multivariate sense, then
the still lower order marginals or smaller subsets should be checked. It may
be that only one of the lowest order marginals (a one dimensional distribu-
tion) is the non-normal. If this is the case, then transformation of this
distribution should be sought.

The above does not imply that it is impossible to use the non-normal dis-
tributions. It does imply that the distributions must be normally distributed
or mixed normal in their distribution if the techniques discussed in this
paper can be used to provide valid results.

There may be some cases where appropriate normality cannot be achieved
through any transformation (Graybill, 1961, pg. 318).

For those distributions that can be normalized, the transformation process
is often carried forward in graphical procedures. Boehm (1974) uses this
procedure and has an electronic computer program to provide such trans-
normalization.

The methods of factor analysis, principal components, and discriminant
functions are techniques to linearize a multidimensional situation to a

21

single univariate situation. Hopefully the final distribution or distribu-
tions in the components, factors, or clusters will allow linearization within
these to the univariate problem.

There are many tests for univariate normality. Some of these (Graybill,
1961) are (1) likelihood ratio tests associated with transformation toward
normality, (2) skewness and kurtosis tests, (3) omnibus tests, and (4) normal
probability plots.

A useful transformation to improve normality is that proposed by Box and
Tidwell (1962) for estimating a shifted power transformation (x + ?)^ of a
single variable. This is a generalization of many tests developed through
the years [Tukey (1957) and Moore (1957)]. Simultaneous transformation of a
multiple set of variables has been the subject of much research. Andrews et
al. (1971) discuss the problem in more detail for the bivariate distribution.
Extensions to the multivariate case are indicated.

Transformations are not discussed further here as the climatological
examples used in this report do not require transformation to normality. The
clustering technique used here will be applied to other climatological and
geophysical problems. Many of the data distributions of the future study
will require transformation to normality. The two techniques described above
then will be treated in detail.

22

6. SEPARATION OF MIXTURES

6.1 Univariate Mixtures

Karl Pearson (1894), in his paper on "Contributions to the Mathematical
Theory of Evolution," gives the procedure to mathematically dissect a mixture
of two univariate normal distributions. In the general case the five param-
eters to be estimated are two means, two variances, and a proportionality
factor. It is necessary to find a particular solution of a ninth degree
polynomial equation, a nonic.

The following list is neither exhaustive nor the most important but it does
provide sufficient references for the reader to pursue the subject further:
Charlier (1906), Charlier and Wicksell (1924), Burrau (1934), Stromgren (1934),
Essenwanger (1954), Schneider-Carius and Essenwanger (1955), Cohen (1965),
and Cohen and Falls (1967). Essenwanger's work is applied to meteorology
while Cohen's and Fall's work provides electronic computer routines for dis-
section of heterogeneous mixtures.

Hald (1952) discusses the subject of heterogeneity in the univariate case.
Graphical as well as analytical techniques are discussed. The general case
may be written as

m

f(x) = E a^. f^. (x); la. = 1 ,

i = i

where

fi(x) = (2710^.)"'-^ exp{-[(X-vi^.)/a^.]^2}

Even for a mixture of two distributions the solution of the nonic is a big
task. Cohen and Falls (1967), using procedures developed by Charlier and
Wicksell (1924) and a modern electronic computer, provide procedures to
effect the requisite dissection and to obtain estimates of the parameters.
Discriminant criteria can then be developed to enable classification of a new
measure to one of the two groups or to classify each datum from the original
data set.

The computer program, kindly lent to the authors by Cohen and Falls, has
been used to dissect a mixed distribution in Section 9.

6.2 Bivariate Mixtures

Hartley (1959) provides techniques to separate two mixed bivariate normal
wind distributions. Crutcher and Clutter (1962) apply these to wind distri-
butions but encounter difficulty when the covariance matrices are singular or
near singular or when the variances or covariance matrices are assumed to be
different.

In the bivariate case of a unimodal distribution there are five parameters
to be estimated. These are the two means, the two variances, and the corre-
lation. When there is a mixture of two bivariate normal distributions, there

23

are eleven parameters to be estimated. These are the two means in each group,
the two variances in each group, the correlation in each group, and the mix-
ture parameter. If two assumptions are made, (1) the variances are all equal
one to each of the other three and (2) the correlation is. zero, then the
simplest mixture of two circular normal distributions is obtained. If the
component variances within each are equal, but unequal from one group to the
other, the more complicated mixture of two circular normal distributions with
different variances is obtained.

If there are "a priori" reasons to suspect that there are two groups and if
the assumption of circularity provides meaningful and useful results, then it
is suggested that Hartley's procedures be used. Computing time will be much
less than in the clustering program discussed here.

The authors know of no other computer program available specifically to take
a set of mixed bivariate normal data and dissect it even for the simplest case
of two subsets with unequal means but equal variances, component and vector,
and zero correlation. The cluster program used here will do the above as well
as the general case. But this clustering program is far from an optimum one
in terms of least time and cost. For the one or two time case, the clustering
program could be used. However, for repetitive processing of many many sets
of data it would be advisable to develop an optimized specific program based
on Hartley's procedures. This, the authors plan to do.

The mathematics may be written for the simplest bimodal bivariate circular
case as

2 2

f(x,y) = I a.f.(x,y); E a. = 1 ,
i=i i=i

where

fi(x,y) = {Zttc!)-' exp{-[(X-y^.)'/a^^ + (Y-yy.)2/ay^]2"'}

\

and the a. are the proportionality factors. In the slightly more general case,
fi(x,y) = [27Ta^a^,(l-p^p]~^^ exp{-[((X-y^.)Va^?)

6.3 Multivariate Mixtures

The general multivariate-multimodal case may be written as

n
f(x,y,...) = z a.f .(x,y,...); Za. = 1 ,

24

where

fi(x,y,...) = [(2TTa^.,o^.,...)|R|]"^' exp-(i|; ID ,

a is a proportionality factor, R is the correlation matrix, |R| is the
determinant of the correlation matrix, R. . is the respective cofactor, and

Sample estimates such as X, Y,..., s ., s .,..., r , ..., replace the popula-
tion parameters y ., y .,..., a . , a .,..., p ,..., where the bar represents
XI y I XI y I xy

an averaging process.

25

7. WOLFE - NORMIX 360 COMPUTER PROGRAM

Culminating over a decade of work, Wolfe (1971b) published his NORMIX com-
puter program. Earlier versions and other pertinent papers by Wolfe appeared
in 1965, 1967 a and b, 1968, 1970, and 1971a. He provides background dis-
cussion, the necessary programs, and an example. The example chosen is the
old standby example of Fisher (1936) which so many investigators use.

7.1 Maximum Likelihood Estimation

Unless "a priori" conditions indicate otherwise, the first hypothesis made
is that there is a specified number of types and the probability of a datum
being assigned to any one of the types is the same as to any other type. With
two groups, the probability of being assigned to one of the two types is 0.50.
With four groups, the probability of being assigned to one of the four types
is 0.250. With no "a priori" indication, a default option of the program then
provides equal mixing proportions as a first guess.

As shown by Wolfe (1970), the maximum likelihood estimates of the mixing
proportion (a ), mean (y^)^ and covariance (a ) of the cluster (type s) in a

mixture satisfy the following conditions:

n .
L = (1/n) z P(SIX. )
^ k=i ^

n
yg = (V(nx^)) E X,^ P(S|X^)

K~" 1

^3 = 0/{nX^)) I (Xj^-y^) (X,^-;^)^ P(S|X^)

where the prime indicates a transpose, X. is the vector of observations for
the kth point in the sample, and P(S|X. ) is the probability of membership of
X. in clusters and is equal to x times the ratio of the normal density of
type s at X. to the density of the mixture. In the special case where the
clusters have a common covariance matrix, a is a and may be written as

o = {^/{nX^)) E X^ X^ - y^ y^.

K~ 1

As the unknown parameters appear on both sides of the equation, it is
necessary to use an iterative process to solve the equations. For this
reason, if "a priori" estimates are used the iterative processing is
diminished. Computing costs therefore are much lower. Several sets of
values may satisfy the equations, and the results may depend on the starting
values for the iteration process. As indicated previously, a change in the
order of the input data will usually change the initial estimates obtained.

26

Local or relative maxima or saddle points may be obtained. The procedure
does not guarantee to find an absolute maximum. The initial estimates are
simply modified and improved upon. Bad initial estimates or diverging itera-
tions can cause strange estimates in the parameters such as negative variances
or singular correlation matrices. When this happens, re-initialization or
change of initial estimates may resolve the problem. The program also attempts
to resolve the problem of singular matrices by adding a small normal random
number to the diagonal terms.

7.2 Initial Estimation

It is assumed that the problem involves clustering within a multimodal
multivariate data set. All measurements are standardized. That is, estimates
of the individual component means and variances are computed. The squares of
the deviations from the means divided by the variances produce the square of
a standardized deviate. Thus, the set is reduced to a multimodal multivariate
set with a zero mean and a variance of one. This is the basic Mahalanobis
distance technique.

Within the above framework, if initial estimates of the cluster means and
covariances are not provided, the program generates initial estimates in a
KMEAN subroutine which was adapted from Ward, Hall, and Buchhorn's (1967)
hierarchical grouping for minimum variance. The variance which is minimized
is the sum of Mahalanobis distances between points within a cluster given by

'ij' = ('<r^j'' c;' (^i-^j) '

where X. and X. are points in the same cluster and Ci is the covariance

matrix. The prime indicates a transpose and the superscript indicates an
inverse.

The subroutine first transforms the data to principal axis factor scores,

■ ' , Y = a"^ F'X ,

where F is the eigenvector matrix of C^ and A is the diagonal matrix of
eigenvectors. (See discussion on principal component analysis.) Then

d./ = (Y.-Yj)- (Y,-Yj)

which is easier to compute.

After hierarchical grouping, the within-group covariance matrix C2 for ten
clusters is determined. The number ten is arbitrary and could be changed.
With this modified matrix, hierarchical grouping is performed again. A third
covariance matrix C3 within the ten clusters is obtained. New distances are
obtained which are used for the third and final hierarchical grouping.

The program permits the assignment of an arbitrary number of clusters or
means known as KMEANS. If the number setup is 150 or less, the procedure
uses the number established. If the number of input data is greater than 150,

27

a leading subroutine for the KMEANS (MacQueen, 1967) precedes the hierarchical
grouping. The first 150 input data points form the centroids of 150 clusters.
The two closest points (clusters) are merged into a two-point cluster with a
cluster mean in the location of the first of the two points read. The 151st
datum is moved into the place vacated by the second of the two points. The
next two closest cluster centroids are merged and the process continues until
all data points have been collected into 150 clusters which then are grouped
hierarchically.

7.3 Significance Tests for Number of Clusters

The user has some idea that clusters of data exist in the data set or he
would not be concerned with this report or its uses in his field of endeavor.
Suppositions or hypotheses concerning the number of types in the sample are
listed and control is placed on each of these in the program. For example,
he suspects two groups at least or perhaps four at most. Just to make sure,
as far as this technique is concerned, he might place an upper limit of six.
The program reaches a solution in each case which provides a relative (local)
maximum to the likelihood function. The ratio of the likelihoods for any two
of the hypotheses above 2, 4, or 6 provides a basis for a significance test
for rejecting the null hypothesis that there is no difference between the two
hypotheses, i.e., the smaller number of types against the alternative of a
larger number of types.

7.4 Strategy of Use

As computing time increases with number of variates, number of input data,
and null hypotheses to be checked, care should be exercised to keep time,
which is transformable to cost, to a minimum. There are certain strategies
for use which Wolfe (1971b) discusses.

7.5 Usage

It is appropriate that, since the program can be obtained from Wolfe (1971b),
a few usage comments from his program be placed here (with his permission).

7.5.1 Storage Requirements

Storage requirements are variable, depending on the data. The arrays are
dimensional at execution time. A minimal problem would require 120,000 bytes
of storage (IBM 360). The first two lines of the printout give the storage
required for a particular problem.

7.5.2 Restrictions

The number of types or clusters must not exceed 20. There are no fixed
limits on the number of variables or sample size. However, all data and
arrays must fit into core.

7.5.3 Error Messages

Bad initial estimates or diverging iterations can cause strange estimates

28

I

of the parameters, such as negative variances or singular correlation matrices
which result in system diagnostics. In such a case, the user may have to
specify his own initial estimates on the input form or rescale the data to
eliminate dichotomous variables.

7.5.4 Input Deck

Input Deck set up cards are:

Input Form, Cards 01-11

Data Cards (omit if data are on tape)

Input Form, Card 12

Initial estimates, if any, for one hypothesis

Input Form, Card 12

Initial estimates, if any, for a greater number of types

Blank Card

For example:

Card 1 provides the user and the data used.

Card 2 provides the title.

Card 3 provides room for comments.

Card 4 provides room for comments.

Card 5 provides for the number of variables, the sample size, and input

tape.
Card 6 provides hypotheses for number of clusters.
Card 7 provides space for assumption of same or different covariance

matrices and the minimum number to be accepted into a cluster.
Card 8 provides for the significance level for rejection of the null

hypothesis.
Card 9 provides for the entry of the number of KMEANS to be first

examined and whether the iterations are to be printed.
Card 10 provides for the establishment of a time limit and a limit on

the number of iterations.
Card 11 provides for the data format.
Card 12 provides for the initial estimates, if any, for the number of

types J the means, the standard deviations, and the correlations.

7.5.5 Validation Examples

Wolfe (1967, 1971) uses two examples to test his program. The first is the
Fisher (1936) treatment of Anderson's (1935) measurements on Irises. Nearly
all discrimination papers or clustering papers refer to this classic paper.
The second example is an artificial set. These and another set of data on
azotobacter, the nitrogen fixing bacteria [Cox and Martin (1937)], were used
to validate the Wolfe program adapted for use at the National Climatic Center.

The NORMIX program has been adapted for use at the National Climatic Center.
Considerable difficulty was experienced in adapting the program for use on the
Univac Spectra 70/45 due to language configurations. One adaptation is an

29

output routine to furnish a sequential leading copy of the input data. This
permits ready referencing as to a datum and its assignment to a cluster. Some
modification of the output sequences were also made.

Though this point is made by Wolfe, it is given here again for emphasis.
The techniques apply to only normally or near normally distributed variates.
Any significant departure from normality will invalidate the probability de-
cision levels. Therefore, if data distributions cannot be approximated by
the normal distribution, transformations to normality or near normality of
distribution must be made before input to the program.

A brief summary by Wolfe (1971b) is quoted below:

"The problem was to develop a computer program for cluster analysis
and unsupervised pattern recognition of types with different cluster
shape

"The program will cluster a sample of thousands of objects measured
on many continuous variables....

"The approach seeks maximum likelihood estimates of the parameters
of multivariate distributions. The likelihood equations are solved
iteratively by continually re-estimating the probability membership
of each sample point in each cluster until the likelihood reaches a
relative maximum. The initial estimates are derived from a minimum
variance hierarchical grouping subroutine, which itself is itera-
tive in seeking an appropriate distance function. The program prints
out the means, standard deviations, and intercorrelations of the
variables within clusters and the proportions of the population for
each cluster. The probabilities of membership for each cluster are
also printed."

Briefly some points mentioned in the foregoing section are iterated. In
clustering procedures, unless there is "a priori" knowledge as to the char-
acteristics of the clusters in the data set, the data themselves determine
the clustering. This is called the unlearned procedure or learning process.
The use of "a priori" estimates is termed the learned procedure, there is,
then, the necessary decision as to just where to start. This is true of any
discriminating procedure whether it be any of the allied techniques of analysis
such as discriminant function, factor, principal component, or any other
analysis.

Any change in the order of the input data will produce a difference in the
output cluster characteristics (MacQueen, 1967, p. 290). Therefore, one run
only provides an estimate of the clusters. A number of runs with a different
ordering of the data will produce different results. Hopefully, these will
not be too different and usually are not too different even though their co-
variance matrices may be unequal. The more distinctly different the clusters
are, the morti the various runs will provide the same estimates. This is not
unexpected. There will be some data which will not be assigned to the same
cluster each time.

30

8. EXAMPLES

8.1 Introduction

The techniques used here are applicable in any field as they reduce all
problems to non-dimensional ones in the sense of units. Here, climatological
(weather) observations are used. These were obtained from the archives of
the National Climatic Center (NCC) in Asheville, N.C.

We present six data sets from among the many that could have been used.
The first four present situations which are recognized as producing mixed
distributions. Obviously, these are selected to demonstrate the usefulness
of this procedure.

Output of this program for the six data sets used for this paper is not
presented in total. The data sets are:

1 . Land-sea breeze data

2. Tropical stratospheric wind data

3. Mid-latitude tropospheric wind data

4. Mountain pass wind data

5. Marine surface data

6. Radiosonde and rawinsonde data

In the first and second data sets, only selected tabular output data and
computer-drawn, 0.50 probability ellipses are shown. In the third data set,
a total input-output of the Wolfe (1971b) program adapted for use at the NCC
is shown. Except for the input data and the modified eigenvalue-eigenvector
output, the format is essentially that of the Wolfe program. Also, a con-
siderable number of 0.50 probability ellipses are shown. The computer plot
routines for these exist in programs written for the CALCOMP drum plotter and
Computer Output Microfilm and the Hewlett-Packard desk-top plotter at the
National Climatic Center. The computer print plot of the discriminant function
for the common covariance assumption is part of the Wolfe program. The last
three data sets simply illustrate selected and re-arranged output as well as
the 0.25 probability ellipses.

Winds are used in all six data sets as upper winds usually are distributed
in the bivariate normal sense. In the second, fifth, and sixth data sets,
additional weather elements are used. Where mixtures are evident, the usual
unimodal distribution cannot be used. These mixtures occur in the planetary
boundary layer, in the trade-wind inversion region, in the tropopause, or in
the monsoon change region. It is precisely the advisability of separating
these and other mixtures that prompted the investigation of the problem and
the publication of these results.

8.2 Brief Descriptions of Data and Locations

8.2.1 Land-Sea Breeze Data

San Juan, Puerto Rico, U.S.A., is a seaport located on the northern shores
of the Island in its eastern portion. It is a sub-tropical station at 18°26'

31

north latitude and 66°00' west longitude. The data are taken from the obser-
vations made at the airport at Isle Verde. Its elevation is 20 m. This
location was selected because it is a logical one for a land-sea breeze effect.
The hours selected, 0600-0800 and 1200-1400, are in local standard time. The
month of October was selected though any other period could have been used.
One hundred observations for each hour group are used for October 1955 and
the first portion of the first 3 days of October 1956. The early morning hours
should show the land breeze or a balanced condition. The 1200-1400 observation
hour should begin to show the effects of the sea breeze though perhaps not as
strongly as later hours. The hours of 0900-1100 were not used so as to elimi-
nate some of the overlapping of the land-sea breeze effect if any existed.
This would provide a clearer separation for the technique demonstration.

8.2.2 Tropical Stratospheric Wind Data

Canton Island is in the Southern Hemisphere at latitude 2°46' and longitude
171 °43' west, elevation 4 m. This location was selected because of its known,
distinct quasi-biennial wind oscillation in the stratosphere. During the
preparation of Technical Paper 34 (Crutcher, 1958) and U.S. NAVAER 50-1C-535
(Crutcher, 1959), the apparent biennial oscillation of the tropical tropo-
spheric winds had been noted. U.S. NAVAER 50-1C-535 is apparently the first
atlas in chart form to use the elliptical bivariate normal distribution to
describe the distribution of upper winds, particularly those of the strato-
sphere. This followed the very important atlases of the British groups
[Brooks and Carruthers (1950)] using the circular bivariate distribution and
preceded their update, [Heastie and Stephenson (1960) and Tucker (I960)].
The British groups assume the circular distribution to adequately describe an
upper wind distribution. This appears to be a good assumption as a first
approximation if the winds are not mixed. The extension (Crutcher, 1957) of
this assumption to that of the elliptical normal as a second approximation
increased the representativeness of that bivariate normal from about 70 per-
cent of the cases to about 90 percent in the troposphere. It was noted during
the preparation of NAVAER 50-1C-535 that, in the tropics at the higher alti-
tudes (lower pressure levels), the ratio of the major axis to the minor axis
was seemingly extraordinarily large, near four ranging up to ten at times.
An examination of the distributions generally revealed two clusters with the
easterly winds being stronger and with a higher constancy than those from the
west. In the meantime other investigators also were studying the problems.
McCreary (1959) reported that in the stratosphere during October 1956 - July
1957 at Christmas Island, westerly winds overlay the stratospheric easterlies.
This was a reversal of what was considered to be the usual pictures of east-
erlies over westerlies. Graystone (1959) then reported on a year to year
reversal of the equatorial stratospheric winds. In the following year Reed
(1960), Ebdon (I960), and Veryard (1960) provided the impetus for a great
amount of research into the phenomenon now known as the quasi-biennial oscil-
lation (QBO) of the winds in the tropical stratosphere. These same authors
soon produced further work in the field [Reed et al . » (1961) and Reed and
Rogers (1962) in the U.S. and Veryard and Ebdon (1961e. b, and 1963) in the
U.K.]. Later, an oscillation in the temperatures associated with the winds
was soon reported as were oscillations in the tropopause heights, ozone, and
other phenomena. Newell et al., (1974) discuss the development of research

32

in this field in the second volume of an important work on the general circu-
lation of the tropical atmosphere and interactions with extratropical
latitudes.

In these wery important studies, the oscillation was shown to start at the
higher altitude of the 10-mb level and propagate downward through the strato-
sphere to the 100-mb level.

Clayton (1885), Berlage (1956), and Landsberg et al . , (1963) had noted this
feature in surface phenomena. This feature now occupies a rather solid part
in the literature of the atmospheric circulation.

Upper winds during the months of July and January for the years 1954-1964
at Canton Island at the 50-, 30-, 20-, and 10-mb pressure levels were used.
For the period 1957-1967, heights and temperatures were added. Data from this
station have been used many times in studies. The quasi-biennial oscillation
(QBO) is barely evident at 100 mb and becomes strongest at about 30 to 25 mb.
Apparently it weakens at lower pressure levels (higher altitudes), such as
10 mb, though this effect may be due to loss of observations. There is a time
dependence as noted by Reed et al . , (1961), Reed and Rogers (1962), Veryard
and Ebdon (1961a, b), Angell and Korshover (1962), and later works.

The purpose of this paper is to provide the techniques to dissect such dis-
tributions. Dynamic or synoptic considerations are not made here except in
the sense that these do effect the QBO. Newell et al., (1974) discuss the
dynamic properties of the phenomenon. Dynamic considerations do create the
separate distributions insofar as the calendar year dating system is involved.
The results may be used to assess the extent of these effects. Only the
horizontal winds and temperatures are used at the individual levels. An ana-
lytic solution or more detailed examination of the atmospheric circulation and
weather is not attempted here. For example, the problem could have been made
much more multidimensional by including all stratospheric levels and concomi-
tant surface weather. The results of this section clearly demonstrate the
usefulness of this separation technique.

8.2.3 Mid-Latitude Tropospheric Wind Data

A continental U.S. station was needed. Rantoul , Illinois, was selected
since, in an earlier unpublished work, Crutcher and Clutter (1962) experienced
some difficulty in applying the techniques of Hartley (1959). The difficulty
involved singularity problems in the data when assumptions of unequal variance
in the groups were made. Its latitude is 40°18' north, longitude is 88°09'
west, and elevation is 227 m. There are some data sets which produce such
near singular matrices that the techniques cannot be used in their present
form. This is occasionally true of the present technique here. Ordinarily,
the singularity problem is resolved in the technique by the addition of a
random small amount to the matrix diagonal elements. Even then, negative
variances may be encountered. Re-initialization or change in order of data
entry may resolve the problem.

The period of record of data used is the month of October for the years
1950-1955. The upper wind data used are those at the 700- , 500- , and 300-mb

33

levels. Both input and output data are shown to better illustrate the pro-
cedures. The first two data sets described were two-dimensional. This data
set example illustrates a distribution in six dimensions, two for each of
three levels. For this purpose there had to be a simultaneous wind observa-
tion at each of the three levels. Therefore, each input data vector given
was composed of the vector formed by the zonal and meridional component of the
wind at each level, i.e., a vector composed of six components. The two-
dimensional illustration for a given level then is comparable to that of
another level for each datum at the given level having a matching datum at
the other level .

As more than three space is difficult to illustrate except for linear
reduction to three space or less, output for the six-dimensional forms is
given in eigenvector and covariance (correlation matrix) form.

No attempt is made here to correlate the wind data with current or later
weather. This can be done by simply extending the six-dimensional vectors to
a greater number of dimensions by including the desired surface weather data.

8.2.4 Mountain Pass Wind Data

Stampede Pass, Easton, Washington, U.S.A., was selected to demonstrate the
existence of two or more distinct clusters of wind, and of wind with tempera-
ture regimes, in a constricted geographic location. Its latitude is 47°17'
north, its longitude is 121°20' west, and its elevation is 1206 m. It is
almost in the lowest part of a saddle-back with a ridge running north-south
upwards from the station location to a height of about 200 m above the station
at a distance of about 3 km. North of the saddle, the ridges rise in an east-
west fashion to heights near 3,000 m at distances ranging from 20 to 80 km.

8.2.5 Marine Surface Data

Weather observations over oceanic areas constitute an important part of the
archives at the National Climatic Center. These have been the basis for the
Marine Climatic Atlases program of the U.S. Navy since 1950. In some regions
the weather is composed of quite distinct weather regimes over the year.
Within shorter time periods there may or may not be such distinctness. There
may or may not be distinct separation of weather factors between the traveling
cyclones and anti -cyclones.

Observations from transient ships offer no real opportunity for time studies
For a particular spot, over time, this may or may not be important. For
reasons of time continuity and a measure of pressure gradients, however, an
ocean station vessel location was selected. The ocean station vessel is OSV
"C" (Charlie) at 52''45' north latitude and 35°30' west longitude. The month
of February was selected for the years of 1965-1970. The 1200Z observations
are used. The elements selected are the surface winds in m-s" , the tempera-
tures (°C), the sea level atmospheric pressures (mb), and the dew-point
temperatures (°C). Other elements such as wave height, wave frequency, visi-
bility, cloud, cloud height, precipitation, and other obstructions to vision
can be used. However, some of these have a lower bound of zero or are
dichotomous, i.e., a yes or no condition. Transformation to near normality

- 34

of the distributions of these elements should be made before they are used
with this technique.

The input data are not shown. Only the pertinent output data are shown,
i.e., the means, variances or standard deviations, the correlation matrices,
and the eigenvalue-eigenvector and mixture proportions.

8.2.6 Radiosonde and Rawinsonde Data

The foregoing examples clearly illustrate the usefulness of Wolfe's NORMIX
and NORMAP techniques to separate mixed multivariate normal distributions.
Two- or higher-dimensional vectors have been used. A radiosonde observation
is composed of observations at levels in the atmosphere where significant
changes are made from level to level. Also included are observations at
mandatory levels so that all stations transmit data for the same levels.
These can be used to produce synoptic charts of the data. Ordinarily, these
charts can be studied by element and level by level or by changes within and
between levels. Also, a complete observation can be studied as one vector in
multidimensional space. Only the time and cost limitations of the computers
restrict the problem. For example, if the winds, temperatures, dew points,
and heights of selected pressure levels are used for 30 levels, an observation
may be characterized by a vector of 150 components, i.e., a 150-dimension
problem. Extension of this to other levels and features as well as differences
level to level such as lapse rates, shears, etc., would extend the vector to
still higher dimensions.

As an illustration, radiosonde and rawinsonde observations at Balboa, C.Z.,
Panama, are used. The 1500Z observations in the month of July for the years
1961-70 were selected. The levels selected were the surface, 950-, 850-, and
700-mb. The data used are winds in m-s"^, dry bulb and dew-point temperatures
in °C, and heights in gdkm. Each vector then is a 4 x 5 or 20-dimensional
vector. Assumption of both equal and unequal covariance matrices is made.
No input data are shown. Only the output data in terms of means, variance
(standard deviations), covariance (correlation) matrices, eigenvalue and
eigenvector, and mixtures are shown.

8.3 Selected Data

8.3.1 Land-Sea Breeze Data Set

8.3.1.1 Input Information.

a. San Juan. Puerto Rico

b. The period of record is October 1-31, 1955, and
October 1-3, 1956.

c. The data are surface winds.

d. The number of variables is two; these are zonal and
meridional wind components.

e. The number in the sample is 200; the first 100 are from
the 0600-1800 local standard time (l.s.t.) while the
second 100 are from the 1200-1400 l.s.t.

f. The minimum number to be accepted into a cluster is three.

35

g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.

h. The first 40 two-dimensional vector entries are .set up as
the 40 means of 40 separate and individual clusters. These
are 40 points in two dimensions.

i. Two assumptions are made. The first is the equality of co-
variances while the second is the non-equality of covariances.

8.3.1.2 Tables - Output Information. The program computes the necessary
statistics for the three-cluster versus two-cluster including the discriminant
functions and classification of entries into the clusters.

The output statistics are now shown in tables 1 and 2 for the two-cluster
and the three-cluster even though the hypothesis that three types and two
types were not significantly different was not rejected.

8.3.1.3 Figures and Discussion. San Juan, Puerto Rico, was selected because
this is a known land-sea breeze effect location. The hour groups of 0500-0800
and 1200-1400 were selected when the land-sea breeze effect would most likely
be operating. The periods of October 1-31, 1955, and October 1-3, 1956, were
selected so as to provide 100 observations of the wind at each of the hour
groups. The wind direction and speed are used to provide the zonal and merid-
ional components of the wind. The first variable is the x-component or zonal
(west-east) component of the wind while the second variable is the y-component
or meridional (south-north) of the wind. Minus signs indicate components from
the east or north.

The procedural techniques of the Wolfe NORMIX-NORMAP electronic computer
routines (1971) worked well in this case.

Figure 8a shows a two-dimensional representation for the entire group of
200 observations. This may or may not be an adequate representation. No plot
is made of data to visually assess the fit of the mathematical elliptical form.
This form is obtained under the assumption that there is one single group.
Here it is realized that such an assumption is not valid, yet it is shown for
illustrative purposes. Under the assumption of two groups, the land and the
sea breezes, and the equality of covariances, the breakout into two groups is
shown. Elliptical error probable (e.e.p.) ellipses are shown. These are the
0.50 probability ellipses. When the axes of the ellipses are equal, the
ellipses are circles and the 0.50 probability circle is called the circular
error probable (c.e.p.). Figure 8b then shows a breakout into three clusters.
The assumption of equality of covariances is exemplified by the fact that the
ellipses all have the same shape, size, and orientation. This is not the case
as shown in figures 8c and 8d where the covariances are assumed to be different.
In these figures, the shape, size, and orientation may be different from ellipse
to ellipse. The authors believe that assumption of unequal covariances is
valid. The component means of the two-cluster configuration under the assump-
tion of equal versus unequal covariance matrices are in the first type -0.4620
versus -0.5764 and 1.4574 versus 1.3866 and in the second type -4.7149 versus
-4.7913 and -2.3420 versus -2.4596 m-s"^. The proportions assigned to the

36

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10

L 10

Figure 8 San Juan, Puerto Rico, surface wind distributions; period of record
October l-Sl, 1955, and October 1-3, 1956, hours 0600 and 0800 and
1200-1400 local standard time; n = 200, 100 from each period; units,
m-s"^; separation shown for two and three types with covariances
assumed equal and then unequal .

(a) Total and two types, equal covariance matrices.

(b) Three types, equal covariance matrices.

c) Total and two types, unequal covariance matrices.

d) Three types, unequal covariance matrices.

An

first type are 0.5887 versus 0.612 and the second type are 0.4113 versus 0.388.
The comparisons indicate similarity. Under the assumption of unequal co-
variance matrices, the second type shown above (though losing a part to the
first) then breaks down into two clusters which are proportionally 0.255 and
0.110.

Quite clearly the unequal covariance ensemble breaks down into at least
three clusters, a south-southeast wind and an east-northeast wind, the early
morning wind versus the noon wind, and the land breeze versus the sea breeze.
The mean wind is from the east by southeast with components -2.2114 and -0.1054
m-s"^. This is the easterly trade though it is somewhat less than the average
of 3.5 m-s"i for that region [Crutcher, Wagner, and Arnett (1966)].

The land-sea breeze situation illustrates the output of the technique in
tabular form and in two-dimensional illustrations with ellipses. Two assump-
tions are shown. First, there is the assumption that the underlying statistics
have the same covariance matrix; i.e., the underlying physical bases are
operating the same. Second, there is the assumption that the underlying sta-
tistics have differing covariance matrices; i.e., there is some reason to
believe that the underlying physical bases operate differently. Whether the
assumptions are correct in any particular case is not known for these are not
tested here. Both assumptions are made. The results are presented. The
reader can make his own assessment and choose the assumption he pleases. How-
ever, statistical tests are used to reach decisions as to whether there are
two groups or less, three groups or two, etc. As with all other examples, the
decision level chosen to work with is the probability level of 0.01. The null
hypothesis is that the distribution of (k + 1) groups is not different from
(k) groups; that is, rejection of the null hypothesis that there are (k + 1)
groups rather than (k) groups is sought.

Output or input data in computer format and the intermediate steps are not
shown in this example. For these the reader is referred to Wolfe (1971b)
whose electronic computer program is adapted for use here. The mid-latitude
tropospheric wind data set (paragraph 8.3.3) does contain some of the inter-
mediate steps.

8.3.2 Tropical Stratospheric Wind Data Set

8.3.2.1 Input Information.

a. Canton Island, South Pacific, (U.S.A. and Great Britain)

b. The periods of record are the months of July and January,
1954-1964, and for 1957-1967.

c. The data are stratospheric winds, heights of pressure
surfaces, and temperatures. The pressure levels are 50-,
30-, 20-, and 10-mb.

d. The number of variables is two for the first period and four
for the second. These are for the first period, zonal and
meridional components of the wind, positive from the west and
south. For the second period, these are the winds, heights,
and temperatures. The units are m-s"^, m, and °C.

e. The number in the samples vary from level to level because at

41

times the balloons failed to reach the higher altitudes.
The numbers are 263, 244, and 162, respectively, for the
first three levels above. Though the 10-mb data were
processed, only one cluster was determined. The results
are not shown.

f. The minimum number to be accepted into a cluster is three
for the first period and five for the second.

g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.

h. The first 40 two-dimensional vector entries in each level are

set up as the 40 means of 40 separate and individual clusters.

These are 40 points in two dimensions,
i. Two assumptions are made. The first assumption is the equality

of covariances. The second assumption is the non-equality of

covariances.

8.3.2.2 Tables for Wind Configurations (1954-1964). Tables 3 through 10
provide the output data in tabular form provided by the Wolfe (1971b) NORMIX-
NORMAP computer routine for the first period. An asterisk indicates the re-
jection of the null hypothesis that (k + 1) types are not significantly
different from the (k) types. Thus, under the following assumption of equal
then unequal covariances and at the 0.01 probability level, the greatest
number of clusters (types) for the wind distributions 1954-1964 is indicated
below:

July

Equal Unequal (Covariances)

50-mb

2

3

30-mb

at least 4

3

20-mb

3

2

10-mb

1

1

Janua

ry

Equal

Unequal

2
1
1
1

2
1

1
1

The statistics for the single clusters are not shown.

42

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56

8.3.2,3 Figures and Discussion for Wind Configurations (1954-1964). Figures
9 through 19 show various combinations of the 0.50 probability ellipses as
well as the wind plot diagrams to which these pertain. For example, figure
10a gives the 0.50 probability ellipse for the total group and a breakout into
two groups. This 0.50 probability ellipse is the ellipse estimated to contain
one-half of the winds from the cluster (type) to which it pertains. Figures
10b and 10c are similar illustrations for the 30- and 20~mb levels. Comparison
between the results of the assumptions of equal and unequal covariance matrices
are possible. Under the assumption of equal covariance matrices, the size,
shape, and orientation of the cluster breakout are the same but not necessarily
the same as the complete (total) distribution. That is, the flux indicated by
the total may be quite different from that within the individual clusters
though the individual clusters imply that the flux across the pertinent air-
stream in each cluster is the same as in another. Under the assumption of
unequal covariance matrices, the various clusters may show different size,
scale, shape, and orientation.

In addition, other figures in the above group permit comparison from level
to level of the totals and of the various types. These tabular and graphical
representations quite clearly indicate that elongated elliptical distributions
computed from ordinary bivariate normal statistical routines should be backed
up by plot or scatter diagrams. If a plot is not available, then a wind rose
of some type should be available for examination. The wind rose usually re-
ferred to is the WBAN-120 (revised) available from the National Climatic Cen-
ter (1958). This is based on the work by Crutcher (1957). A decision can
then be made as to whether the computed bivariate statistics are valid. These
assume a unimodal bivariate distribution. In the output statistics of WBAN-
120 revised format, for example, a ratio of the major to minor axes in excess
of four should indicate a need for study to see whether clustering is evident.
See table 9 for such a comparison. Both plots and analytic procedures such
as are available in discriminant function analysis, factor analysis, or prin-
cipal component analysis or other clustering routines can be used.

"A priori" considerations may also indicate that clustering techniques
should be used. That is, previous research may indicate that the use of the
total distribution as a unimodal bivariate (or multivariate) distribution is
unwarranted and that probabilistic statements from such a model may be erro-
neous.

The previous example of the land-sea breeze effect at San Juan, Puerto Rico,
and the present example of the stratospheric winds at Canton Island are "a
priori" types.

The procedures also are helpful in establishing the quality assurance of the
data. Clusters composed of only one or a few observations and far distant from
the main cluster or clusters may be examined for validity. This feature will
be discussed in a later section. In particular, in tables 6 and 7 and in
figures 17a and 17b for the four-cluster breakout, the wery low proportions in
one of the clusters and the wery large ratio of the major axis to minor axis
indicate that these groups may be suspect for one or more reasons. The non-
rejection of the null hypothesis, four clusters versus the three clusters,
implies that the four-cluster breakout was not significantly different from

57

the three-cluster breakout. Thus, the investigator can simply stay with the
three-cluster configuration while examining the isolated questionable groups
indicated in the four-cluster configuration.

The equatorial stratospheric winds situation illustrates the output of the
technique in tabular form and in two-dimensional depiction with 0.50 ellipses.
Two assumptions are used. First, there is the assumption that the underlying
distributions are the same, i.e., that the statistics have the same covariance
matrices (that the underlying bases are the same). Second, there is the alter-
native assumption that the statistics represent different physical and dynamic
situations, i.e., the covariance matrices are not the same. There is some
reason to believe that the underlying physical bases operate differently. The
reader can select the assumption that best fits his knowledge and experience.

Under the assumption of equal covariance matrices, one of the outputs is a
plot of discriminant function assignments, such as one versus two or one versus
three, etc. This is a computer tabulation type of two-dimensional plotting
with each individual data point carrying the number of the cluster type to
which it is assigned by the classification (discrimination) procedure. Figures
13, 14, and 15 illustrate the output for Canton Island during July at the 50-mb
and 30-mb levels. Figure 13 for the July 50-mb level shows the print plot of
discriminant function 1 and 2 where three types are computed. Figures 14 and
15 for the July 30-mb level show the print plots of discriminant functions 1
and 2 for the first three-type dissection and then four-type dissection. It
appears here that type 2 of 3 becomes type 3 of 4 while type 3 of 2 breaks down
into types 2 and 4 of 4.

58

6O703 7 50
« 36 32 2S M 20 It 12

■12 16 -20 -24 28 -32

— [

1

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(b)

—

X

1

Figure 9 Canton Island, U.S.A. and U.K.; upper wind distribution plots;
period of record, the months of July 1954-1964; pressure levels
(a) 50-, (b) 30-, (c) 20-, and (d) 10-mb; units, m-s"i.

59

60703 7 20
40 36 32 28 24 20 16 12

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1

Figure 9 (continued)

60

Figure 10 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, amd 20-mb;
units, m-s"^; wind plot shown in figure 9; separation shown for
two types, assumption of unequal covariance matrices; elliptical
error probable, (e.e.p.), 0.50 probability ellipses.

(a) Total distribution and two types at 50-mb, n = 263

(b) Total distribution and two types at 30-mb, n = 244

(c) Total distribution and two types at 20-mb, n = 162

61

(a) Three types at 50-mb, n=263.

(b) Three types at 30-mb, n=244.

(c) Three types at 20-mb, n=162.

(0

Figure 11 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s"^; wind plot shown in figure 9; separation shown for
three types, assumption of equal covariance matrices; elliptical
error probable, (e.e.p.), 0.50 probability ellipses.

(a) Three types at 50-mb, n = 263.

(b) Three types at 30-mb, n = 244.

(c) Three types at 20-mb, n = 162.

62

SOHB 4 TYPES

CANTON IS- SRHE COV

1 L

T r

"1 r

-J i_

Figure 12 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure level, 30-mb; units, nrs"^;
n = 244; wind plot shown in figure 9; separation shown for four
types, assumption of equal covariance matrices; elliptical error
probable, (e.e.p.), 0.50 probability ellipses.

63

SAMPLE p CANTON ISLAND
JULY

~T0 — PE
X AND Y WIND COMPONENTS

"DISCRIMINANT PlINCTIDM 2

NUMBER DF TV|>PS» 3

r
1 I

1 rr

1 1

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"31 Wt Wt 6^ 7^ 37

DISCRIMINANT FUNCTION 1

Figure 13 Plot of Discriminant Functions 1 versus 2 for Canton Island, U.S.A.
and U.K.; July winds shown in figure 9 for winds shown in figure 11,
based on assumption of equal covariance matrices; period of record
1954-1964; elliptical error probable, (e.e.p.), 0.50 probability
ellipses.

(a) Three types at 50-mb.

(b) Three types at 30-mb.
■ (c) Four types at 30-mb.

(d) Three types at 20-mb.

64

SAMPLE • CANTON ISLAND

JULY 50" MB

X AND Y WIND CnMRONENTS

DISCRIMINANT FUNCTIUN 2

NUMBER PP TYPES" 3

3 3

3 3 3

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Hill

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DI$CRI^'.I^ANT FUNCTION 1.

"t: 5"; 6^ T, 57

Figure 13 (continued)

65

SAMPLE

CANTON ISLAND

JULY W MB

X AND Y WIND COMPONENTS

DISCRIMINANT FUNCTION 2

NUMBER OF TYPES* 4

T

6

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Ti t; 87

Figure 13 (continued)

66

SAMPLE ■ CANTON ISLAND

JULY 20 MB

X AND Y WIND CHMPONEVTS

DIStKIMINANT FUNCTIUN 2 NUMBER OF TYPES- 3

7.

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TT 87

Figure 13 (continued)
67

CfiNTON rS, OIFF COVIH.

T

JULY TYrC 1

— 1 1 1 r-

"T 1 1 r-

-1 1 J ] I u

(b)

10 1-

-1 1-

JULT TfPE 2

CRNTQN IS. UIrr COVIN.

Figure 14 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s~^; n = 263, 244, and 162, respectively, assumption of
unequal covariance matrices; elliptical error probable, (e.e.p.),
0.50 probability ellipses.

(a) Total for the three levels.

(b) Type 1 for the three levels.

(c) Type 2 for the three levels.

68

JUL SOPie 2 TTPeS chnton is. srme cqvrr.

JX aOfIB 2 TTPES CSNTON IS. SfinE COVflR

1 1 r r —

(b)

^^^ ..^^A^QO...

JUL 20ne 2 TTPES CfiNTON 18. 3fine CQVflR.

n r

T r

li .mn

(c)

Figure 15 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s"^; n = 263, 244, and 162, respectively; separation shows
two types; assumption of unequal covariance matrices; elliptical
err-or probable, (e.e.p.), 0.50 probability ellipses.

(a) Total and 2 types at 50-mb.

(b) Total and 2 types at 30-mb.

(c) Total and 2 types at 20-mb.

69

~I 1 1 1 1 —

JUl 60t«. 3 TYPES CflMTON [S. DIFF CQVIM.

- - -10 -[■

Figure 16 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s~^; n = 263, 244, and 162, respectively; separation shows
three types, assumption of unequal covariance matrices; elliptical
error probable, (e.e.p.), 0.50 probability ellipses.

(a) Three types at 50-mb.

(b) Three types at 30-mb.

(c) "hree types at 20-mb.

70

JUL SOHB. « TYPES CflllTON IS. OIFF COVIN.

(a)

-I , 1 , 1 ^

-1— 1 ] L.

-1 L.

_l L_

_i J -L- L.

JUL 30H8. < TYPES CANTON IS- OIFF COVIU.

Figure 17 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50- and 30-mb; units,
m-s~i; n = 263 and 244, respectively; separation shows four types,
assumption of unequal covariance matrices; elliptical error probable,
(e.e.p.), 0.50 probability ellipses.

(a) Four types at 50-mb.

(b) Four types at 30-mb.

71

60703 1 60

24 20 16 12 e

-16 -20 -24 -28 -32 -36

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1

(b)

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: i

i 1 ix j

1
> ; 1

i

1

-12

Figure 18 Canton Island, U.S.A. and U.K.; upper wind distribution plots;
period of record January 1954-1964; pressure levels, (a) 50-,
(b) 30-, (c) 20-, and (d) 10-mb; units, m-s'i.

72

60703 1 20
24 2D 16 12

-12 -16 -20 -24 -28 -32 -36 -40

X

X

X

X

X

X

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X

X

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Xx

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X

X

(c)

1

-16

12

60703 1 10

20 16 12 8 4 0-4-8 12 -16 -20 -24 -28 -32 36 -40

[

u -^

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X

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16

Figure 18 (continued)

73

JfiN 50MB 2 TYPES

r

CANTON IS. SOME COVRR.

JfiW 50f1B. 2 TYPES

MNTON IS. nrFF COVIN.

-1 1 r -10

Figure 19 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record January 1954-1964; pressure level, 50-mb; wind plot
shown in figure 18; units, m-s"^; separation shows for two and
three types with assumption of equal then unequal covariance
matrices; elliptical error probable, (e.e.p.), 0.50 probability
ellipses.

(a) Total and two types with equal covariances.

(b) Total and two types with unequal covariances.

(c) Three types with equal covariances.

(d) Three types with unequal covariances.

74

JAH 50MB. B TYPES CANTON IS. aiFF COVIN,

r 1 1- 1 r -^° r

Figure 19 (continued)
75

8.3.2.4 Tables and Discussions for Height, Temperature and Wind Configura-
tion (1957-1967). The foregoing sections discussed wind configurations only
for the 50-, 30-, 20-, and 10-mb levels for January and July 1954-1964. The
height, temperature, and wind configurations now are discussed for the 30-mb
level during the months of January and July 1957-1967.

Veryard and Ebdon (1961b) in their study of the quasi-biennial oscillation
with tropical stratospheres reported that the westerly regimes were warmer
than the easterlies. There was a slight lag in the temperatures behind the
wind changes. Most of the data examined were for the 80- to 50-mb levels.
Substantiating this feature for July but not for January are the statistics
presented in tables 11 and 13 for the 30-mb level. Here, the temperature for
the westerlies and easterlies was -56.8 and -56.7 during January and -51.0
and -55.0 during July. In a further breakdown, not illustrated here, one
cluster in January has a temperature of -59.1 though the westerly component
for the cluster is only +4.2 m-s"^.

Tables 11 and 13 have been assembled differently from previous material.
This permits a better reference for this discussion. For example, there are
essentially only three clusters for the January clustering and only four
clusters for the July clustering. In both months, except for a fourth cluster
in July, the total sample breaks down into westerly regimes with slightly
greater heights than the easterly regimes; but while there are no temperature
differences in January, there are differences in July.

In January, the easterly cluster characteristics remain essentially fixed
in all characteristics while the more variable westerly cluster breaks down
into two clusters and then three. The last breakdown is not significant.

During July, the easterly cluster characteristics remain essentially fixed
throughout subsequent breakdown of the distribution. The westerly group
breaks into three significantly different groups. The difference between
the most westerly group and the first easterly group is 41 m-s"^.

Examination of the standard deviations is revealing during both months. The
ratios of the major axis to minor axis deviation ranges from about five for
the total sample to two for the clusters.

A look at tables 12 and 14 is interesting. These tables present the corre-
lation coefficients among the four variables, height, temperature, zonal wind
component, and meridional wind component. It is difficult to assess the de-
grees of freedom in each case. From the QBO basis, a January or a July is
expected to be consistent within itself. Therefore, a maximum of eleven points
are available for the 1957-1967 period. The correlation coefficients are ex-
pected to be large, Jarge enough to be significant. This appears to be the
case for July but not for January except for the heights and the zonal winds.

The heights and the zonal winds appear to be significantly correlated for
all cluster breakouts except one in each month where the correlation shifts
to the meridional winds. During January there is no definite correlation
between heights and temperatures at the 30-mb level. During July there is a
definite correlation. This does not indicate that there is no correlation

76

during January or July between the heights and the temperature structure
between the surface and 30 mb. This has not been examined here.

The large correlation coefficients and the large ratio of standard devia-
tions for the total samples may be considered to be a necessary indication
but not a sufficient basis for the conclusion of a two-cluster existence.
Conversely, small correlation coefficients and ratio of standard deviations
of one are indications of only one cluster, i.e., the total sample is a
homogeneous cluster.

77

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81

8.3.3 Mid-Latitude Tropospheric Wind Data Set

8.3.3.1 Input Information.

a. Rantoul , Illinois, U.S.A., latitude 40°18' north,' longitude
88°09' west, elevation 227 m.

b. The period of record is the month of October for the years
1950-1955.

c. The data are the zonal and meridional components at the
700-, 500-, and 300-mb levels, m-s"i.

d. The number of variables is six.

e. The number in the sample is 503.

f. The minimum number to be accepted into a cluster is seven,
one more than the number of variates.

g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.

h. The first 40 six-dimensional vector entries are set up as
the 40 means of 40 separate and individual clusters. These
are 40 points in six dimensions.

i. The assumption is made that the covariance matrices are not
equal .

8.3.3.2 Tables. Table 15 shows the sequential input data with six variates.
The variates are the zonal and meridional components of the 700-mb wind, the
zonal and meridional components of the 500-mb wind, and the zonal and meridi-
onal components of the 300-mb wind in m-s"^.

Table 16 shows the computer output after the hierarchical grouping and the
subsequent grouping into two clusters after 39 iterations. The two clusters
with proportions of 0.171 and 0.829 comprise the total set (1.000). First,
the characteristics of the total group are shown. This would be the assumption
of a unimodal six-variate model. Then the characteristics of the two clusters
are provided. Then, through the use of the discriminant function, assignments
of each datum to the clusters may be made as shown. The probability of assign-
ment to each cluster is printed. For example, the first datum of table 15 is
assigned to cluster 2 with a probability of 0.935 versus 0.065 for cluster 1.
The previous section illustrated computer plots for discriminant functions.

Table 17 provides, from the same input data, the output for three clusters
rather than two. The number of iterations required in this case is 101. The
probability of the null hypothesis being rejected is 0.00000349. The three-
cluster configuration is judged to be better than a two-cluster configuration.
Therefore, the program continues further to test a four-cluster configuration
versus the three-cluster form.

Table 18 provides, again from the same input data, the output for the four-
cluster configuration. The initial computing output, i.e., the first estimate
of the learning process, is shown as the zero iteration. This set then becomes
the initial estimation in the iteration process leading to 101 iterations be-
fore the results converged. Please note the changes from the initial estimates

82

to the final estimates. For example, for the zeroth iteration, the estimate
for the 700-mb zonal component in the first cluster is 11.2762 and after 101
iterations the estimate is 11.1041. The respective standard deviations are
5.5488 and 7.2047. Though the means do not appear to be yery different, this
is almost a forty percent reduction in variance in the cluster.

The probability of rejection of the null hypothesis in the above case is
0.00980137. This is near the decision level of 0.01. Therefore, no further
calculations are shown here.

83

Table 15 A multivariate (6) set of Rantoul , Illinois, October 1950-55,
upper wind components, zonal and meridional, at the 700-, 500-,
and 300-mb levels. For example, input variables 1 and 2 are
the zonal and meridional components, respectively, at the
700-mb level. The units are m-s"^.

RANTOUL
OCTDnEP

700/500/AND 300MB
X 4N0 V COMPONfNTS

NV8L6S- 6 NSAMPf 503 DIFF (Q\l fATRIx

MiN Cluster size- t hvputhesis test.o.'>oo

INITIAL KMEANS» 40 TIME LIMIT. *n ITER LIMIT. 100
NO. OF TYPES* 1234560100000
STORA&E BEOUIREMENT. 111932 DFCIMJL BYTES
OIMEHSION A( 3951 )

ANTICIPATED EXECUTION TIME • 21.11 MINUTES

PROGRAM IMDRMIX
WQLFJ NORMAL MIXTURE ANALYSIS PROC EDURF < 1 974 REVISION)

INPUT VARIAdLFS

£0

1

2

3

4

5

6

1

7,071

7.071

2.121

-2.121

5.563

2,248

2

4.296

10.126

1.9S4

4.603

10.000

0,000

3

-0.000

14.000

4.2*3

4.243

5,523

-2.344

4

-0.000

10.000

20.398

8.241

8.485

-6.485

5

-0.000

16.000

1.9il4

4.603

11.046

-4.689

6

5.470

12.887

-o.ono

1 1.000

5.657

5.657

7

4,636

1.873

15.000

0.000

24.107

9.740

S

7,000

0.000

19.331

-8.205

20,229

-50.068

9

11,3J4

-11.314

21.172

-8.987

31,297

-13.285

10

12.887

-5.470

14.142

-14.142

16.108

-39,869

11

9,192

9.192

-3.1?6

-7.364

8.285

-3.S17

12

2,828

2.828

4.636

1.873

2.828

2.828

13

3.536

3.536

2.715

6.444

7.778

7,778

14

-1.124

2.782

2.715

6.444

7.778

7.778

IS

1,954

4.603

3. l?6

7.364

34,306

13.860

16

0,000

5.000

12.053

4.870

13.285

31.297

17

4.243

-4.24S

4.243

-4.243

29.000

0,000

18

5.994

-14.835

7,778

-7.778

21.000

0,000

19

-3,126

-7.364

11.3U

-11.314

17.490

-7. 424

20

2,622

-6.490

19.092

-19.092

23.160

9,365

21

0,000

2.000

8.000

0.000

21.000

0.000

22

2.000

0.000

3.000

0.000

18.385

18.385

23

6.490

2.622

11.046

-4.689

30,597

12.362

24

5,523

-2.344

3.371

-8.345

11.046

-4.689

25

7.000

n.ooo

12.000

0.000

21.000

0,000

26

11.314

-11.314

14.7?8

-6.252

-36,000

O.OOU

27

15.762

6.368

20.000

0.000

20,251

-8.596

28

4.636

1.873

12.887

-5.470

28,536

-12.113

29

3,536

-3.536

17.490

•7.424

21,213

-21.213

30

8.285

-3.517

O.UOO

-14.U00

9.740

-24.10"

31

9.272

3.74»

11.046

-4.689

4.870

-12.053

32

17.000

0.000

11.9*7

-5.079

6,364

-6,364

33

11,000

0.000

12.000

0.000

13,908

5.619

34

16,000

0.000

13.808

-5.861

9.000

0.000

35

18,544

7.492

15.649

-6.642

5.523

-2.344

36

9.000

0.000

11.000

0.000

7,364

-3.126

37

4.636

1.873

12.0no

0.000

13.000

0,000

38

5,657

5.6!7

8.000

0.000

13,808

-5,861

39

6,490

2.622

a. 345

3.371

18,000

0,000

40

1,954

4.603

9.272

3.746

19,000

0,000

♦ I

12,728

12.728

10.000

0.000

20,398

8.241

♦ 2

6.364

6.364

8.485

8.485

18,544

7.492

4}

12.981

5.245

17.617

7.118

12.021

12,021

44

11,000

0.000

12.053

4.870

19.471

7,867

45

16.689

6.743

14.835

5.994

22.252

8.991

46

16,000

0.000

le.ono

0.000

32,451

13.111

47

13.908

5.619

15,7*2

6. 368

26.888

10.864

48

0.000

-3.000

7.118

-17,617

18.000

0.000

49

7.071

-7.071

9.900

-9.899

9.192

-9.192

50

6.364

-6.364

10.607

-10.607

12.887

-5.470

51

2,248

-5.563

5,619

-13.908

7. lie

-17,617

52

3,371

-8.345

-6.292

-14.728

0.000

-26.000

53

0.000

-9.000

0.000

-17.000

0,000

-26,000

54

-4.950

-4.950

-20.506

-20.506

-7.815

-18,410

55

0.000

-9.000

0.000

-13.000

5.994

-14,835

56

2.622

-6.490

4. 495

-11.126

6,364

-6.364

57

3.371

-B.345

0.000

•8.000

0,000

-4.000

58

4,243

4.243

0.000

-5.000

0,000

-5.000

59

1.124

-2.782

1.498

-3.709

-11.126

-4,495

60

-1.854

-0.749

0.000

-2.000

-7.417

-2.997

84

Table 15 (continued)

INPUT VARIABLES

SEQ

1

2

3

4

5

6

61

-0.921

0.

391

-0.921

0.

391

-8.3*5

-3,

371

fr2

0.000

5.

000

-2.997

7

417

-0.749

1.

654

63

0.000

5.

000

2.3**

5

523

2.735

6

44*

5<.

0,000

3

000

0.000

2

000

l.*l*

1

*1*

65

12.000

0.

000

10.000

0

000

15.6*9

-6

6*2

66

7.071

7.

071

11,000

0

000

11.0*6

-4

689

67

6.36*

6

36*

6.0fl0

0

000

5.657

-5

657

68

8,3*5

3

371

8,3*5

3

371

11.000

0

000

69

7.*17

2

997

9. OOO

0

000

13.000

0

000

70

8.3*5

3

371

8,3*5

3

371

12.053

4

670

71

3.907

9

205

7.417

2

997

12.981

5

2*5

72

3.907

9

205

7.071

7

071

15.762

6

368

73

3.126

7

36*

12.0<3

4

870

15,000

0

000

7<,

5.657

5

657

7.071

7

071

13.908

5

619

75

«.3'i5

3

37l

9.899

9

900

16.689

6

7*3

76

8.*85

8

*35

10.607

10

607

13.*35

13

*35

77

7.778

-7

778

13.000

C

000

19.331

-8

205

7a

11.000

0

000

16.569

-7

033

26,000

0

000

79

12.981

5

245

18.5*4

7

492

25.000

0

000

80

11.31*

11

31*

16.6^9

6

7*3

27.816

11

238

61

10.607

10

617

13.908

5

619

1*.1*2

1*

1*2

8a

9.192

9

192

11.314

11

31*

20,398

8

2*1

93

«.*d5

8

*S5

16.263

16

263

30,597

12

362

S'.

8.*S5

-8

*95

22.092

-9

378

31,297

-13

2B5

85

*.603

-1

95*

17,490

-7

*2*

21,172

-8

987

86

11.126

*

*95

12.000

0

000

22,000

0

000

37

12.981

5

2*5

13.908

5

619

25,961

10

489

US

e.*85

8

*85

11.314

U

31*

19,*71

7

867

89

11.000

0

f^Cj

21.000

0

000

35,000

0

000

vo

17.000

0

ooo

25.000

0

000

49.000

0

000

91

9.205

-3

907

20.251

-8

596

38,000

0

000

92

9.000

0

000

17.000

0

000

28.743

11

613

93

-2.622

6

*90

13.435

13

*35

9,899

9

900

9<,

6.***

-2

735

4.5*6

1

873

24,000

0

000

95

*.000

0

COO

12.887

-5

*70

28.000

0

000

96

13.000

0

000

4.2*3

• 4

2*3

18.000

0

000

97

16.689

6

7*3

13.000

0

000

10.126

-4

296

98

11.31*

-11

31*

IB, 410

-7

815

24,85*

-10

550

99

9.205

-3

907

14.8*9

-14

8*9

23.933

-10

159

100

13.806

-5

861

15,649

• 6

6*2

17,490

-7

42*

101

1?.806

-5

861

14.728

• 5

252

13,808

-5

651

102

8.*85

-8

4S5

21.000

0

000

27.816

11

238

103

7.071

-7

07i

15.6*9

-6

6*2

11,987

-29

670

10*

8.*85

-8

*e5

9.192

-9

192

15.649

-6

6*2

105

6.***

-2

735

13.000

0

000

17.000

0

000

106

6.36*

-6

36*

7.36*

-3

126

0.000

-12

■ 000

107

2.762

-1

172

0.000

-9

000

-14.8*8

-34

979

103

2.828

-2

828

0.000

-6

000

-7.*2*

-17

*90

109

0.000

-*

000

0,000

-3

000

-*.243

*

2*3

110

l.*98

-3

709

2.762

-1

172

-3.000

0

000

111

0.000

-*

000

4.9?0

-4

950

10.!99

*

121

112

-3.126

-7

36*

0.000

-6

000

0.000

-5

000

113

-7,773

-7

778

-*.6»9

-11

0*6

-6,36*

-6

36*

lU

-10,607

-10

507

-12.0?1

-12

021

*,*95

-U

126

Us

-3.536

-3

53»

0.000

-6

000

0,000

-2*

000

116

l.*98

-3

709

3,7*6

-9

272

11.31*

-U

31*

117

3.536

-3

53»

11.000

0

000

13,808

-5

861

iia

6.***

-2

735

2.7*2

-1

172

21.000

0

000

119

11.0*6

-*

589

10.1?6

-4

298

20.000

0

000

120

6.***

-2

735

9.000

0

000

11.0*6

-4

689

121

10.126

-*

298

19.000

0

000

33,000

0

.000

122

18.000

0

000

18.410

-7

815

17.*90

-7

42*

123

18,000

0

000

22.000

0

000

28,000

0

.000

12*

11.000

0

000

27.000

0

.000

28,000

0

,000

125

13.000

0

.000

30,0(10

0

.000

30.000

0

.000

126

16.000

0

000

22,002

-9

,378

37.000

0

,000

127

5.657

-5

557

22.0no

0

.000

21,000

0

iOoo

128

10.126

-*

298

17.490

-7

.*2*

0,000

-22

.000

129

13.808

-5

861

14.7J8

-6

252

0,000

-28

000

130

O.OOO

-10

000

5.619

-13

.908

15.556

-IS

555

131

*.2*3

-*

.2*3

2.6?2

-6

.*90

0.000

-5

000

132

3,682

-1

563

3.000

0

000

-10,000

0

000

133

2.828

-2

828

3,010

0

.000

-8.*85

8

485

13*

1,873

-*

536

4.603

-1

95*

6,*90

2

622

135

3.000

0

,000

9.205

-3

907

9.272

3

,746

136

3,000

0

000

3.5'(5

-3

535

7,*17

2

,997

137

i.285

-3

.517

8.2S5

-3

.517

15.689

6

.7*3

138

8,285

-3

517

11.0*5

• 4

689

*.950

-4

950

139

11,967

-5

.079

11.31*

-11

31*

*.*95

-U

126

1*0

0,000

-9

000

0.000

-13

000

-16,*ll

-3b

661

1*1

0,000

-8

000

0.000

-16

000

-33.9*1

-33

9*1

1*2

9,205

-3

.907

5.2*5

-12

981

4.870

-12

053

1*3

8.*85

-8

.485

6.7*3

-16

.589

14,235

-35

.233

1**

12.887

-5

.*70

7.778

-7

778

11.314

-U

314

1*5

12.887

-5

.470

16,000

0

.000

15,649

-6

642

1*6

7.778

-7

.778

16,569

• 7

.033

27,615

-11

722

1*7

5,657

-5

.5!7

11.31*

-11

.31*

26.69}

-11

331

1*8

*.2*3

-*

.2*3

14.7S8

•5

.252

23.013

• 9

,766

1*9

8,285

-3

.517

10.1?5

• 4

.298

26.000

0

.000

150

10,199

*

.121

1*.000

0

.000

2*. 107

9

,7*0

151

17,000

0

.000

20.251

• e

.596

22.000

0

,000

85

Table 15 (continued)

INPUT VARIABLES

SEQ

1

2

3

i

5

6

152

17,000

0.000

1*.7?8

-6.252

30,000

0.000

153

12.72B

-12,728

12.0?1

-12.021

27.615

-11.722

15*

15.6*9

-6.6*2

8.616

-21.325

10.86*

-26.888

155

1*.728

-6.252

13,*55

-13.*35

21,213

-21.213

156

13.808

-5.861

17.*90

.7.*2*

21,172

-8,987

157

13.9C8

5.619

15.7*2

6.368

26.000

0.000

158

1*.000

0.000

l*,000

0.000

19,000

0.000

159

18.*10

-7,815

17,*90

.7.*2*

17,000

0.000

160

2.622

-6.*90

6.***

-2.735

15.6*9

-6.6*2

161

-2.3**

-5.523

*.910

-*.950

13.808

'5.861

162

-6.*90

-2.622

3.536

.3.536

18.*10

-7,815

163

-7.36*

3.12*

3.000

0.000

9.900

-9,899

16*

-1.873

*.63»

6.*90

2.622

11,967

-5,079

165

1,172

2.762

3,536

.3.536

11,0*6

-*,689

166

1.85*

0.7*9

3,7-59

l.*98

12,887

-5,*70

167

*,2*3

*.2*3

*,2*3

*.2*3

17,*90

-7,*2*

168

2.3**

5.523

*,603

-1.95*

16.569

-7.033

169

3.536

3.53*

6.*<»0

2.622

11.967

-5,079

170

5.657

5.657

10.199

*.12l

20.000

0,000

171

7.071

7.071

11.1?6

*.*95

2*. 000

0.000

172

7.071

7.071

5,079

11.967

16.000

0.000

173

7.071

7.071

16,6H9

6.7*3

2*. 107

9,7*0

174

12.000

0.000

15,000

0,000

31,000

0,000

175

3.000

0.000

10.000

0.000

25,961

10,*89

176

6.000

0.000

13,908

5.619

31,52*

12,737

177

8.000

0.000

20.000

0.000

31.52*

12,737

178

8.285

-3.517

9.205

-3.907

18.000

0,000

179

8.*85

-8.*85

0.707

-0.707

7.778

7,778

180

13.*35

-13.*35

26,000

0.000

29.*56

-12,503

IBl

12.021

-12.021

29.*56

-12.503

28.991

-28.991

182

0.000

-15.000

10.11*

-25.03*

20.229

-50,068

183

0.000

-1*.000

11.238

-27.815

16.*8J

-♦0,796

16*

5.2*5

-12.981

10.*89

-25.961

22.627

-22.627

185

*,*95

-11.126

8.*85

•8.*85

19.799

•19.799

186

3.7*6

-9.272

6.7*3

-16.689

10.11*

-25.03*

187

8.285

-3.517

12.8117

.5.*70

20.251

-8,596

188

7.36*

-3.12*

12.000

0.000

21.000

0.000

189

7,000

0.000

13.000

0.000

19.000

0,000

190

6.*90

2.622

9.272

3,7*6

9.192

9.192

191

3.536

-3.53*

11.31*

-11,31*

19.799

-19.799

192

13.808

-5.851

12.728

-12,728

10,11*

-25.03*

193

7.000

0,000

8.000

0.000

7,36*

-3,126

19*

5.563

2.2*8

10.000

0,000

9,205

-3.907

195

7.35*

-3.12*

11.9*7

-5,079

9,205

-3.907

196

*,950

-*.950

0.000

-10,000

-11,331

-26,695

197

5.000

0.000

0.000

-1*,000

0,000

-22,000

198

0.000

-5.000

*.87o

-12,053

0.000

-I*. 000

199

7.36*

-3.12*

10.607

-10,607

9.900

-9,S99

200

*.950

-*.950

11.31*

-11,31*

16.263

•16,263

201

13.808

-5.861

19.331

-8,205

8.991

-22.252

202

7.778

-7.778

23.335

-23,33*

2*. 7*9

-2*, 7*9

203 .

5.619

-13.906

9.365

-23,180

8.991

-22,252

20*

0.000

-17.000

17.000

0.000

13.*35

-13, ♦SS

205

.5.080

-11.967

0.000

-12.000

13,808

-5.861

206

0.000

-7.000

7.36*

-3.126

16,000

. 0,000

207

-2.3**

-5.523

2.997

-7,*17

13.000

0.000

208

-3.907

-9.205

2.622

-6.*90

U.OOO

0.000

209

-9.192

-9.192

0.000

-13.000

8.285

-3.J17

210

.6.36*

-6.36*

0.000

-7.000

7.36*

-3,126

211

-5.657

-5.657

-8.*"5

-e.*e5

7.778

-7,778

212

•6,36*

-6.36*

-3.517

-8.285

9.205

-3.907

213

-7.071

-7.071

-3.536

3.536

6.36*

-6.36*

21*

2.622

-6.*90

1.873

.*.636

-8.203

-19,331

215

2.622

■-6.*90

0.000

-9.000

-8.987

-21,172

216

7.778

-7.778

.*.298

-10.126

-16.971

-16.971

217

3.7*6

-9.272

-3.907

-9.205

-13.*35

-13. ♦35

218

*.*95

-11.12*

0.000

-6.000

-12.021

-12,021

219

2.997

-7.*i7

-3.517

-8.285

-*.298

-10,126

220

0,000

-10.000

0.000

-5.000

-3.517

-8,285

221

-1.8*1

0.781

l.*98

-3.709

-1.12*

2.782

222

-l.*l*

-l.*l*

l.*l*

-l.*l*

2.735

6.***

223

2.121

-2.121

3.682

-1.563

3.126

7,»6*

22*

3.536

-3.536

5.523

-2.3**

5.657

5.657

225

*.603

-1.95*

7,000

0.000

*.2*3

*,2*3

3.7*6

226

5.000

0.000

8,000

0,000

9,272

227

6.000

0.000

3.709

1,*98

7.778

7,778

228

2.3**

5.523

8.3*5

3,371

9,000

0,000

229

*.2*3

*.2*3

*.950

*.950

9.000

0.000

230

6.*90

2.622

5.523

-2.3**

20.000

0,000

231

7.*17

2.997

0.921

-0.391

16.000

0.000

232

-0.391

-0.920

3.000

0.000

18.*10

-7,815

233

0.000

3.000

-2.000

0.000

15.6*9

-6,6*2

23*

2.735

6,***

0.000

3.000

12,021

-12,021

235

0.000

2.000

3.000

0,000

10.607

-10,607

236

-2.622

6.*90

l.*l*

-1,*1*

*.870

-12,053

237

0.000

7.000

-1,000

0,000

-*.689

-11,0*6

238

-0.000

9.000

-1,873

*.636

0.000

-12.000

239

-3.7*6

9.272

-1.873

*.636

0.000

-10,000

2*0

-2,997

7.*17

.0,921

0.391

10.607

-10,607

2*1

0,391

0.921

l.*l*

l.*l*

12.728

-12,728

2*2

0,000

5.000

*.*95

-11.126

23.933

-10.J59

86

Table 15 (continued)

INPUT VARIABLES

SEQ 12 3 4 5 6

2«3 «.«S0 4.930 12.000 0.000 19.331 -8.205

244 11.126 4.495 14.000 0.000 26.000 0.000

24J 10.000 0.000 21.000 0.000 24.000 0,000

246 15.000 O.OnO 19.000 0.000 22.092 -9,376

247 10.607 -10.607 23.335 -23.334 53.369 -22,662

248 7.492 -18.544 10.489 -25.961 20.506 -20.506

249 0.000 -14.000 7.492 -18.544 16.971 -16,971

250 0.000 -12.000 7.118 -17.617 15.556 -15,556

251 3.746 -9.272 8.485 -8.485 18.410 -7,815

252 3.000 0.000 8.000 0.000 15.000 0.000

253 4.636 1.873 11.126 4.495 20.000 0.000

254 3.536 3.536 4.950 4.950 11.314 11,314

255 3.517 8.285 14.142 14.142 16.971 16.971

256 1.485 8.485 8.205 19.331 21.920 21.920

257 8,485 8.485 14.142 14,142 23.335 23,335

258 4,243 4.243 11.314 U.314 11.722 27,615

259 0.000 6.000 9,899 9.900 8.987 21,172

260 1.563 3.682 3.90? 9.205 8.967 21.172

261 -3.000 0.000 2,622 -6.490 0,000 -3,000

262 5.657 -5.657 8.000 0.000 12.887 -5,470

263 9,000 0.000 9.272 3.746 10.199 4.121

264 5.657 -5.657 6.490 2.622 4,950 4,950

265 2,997 -7.417 4,603 -1.954 0.000 -7.000

266 -5,080 -11.967 -5.080 -11.967 -10,550 -24.854

267 -6.642 -15.649 -10.550 -24.854 -16.411 -38.661

268 -4.689 -11.04* -10. H9 -23.933 -13.285 -31,297

269 -1.563 -3.682 -8.205 -19.331 -14.848 -34,979

270 0.000 -9.000 0.000 -14.000 0.000 -24,000

271 4,950 -4.950 4.1J1 -10.199 0.000 -18,000

272 2.000 0.000 4.603 -1.954 3.746 -9.272

273 8,000 0.000 8.285 -3.517 6.000 0.000

274 7.000 0.000 4.910 4.950 13,908 5,619

275 9,272 3.746 13.000 0.000 13.908 5,619

276 8,345 3.371 12.053 4. 870 17.678 17.678

277 12,000 0.000 14.655 5.994 16.263 16,263

278 13.000 0.000 17.000 0.000 35.233 14.235

279 9.272 3.746 16.000 0.000 38.000 0.000

280 11.126 4.495 14.835 5.994 34.000 0,000

281 7.778 7.778 15,762 6.368 26.000 0.000

282 7.417 2.997 18.000 0.000 28.000 0.000

283 2.828 2.828 15.000 0.000 33.000 0,000

284 9.000 0.000 16.000 0.000 24.854 -10.550

285 7.000 0,000 15.000 0.000 32.218 -13.676

286 11.000 0.000 15.742 6.368 27.515 -11.722

287 14.835 5.994 12.8§7 -5.470 31.297 -13.285

288 16.000 0.000 15.649 -6.642 28.000 0,000

289 21.325 8.614 21.000 0.000 23.013 -9,768

290 12.887 -5.470 19.000 0.000 26.000 O.OOC

291 22.000 0.000 25.000 0.000 14.835 6.994

292 14.000 0.000 15.649 -6.642 21.000 0.000

293 13.908 5.619 20.251 -8.596 24.107 9.740

294 14,000 0.000 12.053 4.870 29.000 0,000

295 12.981 5.245 15.589 5.743 32.000 0,000

296 18,000 0.000 25.000 0.000 29.670 11.987

297 17.000 0.000 23.013 -9.756 43.000 0,000

298 21.000 0.000 26.000 0.000 27.000 0,000

299 10.607 -10.607 19.000 0.000 34.059 -U.i,57

300 7.071 -7.071 19.331 -8.205 35.820 -15,629

301 4.495 -11.126 19.331 -6.205 38.661 -16.411

302 0,000 -4.000 5.619 -13.908 22.627 -22,627

303 2.000 0.000 11.045 -4.689 18.385 -18,385

304 6.000 0.000 12.000 O.OOO 19.331 =8.205

305 7,417 2.997 21.000 0.000 24.000 0,000

306 18.000 0.000 17.4<S0 -7.424 18.385 -18.385

307 15,649 -5.642 14.7?8 -5.252 15.556 -15,556

308 11.046 -4.689 12.887 -5.470 21.172 -8.987

309 9,272 3.746 13.000 0.000 20.251 -8,596

310 16,000 0.000 16.000 0.000 12.000 0.000
3il 14.728 -6.252 11.967 -5.079 21.000 0,000

312 12.961 5.245 17.000 0.000 17.490 -7,424

313 15.762 6.368 21.000 0.000 18,410 -7,815

314 18.544 7.492 22.000 0.000 16.000 0.000

315 18,544 7.492 24.000 0.000 24.000 0,000

316 24.107 9.740 15.5S6 15.556 32.451 13.111

317 20.398 8.241 28.7*3 11.613 20.506 20.506

318 16.689 5.743 20.308 8.241 25.456 25.456

319 21.213 21.213 21.213 21.213 18.385 18.385

320 15.762 5.368 13.000 0.000 32.451 13,111

321 12.981 5.245 2*.0oo 0.000 26.000 0,000

322 9.192 9.192 34.000 0.000 21.325 8,616

323 5.079 11.957 15.619 6.743 21.325 8.616

324 15.556 15.556 5.470 12.887 21.325 8,616

325 13.435 13.4SS 5.079 11.967 10.159 23.933

326 16.971 16.971 20.506 20.506 12.113 28,536

327 11.385 18.385 29.698 29.699 21.881 51.548

328 10.199 4.121 14.457 34.059 -0.000 78.000

329 9.272 3.746 13.000 0.000 -0.000 72,000

330 7.364 -3.126 5.563 2.248 -23.226 57,485

331 5.657 -5.657 3.371 -8.345 -4.689 -11.046

332 3.371 -B.34S 0.000 -32.000 0.000 -41.000

87

Table 15 (continued)

INPUT VARIABLES

SEQ

1

2

3

4

5

6

333

4,870

-12.053

16.897

-♦1.723

20.229

-90.066

334

12.7J8

-12.728

24.7*9

-2*. 7*9

36.184

-38.184

33S

19-331

-3.205

32,216

-13.676

92.469

-22.272

336

14.728

-6,252

28,536

-12.113

38.184

-38.184

337

12.887

-5,470

23.933

-10.159

49.707

•21.099

33S

7.778

-7,778

19,799

-19.799

26.870

-26,870

339

7.778

-7.778

18.395

-18.385

15.73*

"38.942

3*0

-6.364

-6.364

0.000

-15.000

7.867

-19.471

3*1

0.000

-8.000

0.000

-15.000

!3,*39

-13,435

3*2

.4.2*3

-4.2*3

0.090

-12.000

4.368

-19,762

3«.3

-5-657

-5.697

-3.517

-e.2es

-15.762

-6.366

3*'.

-3.536

-3.53*

-4.990

-4,950

-9,899

-9.899

345

0.000

-6.000

o.oco

.9.000

-8.34J

-3,371

346

0.000

-6,000

•3.126

-7.364

-7.778

-7,776

347

-6.000

0,000

-9.697

-5,657

-5.657

-5.697

348

-1 .«54

-4,603

-1.5f3

-3,682

•7;o7i

-7.071

349

-4.243

-4.243

-6,3»*

-6,364

-17.000

0.000

350

-1.954

-4.603

-7.417

-2.997

•13.906

-5.619

351

-5.000

0.000

.2.826

.2.628

-12.021

-12.021

352

1.172

2,762

-4,603

1.954

-22.252

-8.991

353

0.000

3.000

-0.375

0.927

-12.053

-4,870

354

1.172

2.762

2,3**

5.523

-6.490

•2.622

355

4.2'<3

4.2*3

4.689

11.046

. 3.682

1,563

356

1.563

3.682

5,079

11.967

-3.371

8,3*5

357

2.782

1.12*

J. 907

9.205

-0.000

15.000

339

2.121

2,lSl

4.298

10.126

-0.000

19,000

359

1.563

3.682

6.364

6.364

7.819

18,*10

360

15,000

0,000

9.192

.9,192

30.377

-12,89*

361

10.126

-4.298

13.000

0.000

26.936

-12.113

362

11.000

0.000

22.000

0.000

23.180

9.365

363

22.000

0.000

25.000

0.000

25.961

10. 489

364

18.410

-7. 815

23,013

.9.768

44.000

0.000

365

14.849

-14.649

18,385

-16.385

9.740

-24.107

366

3.746

-9.272

7.071

.7.071

12.887

-5.470

367

10.126

-4.298

11.126

4.495

23.000

0.000

368

7.778

7.778

9.192

9.192

31.52*

12.737

369

12.887

-5.470

14.7J8

•6.252

24.854

-10,550

370

21.000

0,000

30.507

12.362

54.704

22.102

371

19.000

0.000

22.29?

8.991

28.743

11.613

372

12.887

-5.470

8.241

-20.398

12.728

-12.728

373

9.192

-9.192

17.678

-17.678

23,335

-23.334

374

11.046

-4.689

21.172

•8.987

33.138

-14.066

375

6.364

-6.36*

13.808

-5.861

31.297

-13.289

376

6.444

-2,735

10.126

•4.298

15,762

6.368

377

8.285

-3,517

6.489

•8.485

12.887

-5.470

378

1.485

-8.485

7,071

"7.071

11.967

-5.079

379

5.657

-5.657

9.192

.9.192

11.046

-4.689

380

9,192

-9.192

12.021

-12.021

15.556

-15.556

361

14,835

5.99*

26.000

0,000

31.000

0,000

382

17.000

0.000

21,000

0,000

25.000

0,000

383

12,887

-5.470

19,000

0.000

23.000

0.000

384

8,485

-8.485

18,000

0,000

25.000

0.000

385

7.778

-7.778

15,000

0.000

23.000
23.93J

0-000
-10.159

386

7.417

2.997

15,000

0,000

387

9,27?

3.7*4

17.000

0.000

24.654

-10.550

388

5.523

-2.344

18,000

0.000

26.695

-11.331

389

3.682

-1.563

10.126

-4.298

2*. 85*

-10.J50

390

3.709

1.498

14.000

0.000

23.013

-9,769

391

1,485

8.485

14.000

0.000

22.000

0,000

392

7.776

T.778

10.000

0.000

26.000

0,000

393

6,364

6.364

3.536

3.5S6

22.000

0.000

394

9.272

3.746

10.000

0.000

24.000

0,000

395

3.907

9.205

6.000

0.000

28.000

0.000

396

7.071

7.071

7.417

2.997

20.000

0.000

39?

5.657

5.697

13,908

5.61?

21.000

0.000

398

9.192

9.192

16.000

0.000

26.000

0,000

399

12.981

5.2*5

12.981

3.245

16..69f

6.743

400

14.639

5.994

18.410

.7.815

22.000

0,000

401

13,908

5.619

19,000

0,000

16.000

0.000

402

16.000

0.000

12,991

5.245

U'.OOO

0,000

403

17.617

7.118

19.471

7.867

21.325

8,616

404

17.000

O.QOC

16.971

16.971

23.339

23.339

405

22.252

8.991

15.238

35.900

16.41V

38.661

406

14,849

14,849

17.1«2

40.502

35. 35 J

35.355

407

17.617

7.116

21.881

51.548

22,662

93,389

408

21,000

0,000

44.000

0.000

38.18*

36,184

409

U.046

-4,689

56.000

0.000

54.70*

22,102

410

14,1<>2

-14.142

27.619

-U.722

53,000

0.000

'. U

8,485

-8.485

31.297

-13,285

35.900

-15,238

412

7.364

-3.126

21.920

-21.920

22.627

-22,627

413

. 4.243

-4.2*3

7,071

.7.071

22.627

-22,627

414

6.000

0.000

U,967

-5.079

28.536

-12.113

415

8,000

0.000

14.728

.6.292

12.728

-12,T28

416

7.000

0.000

12.887

•5.470

8.48}

-8.499

417

4.000

0.000

11,046

.4.689

6.46!

-8, 485

418

6.490

2.622

4,602

-1.994

5.697

-9.657

419

9.000

0.000

9.000

0,000

4;634

1.873

420

7.364

-3.126

9,000

0.000

10.607

10,607

421

5.563

2.248

7,000

0.000

10.199

4.121

422

9.272

3.7*6

9,000

0.000

10.000

O.QOO

88

Table 15 (continued)

INPUT VARIABLES

SEQ

423
*2«
429

426
♦ 27
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
44S
449
450
451
452
453
454
455
456
457
498
459
460
461
462
463
464
469
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
489
486
487
488
489
490
491
492
493
494
499
496
497
498
499
900
901
902
903

4.835
4.142
5.762
2.292

1.000

2.981

1.0'i6

7.364

7.071

4.000

1.124

-1.95*

-0.927

0.000

0.000

1.854

5.523

•2.828

-5.6'7

0.000

0.000

0.000

0.000

0.000

0.000

3.371
0.000
2.997
3.371
4.870
5.994
9.619
7.071
7.778
11.967
6.444
13.000
12.000
18.000
13.808
6.364
6.444
6.000
7.000
9.000
3.908
2.728
0.398
5.034
32.451
5.000
6.263
1.213
5.556
5.649
2.728
5.649
6.569
9.192
9.192
7.364
9.205
5.657
9.899
14.849
6.364
5.470
4.298
7.071
8.596
9.192
4.000
2.053
1.126
6.689
0.199
2,000
4.728
8.410
4.726
7.490

5.994

14.142

6.368

8.991

0.000

5.245

-4.689
-3.126

-7.071

0.000

-2.782

-4.603

-0.375

5.000

6.000

0.749

-2.344

-2.828

-5.6S7

-9.000

-12.000

-17.000

-13,000

-10.000

-13.000

-8.345

-9.000

-7.417

-8.345

-12.053

-14.835

-13.908

-7.071

-7.778

-5.079

-2.735

0.000

0.000

0.000

-5.861

-6.364

-2.735

0.000

0.000

0.000

5.619

12.728

8.241

10.114

13.111

0.000

•16.263

-21.213

-15.556

-6.642

•12.728

-6.642

-7.033

-9.192

-9.192

-3.126

-3.907

5.657

9.900

14.849

6.364

12.887

10.126

7.071

20.251

9.192

0.000

4.870

4.495

6.743

4.12J

0.000

-6.252

-7.815

-6.252

-7.424

9.899
17.617
10.607
21.920
15.762
20.338
18.544

8.000

8,000
10.000

4.603

.090
.8*1
,3*4
,000
,4*4
,657
,245
,000
0.000

o.o'oo

-5.8*1
•5.3*1
0,000
-7.033
4,495
0.000
0.000
4.870
0.000

0.000

9.3*5
11.314

7.118
10.607
10.607
19.000
19.000
23.000
27,000

6,364
10,000

6,000
10.126
IS.fOB
11.9*7
18.544
29.034
41.723
22.627
49.432
34.648
27.577
22.627
19.799
16.2*3
20.506
16.263
20.291
14.849
10.607
11.9*7
15.7*2
19.956
17". 67 8
17.617

8.205

8.987
12.894

-0.000

5.861

9.899

17.617

8.48!>

9.657

6.3*4

9.900

14.142

20.506

18.385

17.000

9.900

7.118

10.607

21,920

6.36B

8.2*1

7.492

0.000

0,000

0.000

-1.954

0.000

-0,781

•3,126

0,000

-2,735

-5,697

-12.981

-12.000

-13.000

-13.000

-13,808

-13,808

-16.000

-16,569

-11,126

-6.000

-12.000

-12,053

-16,000

-26.000

-23.180

-11.314

-17.617

-10,607

-10,607

0,000

,000

,000

.000

.3H>

,000

.000

,298

,861

.079

.492

.114

•5.

-5.

7.
10,

16.857

22.627

18.356

-34.648

-27.577

-22.627

-19.799

-16,263

-20.906

-16.263

-8.596

-14.849

-10.607

•5.079

6.368

15.596

17.678

7.118

19.331

21.172

30.377

25.000

13.808

9.900

7,118

8.485

5.697

•6.364

•9.899

-14.142

-20.906

-18.385

0.000

16.689

23.000

20.398

18.544

19.799

25.4"56

15.238

12.000

16.000

14.835

12.981

1.873

8.485

10.126

10.607

14.1*2

15.359

31.113

22.627

0.000

-10.199

-37.087

10.489

0.000

5.619

11.0*6

•4.000

0.000

10.489

0.000

11.238

11.238

9.7*0

10.489

16.263

16.569

22.000

28.000

33.000

33.000

26.69S

11.046

13.435

19.649

21.920

27.619

32.000

4 3.000

46.000

48.214

60.104

41.719

28.991

28.284

22.627

24.854

22.627

21.213

19.799

14.142

13.000

17.490

20.000

23.961

21.329

28.74)

26.163

8.996

28.991

-0.000

5.861

5.079

5.470

9.470

1.414

7.778

12.021

18.383

37.477

36.820

39.900

6.743

0.000

8.241

7.492

19.799

25.456

35.900

0.000

0.000

5.994

5.245

-4.636

-8.485

-4,298

-10.607

-14.1*2

-38.015

-31.113

-22.627

-14.000

-4.121

-14.984

-25.961

-22.000

-13.908

-4.689

0.000

-21.000

-25.961

-43.000

-27.815

-27.815

-24.107

-25.961

•16.263

-7.033

0.000

0.000

0.000

0.000

-11.331

-4.689

-13.435

-6.642

-21.920

-U.722

0.000

0.000

0.000

19.480

60.104

•41.719

•28,991

•28,284

-22,627

•10,550

•22.627

-21,213

•19.799

•14.142

0.000

-7.424

0.000

10,489

8.616

11.613

26.163

20.251

2A.991

58.000

13.808

11.967

12.887

12.887

1,414

•7,778

•12,021

•18,185

-37,477

•19,629

-15,238

89

Table 16 Separation of a multivariate (6) set of Rantoul , Illinois,
October 1950-55, upper wind components, zonal and meridional
mixed distribution, at the 700-, 500-, and 300-mb levels into
two separate distributions. The units are m-s"^.

PROGRAM NORMIX
Mifi NQRMiL MIXTURE ANALYSIS PROCgDURE ( 197* REVISION)

SAMPLE • RANTDUL
OCTOBER

70a/500»AND 300MB
X AND Y COMPONENTS

SAMPLE SIZE • S03

NUMBER OF VARIABLES • »
NUMBER HE TYPES • f

ITERATION NUMBER 39

ClKELIHDDO OF 2 TYPES IN THIS SAMPLE • -0.73297066D 0*

CHARACTERISTICS OF THE WHOLE SAMPLE

MEAKS
12 3*56

7.3252 -0.9*86 10.6700 -2.570* 15.*0*9 .3.6999

STANPA»0 DEVIATIONS
1 2 3 * 5 *

6.6262 6. 8921 8.959* 10.31*9 13.70B2 16.0056

CORRELATIONS

1

2

3

*

5

6

I. 0000

0.2215

0.752S

0.2313

0.5589

0.2311

0.2215

1.0000

O.U3«

0.7581

0.1265

0'.5757

0.752!

0.1138

I. 0000

0.1587

0.7513

0.1977

0.2313

0.7581

0.1587

1.0000

0.1200

0.7856

0.5589

C.1265

0.7513

0.1200

I. 0000

0.1266

0.2311

0.5757

0.1977

0.7856

0.1266

I '.0000

CHARACTERISTICS OF TYPE 1

THE PROPORTION OF THE POPULATION FROM THIS TYPE. 0.171

MEANS
12 3*'?

10.*5*5 -2.0123 16.2238 -2.6858 18.763* .3.325*

1 2

STANDARD DEVIATIONS

3*5

5 *

7.*317 9.0069 12.5099 17.*952 20.1227 ?9.783*

CORRELATIONS

lloOOO 0^3578 o!607* 0.**01 0.5639 0.3522

0.3578 1.0000 0.0J97 0.7*69 0.0730 '•""

0 6076 i.0297 1.0000 0.1637 0.7*20 0.1966

O.**0l 0.7*69' 0.U37 1.0000 0.11*8 °'l*.lt ■

0.5639 0.0730 0.7*20 0,11*8 1.0000 0.0799

o!3522 0.6333 0.1966 0.788* 0.0799 1.0000

EIGENVALUES EIOENVEeTORS 3*56

,164 8027 o!l01* o!l798 0.1382 0.2729 o;.*299 0-"*0

iSV??79 0 86* .0.0377 0.38*0 -0,17*9 0.7770 .0.*266

!oe *213 0 1060 0.**67 -0.20*3 0.7783 0.0*92 -O-^Ol

5^7679 0 **98 -0.0*90 0.7*97 0.199* .0.*398 -0.0070

lo'Jell o'.ull 0 8592 0.0*71 .o.*e2* -o nss 0.0091

22 0357 0 8529 -0.161S .0.*76e -0.1293 0.0181 0.0*J7

CHARACTERISTICS OF TYPE 2

THE PROPORTION OF THE POPULATION FROM THIS TYPE. 0.829

MEANS
12 3*56

6. 6783 -0.7287 9.5n8 -2.5*66 1*.7107 -J. 7773

STANDARD DEVIATIONS
12 3*5*

6.25*8 6.3*57 7.5377 8.0693 11.8*22 11.2131

90

Table 16 (continued)

CnnRELATIONS

1.0000
0.2039
0.8038
0.1453
0.5563
0.1878

EICFNVALUES

236.2003

155.9890

31.5369

29.5496

8.3730

6.7008

0.2039

l.oono

0.1937
a.79<.3
0. 1673
0.5'''t9

0.8038
0. 1937
l.UOOO
0.1721
0.7*26
0.2195

EIGENVECTORS

1
0.2*83
0.2529
0.3590
0.3591
0.5713
0.5387

-0.2052
0.2350

-0.3U0
0.4036

-0.5859
0.5457

0

7943

0

1673

0

1721

0

7625

1

0000

0

1260

0

1280

1

0000

0

7914

0

1756

}

4

0

6858

-0

0776

0

1020

0

6736

0

4636

-0

0919

0

0176

0

4196

0

5400

0

0860

0

1120

-0

5902

0.5849

0.2195

0.7914

0.1756

1.0000

5

6

0.3252

0

5605

0.5317

-0

3660

0.4232

-0

6083

0.6195

0

3846

0.0879

0

1540

0.2024

-0

1008

SAMPLE • RANTOUL
OCTOBER

700/500*4NB 300MB
X ANO y COMPONENTS

PROBABILITIES
OF TYPE MEMBERSHIP
1 2

0.065 0.935
0.011 0.989
0.015 0.985
0.996 0.004
0.022 0.979

6 0,037

7 0.026

8 1.000

9 0.119

0 0.851

1 0.981

963
974
000
881
149
019

0.005 0.995
0.007 0.993
0.003 0.997
0.122 0.878

0.976
0.034

8 0.054

9 0.088

0 1.000

1 0.003

2 0.125

3 0,269
0,011
0.005

0,024
0.966
0.9*6
0.912
0.000
0,997
0,875
0.731
0.989
0.995

1,000 0,000
0.028 0,972

987
706

0.013
0.294

0,218 0.782

0.029 0.971

0.034 0,966

0.011 0,969

0,037 0,963

0,407 0.593

0.011 0.989

37 0.008 0.992

8 O.OOS 0.995

9 0.005 0,995

0 0,003 0,997

1 0,323 0.677

0.006 0.994
0.0*3 0.957

0.017
0,025
0.071

0,983
0,975
0,929

5

6

7 0,016 0,9S4

8 0,916 0,084

9 0,010 0,990

0 0.013 0.9J7

1 0,013 0,9S7

2 0,181 0,819

3 0,020 0.960

4 0.996

5 0,005

6 0.009 0,991

7 0.012 0.968

8 0.030 0,970

9 0.028 0,972
0,008 0,992

0.004
0.995

0.008 0.
0.008 0,
0.004 0,
0.003 0,
0,017 0,

992

992
996
997
983

0.993
0.995
0,979

,991
.978
,866
,919
.968
,766

PROBABILITIES
OF TYPE MEMBERSHIP

1 2

66 0,010 0,990

67 0,009 0,991

68 0,007 0.993

69 0,005 0.995

70 0.007 0.993

71 0,007 0,993

72 0,004 0.996

73 0.007

74 0.005

75 0.021

76 0.010 0.990

77 0,093 0.907
76 0,024 0.976

79 0,025 0,975

80 0.014 0,986

81 0,036 0.962
8? 0,010 0,990
63 0,050 0.950

84 0,163 0.837

85 0,066 0.934

86 0.008 0,9y2
0,016 0.984
0.009
0.022
0.114
0,081
0,032
0,234

9» 0,048 0,952

95 0,014 0.986

96 0.205 0,795

97 0.031 0.969

98 0.058 0.942

99 D.O55
lOP 0.013

101 0.015

102 0.403

103 0.873

104 0.009

105 0.009

106 0,045

107 0.875

108 0,028 0.972

109 0.014 0,986

110 0,010 0,990
in 0,008 0,992

112 0,004 0,996

113 0.009 0.991

114 0,005 0,995

115 0,026 0,974

116 0,004 0,996

117 0,013 0,987

118 0.037

119 0.013

120 0,009

121 0,034

122 0,023 0.977

123 0.019 0.981

124 0.069 0.931

125 0,685 0,315

126 0.106 0.694

127 0,615 0,385

128 0.811 0,189

129 0,892 0.108

130 0,006 0,994

87
86
89
90
91
92
93

.945
0.987
0.985
0.597
0.122
0.991
0.991
0.955
0.125

0.963
0.987
0.991
0.966

PROBABILITIES
OF TYPE MEMBERSHIP
1 2

131

0

009

0

991

132

0

037

0

963

133

0

075

0

925

134

0

006

0

994

135

0

019

0

981

136

0

008

0

992

137

0

018

0

982

136

0

016

0

984

139

0

027

0

973

140

0

926

0

074

141

0

998

0

002

142

0

030

0

970

143

0

240

0

760

144

0

023

0

977

145

0

069

0

931

146

0

027

0

973

147

0

on

0

989

146

0

016

c

984

149

0

010

0

990

150

0

020

0

980

151

0

061

0

939

152

0

056

0

944

153

0

047

0

953

154

0

272

0

728

155

0

033

0

967

156

0

014

0

986

157

0

019

0

981

156

0

on

0

989

159

0

038

0

962

160

0

006

0

994

161

0

003

0

997

162

0

002

0

996

163

0

004

0

996

164

0

003

0

997

165

0

004

0

996

166

0

004

0

996

167

0

015

0

985

168

0

005

0

995

169

0

004

0

996

170

0

.004

0

996

171

0

,005

0

995

172

0

092

0

908

173

0

on

0

989

174

0

0:3

0

987

175

0

013

0

987

176

0

023

0

977

177

0

098

0

902

178

0

■ 007

0

993

179

0

.311

0

689

180

0

998

0

002

181

0

997

0

003

182

0

977

0

023

183

0

.857

0

1*3

184

0

.204

0

.796

185

0

,043

0

,957

186

0

,020

0

.980

187

0

.007

0

.993

138

0

,008

0

.992

189

0

,006

0

.994

190

0

,010

0

.990

191

0

.020

0

.980

192

0

,063

0

.937

193

0

.007

0

.993

194

0

.006

0

994

195

0

.012

0

988

91

Table 16 (continued)

PROBABILITIES
OF TYPE MEMBERSHIP
1 2

19t, 0,208 0.792

197 0, 121 0.B79

199 0.023 0.977

199 0,015 0.985

200 0.013 0.987

201 0.397 0.603

202 0.983 0.017

203 0.073 0.927
20* 1.000 0.000

205 0.005 0,095

206 0.006 0.99*

207 0.005 0.995

208 0.005 0.995
20' 0.021 0.979
2^0 0,003 0.997

211 O.OOe 0.992

212 0.003 0.997

213 0.022 0.978

214 0.107 0.893

215 0,0*5 0,955
214; 0,371 0.629

217 0.066 0.93*

218 0.131 0.669

219 0.019 0.961

220 0.01* 0.986

221 0.01* 0,986

222 0,007 0,993

223 0,013 0,987
22* 0.011 0.989

225 0.010 0.990

226 0.006 0.99*

227 0.013 0.987

228 0.005 0.995

229 0,005 0,995

230 0.011 0.989

231 0.062 0.938

232 0.00* 0.996

233 0.011 0.989
23* 0.035 0.965

235 0.00* 0.996

236 0.006 0.99*

237 0.015 0.985

238 0.025 0.975

239 0.007 0.993
2*0 0.005 0.995
2*1 0,012 0.988
2*2 0.181 0.819
2*3 0.007 0.993
24* 0.008 0.992
2*5 0.037 0.963
2*6 0.03* 0.966
2*7 0.922 0.078
2*8 0,08* 0,916
2*9 0,020 O,980

250 0,017 0,983

251 0,006 0,99*

252 0,003 0,997

253 0,006 0.99*
25* 0.006 0.99*

255 0,026 0.97*

256 0.213 0.787

257 0,033 0,967

258 0,211 0.769

259 0.063 0.937

260 0.025 0.975

261 0.017 0.983

262 0.016 0.96*

263 0.012 0.988
26* 0.033 0.967

265 C.023 0.977

266 0.028 0,972

267 0,200 0,800

268 0.197 0.803

269 0.6*9 0.351

270 0.013 0.987

271 0.015 0.965

272 0.005 0.995

273 0.011 0.989
27* 0.018 0.982

275 0.012 0.988

276 0.031 0.969

277 0.057 0.9*3

278 0.087 0,913

279 0,017 0.983

280 0,029 0.971

281 0,009 0.991

282 0.013 0.967

283 0.012 0.988
28* 0.023 0,977
283 0.058 0.9*2

PROBABILITIES
OF TYPE MEMBERSHIP
1 2

286 0.566 0.43*

287 0.1*5 0.855

288 0.032 0.968

289 0.056 0,9**

290 0.0*1 0.959

291 0.132 0,868

292 0,020 0,960

293 0.929 0.072
29* 0.1*5 0.855

295 0.028 0.972

296 0.088 0,912

297 0.195 0.805

298 0.039 0.961

299 0.933 0.067

300 0.103 0.997

301 0.*15 0,585

302 0,023 0,977

303 0,02* 0,976
30* 0,008 0,992

305 0,069 0.931

306 0.0*6 0.95*

307 0.051 0.9*9

308 0.010 0.990

309 0,009 0,991

310 0,020 0,980

311 0,030 0,970

312 0,016 0,98*

313 0,038 0.952
31* 0,050 0,950

315 0,035 0,965

316 0,96* 0,036

317 0.*e6 0,51*

318 0,22* 0.776

319 0.171 0.829

320 0.36* 0.636

321 0.166 0.83*

322 1,000 0.000

323 0,023 0,972
32* 0,811 0, 189

325 0,6*3 0,357

326 0,367 0,633

327 0,999 0,001
320 1,000 0,000

329 1.000 0.000

330 1,000 0,000

331 0.018 0.982

332 0.980 0.020

333 1.000 0.000
33* 0,931 0.069

335 0.796 0.20*

336 0.998 0.002

337 0.635 0.*65

338 C.*76 0.52*

339 0.9*0 0.060
3*0 0.009 0.991
3*1 0.011 0.989
3*2 0.005 0.995
3*3 0.037 0.963
3** C.006 0.99*
3*5 0.026 0.972
3*6 0.011 0.989
3*7 0.005 0.995
3*8 0,007 0,993
3*9 0,0*7 0.953

350 0,019 0.981

351 0.013 0.967

352 0.206 0.792

353 0.021 0.979
35* 0.020 0.980

355 0.080 0.920

356 0.053 0,9*7

357 0.036 0.96*

358 0.055 0.9*5

359 0.02* 0.976

360 0.310 0.690

361 0.130 0.870

362 0,111 0.869

363 0,069 0.931
36* 0.139 0.861

365 0,193 0.807

366 0.007 0.993

367 0,101 0,899

368 0,031 0.969

369 0.015 0.965

370 0.95* 0.0*6

371 U.168 0,912

372 0.386 0.612

373 0,090 0.910
37* 0.0*8 0,952
375 0.026 0,97*

PROBABILITIES
OF TYPE MEMBERSHIP
1 2

376 0.016 0-98*

377 0,008 0,992
3/9 0,011 0.989

379 0.010 0.990

380 0.012 0.989

381 0.06* 0.936

382 0,017 0.983

383 0.04* 0,956
38* 0,119 0,881
365 0,0*7 0,953

386 0,01* 0,986

387 0.020 0.980

388 0.107 0.893

389 0.006 0,99*

390 0i0l5 0.985

391 0.012 0.988

392 0,013 0.987

393 0.027 0.973
39* 0,008 0,992

395 0.028 0.972

396 0.007 0.993

397 0.007 0,993

398 0.017 0.983

399 0.011 0.989
*00 0.207 0.793
*01 0.02* 0.976
*02 0.087 0,913
*03 0,021 0,979
*0* 0,9*2 0.056
*05 I. 000 0.000
*06 1,000 0,000
»07 1,000 0.000
*08 1.000 0.000
»09 1.000 0,000
*10 0,816 0,18*
*ll 0,996 0,00*
*12 0,991 0,009
*13 0,0*5 0,955
*1* 0,010 0.990
*15 0.025 0,975
*16 0,020 0,980
*17 0.01* 0.986
*18 0,007 0.993
*19 0,012 0,9S8
*20 0,02* 0,976
*21 0,006 0,99*
*22 0,008 0.992
*23 0,099 0,901
*2* 0,021 0,979
*25 0,157 0.843
*26 0,992 0.008
*27 0,09* 0.906
*2e 0,238 0,762
*29 0,997 0.003
*30 0,008 0.992
*31 0,023 0.977
*32 0.007 0.993
433 0.005 0.995
*3* 0,006 0,99*
*35 0,003 0.997
*36 0,010 0.990
437 0,005 0,9«S
*36 0,006 0,994
4J9 0,980 0.020
4*0 0,292 0,708

441 0,015 0.985

442 0,007 0,993

443 0.062 0.918

444 0,967 0.033

445 0.161 0,839

446 0.011 0,989

447 0.055 0.945

448 0.009 0«'9l

449 0.015 0.985
490 0,015 0,965

451 0,025 0.975

452 0,914 0.086

453 0,183 0,817
4}4 0,061 0,9J9
4J5 0.048 0.9S2

456 0,033 0.967

457 0,015 0,985
453 0.014 0,986
439 0.014 0.986

460 0.012 0,988

461 0,023 0,977
*62 0.37* 0.626
*63 0,028 0.972
464 0,C10 0.990
*65 0.U21 0.979

92

Table 16 (continued)

PROBABILITIES

OF TYPE MEMBERSHIP
1 2

466 0,009

0.995

OftT 0.055

0.945

♦66 0.071

0.929

»69 0.019

0.981

*70 0.265

0.735

'.71 U.999

0.301

*72 1.000

0.000

*73 1.000

0.000

<.7» 1.000

0,000

*73 0.556

0,444

»76 0.241

0,759

477 0.132

0:868

478 0.030

0.970

PROBABILITIES PROBABILITIES

OF TYPE MEMBERSHIP OF TYPE MEMBERSHIP

12 12

47? 0.194 0.806

480 0,037 0.963

481 0.362 0.638

482 0,032 0.968

483 0,051 0,949
46^ 0,007 0.993

485 0.015 0.985

486 0,029 0.971

487 0,067 0,933
48H 0,015 0,985

489 0,054 0.946

490 0.087 0.913 503 0.955 0,045

491 0.998 0.OO2

492

1

000

0

000

49=<

0

045

0

955

494

0

312

0

688

495

0

!71

0

829

4S6

0.J41

0

959

497

0

332

0

66S

498

0

035

0

965

499

0

023

0

977

500

0

.023

0

977

501

&

,785

0

215

502

0

.149

0

851

LOGARITHM OF LIKELIHOOD RATIO OF J TO I TYPtS • 0.169250330 03
CHI-SOUARE WITH 54 DEGBEES OF FRFETOi'. 372,86
PI<a8'BILITY OF NULL HYPOTHESIS, 0.00000000

9?

Table 17 Separation of a multivariate (6) set of Rantoul , niinois,
October 1950-55, upper wind components, zonal and meridional
mixed distribution, at the 700- , 500- , and 300-mb levels into
three distinct distributions. The units are m-s"^.

PROGRAM NORMIX
WOLFE NORMAL MIXTURE ANALYSIS PROCEDURE (1974 REVISION)

SAMPLE • RANTOUL
OCTOBER

700/500/ANO 300M8
X 4ND Y COMPONENTS

SAMPLE SHE ■ SOS

NUMBER DF VARtiBLFS • 6
NUMBER OF TYPES ■ 3

ITERATION NUMtER 101

LIKELIHOOD OF 3 TYPES IN THIS SAMPLE . -0,727177290 0*

CHARACTERISTICS OF THE WHOLE SAMPLE

MtANS

1

2

3

4

5

6

7.J232

-0.9'i86

10.6700

.2.570*

IJ.4049

.3'. 6999

STANDARD OBVIATIONS

1

2

S

*

5

6

6.62D2

6.6921

8.9S9A

10.31*9

13.70S2

16'.0OS8

CPRRELATIONS

1

2

3

*

5

6

I. 0000

0.2215

0.7525

0.2313

0.5389

0'.231l

0.2215

1.0000

0.113S

0.7581

0.126S

0-.5TJ7

0.7525

0.1138

1.0600

0.1587

0.7513

0'.1977

0.2313

0.7581

0.1587

1.0000

0.1200

0'.7I56

O.SJJ?

0.1265

0.7513

0.1200

1.0000

0.1266

0.2311

U.5757

0.1977

0.7B56

0.1266

r.oooo

CHARACTERISTICS OF TYPE 1

THE PROPORTION OF THE POPULATION FROM THIS TYPE» 0.182

1

2

3

4

J

6

10.5173

-1.7616

16,0**7

-2.8166

19.4572

-3.J912

STAi^DARD DEVIATIONS

1

2

3

4

5

*

7.<.715

9.10*3

12.1651

17.3130

19.5*70

29.2311

correCations

1

2

3

4

5

6

I. 0000

0.2932

0.6046

0.3978

0.5377

0.3250

0.2932

1.0000

-0.0231

0.7480

0.05*0

0.6252

0.60«6

-0.0233

1.0000

0.1336

0.719*

o;iTi9

0.3978

0.7480

0.1S36

1.0000

0.0917

0.790*

0.5377

0.05*0

0.7194

0.0917

1.0000

O'.0S73

0.3250

0.6252

0.1719

0.7906

0,0573

I'.OOOO

EIGENVALUES

EIGENVECTORS

1

2

3

*

3

6

1122.2211

0.0936

0.1893

0.0998

0.3370

o;si3*

0.7539

481. ^OS*

0.1910

-0.0S7J

0.4161

-0.1832

0.72*1

-0.4807

107.0029

0.0873

0.447S

-0.2426

0.733*

0.0*23

-0.4409

59.8619

0.4557

.0.0371

0.7255

0.2552

-0'.*»*4

0.0433

29.3601

0.0895

0.8629

0.0824

.0.*746

.0.1111

0.0303

23.1239

0.8353

.0.128*

-0.4742

-0.1571

0;0274

0.0403

CHARACTERISTIeS OR TYPE 2

THE PRDPCtRTION OF THE POPULATION FROM THI$ TVPB» 0.183

MEANS
^23456
5.9298 1.1741 11.3*42 0.4841 19.9322 .4.8802

4.0432

STANDARD DEVIATIONS
2 3 4 J

6.0597 6.8744 4.7102 8.3419

6
7.2*47

94

Table 17 (continued)

CBRdELATinNS

; ' 1

2

S

4

5

t,

1.0000

•0.3849

0

7454

-0.0248

0.7052

0'.3055

■ -0.3849

1.0000

-0,

5189

0.6927

-0.5375

0;4247

0.74 J*

-0.91S9

1.

0000

-0.3651

0.8079

0.0263

-0.0248

0.6927

-0.

3451

I. 0000

-0.3992

o;7287

0.70S2

-0.5375

0.

8079

-0.3992

1.0000

-0.1160

0.30S5

0.4247

0.

0263

0.7287

-0.1160

r.oooo

EIGENVALUES

EIOBNVECTOR!

2

3

4

1

6

J37.9714 0

234S

0.2674

.0

1184

-0.0908

0'

5475

0.7426

71.7321 .0

3902

0.2424

0

7711

0.3460

-0

0«10

0.2609

16.9294 0

5133

0.2587

.0

0625

0.7630

0

1223

-0.2621

9.8691 -0

2481

0.3*73

0

1214

-0.2410

0'

6562

-0.5478

5.1576 0

6998

0.2014

0

4947

-0.4725

-0'

2137

-0.1023

3.2296 -0

1801

0.7921

.0

3577

-0.0925

-0'

4499

0.0351

CHARACTERISTICS Or TYPE 3
THE PROPORTIDN OF THE POPULATION FROM THIS TYPE* 0.635
MEANS

1

2

3

4

5

6

6.8098

-1.3264

8.9194

-3.3796

12.9361

-3.3911

STANDARO DEVIATIONS

1

2

S

4

5

6

6.6829

6.2294

7.6649

8.4692

12.2260

11.9304

CORRELATIONS

1

2

S

4

5

*

I. 0000

0.3441

0.8404

0.1994

0.5799

0.1907

0.3441

I. 0000

0.3814

0.8169

0.2775

0.6420

0.8404

0.3814

1.0000

0.2586

0.7613

0.2987

0.1994

0.81*9

0.258*

1.0000

0.1608

0.8260

0.5799

0.2775

0.7613

0.1608

1.0000

0.2*43

0.1907

0.6420

0.2*87

0.8260

0.2643

1.0000

EIGENVALUES

EIGENVECTORS

1

2

3

4

5

6

278.7235

0.2436

-0.2469

0.6157

-0,2932

0.250*

0.5932

154.5198

0.2676

0.1768

0.2977

0.5868

0.6060

-0.3113

37.4431

0.3437

-0.3103

0'.4015

-0.2494

-0.3559

-0.6599

22.2415

0.3694

0.4038

0.1879

0.4234

-0.6307

0.2970

6.6393

0.5399

-0.6093

-0.4)85

0.2521 .

0'.0212

0.1577

6.1806

0.5683

0.526*

-0.2955

-0.5145

0".2l24

-0.051S

PROCRAH NORMIX
WQLPE NORMAL MIXTURE ANALYSIS PROCEDURE ( 1974 REVISION)

SAMPLE ■ RANTQUL
OCTOBER

700<500/AND 300HB
X AND Y COMPONENTS

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

1 0,249 0.001

2 0.004 0.944

3 O.OOl 0.993

4 1.000 0.000

5 0.001 0.998

6 0.015 0.942

7 0.032

8 1.000

9 0.027
lO 0,90*

0.750
0.052
0.005
0.000
0.001
0.043
0.959

0,009

0.000 0.000

0.960 0.013

.. -. __ 0.000 0.094

11 1,000 0,000 0.000

12 0,007 0.062 0.932
0.196
0.006
0.000
0.000
0.000
0.001
0.001
0.000
0,052
0.000

13 0,008

14 0,005

15 0,241

16 0,960

17 0,041

18 0,116

19 0,198

20 1.000

21 0,008

22 0.102

23 0,219

24 0.014

25 0.003

26 1.000

27 0,034

28 0,013

29 0,087

796
990
759
040
959
8(3
.802
000
0.941
0.898

0.000 0.781
0.000 0.986
0.614 0.383

000 0.000
000 0.966
768 0.219
900 0.012
30 0,267 0.000 0.733

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

31 0.033 U.OOO 0.967

32 0,028 0.000 0.972

33 0.012 0.011 0.978

34 0.039 0.000 0.9*1

35 0.495 0.000 0.509

36 0,011 0.006 0.983

37 0.006 0,535 0.459
3» 0.005 0.675 0.321

39 0,002 0.772 0.225

40 0.004 0.618 0.377

41 0,740 0.000 0.2*0

42 0.006 0.536 0.458

43 0,037 0,009 0-95*

44 0,014 0.379 0.607

45 0.028 0.005 0.967

46 0,065 0.001 0.934

47 0,016 0.125 0.858
4e 0,907 0.000 0.093

49 0,011 0.000 0.989

50 0.014 0.000 0.986

51 0,014 0.000 0.986

52 0,164 0.000 0.836

53 0.015 0.000 0.985

54 1.000 0.000 0.000

55 0,004 0.000 0.996

56 0.008 0.000 0.992

57 0,011 0.000 0.989

58 0,089 0,000 0.911

59 0,019 0.000 0,981

60 0.008 0.000 0.991

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

61 0,009

0.001 0.991

62 0.013

0.379 0.607

63 0,006

0.081 0.913

64 0,006

0,088 0.907

65 0,023

0.014 0.9*3

66 0,017

0.205 0.778

67 0.019

0.038 0.943

68 0,007

0,224 0.7*9

69 0,006

0.183 0.811

70 0,007

0.190 0.803

71 0.007

0.716 0.277

72 0.003

0.722 0.275

73 0.004

0.864 0.133

74 0,004

0,636 0.3*1

75 0,019

0.504 0.477

76 0.011

0.227 0.7*2

77 0,018

0.955 0,047

78 0,024

0.003 0.973

79 0,050

0.206 0.7*3

SO 0,021

0.166 0.813

81 0.053

0.067 0.881

82 0.013

0.»38 0,549

83 0.203

0,000 0.796

84 0,022

0,969 0,009

85 0,031

0,844 0.125

86 0,011

0.106 0.883

87 0,017

0.138 0.849

88 0,013

0.424 0.563

89 0,017

0.754 0.229

90 0,273

0.029 0.»98

95

Table 17 (continued)

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

91

0,

108

0.

004

0.

386

92

0.

032

0.

008

0.

960

93

0,

663

0.

000

0.

337

9*

0,

0*5

0.

536

0,

419

95

0,

020

0,

on

0.

9»9

96

0.

3*0

0,

000

0,

659

97

0,

030

0.

000

0.

970

98

0,

060

0.

811

0.

129

99

0.

053

0,

000

0.

947

100

0,

020

0.

ooo

0,

980

101

0,

020

0.

000

0.

980

102

0.

773

0.

038

0.

189

103

0,

969

0.

000

0.

on

10*

0

010

0

001

0

989

105

0,

007

0

629

0,

364

106

0,

0*8

0

000

0

952

107

0,

777

0

000

0

223

loa

0,

021

0

000

0.

979

109

0,

012

0

000

0.

968

110

0.

008

0

000

0

991

111

0,

007

0.

000

0.

993

11?

0,

003

0.

000

0.

997

113

0,

006

0.

000

0,

994

u*

0,

003

0.

000

0.

997

115

0,

071

0.

052

0.

877

116

0.

005

0.

006

0.

988

117

0.

009

0.

781

0.

210

ue

0,

052

0.

107

0,

6*1

119

0.

016

0.

009

0.

976

120

0.

009

0.

3*0

0.

652

121

0

01*

0

901

0

OS*

122

0

019

0

000

0

981

123

0

028

0

006

0

9*6

12'.

0

1*9

0

018

0

833

125

0

284

0

702

0

014

126

0

082

0

000

0

918

U7

0

360

0

625

0

015

128

0

655

0

000

0

145

129

0

868

0

000

0

132

130

0

009

0

001

0

991

131

0

003

0

000

0

992

132

0

025

0

000

0

975

133

0

059

0

000

0

9*1

13*

0

006

0

.002

0

.992

135

0

,021

0

.005

0

.974

136

0

010

0

001

0

989

137

0

018

0

001

0

9Pi

138

0

016

0

000

0

984

139

0

023

0

000

0

977

UO

0

835

0

000

0

.165

Ul

0

985

0

000

0

01!

1*2

0

030

0

,000

0

970

1*3

0

272

0

,000

0

.728

14*

0

02*

0

,000

0

976

1*5

0

166

0

,050

0

764

1*6

0

008

0

9*3

0

0*9

1*7

0

015

0

.013

0

.971

1*8

0

,007

0

,870

0

,12?

1*9

0

.012

0

.030

0

,958

130

0

,021

0

,018

0

.962

151

0

,058

0

.000

0

.942

152

0

,05*

0

.000

0

.946

153

0

,097

0

.001

0

.902

154

0

,179

0

.000

0

.821

155

0

,033

0

.000

0

.9*7

156

0

,025

0

.001

0

.973

157

0

,032

0

.158

0

.810

158

0

.013

0

.010

0

.977

159

0

.077

0

.000

0

.923

160

0

.008

0

.4*8

0

.54*

161

0

.005

0

,007

0

.988

162

0

■ Oil

0

000

0

.988

163

0

.039

0

.123

0

.833

16*

0

.006

0

.723

0

.271

165

0

.007

0

.2*1

0

.752

166

0

.002

0

.77*

0

.22*

167

0

,00*

0

.9*7

0

.049

168

0

.00*

0

.831

0

.164

169

0

.002

0

.837

0

.1*1

170

0

.002

0

.838

0

.1*0

171

0

.003

0

.809

0

.197

172

0

.106

0

.672

0

.222

173

0

.015

0

.292

0

.692

17*

0

,013

0

.399

0

.588

175

0

.012

0

.000

0

.987

176

0

,036

0

.000

0

.963

177

0

.115

0

.003

0

.881

178

0

.008

0

.025

0

.967

1T9

0

.435

0

.000

0

.564

100

0

.938

0

.062

0

.000

lei l.uuu u.ouu u

182 0.999 0.000 0

183 0.954 0.000 0

!64 0.149

185 0.033

186 0.023

167 o.ooa

168 0.005

189 0.004

190 0.010

191 0.042

192 0,049

000
824
000
292
672
635
062
262
000

001
046
S51
.143
,977
700
.323
.3*2
.929
.695
,951

193 0.007 0.042 0.951

194 0,007 0.271 0

195 0.014

196 0,100

197 0.197
199 0.02*

199 0,016

200 0.021

201 0.556

202 0.996

203 0.070

20*
205
206
207
20B
209
210

000
004
010
005
004
013
003

005
000
000
000
000
025
,000
,000
.000
.000
,000
.n03

722
981

900
603
976
,984
,95*
,444
,00*
.930
.000
,996
,987

,000 0.995

.000 0.996

,000 0.987

.000 0.997

211 0,010 0.000 0,990

212 0,002 0.000 0.998

213 0.073 0.000 0,927

214 0.078 0.000
0

215 0.026

216 0.255

217 0.039

218 0.084

219 0.017

220 0,012

221 0,019

222 U,007

223 0.012

224 0.010

225 0.010

226 0,006

227 0,015

,000
,000
,000
,000
.000
.000
.000

922

974
745
961
916
983
988
961

,000 0.993

0.988
0.989
0.985
051 0.943
010 0,975

,000
,000
,006

228 0.003 0.708 0,269

229 0,003 0,671 0.326

230 0,019 0.070

231 0,139

232 0,009

233 0.015

234 0.009

235 0,002

236 0.003

237 0,056

238 0,032

239 0,004

240 0.002

,124
,485
,725
,965

9U
737
306
2*0
026

672 0.126
942 0.055
112 0,831
839 0.129
95! 0.0*0
977 0.022

241 0.005 0.927 0,0*9

242 0,563 0.001 0.436
0,002 0.923 0,075

0.206 0.784

0.870

0,003

2*3

2*4 0,010
2*5 0.013
246 0,067
0.957

247

246 0.093

0.000
0.000

0.117
0.930
0.0*3
0.907

2*9

250

0,026
0.019

0.000 0.974
0.000 0.981

251 oiOlO 0,034 0.9J6

252 0.003 0.336 0.6*0

253 0,004 0.7*7 0.2*9

254 0,008 0.027 0,9*4

255 0,050 0.000 0.949

256 0.340 0.000 0.6*0

257 0.035 0.001 0.9*4

258 0,1*9 0.000 0.851

259 0,062 0.000 0.938

260 0,023 0,000 0,977

261 0.025 0,001 0,97*

262 0,012 0.653 0.335

263 0,012 0.108 0.880
26* 0.039 0.015 0.9*6

265 0,026 0,003 0.971

266 0.017 0.000 0.983

267 0.057 0,000 0.9*3

268 0,115 0,000 0,885

269 0,686 0.000 0.31'>

270 0,011 0.000 0.989

271 0.012 0,000 0.988

272 0,007 0,122 0,871

273 O.OU 0.000 0.988
27* 0,015 0,417 0.5*8

275 0.01* 0,036 0.948

276 0,029 0.006 0.9*5

277 0.060 0.032 0.908
276 0,068 0,000 0,932

279 0,036 0,070 0.893

280 0.055 0.464 0.481

281 0.010 0.725 0.2*5

282 0.008 0.784 0,208
263 0,051 0,025 0,925
28* 0,012 0,679 0.108

285

0

042

0

907

0

051

286

0

865

0

070

0

045

287

0

269

0

000

0

731

286

0

029

0

000

0

971

269

0

041

0

000

0

959

290

0

016

0

888

0

096

291

0

150

0

000

0

850

292

0

020

0

000

0

9B0

293

0

941

0

000

0

059

294

0

105

0

646

0

249

295

0

053

0

366

0

582

296

0

107

0

008

0

685

297

0

132

0

000

0

8*6

298

0

059

0

000

0

941

299

0

310

0

,690

0

,000

300

0

034

0

945

0

,021

301 0,292 0.702 0.006

302 0,065 0.026 0.907

303 0.008 0,947 0.0*4

304 0,003 0,902 0.095

305 0.016 0.919 0.0*5

306 0,037 0.000 0.9*3

307 0.076 0.000 0.922
306 0,015 0.058 0.927

309 0,012 0.406 0.582

310 0.019 0.000 0.981

311 0,043 0.000 0.957

312 0,022 0,001 0.977

313 0,043 0,000 0.957

314 0.036 0,000 0.9*4
3l4 0,030 0,000 0.970

316 0.980 0.000 0.020

317 0.39S 0.000 0.604

318 0,175 0.001 0,823

319 0,133 0.000 0.8*7

320 0.432 0,000 0.5*8

321 0,207 0.338 0.4J5

322 1,000 0.000 0.000

323 0.033 0.664 0,303

324 0,977 0,003 0.020
323 0,926 0,002 0.072

326 0.227 0,000 0,773

327 0.997 0,000 0,003

328 1,000 0,000 0.000
929 1,000 0,000 0,000

330 1,000 0,000 0,000

331 0,013 0,000 0.987

332 0.968 0.000 0.032
3J3 1.000 0.000 0.000

334 0.993 0.000 0.007

335 0.989 0.000 0.011

336 1.000 0.000 0.000

337 0.928 0.021 0,050

338 0.773 0,000 0.227

339 0,990 0.000 O.OIO

340 0,015 0.000 0.965

341 0.009 0.000 0.991

342 0.009 0.001 0.991

343 0,024 0,000 0,976
3*4 0.006 0.000 0,994
345 0,022 0.000 0,978
3*6 0,008 0,000 0.992
347 0,010 0,000 0,990
3*8 0,005 0,000 0.995

349 0,041 0,000 0.959

350 0.014 0,000 0.986

351 0,017 0,000 0,982

352 0,142 0,000 0,856

353 0,023 0.000 0.977

354 0,021 0.004 0.975

355 0,079 0.002 0.919

356 0.050 0.001 0.9*9

357 0,029 0,001 0.970
356 0.041 0.000 0.959
359 0,023 0.000 0.977
260 0.380 0.000 0.620

96

Table 17 (continued)

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

361 0.032

0

.937

0

.031

362 0.117

0

,295

0

,588

363 O.oa')

0

000

0

916

36* 0.199

0

000

0

801

365 O.*30

0

000

0

570

366 0.009

0

oil

0

980

367 0.020

0

926

0

.053

368 0.072

0

005

0

922

369 0.027

0

032

0

9*0

370 0.991

0

000

0

009

371 0.3H

0

120

0

565

372 0.»29

0

000

0

571

373 0.193

0

000

0

802

37* 0.072

0

611

0

317

375 O.UU

0

926

0

063

376 0.015

0

001

0

98^

377 0.009

0

000

0

991

378 0.01*

0

001

0

986

379 0.010

0

000

0

990

3«0 0.018

0

000

0

982

381 O.lOO

0

102

0

798

382 0.02*

0

006

0

970

383 0.031

0

785

0

184

38* 0.027

0

951

0

022

385 0.019

0

905

0

076

386 0.007

0

888

0

105

387 0.025

0

68*

0

291

388 0.027

0

961

0

012

389 0.005

0.

786

0

209

390 0.005

0.

957

0.

038

391 0.01*

0

.51*

0

*72

392 0.025

0

■ 357

0

618

393 0.022

0

.801

0

177

39* 0.010

0

.295

0

695

395 0.1*9

0

,138

0

712

396 0.005

0

.78*

0

212

397 0.005

0

813

0

182

398 0.023

0

.*81

0

*95

399 0.012

0

.o^e

0

939

*00 0.238

0

.000

0

7*2

*01 0.02*

0

001

0

975

*02 0.106

0

,001

0

89*

*03 0.019

0

001

0

980

*0* 0.97*

0

001

0

02*

♦05 1.000

0

000

0

800

*06 1.000

0

.000

0

000

*07 1.000

0

.000

0

.000

♦08 I. 000

0

.000

0

.000

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

♦ 09

1.000 0.000 0.000

*10

0.981 0.000 0.019

*11

0.702 0.298 0.000

*12

0.997 0.000 0.003

*13

0.0*^ 0.772 0.18^

*1*

0.010 0.719 0.271

*15

0.041 0.010 0.9*9

*16

0,026 0.005 0.968

*17

0.021 0.13* 0.e45

*lG

O.Oji 0.008 0.981
O.0I2 0.001 0.988

*19

*20

0.02* 0.002 0.975

*21

0.007 O.OSK 0.935

*22

0,010 0.012 0.978

*23

0.136 0.039 0.825

*2*

0.053 0.004 0.9*3

*25

0.222 0.058 0.720

*26

0.993 0.000 0.007

*27

0.096 0.010 0.89^

♦ 28

0.183 0.002 0.816

*29

0.999 0,000 0.001

*30

0.009 0.158 0.833

*3l

0.022 0.473 0.505

*32

0.008 0.0*7 0.945

*33

0.005 0.001 0.995

*3*

0.008 0.013 0.980

*35

0.003 0.51* 0.483

*36

0.012 0.652 0.336

*37

0.002 0.956 0.043

*3e

0.003 0.878 0.119

*39

0.998 0.000 0.002

**0

O.89I 0.022 0.086

**1

0.072 0.002 0.925

**2

0.005 0.000 0.995

♦ ♦3

0.06* 0.000 0.936

***

0.896 0.000 O.lO*

**5

0.185 0.000 0.81*

**6

0.008 0.000 0.992

*«7

0.038 0.000 0.9*2

♦ *8

0.007 0.000 0.993

*«9

O.Oll 0.000 0.989

*50

0.012 0.000 0.968

*51

0,0*3 0.002 0.955

*52

0.89* 0.000 O.IO6

*51

0.101 0.000 0.899

*5*

0.062 0.000 0.938

♦ 55

0.082 0.000 0.918

♦ 56

0.027 0.000 0.973

PROBABILITIES

OF TYPE MEMBERSHIP

1 2 3

457

0

.015

0

■ 000

0

■ 9(5

45E

0

.015

0

.001

0

.98*

459

0

.018

0

■ 197

0

■ 785

460

0

,008

0

■ 656

0

.335

461

0

.038

0

■ 032

0

■ 9J0

462

0

.039

0

■ 957

0

.00*

*63

0

.039

0

■ 30*

0

■ 658

*6*

0

,010

0

■ 353

0

■ 637

465

0

.023

0

.583

0

■ 394

466

0

,006

0

■ 126

0

868

467

0

.177

0

■ 029

0

.794

468

0

.135

0

000

0

865

469

0

,075

0

■ 063

0

■ 861

470

0

.577

0

■ 000

0

423

471

1

,000

0

■ 000

0

000

472

1

,000

0

000

0

000

473

1

■ 000

0

000

0

000

47*

I

,000

0

000

0

000

*75

0

,967

0

000

0

.033

*76

0

636

0

000

0

364

*77

0

■ 122

0

000

0

878

*78

0

056

0

000

0

944

*79

0

■ 183

0

000

0

817

*80

0

032

0

000

0

968

*81

0

■ 70*

0

206

0

090

482

0

048

0

000

0

952

*83

0

052

0

000

0

948

*8*

0

009

0

070

0

921

485

0

015

0

766

0

220

480

0

091

0

015

0

894

487

0

129

0

000

0

871

486

0

032

0

024

0

943

489

0

123

0

000

0

877

490

0

119

0

000

0

881

491

I

000

0

000

0

000

492

1

000

0

oon

0

000

493

0

061

0

089

0

850

*9*

0

323

0

001

0

476

*95

0

135

0

001

0

864

*96

0

041

0

006

0

952

*97

0

419

0

000

0

581

498

0

063

0

000

0

937

499

0

023

0.

000

0.

977

50U

0

020

0.

000

0

980

501

0,

8*7

0,

000

0,

153

502

0,

113

0.

000

0.

887

503

0,

989

0.

008

0.

003

LQCARITHM OF LIKELIHOQO R4T10 HF 3 TO 2 TYPES ■
CHI-SOOARE WITH 54 0F6HEES OF FRFEeOH» 114.02
PHOBABILITV OF NULL HYPOTHESIS. 0.000003*9

0.579336430 02

97

Table 18 Separation of a multivariate (6) set of Rantoul , Illinois,
October 1950-55, upper wind components, zonal' and meridional
mixed distribution, at the 700-, 500-, and 300-mb levels into
four distributions. The units are m-s~^.

PROCRAM NQRMIX
WDLFf NORMAL MIXTURE ANALYSIS PROCEDURE ( 197<. REVISIONI

SAMPLE • fANTnuL
OCTOBER

700/SOO,AND 300MB
X AND Y COMPONENTS

SAMPLE SIZE • 503

NUMBER OF VARIABLES • 6
NUMBER OF TYPES • ♦

ITERATION NUMBER 0

LIKELIHOOD OF « TYPES IN THIS SAMPLE • -0.727177290 0*

CHARACTERISTICS Q' THE WHOLE SAMPLE

MEANS

1.0000
0.2215
0.7525
0.2313
0.5589
0.2311

11.2762

5. 5488

1
1.0000
0.1304
0.6960
0.1793
0.*565
0.0233

3.7585

5.5489

-0.9486

6.8921

2

0.221!
1.0000
0.1136
0.7581
0.1265
0.5757

10.6700

-2.570*

STANDARD DEVIATIONS
3 *

8,959* 10.31*9

CBRRELATIONS

3 *

0.7)25 0.2313

0.1139 0.7581

1.0000 0.1567

0.1)87 1.0000

0.7513 0.1200

0.1977 0.7856

19.*0*9

13.7082

5
0.5589
0.1265
0.7513
0.1200
1.0000
0.1266

6
.3.6999

16.0058

6
o;23U
0'.5T57
0'. 1977
0'.78J6
o;i266
1.0000

CHARACTERISTICS OP TfPE 1
THE PROPORTION OF THE POPULATION FROM THIS TYPEp 0.*06

MEANS

2.*46*

0.130*
l.OOOO
0.0569
0.6547
-0.0423
0.4006

1*.1*60

1.2900

STANDARD DEVIATIONS
3 *

7.8*21 8.8618

CORRELATIONS

3 *

0.6960 0.1793

0.0)69 0.65*7

l.OOOO 0.1*98

0.1*98 1,0000

0.6851 0.0*92

0,092* 0.7*51

20.3731

10.7092

5

0.*S65
-0.0*23
0.68S1
0.0*92
l.OOOO
-0,03S*

6
5.8172

6
i3'.*027

0'.0233
0.*006
0.092*
0.7*51
-0.035*
l.OOOO

CHARACTERISTICS Of TYPE 2
THE PROPORTION OF THE POPULATION FROM THIS TYPE* 0.189

MEANS

4.1B69

3

7. 5600

*
2.5*86

1

2

1.0000

0.130*

0.1304

1.0000

0.6960

0.0569

0.1793

0.6547

0.4565

-0.0423

0.0233

0.4006

STANDARD DEVIATIONS
3 *

7,8*21 8.8618

chrreCations

3 *

0,6960 0.1793

0.0)69 0.65*7

l.OOOO 0.1*98

0,1*98 1,0000

0,6851 0.0*92

0.092* 0,7*51

13.1635

10.7092

5

0.*56S
-0.0*23
0.6891
0.0492
l.OOOO
-0.03S*

6
.3.0*06

13.4027

6
0.0233
0.4006
0.092*
0'.7*51
■0.039*
l.OOOO

CHARACTERISTICS OP TYPE 3
THE PROPORTIPN OF THE POPULATION FROM THIS TVPg» 0.310

MEANS

2 3 *

-7.19*0 11.4*81 .9.710*

16.9286

-1*.19*7

98

Table 18 (continued)

5T4NBAK0 OfVIiTIONS

1

2

3

*

5.:<.8e

*.B'?68

7.8*21

8.S6ie

CnRRELATIDNS

1

2

3

*

1.0000

0.130*

0.6'>60

0.1793

0.130*

1.0000

0.0969

0.65*7

Q.bfhO

0.0569

I. 0000

0.1*9B

0.1793

0.65*7

0.1*98

1.0000

0.*565

-0.0*23

0,6851

0.0*92

0.0233

0.*006

0.092*

0.7*51

10.7092

5
0.*565

-0.0*23
0.6831
0.0*92
1.0000

-0.03S*

CHARACTERISTlej Of TYRE *

THE PROPORTION OF THF POPULATION FROM THIS TYPE.

MEANS

5
■10.18*0

5

10.7092

5

0.*565
-0.0*23
0.68S1
0.0*92
1.0000
-0.035*

13.*027

0'.0233

0,*006
0.092*
0.7*51
-0.035*
1.0000

1

2

3

*

0.

7826

-5.2503

-0.*77l

.5.9032

STANOA'D

DEVIATIONS

1

2

3

4

5.

5*88

*.e968

7.6*2

6.8618

CORRELATIONS

1

2

3

*

!■

0000

0.130*

0.6460

0.1T93

0.

130*

1.0000

0.0969

0.65*7

0.

6960

0.0569

I. 0000

0.1*98

0.

1793

0.65*7

0.1*98

i.ooon

0.

*565

-0.0*23

0.6851

0.0*92

0.

0233

0.*006

0.092*

0.7*51

ITERATION

1

LOG

LIKELIHOOD

OF

TYPES

ITERATION

2

LOO

LIKELIHOQB

OF

TYPES

ITERATION

3

LOG

LIKELlHOOB

OF

TYPES

ITERATION

*

LOG

LIKELIHHOD

OF

TYPES

ITERATION

5

LOG

LIKELlHOOB

OF

TYPES

ITERATION

6

LOG

LIKELlHOOB

OF

TYPES

ITERATION

7

LOG

LIKELlHOOB

OF

TYPES

ITERATION

a

LOG

LIKELIHOOD

OF

TYPES

ITERATION

9

LOG

LIKELlHOOB

OF

TYPES

ITERATION

10

LOG

LIKELIHOOD

OF

TYPES

ITERATION

11

LOG

LIKELlHOOB

OF

TYPES

ITERATION

12

LOG

LIKELlHOOB

OF

TYPES

ITERATION

13

LOG

LIKELIHOOD

OF

TYPES

ITERATION

1*

LOG

LIKELIHOOD

OF

TYPES

ITERATION

15

LOG

LIKELIHOOD

OF

TYPES

ITERATION

16

LOG

LIKELIHOOD

OF

TYPES

ITERATION

17

LOG

LIKELIHOOD

OF

TYPES

ITERATION

18

LOG

LIKELlHOOB

OF

TYPES

ITERATION

19

LOG

LIKELIHOOD

OF

TYPES

ITERATION

20

LOG

LIKELIHOOD

OF

TYPES

ITERATION

21

LOG

LIKELIHOOD

OF

TYPES

ITERATION

22

LOG

LIKELIHOOD

OF

TYPES

ITERATION

23

LOG

LIKELIHOOD

OF

TYPES

ITERATION

2*

LOG

LIKELlHOOB

OF

TYPES

ITERATION

25

LOG

LIKELIHOOD

OF

TYPES

ITERATION

26

LOG

LIKELIHOOD

OF

TYPES

ITERATION

27

LOG

LIKELlHOOB

OF

TYPES

ITERATION

28

LOG

LIKELlHOOB

OF

TYPES

ITERATION

29

LOG

LIKELlHOOB

OF

TYPES

ITERATION

30

LOG

LIKELIHOOD

Of

TYPES

ITERATION

31

LOO

LIKELIHOOD

OF

TYPES

ITERATION

32

LOG

LIKELlHOOB

OF

TYPES

ITERATION

33

LOG

LIKELIHOOD

OF

TYPES

ITERATION

3*

LOO

LIKELIHOOD

OF

TYPES

ITERATION

35

LOG

LIKELIHOOD

OF

TYPES

ITERATION

36

LOG

LIKELIHOOD

OF

TYPES

ITERA-ION

37

LOG

LIKELIHOOD

OF

TYPES

ITER flON

38

LOG

LIKELIHOOD

OF

TYPES

ITEf'-TION

39

LOG

LIKELIHOOD

Of

TYPES

ITERATION

*0

LOG

LIKELIHOOD

OF

TYPES

ITERATION

*1

LOO

LIKELIHOOD

OF

TYPES

ITERATION

*2

LOO

LIKELlHOOB

OF

TYPES

ITERATION

*3

LOG

LIKELlHOOB

OF

TYPES

ITERATION

**

LOG

LIKELlHOOB

OF

TYPE?

ITERATION

*5

LOG

LIKELIHOOD

OF

TYPES

ITERATION

*6

LOG

LIKELIHOOD

OF

TY^ES

ITERATION

*7

LOO

LIKELIHOOD

OF

TYPES

ITERATION

*8

LOG

LIKELIHOOD

OF

TYPES

ITERATION

*9

LOG

LIKELIHOOD

OF

TYPES

ITERATION

50

LOG

LIKELIHOOD

OF

TYPES

ITERATION

51

LOO

LIKELlHOOB

OF

TYPES

ITERATION

52

LOG

LIKELIHOOD

OF

TYPES

ITERATION

53

LOG

LIKELIHOOD

OF

TYPES

ITERATION

5*

LOG

LIKELIHOOD

OF

TYPES

ITERATION

55

LOG

LIKELIHOOD

OF

TYPES

ITERATION

56

LOG

LIKELIHOOD

OF

TYPES

ITERATION

57

LOO

LIKELIHOOD

OF

TYPES

IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN TnIS
IN THIS

IN This

IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS

SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SaMPlE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE

13.*027

6
0,0233
0,*006
0,092*
0,7*51
.0'.035*
1,0000

-0,752736060 0*
-0,7*2l5g89D 0*
-0,737263560 0*
-0,73*15*930 0*
-0,731713750 0*
-0,729716050 0*
-0,728398620 0*
-0,727597600 0*
-0,727053890 0*
-0.7266*5320 0*
-0.726326910 0*
-0.726071*50 0*
-0.725857860 0*
-0.725673520 0*
-0.72551*010 0*
-0.729378530 0*
-0.72526*750 0*
-0.726*00870 0*
-0.72*927*90 0*
-0,72*790960 0*
-0,725576«60 0*
-0,72*6*5810 0*
-0.72*571510 0*
-0,72*825150 0*
-0.72*469*50 0*
-0.72**23910 0*
-0,72*887810 0*
-0.72*293210 0*
-0.72*216810 0*
-0.72*726580 0*
-0,72*015990 0*
-0,723933290 0*
-0.72****610 0*
-0.723802560 0*
-0.7237*89*0 0*
-0.723803760 0*
-0.7237025*0 0*
-0.723682900 0*
-0.723932*50 0*
-0,7236*1900 0*
-0,72361*200 0*
-0.723769090 0*
-0.7235876*0 0*
-0,723568290 0*
-0,723977760 0*
-0.7235*6010 0*
-0.723520*30 0*
-0.7235*1*60 0*
-0.723*9*060 0*
-0.723*82260 0*
-0.7238991*0 0*
-0.723*28970 0*
-0.7233912*0 0*
-0.723699620 0*
-0.72336567D 0*
-0.7233078*0 0*
-0.7236999*0 0*

99

Table 18 (continued)

iTERiTlQN

53

LOG

LiKELIHCOn

OF

TYPES

IN

THIS

SAMPLE •

ITER4TIDN

59

LOG

LIKELIHCOn

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

60

LOG

LIKELIHDOC

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

61

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

62

LOG

LIKELIHCOe

BF

TYPES

IN

THIS

SAMPLE •

ITERATION

63

LOG

LIKELIHPOB

OF

TYPES

IN

THIS

SAMPLE ■

ITERATION

64

LOG

LIHELIHOQO

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

65

LOG

LIKELIHCOB

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

66

LOG

LIKELIHHOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

67

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

68

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

69

LOG

LIKELIHDOn

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

70

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

71

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAHPLt •

ITERATION

72

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

73

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

T,

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

75

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

76

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

77

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

78

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

79

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

80

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

81

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

62

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE ■

ITERATION

83

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE ■

ITERATION

8<i

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

B5

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

86

LOG

LIKELIHOOD

OF

Types

IN

THIS

SAMPLE •

ITERATION

87

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

88

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

89

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

90

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

91

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

92

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

93

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

Sample •

ITERATION

94

LOG

LIKELIHOUD

OF

TYPES

IN

THIS

SAMPLE •

Iteration

93

LOO

LIKELIHOOD

Of

TYPES

IN

THIS

SAMPLE .

ITERATION

96

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

iteration

97

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE .

ITERATION

98

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

ITERATION

99

LOG

LIKELIHOOD

OF

TYPES

IN

This

SAMPLE .

ITERATION

100

LOG

LIKELIHOOD

OF

TYPES

IN

THIS

SAMPLE •

-0,72327*170 0*
-0.723179610 0*
-0.723911560 04
-0.723298950 04
-0.72317J96D 04
-0.72312275D 04
-0.7237J290D 04
-0.723256210 04
-0.723155650 04
-0.723106720 04
-0.72308061D 04
-0.723416780 04
-0.7231463O0 0^>
-0.723087620 04
-0.72306692D 04
-0.723058890 0»
-0,723055380 04
-0.723052980 04
-0.723059780 04
-0.7230SJ590 04
-0.7230J249D 04
-0.723052110 04
-0.723054140 04
-0.72J052290 04
-0.723051950 04
•'•0.723051830 04
-0.72S05197D 04
-0.723051730 04
-0.723051690 04
-0.723051720 04
-0.723051640 04
-0.723051600 04
-0.723051610 04
-0.723051520 04
-0,723051470 04
-0.723051800 04
-0.723051420 04
-0.723051280 04
-0.723051560 04
-0.723051120 04
-0.723050940 04
-0.723050950 04
-0.723030570 04

ITERATION TIME •
TOTAL TIME USED •

3)43.00 SECONDS.
11S62.01 SECONDS.

ITERATION NUMBER 101
LIKELIHOOD OF 4 TYPES IN THIS SAMPLE • -0.72305057D 04
CHARACTERISTIes OF THE WHOLE SAMPLE

7.3232

11.1041

1

1.0000

0.2647
0.5390
0.3324
0.5169
0.2525

MEANS

STANDARD DfVIATIONS
3 4

8.9594 10.3149

CORPELATIONS

3 4

0.7525 0.2313

0.1138 0.7581

1.000b 0.1587

0.1587 1.0000

0.7515 0.1200

0.1977 0.7856

CHAPACTERISTIes OP TYPE I
THE PROPORTION OF TH( POPULATION FROM
MIANS

1

2

1.0000

0.2215

0.2215

1.0000

0.7525

0.1138

0.2313

0.7581

0.5589

0.1265

0.2311

0.5757

15.4049

13.7082

3

0.5589
0.1265
0.7513
0.1200
1.0000
0.1266

-3;6999

6
16'. 003 8

6
0.2311
0'.J757
O; 1977
0'.7e36
0,1266
1,0000

THIS TYPE. 0.193

2

-0.3525

2
9.2886

0.2647

1.0000
-0.1035

0.7443
-0.1470

0.6119

16.6560

4
■1.3465

STANDARD DfVlATIONS
3 4

11.3528 17.2976

CORRELATIONS

3 4

0.5390 0.3324

-0.1035 0.7443

l.OOOP 0.0730

0.0730 1.0000

0.7250 -0.0241

0.0848 0.7911

22.8836

14.8718

5

0.5169
-0.1470

0.72S0
-0.0241

1.0000
-0.1123

6
.0'.6482

27.5962

6
0.2525
0'.6119

0;0948

o'.79ai

■ 0'.lt23
I'. 0000

100

Table 18 (continued)

EIGENVALUES

EIGENVECTORS

3
O.ll"

0.5*60

1021.7*63

0.0*66

0.251''

0.1879

326.0660

0.2057

-O.OSl'!

0.4355

-0.0397

0.6673

104,5090

0.0270

0.5!2«

-0.1959

0.7'i83

-0.0513

'•0.0002

0.<.61T

0.0*27

0.7053

0.1472

-0.<.735

33.*5*5

-0.0'i90

0.7903

0.0619

-0.5999

-0.0621

23.2217

O.B'.7»

-0.0U8

-0.5063

-0,147<,

0'.0608

CH4R4CTFRISTleS QP TvPE 2
THE PROPORTION OF THE POPULATION fPQM T-HIS TYPE. 0.293

CHARACTERISTIf S OP TYPE 3

THE PROPORTION DP THE POPULATION FROM THIS TYPE» 0,<.21

MEANS
12 3 4 5*

6.3830 -2.6212 8.3100 -5.2*61 10.5901 -4.5*39

STANBARD DEVIATIONS
12 3 4 5 6

6.6569 5.5893 7.5718 7.9899 n.9976 12.0984

C8RPELATI0N5

CHARACTERISTICS OF TYPE 4
THE PROPORTION OF THI POPULATION FROM THIS TYPE" 0.094
MEANS

1

2

i

4

5

6

5.6571

-3.2181

6.8328

.3.4952

7.2228

^6.5235

STANDARD DEVIATIONS

1

2

3

4

5

6

6.3812

6.6990

8.7872

8.7555

IB. 2393

16.5218

CORRELATIONS

, 1

2

3

4

5

*

X.OOOO

0.3470

0.9558

0.3597

0.4762

0'.1332

0.3470

1.0000

0.4558

0.9402

0.6151

0.6943

0.9558

0.4550

1.0000

0.4338

0.4895

0'.2214

0.3597

0.9402

0,413»

1.0000

0,5464

0.8140

0.4762

0.6151

0.4895

0.5464

1,0000

0'.3738

0.1332

0.6943

0.2J14

0.8140

0,3738

l.OftOO

6

0.7497
-0.5643
.0.3054

0.1439
-0.0739

0.0017

MEANS

1

2

3

4

5

6

6.7197

1.7879

11.3394

0.7926

20.0058

-3.5488

STANDARD DEVIATIONS

1

2

3

4

5

*

5.2062

5.7219

6.7337

5.1816

8.3725

8.0912

CHRRELATIONS

1

?

3

4

5

*

I. 0000

-0.0490

0.7322

0.2324

0.5236

0;4*35

-0.0490

1.0000

-0.2795

0.6794

-0.3196

0;4835

0.7322

-0.2795

1.0000

-0.1278

0.7437

0'il412

0.2324

0.6794

-0.1278

1.0000

-0.2774

0.7870

0.523*

-0.319*

0.7437

-0.2774

I. 0000

-0.0414

0.4435

0.4635

0.1412

0.7870

-0.0414

1.0000

EIGENVALUES

EIGENVECTORS

1

2

3

4

5

6

120.3379

0.3038

0.2911

-0.2839

0.3728

0.6914

.0.3541

102.2809

-0.2543

0.3305

0.7201

0,462*

-0.1243

•0.2796

20.2525

0.5440

0.1*97

.0.2136

0.5162

.0'.5472

0.2525

13.1203

-0.1758

0.4J19

0.0912

0.0344

0'.3317

0.8145

6.870*

0.7153

0.0421

0.5396

-0.4197

0'. 1338

0.0349

4.7389

-0.0663

0.7*77

.0.2360

-0.4509

-0'.2815

-0.2609

1.0000

0.3045

0.8419

0.1154

0.606S

0'. 1489

0.3045

1.0000

0.3*15

0,7889

0.1293

0.6263

0.8419

0.3415

1.0800

0.1965

0.8310

0.3252

0.1154

0.7889

0.1965

1.0000

0.0264

0.8370

0.6065

0.1293

0,8318

0.0264

1.0000

0'.3023

0.1489

0.6263

0.3252

0.8370

0.3023

1,0000

EIGENVALUES

EIGENVECTORS

1

2

3

4

5

*

266.8513

0.2422

0.2551

0.6124

-0.4722

-0.1*61

0.5007

159.8561

0.2087

-0.1989

0.4059

0.6391

-0.5800

.0.0858

37.9134

0.3663

0.2998

0,3293

-0.1248

0.2361

-0.7721

12.6521

0.3106

-0.4448

0.2560

0.2736

0.7070

0.2558

5.5168

0.5554

0.5782

-0.4125

0.3347

0',0417

0.2707

4.2504

0.5989

-0.5228

-0.3407

-0.4076

-o'.2eo5

-0.0841

101

Table 18 (continued)

EIGENVALUES

EIGENVECTORS

529.7700

20*. 9783

80.2752

2J.726'.

3.3614

1.0011

0
0
0
0
0
0

1370
2'.52
2120
3267
6787
5552

2

-0.1791
0.0753

-0.201S
0.1956

-0.6n8*
0.716*

3
0.5272
0.0656
0.7391
0.1260
-0.3929
-0.0352

-0.1521
0.6255

-0, >3<i7
0.6207

-0.1170

-0.4105

5

0',5265

.0.5210

-0'.<il52

0,5243

0.0120

-0'.063«

PROBABILITIES

PROBABILITIES

OF TYPE MEMBERSHIP

OF TYPE MEMBERSHIP

2 3 4

1

2 3

4

,6091
,5164
.4215
,4232
,0310
06d!

1 0.622 0.051 0.327 n.OOO

2 0.007 0.992 0.001 0.000

3 O.Ooe 0.991 0.002 0.000

4 1.000 0.000 0.000 0.000

5 0,004 0.996 0.000 0,000

6 0,022 0.978 0.000 0.000

7 0.036 0.031 0.933 0.001

8 1.000 0.000 0.000 0.000

9 0.060 0.935 0.005 0.000
10 0,699 0.000 0.008 0.293
U 1.000 0.000 0.000 0.000

12 0.006 0.072 0.837 0.085

13 0.018 0.469 0.321 O-l'l

14 0.015 0.127 0.839 0.019

15 0.002 0.000 0.000 0.998

16 0.991 0,000 0.008 0.000

17 0.017 0.014 0.004 0.96?

18 0.223 0,013 0.7»4 0.000

19 0.366 0.046 0.969 0.000

20 1.000 0.000 0.000 0.000

21 0,011 0.841 0.147 0.001

22 0.059 O.OOO 0.158 0.783

23 0.307 O.OOO 0.692 0.001

24 0,030 0.002 0.9*7 0.000

25 0.006 0.758 0.195 0.041

26 0.000 0.000 O.OOn 1.000

27 0.132 0.124 0.743 0,001

28 0.011 0.984 0.005 0.001

29 0.155 0.844 0.001 0.000

30 0.820 0.000 0.180 0.000

31 0.025 0.00? 0.972 0.000

32 0,015 0.000 0.985 0.000

33 0.C13 0.061 0.69* 0.032

34 0.052 O.OOO 0.948 0.000

35 0.729 0.000 0.271 0.000

36 0.008 0.06"; 0.724 0.164

37 0.011 0,497 0.478 O.OU

38 0,007 0,937 0.054 0,001

39 0.005 0.861 0.024 0.091

40 0.005 0,968 0.024 0.003

41 0.944 0,044 0.012 0,000

42 0.012 0.817 0.025 0.146

43 0.038 0.153 0.580 0.229

44 0.021 0.638 0.236 0.106

45 0,056 0,720 0.224 0.000

46 0.096 0,022 0.869 0,OlS

47 0,032 0.757 0.166 0.025

48 0.962 0.000 0.038 0.000

49 0.009 O.OOI 0.876 0.113

50 0,016 0.001 0,956 0.027

51 0,012 O.OOO O.oeo 0.007

52 0.424 0.000 0.J76 0.000

53 0.007 0,000 0,943 0,050

54 1,000 0.000 0.000 0,000

55 0.004 0.000 0,8*3 0,132

56 0,008 0.000 0.946 0.046

57 0,008 0.000 0.780 0.212

58 0.147 0.007 0.846 0.000

59 0,004 0.000 0,451 0,545

60 0,004 0,001 0.842 0.153

61 0.003 0.003 0.859 0.135

62 0.025 0.826 0.103 0.046

63 0.009 0.151 0.721 0.H9

64 0.009 0,176 0,704 0,111

65 0,045 0.660 0,271 0.024

66 0.020 0.651 0,329 0,000

67 0.027 0,399 0,574 0,000

68 0,009 0.694 0.270 0.027

69 0,009 0,424 0.555 0.012

70 0,011 0.488 0.480 0.020

71 0.015 0.866 0.118 0.000

72 0.007 0.960 0.021 0,013

73 0.010 0.950 0,038 0,002

74 0,008 0,855 0.059 0.078

75 0.038 0.867 0,045 0.050

76 0,025 0.546 0.370 0.059

77 0,035 0.961 0.004 0.000

78 0,028 0.017 0.955 0.0 ''

79 0.051 0.696 0,019 0.

80 0,041 0.834 0,124 0- il

81 0,100 0,255 0,645 0.000

82 0,027 0,872 0.012 0.088
63 0.974 0.020 0.004 0,003

84 0,064 0,933 0,003 0,000

85 0.077 0,838 0,084 0.000

86 0,020 0,786 0.193 0.000

87 0,036 0,786 0.163 0,015

88 0.028 0,803 0,019 0,150

89 0.024 0.932 0.043 0.000

90 0,375 0,406 0.016 0.203

91 0,079 0,029 0,893 0.000

92 0,029 0,019 0.942 0.010

93 0.973 0,001 0.;27 0.000

94 0.044 0,671 0.001 0,264

95 0,032 0.169 0.797 -^.OOI

96 0,945 0,012 0,042 0,000

97 0,040 0.003 0.957 O.OOO

98 0,064 0,893 0.043 0,000

99 0.056 O.OOO 0.943 0,000

100 0.018 0.026 0,845 0,111

101 0,016 0,008 0.881 0.094

102 0,847 0.047 0,106 0.000

103 0,996 0.003 O.OOI 0.000

104 O.OU 0.005 0.864 0.100

105 0.012 0.459 0.328 0.001

106 0.016 0,009 0.689 0,286

107 0,097 0,000 0.882 0,021

108 0,004 0,000 0.957 0.039

109 0,007 0,000 0.529 0.464
HO 0,004 0.002 0.833 0.161

111 0.008 0.000 0.978 0.013

112 0.003 0,001 0,995 0,002

113 0.005 0,000 0.978 0,015

114 0,023 0,000 0.403 0,574

115 0,128 0,414 0,400 0.059

116 0,009 0,035 0,899 0,05''

117 0.014 0,901 0.085 0,000

118 0.083 0.447 0.011 0.458

119 0.031 0.078 0.815 0,076

120 0,010 0,568 0.352 0,070

121 0,028 0,950 0,022 0,000

122 0,019 0,000 0.981 0.000

123 0.032 0.463 0.4iJ 0,071

124 0,093 0,661 0.U2 0.084

125 0,416 0.563 0.001 0.000
125 0.073 0.001 0,926 0.000

127 0.549 0.450 0.001 0.000

128 0.031 0.000 0,002 0,967

129 0,005 0,000 0,002 0,993

130 0,014 0,012 0,973 0.000

131 0.005 0.000 0,949 0.046

132 0.005 0.000 0.544 0.450

133 0.012 0.000 0.109 0.i79

134 0.006 0,007 0,958 0.030

135 0,030 0,007 0.9;)8 0,02*

136 0,014 0,005 0.956 0,024

137 0,026 0,004 0.875 0,095

138 0.009 0,005 0,653 0,333

139 0,019 0,000 0.981 0.000

140 0,003 0,000 0,015 0,963

141 0,000 0,000 0.000 1.000

142 0,024 0.000 0,975 0,000

143 0.540 0.000 0.237 0.203

144 0.030 0,001 0.9*9 0.000
1*5 0.071 0.758 0.143 0.028
1*6 0.017 0.964 0.018 0.000

147 0.037 O.I43 0.808 0.012

148 0.013 0.931 0.055 0.000

149 0.029 0.202 0.458 0.312

150 0.037 0.074 0.887 0.003

151 0,096 0.000 0.904 O.OOO

152 0.236 0.008 0.755 0.000

153 0.258 0.035 0,9*4 u,143

154 0.103 0.000 0,897 0,000

155 0.064 0.000 0,936 0,000

156 0.028 0.073 0.777 0.122

157 0.037 0.871 O.Oll 0.0B2

158 0.016 0.353 0.615 0.014

159 0.U4 0.001 0.880 0.005

160 0,012 0.837 0.151 0.000

PROBABILITIES

OF TYPE MEMBERSHIP

12 3 4

161 0.011 0,228 0.761 0,000

162 0,021 0,891 0.088 0.000

163 0.009 0.984 0,007 0.000

164 0.006 0,978 0.015 0,000

165 0.015 0.593 0.363 0,010
165 0,004 0,919 0.015 0,062

167 0,011 0.879 0,000 0.109

168 0,005 0,989 0,005 0.000

169 0.004 0.919 0.027 0.050

170 0.004 0.952 0.012 0.03?

171 0.006 0.965 0.003 0.027

172 0.232 0,707 0,000 0,062
17i 0,037 0,519 0,332 D,0l2

174 0.017 0,778 0,041 0,164

175 0,022 0.005 0.9*0 0.013

176 0,171 0,060 0,749 0,000

177 0,087 0.010 0,902 0,000

178 0,012 0,078 0,774 0^135

179 0,796 0,005 0.188 0,OlO
160 0,949 0,051 0,000 0,000

181 0,997 0,003 0.000 0,000

182 1.000 0.000 0.000 0.000

183 0,988 0,000 0,012 0,000

184 0,157 0,000 0,843 0,000

185 0.054 0,934 0.012 0.000

186 0.021 0,000 0,935 0,044
167 0,012 0.933 0.390 0.065

188 0.010 0.803 0.160 0.007

189 0.007 0.588 0.284 0,022

190 0,010 0,065 0.778 0,146

191 0,072 0,668 0.258 0.002
19? 0,024 0,000 0.975 0.001

193 0,006 0,158 0,697 0,139

194 0,008 0,411 0,502 0.079

195 0.013 0.024 0.815 0,148

196 0,005 0,000 0,992 0,003

197 0,153 0,000 0,847 0.000

198 0.023 0.000 0.951 0.026

199 0.016 0.001 0.981 0,002

200 0,032 0,095 0.8*1 0,011

201 0,025 0,000 0.011 0.963

202 1.000 0.000 0.000 0.000

203 0.139 O.OOO 0,855 0,006

204 1.000 0.000 0.000 O.OOO

205 0.006 0,000 0,994 0.000
205 0.017 0.056 0.917 0.000

207 0.006 0.000 0,992 0,002

208 0,007 0,000 0,993 0,000

209 0,018 0,000 0.982 0,000

210 0.004 0.001 0,995 0,000

211 0,006 0,001 0,017 0,976

212 0,007 0,001 0.977 0,0l6

213 0,510 0,420 0.070 0.000

214 0,008 0,000 0,943 0,448

215 0,002 0.000 0.734 0.263
215 0.018 0.000 0.982 0.000

217 0,006 0,000 0,982 0,012

218 0,009 0,000 0,592 0,298

219 0.011 0.000 0,973 0,01*

220 0.007 0,001 0,968 0,004

221 0,023 0,001 0.92* 0.052

222 0,007 0,000 0,875 0.118

223 0,010 0,000 0.699 0,291

224 0,009 0,001 0.770 0.220

225 0,007 0.010 0.786 0.1«6

226 0.006 0.054 0.814 0,12*

227 0.028 0.0*2 0.8*3 0.067

228 0.008 0.758 0.217 0.018

229 0.006 0.851 0.077 0.067

230 0.068 0,765 0.1*5 0.002

231 0,240 0,754 0.005 0,000

232 0,005 0,987 0,002 0,006

233 0,008 0,917 0,000 0,07*

234 0,029 0,969 0,000 0,002

235 0,004 0,959 0,008 0,030

235 0,007 0,963 0,010 0,001
237 0,063 0,397 0.540 0,000

236 0.053 0.944 0.002 0.000

239 0.011 0.985 0.002 0.002

240 0.002 0.997 0.000 0.001

102

PROBABILITIES
OF TYPE MEMBERSHIP

1 2

2<,1 0.010 0.923

2*2 Oj79'3 0.177

2*3 0.005 0.982

2** 0,016 0.B63

2*5 0.028 0.901

2*6 0.0$5 0.316

2*7 0.972 0,000 0.02S 0.000
0.000 0.8**
0.000 0.9**
0.000 0.975

3 4

0.000 0.067
0.02* 0,000
0.009 0.00*
0,119 0,003
0.071 0.000
111 0.518

2*8 0.155

2*9 0,036

250 0.025

251 0.017 0. 12*
232 0.006 0.552
253 0.009 0.962

858

0,*15
0,023

001
000
000
001
026
001

25* 0,016 0,085 0,60* 0.295

255 0,236 0,017 0.7*7 0.000

256 0.933 0.012 0

257 0.10* 0.012 0
25fl 0.183 0.000 0

259 0.096 0.000 0

260 0,03* 0.000 0.711
0.002 0.958
0.863
0.31*
0,063

261 0,032

262 0,017

263 0.011
26* 0,035

,119

0
0
0

265 0,017 0,033 0.9*5
i66 0.00* 0.000 0.979
0.00* 0.000 0,67*

7*7 0,000
003 0,052
5*8 0.337
803 O.OU
90* 0,000
0.25*
0.008
0.001
551 0.123
898 0.005

267

268 0.032 0.000 0

269 0.1*3 0.000

270 0.005 0.000

271 0,00*

272 0.007

273 0,010
27* 0,029

275 0,018

276 0.033

277 0,060

278 0.063

279 0,0*3

280 0.039

281 0.019

,968
,857
,766
,966

.005
,016
.323
.000
.000
.229
.029

622 0.15*
982 0.008
0.121
0.005
7*7 0.1*1
699 0.180
786 0.129
0.1*6

088
883

0
0
0.013

0.000
0.217
0.005
0.762
0.095
0.009
0.061
0.006
0.798

0.»20 0.000 0.5*0
_. . - 0.950 0.006 0.026

282 0.013 0.916 0.070 0.000

283 0.0*3 0.91* 0.0*3 0.000
28* 0.022 0.965 0.005 O.OOB

285 0.091 0.909 0.000 0.001

286 0.9*7 0.053 0.000 0.001

287 0.961 0.035 0.00* 0.000

288 0.067 0.OO5 0.929 0.000
0.001

289 0.272

290 0.022 0,936

291 0,*12 0,015

292 0,025

293 0.9B9
29* 0.109

295 0.0*9 _ . _

296 0.122 0,081

297 0.12* 0,001

298 0,078

0,003
0,000
0,832
0,523

727
0*2
525
972
Oil
000
001
751
875
561

0,000
0,000
0.0*9
0.000
0.000
0.059
0.*27
0.0*6
0.000
0.123

_ . 0.238 . _

299 0.806 0.19* 0.000 0.000

300 0,0*2 0.958 0.001 0.000

301 0.125 0.875 0.000 0.000

302 0.162 0.737 0.093 0.008

303 0.018 0.980 0.003 0.000
30* 0.006 0.972 0.016 0.006

305 0,052 0.907 0.0*0 0.000

306 0.059 0.000 0.9*1 0.000

307 0.07* 0.009 0.65* 0.263
0.357 0.*20 0.203

,032
,3**

308 0.020

309 0.01* 0.908

310 0.016 0.026

311 0,083

312 0.051

313 0.126 0.080
31* 0.070

315 0.075

316 0,995 0.005

317 0.299 0.037

318 0.2*5 0.010

319 0.887

320 0.9*3

321 c.ieo

1.000

322
323
32*
325

0.073 0.8*1
0.806 0.19*

_ 0.93*

326 0.5*5

327 0.98*

328 1.000

036 0.0*2

953 0.005

878 0.007

599 0.006

792 0.002

0.008 0.922 0.000

0.066 0.859 0.000

0.000 0.000

0,070 0,593

0,735 0,010

0.096 0.000

0.0*9 0.000

0.057 0.001

0,000 0.000

0.085 0.000

0.000 0.000

0.001 0.000

0,**1 0.000

0.000 0.0l6

0.000 O.OQO

0.017
0.009
0.761
0.000

0.065
0.01*
0.000
0.000

Table 18 (continued)

PROBABILITIES

OF TYPE MEMBERSHIP

12 3 4

329 1.000 0.000 0.000 0.000

330 1.000 O.OOn 0.000 0.000

331 0.00* O.OOO 0.950 0.0*6
33? 0.919 0.000 n.OBl 0.000
333 I. 000 0.000 0.000 0.000
J3* 0.999 0.000 0.001 0.000

335 0.99* 0.006 0.000 0.000

336 1.000 0.000 0.000 0.000

337 0.890 0.110 0.000 0.000

338 0.925 0.00* 0.071 0.000

339 0.999 0.000 0.001 0.000
3*0 0.027 0.009 0.96* 0.000
3*1 0.02* 0.000 0.911 0.065
3*2 0.01* O.Ol* 0.9*6 0.007
3*3 O.OU 0.000 0.*33 0.556
3** 0.002 0.000 0.820 0.178
3*5 0.01* 0.000 0,*02 0.585
3*6 0.002 0.000 0.67* 0.32*
3*7 0.010 0.002 0,9*9 0.0*0
3*8 0.002 0.001 0.883 O.ll*
3*9 0.01* 0.000 0,258 0.729

350 0.00* 0.000 0.882 0.11*

351 0.005 0.002 0.921 0.072

352 0.005 0.000

353 0.005 0.001
35* 0.009 0.027

355 0.07* 0.060
350 0.0*6 0.012
357 0.026

,995
,957
,85*
,82*

,9*1

.000
.036
.110
.0*1
.001
,027

0.003 0.9**

358 0.039 0.001 0.924 0.036

359 0.026 0.001 0.836 0.137

360 0.99* 0.002 0.00* 0

361 0,099 0,89* 0,000 0
0,253 0.500 0
0.0*0

0.392

0.000
0.772
0.000
0.006

829
805
.0*3
,951

.003
,000

,000
,006
,00*
,002
,005
836
003
,000
,895

,328 0.238
,001 0.000

058 o.on

,339 0.000
,655 0.001
,05* 0.000
.002 0.000
913 0.06B

362 0.2*3

36:t 0.129

36* 0.186 0.00*

365 0.120 0.000

36o O.OU 0.03*

367 0.0*9 0.9*8

368 0,037 0.068

369 0,0*2

370 0,999

371 0.158

372 0.661

373 0.338
37* 0.076 0.870

375 0.016 0,982

376 0.017 0.002

377 0.009 0.002 0.987 0.002

378 0.01* 0.007 0.829 0.150

379 0,010 0.001 0.9*2 0.0*6

380 0.018 0.005 0.870 0.107

381 0.101 0.736 0.161 0.002

382 0,025 0.*2* 0.*»9 0.082

383 0.026 0.910 0.0»* 0.000
38* 0.067 0.921

385 0.038 0.920

386 0.013 0.971

387 0.028 0.933

388 0,066 0.93*

389 0.005

390 0.010

391 0.018

392 0.022

393 0.021
39* 0,017

395 0,0*1 0.959

396 0.006 0.990

397 0,011 0,967 0,018 0.00*

398 0,02* 0,922 0,05* 0,000

399 0,016 0,616 0.3*6 0.001
*00 0.*03 0.001 0.596 0.000
*01 0.039 0.139 0.822 O.OOl

0.583
0.501

0.012 0.000
0.0*2 0.000
0.005 O.Oll
0.009 0.029
0.000 0.000
0.98* 0.008 0.003
0.988 0.002 0.000
019 0.000
009 0.000
000 0.000
0*7 0.00*
000 0.000
003 0.000

0.893
0.970
0.979
0.932

*02 0.116

*03 0.033

*0* 0.986 0.011

*05 1.000

*06 1.000

*07 1,000 0.000

*08 1.000

*09 1.000

*10 0,9**

*ll 0,9*1

0,29* 0,007

0.*** 0,002

0.003 0.000

0.000 0.000 0.000

0.000 0.000 0.000

000 0.000

000

000

052

000

0.000

0.000

0.00*

.. -. - 0.059 _ _ ,

*12 1,000 0.000 0.000 ,

*13 0.065 0.686 0.001 0.2*8

*l* 0.011 0.967 0.005 0.017

*15 0,056 0,162 0.735 0.0*6

*16 0.031 0.05* 0.8*3 0.051

0.000
0.000
0.000
0.000

0.000

PROBABILITIES

OF TYPE MEMBERSHIP

*17 0.026
*ie 0.017
*19 0.D08
*20 0.022
*21 0.010
*22 0.013
*23 0.098
*2* 0.136
*25 0.151

( 4

713 0.026

8*6 0.000

93* 0.0*1

71* 0.261

8** 0,028

159 0,827 0,000

891 0,011 0,000

0.03* 0.000

0.003 0.000

0.000 0.000

2
0.23*

0.U7

0.017

Ot003

0.097

0

0,

0

0

830
8*6
*2» 1.000 0.000
*27 0.100 0.013 0.797 0.089
*28 0.162 0.002 0.633 0.203
0.000
0.288

*29 1.000

*30 O.OIO

*31 0.037

*32 0.010

*33 0,007

*3* 0,010

*35 0,005

0,000 0,000

0.550 0.152
0.730 0.230 0.002
0.061 0.906 0.02*

0.007 0.93* 0.051
0.097 0.893 0.000
0.886 0.079 0.0J5
*36 0.025 0.69* 0.281 O.OOO
*37 0.00* 0.990 0.005 0.002
*38 0.006 0.963 O.Oll 0.020
*39 0,*32 0.000 0.000 0.568
**0 0.319 0.680 0.000 0.000
**1 0.0*1 0.958 O.OOl 0.001
**2 0.003 0.000 0.8*7 0.130
**3 0,009 0.000 0.016 0.979
*** 0.000 0.000 0.000 l.OOO
**5 0.0*9 0.001 0.001 0.9*9
**6 0.00* 0.000 0.896 O.IOO
**7 0,158 0.000 0.515 0.327
**8 0.008 0,000 0.915 0.07t
**9 0.009 0.000 0.7*7 0,22*
*50 0,006 0.000
*51 0.0*8 0.021
*52 0.058 0.000 0.005
*53 0.180 0,000 0.820
*5* 0,065 0.000 0.929
*55 0.056 0.005 0.*91
*56 0.019 0.000 0.981 0,000
*57 0,022 0,001 0.975 0.001
*58 0.017 0.003 0.979 0.001
*59 0.015
*60 0.012
*61 0.038
*62 O.IU
*63 0.03*

.92* 0.071
0.306 0.62*
0.9i7
0.000
0.006
0.**9

0.999 0,J08 0,079
0,8*6 0.122 0.020
0.657 0.215 0.090
0.887 0.001 0.000
0.**8 0.00* 0.51*
*6* 0.011 0.583 0.3*7 0.039
*65 0.02* 0.650 0.008 0.319
0.330 0.62i 0.0)6
0.288 0,022 <).*61
. . _ 0.133 O.03* 0.000

*69 0.151 0.8*2 0.002 0.005
*70 0.315 0.002 0.000 0.682
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.002 0.000
*76 0.822 0.000 0.177 0.001
*77 0.13* 0.000 0.8*6 0.000
*78 0.066 O.OOO 0.927 0.007
*79 0.231 0.000 0.7*9 0.000
*80 0.028 0.000 0.9T2 0.000
*81 0.*13 0.582 0.005 0.000
*82 0.080 0.000 0.8(7 0.03)
0.000 0.906 0.001
0.238 0.607 0.1**
0.9*2 0.025 0.000
. . 0.188 0.011 0.32*

*87 0.859 0.076 0.01* 0.052
*88 0.105 0,335 0,959 0,001
*89 0.119 0.000 0,001 0,8S*
*90 0,637 0.002 0.3*1 0.000
*91 1.000 OtOOO 0.000 0<000
*92 1.000 0.000 0.000 0.000
*93 0.097 0.723 O.IIO O.OOO
*9* 0.2*5 0.222 0.5)1 0.002
*95 0.063 0.017 O.lZt 0.791
*96 0.0*6 0.156 0.796 0.002
*97 0.601 0.012 0.3(7 0.000
*98 0,106 0.002 0.893 0.000
*99 0.02* 0.000 0.976 0.000

500 0.018 0.000 0.912 0.000

501 0.995 C.OOO 0.009 O.COO

502 0.1*5 0.000 0.855 0.000

503 0.98* 0.019 0.000 0.001

*66 0,010
*67 0.228
*68 0.833

*71 1,000
*72 l.OOO
*73 1.000
*7* 1.000
*79 0.998

*83 0.093
*8* O.OU
*85 0.032
*e6 0.*78

LOeARITHM OF LIKELIHnOB RATIO OF * TO 3 TYPES •
CHI>50UARE WITH 9* DEGREES OF FREEOONa 81.1*
PHOBABILITY OF NULL HYPOTHESIS- 0.00986137

0.*12672(*D 02

103

8.3.3.3 Figures and Discussion. Figures 20(a) through (f) illustrate the
tabular output of table 16. In (a), under the assumption of only one cluster
at each of three levels, i.e., a unimodal bivariate distribution, the ellipse
shows the relative size of the distribution at the 700- , 500-, and 300-mb
levels. The 0.50 probability ellipses all have more or less the same orienta-
tions. In (b), the 700-mb unimodal assumption is illustrated along with the
two clusters in the assumed bimodal bivariate distribution. The same procedure
is followed in (c) and (d) for the 500- and 300-mb levels. For further com-
parison, (e) and (f) show the level-to-level comparison for the cluster type 1
and the cluster type 2.

Figures 21(a) through (g) illustrate the tabular output of table 17. Figure
21(a) is the same as figure 20(a). Figure 21(b) shows the 700-mb unimodal
distribution of figure 21(a) with the added three-cluster breakout at the 700-
mb level. The same procedure follows through figures 21(c) and 21(d). Figures
21(e), (f), and (g) show the comparison of type distributions through the three
levels. For example, figure 21(e) shows roughly the same orientation but
greatly increased variance with altitudes. The same is true for figures 21(f)
and (g).

Figures 22(a) through (h) follow the same procedural pattern as the previous
figures. Its comparable table is table 18. Here there are four clusters.
Again the initial unimodal distribution is shown in figure 22(a) while (b),
(c), and (d) show the four clusters at each level. The following four show
the cluster types at the three levels. Though the small numbers may be diffi-
cult to read, they appear also in the tables. Of major importance here is the
pictorial display of the orientations and sizes of the ellipses or clusters.

The change in orientation with altitude is noted. Some strong change in
orientation between types is also noted.

Further work will be done with these distributions to determine the relation-
ship among these clusters, their orientation and dispersion and concurrent
weather.

104

I "/rh-^. 1

(a)

■M

tf^l

■ ■ta

(c)

Figure 20 Bivariate distributions of winds in m-s"^ at Rantoul, niinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels. Two
cluster types (1 and 2) are assumed in the total mixed observed
distribution (0.171 + 0.829 = 1.000). (a) Total distribution,
(b) 700-mb mixed, (c) 500-mb mixed, (d) 300-mb mixed, (e) Type 1,
and (f) Type 2.

105

[M

-"

■

(e)

::

;'

Figure 20 (continued)

106

■m

\

1 . ^

(b)

[

/ / "">W"'"/

)

.

(d)

1

Figure 21 Bivariate distributions of winds in m-s"^ at Rantoul , Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels. Three
cluster types (1, 2, and 3) are assumed in the total mixed ob-
served distribution (0.182 + 0.183 + 0.635 = 1.000). (a) Total
distribution, (b) 700-mb mixed, (c) 500-mb mixed, (d) 300-mb
mixed, (e) Type 1, (f) Type 2, and (g) Type 3.

107

<'.

■

(0

■ m

Figure 21 (continued)

108

m

-.fa

r

(b)

.„

,

^

)

■ T ^F

/ . - -

■ ■ \

(c)

'K^-7;,iti;. '•%.,

Figure 22

Bivariate distributions of winds in m-s"^ at Rantoul , Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels. Four
cluster types (1, 2, 3, and 4) are assumed in the total mixed
observed distribution (0.094 + 0.193 + 0.293 + 0.421 - 1.000).
(a) Total distribution, (b) 700-mb mixed, (c) 500-mb mixed,
(d) 300-mb mixed, (e) Type 1, (f) Type 2, (g) Type 3, and
(h) Type 4.

109

■

(f)

..

f .(.fm

(g)

/• ' ' '

Figure 22 (continued)

110

8.3.4 Mountain Pass Wind Data Set

8.3.4.1 Input Information.

a. Stampede Pass, Easton, WA, U.S.A. - latitude 47°17' north,
longitude 121°20' west, elevation 1206 m.

b. The period of record is the month of December for the years
1966-1970. The local standard time hours are 0700 and 1300.

c. The data are surface winds (m-s"M and temperatures °C.

d. The number of variables is two then three. The first two
are the zonal and meridional components of the wind, positive
from the west and south, then third is the temperature in °C.

e. The number in the sample is 310, 155 from 0700 and 155 from
1300 l.s.t.

f. The minimum number to be accepted into any cluster is one
more than the number of variates.

g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.

h. The first 40 two-dimensional vector entries are set up as the
40 means of 40 separate and individual clusters. These are
40 points in two or three dimensions.

i. The assumption is made that the covariance matrices are not
equal .

8.3.4.2 Tables. Table 19 provides the output data for the two-variable wind
component distributions taken in tabular form from the Wolfe (1971b) NORMIX-
NORMAP computer routine. Table 20 provides the output data for the three-
variable temperature and wind components.

In both tables the mixture proportions, by cluster type, the means, standard
deviations, correlation matrices, and the eigenvalue-eigenvector matrices are
given. An asterisk indicates the rejection of the null hypothesis that the
(k + 1) type is not significantly different from the (k) types.

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115

8.3.4.3 Figures and Discussion. Figure 23a shows the single cluster distri-
bution versus the two-cluster breakout, each cluster assumed to be bivariate
normal. In essence, these assumptions are the unimodal versus the bimodal
bivariate distributions. Table 19 provides the statistics for this figure.
Clearly, it is seen that the total distribution is not well represented by the
single unimodal elliptical bivariate normal distribution with the mean at
(-0.4915, 1.1879) m-s"^ with east-west and north-south component standard de-
viations of 5.7123 and 1.6058 m-s"^, respectively. The ratio is almost four
to one.

The total distribution breaks out into two separate distributions, each
assumed to be unimodal. The attempt to determine whether the distribution
might actually be trimodal rather than bimodal met with no success. There-
fore, it is assumed that the bimodal bivariate distribution is a better
representation than a unimodal or trimodal bivariate representation. The two
modes are east-southeast and west-southwest.

Let us now discuss the location and terrain features of Stampede Pass.
Stampede Pass is located in mountainous terrain on the Main Cascade Divide at
latitude 47°17' north and longitude 121 °20' west. The elevation of the ground
at the station is about 1206 m. The wind instruments are approximately 11 m
higher.

East of the station the ground drops abruptly into the Yakima River Valley,
about 600 m down and a little over three km distant. This land and valley
fall towards the southeast. The lowest part of Stampede Pass is 5/8 km north
and 30 m lower. There is a ridge 1-1/4 km south of the station and about 200 m
higher. West of the station the land drops rapidly about 600 m over a distance
of 6.4 km to the Green River Valley. This land and valley fall toward the
southwest. To the north, from west northwest through east, there are ridges
and peaks which rise to 1.2- and 2.7-km.

General winds from the west will be channeled from the west southwest up and
around the ridge nose then turning to the east southeast and thence southeast.
General winds from the east will traverse this same channel but in the opposite
sense. It would seem then that a wind distribution through the pass would have
to be elliptical. In addition, it would seem that traveling weather systems
would create one distribution from the west southwest and one from the east
southeast. This agrees with the tabular values (table 19) and the illustration
(figure 23a).

Though not presented in either the table or the figure, it should be mentioned
that the computer routine did attempt to converge to a solution for the trimodal
assumption. The last estimates did indicate a tendency for the east southeast
mode to break down into two east southeast modes located on the major axis of
the mode shown but centered, one further to the east southeast and one to the
west northwest.

Table 20 and figures 23b, c, and d present the situation when the temperature
arguments are added to those of the wind. The singularity problem noted above
is resolved and the computations indicate definitely that four cluster types
are present. The null hypothesis would have been rejected at the 0.00000264

116

level. The selected decision level was 0.01. Therefore, there is a good
probability that five or even six clusters would have been isolated if the
computer had been allowed to continue. The option selected was to examine
the structure through only four groups.

Figure 23b provides the two-cluster breakout with an illustration of the
0.25 error ellipses of the wind. The mixture proportion as well as the mean
temperature (°C) associated with each group is shown. As progress is made
through the next two subfigures, note that the western group changes only
slightly moving farther away as the breakdown continues. Please note the
breakout of the eastern group into two groups with the easternmost group being
extremely cold (=-25°C). Note that it then remains relatively fixed while the
central group of (b) breaks down into two clusters.

Note again that the easternmost group which comprises only two percent of
the total is extremely cold. This leads to two conjectures:

(1) This group is an outlier and the data are bad.

(2) This group is an outlier but is a valid cluster. The output of the
data permits identification of the individual datum. The two-percent cluster
occurred in the same period of time ciiid actually composed a string of data.
The records were checked. The data are correct. Figure 24 is a copy of a
National Meteorological Center surface analysis chart for 1200 Greenwich time
on December 30. A star marks the approximate location of Stampede Pass. Note
the incursion of the cold air mass from Canada. With its increasing cold and
speed as seen on prior maps, it is worthwhile to read the comments by Phillips
(1969) published in the Climatological Data publication of the National Oceanic
and Atmospheric Administration. The temperatures mentioned are in degrees
Fahrenheit.

"Washington - December 1968

"Special Weather Summary

"Until near the end of the month, weather systems from over the
Pacific moved across the State at frequent intervals. In western
Washington, this resulted in measurable precipitation on 23 to 28
days and on 12 to 18 days in eastern Washington.

"West of the Cascades, rather heavy precipitation was recorded on
several days, falling as rain in lowlands during the first half
of the month, and as snow and rain the latter half. In the moun-
tains, most precipitation fell as snow, with near record depths
for December on the ground at the end of the month.

"East of the Cascades, snow began accumulating on the ground the
first of the month. After the middle of the month, most agricul-
tural areas in southern counties were covered with 1 to 3 inches
of snow. Temperatures were near or above normal for the first
half of the month and slightly below normal from the 15th to the
25th.

117

"An outbreak of yery cold arctic air accompanied by strong
northerly and northeasterly winds began moving into northern
valleys of eastern Washington, and other localities near the
Canadian State on the 26th, spreading over most of the State on
the 27th. Temperatures continued to fall for 3 days, with the
lowest occurring on the 30th. In many respects, this was the
most severe outbreak of cold air since the winter of 1949-50.
Minimum temperatures dropped below previous records at several
stations in eastern Washington and in the Cascades. The -48°
recorded at Mazama and Winthrop 1 SW is a new record for the
State and -43° at Chesaw 4 NNW is also below previous record of
-42° at Deer Park on January 20, 1937. Other stations where
minimums dropped below previously recorded low temperatures were:
Anatone -32°, Chelan -18°, Colfax 1 NW - 33°, Colville Airport
-33°, Dayton 1 SW -25°, Holden Village -32°, Lacrosse 3 ESE -34°,
Leavenworth 3 S -36°, Methow -37°, Pomeroy -27°, Pullman 2 NW -32°,
Republic -38°, Rosalia -29°, Snoqualmie Pass -19°, Stampede Pass
-21°, Stevens Pass -25°, Stehekin 3 NW -21°, and Waterville -33°.

"On the 30th, a warmer moist airmass from over the Pacific began
moving inland over the colder air near the surface. Snow began
falling during the day, becoming heavy at night and continuing
through the 31st. West of the Cascades, snow depths in the lowlands
ranged from 8 to 1 5 inches and 24 to 36 inches or more in foothills.
In numerous localities, highway traffic was at a near standstill on
the 31st and many offices and businesses remained closed.

"Preliminary reports from fruit producing areas indicate the low
temperatures caused extensive damage to stone fruits and perhaps
some damage to other fruit trees. Most of the winter wheat section
was covered with 1 to 3 inches of snow, thus very little freeze
damage is expected."

Note the fourth paragraph. Ludlum (1969) disucsses this particular feature
in Weatherwise on pages 36-37 of the February issue. Dickey and Wing (1963)
also discuss the problem of arctic air flowing into the Pacific Northwest.

The above example and discussion point out the usefulness of a program such
as this for editing and quality control of multivariate data; i.e., data groups
other than one element at a time. Here, an outlier group was isolated but it
was a valid group. The authors did not realize that this particular outlier
group was embedded in the data set used. Simply, a location and a period were
selected where it was thought that the program would successfully and pointedly
demonstrate its capability.

118

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Figure 23 Stampede Pass, Easton, Washington, U.S.A.; winds (m-s"M and
temperatures (°C), December 1966-1970, showing breakdown of
winds only into groups 2 and 3 from group 1 (a) and breakdown
of wind-temperature combination into 2, 3, and 4 groups (b, c
and d).

119

1200Z DEC. 30 1968
69. NMC SFC ANALYSIS
ASUS 69.

Figure 24 Selected area of North American chart, 1200Z, Monday, December 3,
1958, NMC analysis. The star represents the approximate position
of Stampede Pass, Easton, Washington.

120

8.3.5 Marine Surface Data Set

8.3.5.1 Input Information.

a. OSV "C," 52°45' north latitude, 35°30' north longitude

b. The period of record is the month of February for the years
1964 through 1972.

c. The data are the 1200 G.C.T., pressures, temperatures, dew
points, and surface winds.

d. The number of variables is five. The wind is broken into the
zonal and meridional components. The units are mb, °C, and m-s"^.

e. The number in the sample is 251.

f. The minimum number to be accepted into a sample is six.

g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.

h. The first 40 five-dimensional vector entries are set up as the
40 means of 40 separate and individual clusters. These are 40
points in five dimensions.

i. Non-equality of covariances is assumed.

8.3.5.2 Tables - Output Information. The program computes the necessary
statistics for two clusters versus one cluster, three clusters versus two
clusters, and four clusters versus three clusters. These are taken from the
tabular outform of the Wolfe (1971b) NORMIX computer routine.

The output statistics are now shown for the above in tables 21, 22, and 23
even though the null hypothesis is rejected for the last case.

The first variable is the surface atmospheric pressure in mb, the second
variable is the dry-bulb air temperature in °C, the third variable is the
dew point temperature in °C, while the fourth and fifth variables are the
zonal and meridional components of the wind in m-s"^.

121

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8.3.5.3 Figures and Discussion. Figure 25 is an attempt to illustrate a part
of the output of table 21. The basic two-dimensional representation shews the
decomposition of the assumed unimodal distribution into two modes or groups.
The probability ellipses represent the area of the central 25 percent of the
wind vector origins if the assumption which they represent is valid. As shown
in table 21, the probability of the null hypothesis for two groups versus one
group not being rejected is wery low, i.e., 0.00000002. Therefore, the more
valid assumption is that two groups better represent the data set than does
one group. The single group centered at (2.0414, 1.0315) is shown for reference.

At each point, the following concurrent values are printed, the mixture
proportion, the mean pressure, the mean temperature, and the mean dew point.
The two groups appear to have not too different mean pressures but considerably
different temperature and dew points.

Figure 26 prepared from table 22 similarly portrays the decomposition of the
total group into three groups. It is quite apparent that group 1 is not much
different from group 1 of figure 25, It is just as apparent that group 2 of
figure 25 really breaks down into groups 2 and 3 as shown in figure 26. The
pressures in groups 2 and 3 are quite different though the temperatures and dew
points are not too different. The appearance here implies, as does the proba-
bility of non-rejection of the null hypothesis being small, that the trimodal
representation is better than the bimodal which is in turn better than the
unimodal representation.

Figure 27 prepared from table 23 depicts the further decomposition of the
data set into four groups. Again, group 1 retains essentially the same mixture
proportions and characteristics. The northerly group (group 4) remains almost
the same as in the last decomposition. It is group 2 of figure 26 that now
breaks down into groups 2 and 3 of figure 27. The pressure differences are
remarkable, providing the maximum and minimum pressures for the entire four-
cluster configuration. Group 3 has the highest temperatures and dew points of
the entire ensemble. The 10-percent mixture proportion of group 2 leaves the
impression that this is a real group though the null hypothesis is not rejected.
This non-rejection probability of 0.12 implies that at our decision level of
0.01 probability, the trimodal representation is a better representation than
the four-cluster configuration. However, it is interesting to conjecture that
the differences shown are the difference between storm and non-storm situations
with winds from the southeast quadrant.

No other representations are made though any pairing or triplets could be
graphed in two and three space, respectively, and could be labeled with the
fourth and fifth variable mean values.

The authors consider this to be a good representation of the clustering
techniques of the Wolfe NORMIX (1971b) computer routine. It was hoped that
this routine would isolate a cluster of six or more data which might be con-
sidered to be an outlier cluster which could be examined. There were 251 data.
Ten percent is essentially 25 or 26 data in the clusters. The instructions
provided to the computer were to collect no less than six data in a cluster.
This is a default option which sets the minimum number in any cluster to be
one more than the number of variables. In this case, five plus one is six.

129

Therefore, (a) single, doublet or triplet outlier(s) would not be isolated.
In this case, if the mixture proportion ran as low as 2.5 percent, the group
would be considered as an outlier group and would be examined to see whether
the data were bad because of bad sensors, bad recording, bad entry of data
into the archives, or simply a group of valid observations out of a yery rare
weather situation.

130

-2D

LEGEND
D5Y G FEB. 19EH-72/ Ml DBS.
BlYRRIflTE mmi DI5TR1BUT!DN5 DF U mh Y

PRESSURE IN MB

TEMPERBTURE IN 5EBREE5 (

m POINT IM DEGREES (

MIKTURE RflllD

la

-10

Figure 25 OSV "C" surface distribution of pressure (mb), temperature (°C),

dew point (°C), and wind components (m-s~M» February, 1200 G.C.T.,
1964 through 1972; n = 251. Covariances are assumed to be unequal.
The 0.25 probability ellipses are shown for the wind distribution.
The total distribution and the breakout into two clusters are shown
The mixture proportion and the averages, the pressure, the tempera-
ture, and the dew point data within each cluster are shown.

131

1-20

Figure 26 OSV "C" surface distribution of pressure (mb), temperature (°C),

dew point (°C), and wind components (m-s"M> February, 1200 G.C.T.:
1964 through 1972; n = 251 . Covariances are assumed to be unequal.
The 0.25 probability ellipses are shown for the wind distribution.
The total distribution and the breakout into three clusters are
shown. The mixture proportion and the averages, the pressure, the
temperature, and the dew point data within each cluster are shown.

132

Figure 27 OSV "C" surface distribution of pressure (mb), temperature (°C),

dew point (°C), and wind components (m-s"^), February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed to be unequal,
The 0.25 probability ellipses are shown for the wind distribution.
The total distribution and the breakout into four clusters are
shown. The mixture proportion and the averages, the pressure, the
temperature, and the dew point data within each cluster are shown.

133

8.3.6 Radiosonde and Rawinsonde Data Set

8.3.6.1 Input Information.

a. Balboa (Albrook Field), Canal Zone

b. The period of record is the month of July 1961-1970.

c. The data are pressures or height (mb or m) temperatures (°C),
dew points (''C), east-west (u) wind components, and north-south
(v) wind components, m-s"^ positive from the west and south.

d. The number of variables are 20; the above elements for the
surface and the 850- , 700- , and 500-mb levels.

e. The number in the sample is 259.

f. The minimum number to be accepted into a sample is 21.

g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.

h. The first 40 twenty-dimensional vector entries are set up as the

40 vector means of 40 separate and individual clusters. These

are 40 points in twenty dimensions,
i. Two assumptions are made. The first assumption is the equality

of covariance matrices. The second assumption is the non-equality

of the covariance matrices.

8.3.6.2 Tables and Discussion. Table 24 provides selected output data of the
Wolfe (NORMIX) computer routine. The logarithm of the likelihood ratio of 2
to 1 types is 676.30485, the chi-square with 40 degrees of freedom is 1240.33,
and the probability of the null hypothesis not being rejected is to seven
decimals, 0.0000000. Therefore, the two-cluster configuration is not rejected.
The computation failed to converge for the three-cluster versus the two-cluster
configuration. Convergence under the assumption of unequal covariance matrices
also failed. No figures are provided here as the number of variables is too
high. A number of two-dimensional figures, however, could be made from the
statistics provided.

In order to determine the capability of the NCC computer (a Univac Series 70)
and in view of the previous work, a multivariate problem with 40 element vectors
was chosen for Balboa, C.Z. There were eight levels of each radiosonde with
five elements, pressure (or height), temperature, dew points, and the east-west
and north-south components of the winds. The levels were the surface and the
950- , 900- , 850- , 800- , 700- , 600- , and 500-mb levels. The period chosen was
the months of July during 1951-1970. The assumption was made that the covari-
ance matrices were different. The computation failed to converge for a two-
group separation. The number of vector elements was reduced to 20. Again the
computation failed to converge.

The assumption of different covariance matrices was then replaced by the
assumption of equal covariance matrices. The computation converged for a
breakout of two groups but failed on three groups when the vectors were com-
posed of 20 elemeits, five elements from each of four of the levels above,
namely the surf a _., and the 850- , 700- , and 500-mb levels as these were the

134

only elements believed to have moisture measurements in sufficient quantities
to assure enough input data.

Examination of table 24 shows that though there does not appear to be too
much difference, there is some. About 95 percent of the data comprise one
(Type 1) cluster while 5 percent of the data comprise the other (Type 2)
cluster. Cluster 1 is cooler than cluster 2 at all levels through 500 mb, an
altitude of roughly 5854 m. Cluster 1 is more moist than cluster 2 at alti-
tudes above about 3,100 m and probably above 2000 m. Cluster 1 exhibits lower
wind speeds than does cluster 2 from the surface upwards. The difference
ranges from 1.2 m-s"^ at 1500 m to 1.6 m-s~^ at 5854 m. Cluster 1 winds shift
from east northeast to east by southeast at the highest level while the winds
in cluster 2 remain east by northeast throughout the layer. The surface pres-
sure of cluster 1 is about 0.6 mb higher than that of cluster 2.

The types of weather that accompany these groups have not been investigated
here. It would be interesting to do so. Balboa, C.Z., data were selected

(1) to provide an insight into the lower level atmospheric characteristics,

(2) to further understanding of the capability of this program, adopted for
use on the Univac 70 at the NCC, to handle multidimensional problems, and

(3) to demonstrate the utilization of such a program to consider each radio-
sonde observation as a point in multidimensional space.

From the above experience it appears that the program, restricted by the

present Univac 70 configuration, can handle 300 input 20-dimensional data

problems. The first assumption made should be the assumption of equal co-
variance matrices.

Use of the program for data at other stations where greater differences may
be expected may permit the use of more input data, greater dimensions, and the
assumption of unequal covariance matrices.

The minimum number to be accepted into a cluster is always one more than the
number of dimensions. In this case the minimum number is 21. This is about
8.1 percent. Cluster Type 2 has only about 5.4 percent. Therefore, it would
be advisable to check those observations assigned to Type 2 by the discrimina-
tion function for the possibility of all or some of these being outliers.
This is not done here as this is the basis for work beyond the scope of this
paper.

135

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136

9. MULTIVARIATE QUALITY ASSURANCE AND CONTROL

The capability of the NORMIX program permits a quick elementary view of
principal component analysis and of multivariate quality assurance and control.
The techniques of assurance and control are the same. Assurance is for in-
coming data or material, while control is for data or material processing.
Essentially, these are filtering techniques.

Here, we look at the separation of homogeneous subsets out of mixed sets
which permits the isolation of "outliers" for examination.

Figure 28 (Crutcher, 1966) is an example of a two-tailed Gaussian filter
operating on a set of univariate heterogeneous data to isolate outlying data.
This essentially sets up two or more groups or subsets. The progress of the
sub-figures (a) through (e) schematically shows how the procedure works on a
univariate distribution. The alpha level of rejection is 0.05. In this
illustration the main group is isolated and the statistics estimated. The
other groups have been isolated for further study as to whether they are valid
data.

The NORMIX program clustering techniques essentially go through the same
technique. However, the decision (alpha) level chosen for the previous
examples was 0.01 probability rather than 0.05.

Use can be made of the univariate Gaussian filter or the NORMIX filter can
operate on any univariate set of mixed normal distributions. Refer to the
Canton Island, July, 30-mb data set. Figure 15 illustrates by means of 0.50
probability ellipses the breakdown of the total set of data into two subsets
with quite different characteristics. (See section 8.3.2.3.) A univariate
Gaussian filter could be used on the zonal or meridional components. To
demonstrate the application of the NORMIX clustering technique as well as
principal component techniques, zonal and meridional components are transformed
to uncorrelated pairwise data along the major and minor axes, the two principal
axes of the distribution. Prior to and after the rotation, the components are
standardized to a zero mean and a variance of 1. See table 25.

Figures 29a and 29b show the frequency distribution computer printer plot of
the components along the two principal axes. Figure 29a clearly delineates
the two clusters. In the latter case, separation would be most difficult on
a one by one basis, but at least the mean and variance can be estimated. The
existence of groups with equal mean and variance would not be indicated.
Clearly indicated in Figures 29a and 29b are the potential invalid outliers at
+1.95 on the major axis and beyond 4.16 on the minor axis. These are simply
pointed out to the reader but are not examined for validity.

Figures 30a, 30b, and 30c (Crutcher, 1966) illustrate a Gaussian filter
technique in two dimensions. Figure 30a illustrates a sample drawn from a
homogeneous bivariate distribution contaminated by three subsets of data. One
of the subsets consisting of only a datum is first isolated, eliminated from
further immediate consideration, but set aside for investigation as to its
validity.

137

The clear separation along the major axis exhibited by the frequency diagram
of figure 30a is shown by the above statistics. The separation along the minor
axis is not so clearly demonstrated.

Refer to figure 29a and tables 25 and 26. Table 25 provides the statistics
for the components along the major axis of the distribution of winds at Canton
Island, U.S.A. and U.K., for July 1954-1964. These are standardized trans-
formed variables and as such are dimensionless in the terms of units as in all
figures in this section. The mean of the total distribution is zero and the
variance is about 1.0006 to four decimals. As this is a standardized set of
data, the mean should be zero and the variance one. The error in the fourth
place is a numerical rounding error. The set breaks down into two clusters.
The mean of the first group is -1.0420 with a standard deviation of 0.2442.
The respective values for the second group are +0.7923 and 0.5127.

The output from the Cohen-Falls (1967) program provides the data in table
26. Note the output from ungrouped data as used in the NORMIX program, table
25. There are slight differences which are attributed to the use of ungrouped
data and then grouped data.

The one outlier near 1.95 in figure 30a would not be set aside as a cluster
or outlier by the NORMIX program. As indicated elsewhere, the minimum cluster
size is always one more than the number of variates.

Table 26 also shows the output of the Cohen-Falls program for the dimension-
less wind components along the minor axis as illustrated in figure 29b. It is
recognized that, with the exception of the outliers, the mixed distribution has
degenerated into a single standardized univariate unimodal normal distribution
with a zero mean and variance of one. Note the proportion of two groups 0.998
and 0.002 with only 244 observations. Note the mean of -0.009617 in the first
group with the extraordinarily large mean of 7.410603 for the second group.
Remember the standard deviation of the entire set is only one. Then look at
the untenable negative variance obtained which is printed as x.xxxxx. However,
this negative variance indicates the difficulties in arriving at a solution
and clearly implies that it is the simple case of a symmetric mixed or compound
distribution with equal means and equal variances with proportions equal.
Therefore, the decision will be to use the estimates for the total sample as
estimates of the two groups {a mean of zero and a variance of one in each case).
This is the trivial degenerate case which does provide some computing difficulty
but no interpretative difficulty.

The elimination of the outliers along the minor axis and the re-standardiza-
tion of the data will provide a mean of zero and a variance of one rather than
the values given in table 26.

Figures 31a through 31c show the frequency distributions of the components
along three principal axes of the Stampede Pass, Easton, Washington, wind and
temperature standardized data. These refer to the figure 23. The existence
of outliers is clearly demonstrated. Outliers lie beyond -2.56 on the first
principal axis, beyond -1.68 on the second principal axis, and in both tails
of the third principal axis. Undoubtedly, some of these belong to the extreme
low temperature cluster isolated and discussed in section 8.3.4. The flatness

138

of the distribution on all three axes and the more extreme bimodality exhibited
on the third axis show the existence of the groups or clusters already isolated.

Extension of the Gaussian filter to more than one dimension at a time essen-
tially is the standardization of all components along the major axes so that
the distribution may be assumed to be spherical (Crutcher, 1966). The dis-
tribution of the standardized vectors in n-dimensions then is chi -square with
n degrees of freedom. The appropriate decision probability levels then are
obtained from any standard text book containing chi-square tables. Alterna-
tively, there are computer routines to compute the required values.

For example, the magnitude of the standardized vector in the one-dimensional
case for the rejection of the top and bottom 0.5 percent (0.005 probability)
implying non-rejection of the central 99 percent (0.99 probability) will be
the square root of the 0.99 chi-square value divided by 1 (the degrees of
freedom). This is 2.576 and is, of course, the t-distribution value for a
large sample. In the case of a two-dimensional distribution, the magnitude
of the standardized bivariate vector is the square root of 9.21/2 or 2.146.
For the three-dimensional case, the vector radius would be 1.944.

The point made here is that a datum lying on the main principal axis, the
major axis, has a better chance of being valid than an outlier on only one
of the original coordinate axes. The chance for an observation to be in-
correct in two variables is much less than the chance for an error in the
observation of one variable. An outlying datum on one axis is, of course,
projected as zero on the other transformed axes so that an outlier on one
axis will not be associated with an outlier on another axis. In fact, its
effect is to increase the frequency count at the mean or zero of the other
axes. Another way of viewing this is that the rejection ellipse has a better
chance of enclosing a possible outlier than the rejection bounds on either of
the two original correlated axes.

Figure 30b shows the effect of computing a new filter ellipse once the
outlier shown in figure 30a has been eliminated. Now the two small contami-
nating subsets undetected in the first operation are isolated. Figure 30c
then illustrates the last step where all data points remaining are inside the
filter ellipse. All rejected data are set aside for investigation as to their
validity. Before this check is made, however, these data may be processed to
see whether they constitute one or more groups or clusters.

The procedure, of course, brings up the possible censoring or truncation
problem. Adequate adjustments and better estimates of the parameters can be
made.

There is no discussion here of the applications of higher moments to the
problem of outlier detection, isolation, and removal.

Figures 32a through 32t illustrate the frequency distributions of the radio-
sonde data discussed in section 8.3.6. This discussion indicated only a slight
separation of clusters as contrasted with the stratospheric data of section
8.3.2. The frequency diagrams of figures 32a through 32t also show only a
slight separation potential in the flat of platykurtic curves.

139

Figures 32a through 32t are shown more to demonstrate the outlier problem
than the clustering problem. The mathematical requirements of the clustering
techniques always specify that the minimum cluster size is one more than the
number of variates. Thus, with twenty variables the smallest cluster size is
twenty-one. A cluster of this size would really indicate a true valid cluster
created by weather conditions or a continuing bias in the procedures or operat-
ing conditions. The lone outlier(s) in multi-space might go undetected. How-
ever, the frequency distribution along a principal axis or a clustering tech-
nique along each of the axes would isolate, identify, and permit removal of
the questionable data. Therefore, the frequency distributions of figures 29a
and b and 32a through t can be used to look at the few individual outliers.
Clustering techniques would have a better chance to show this outlier as there
is only one variable or a cluster size of two.

Alt et al . (1973) discuss quality assurance and control in their paper on
the use of control charts for multivariable data. Crutcher and Falls (1976)
discuss the testing of data sets for multivariate normality. The above two
reports and this present report add to the rather meager literature on the
subject of multivariate quality assurance and control.

140

20r

H. 10

(a)

Main body of sample

Outlying subsample

dllh

Outlier

0 10

20

30

40

50

60

70

80

30-

20-

10

(b)

Mean

=

29.

37

Variance

=

177.

74

Standard deviation

=

13.

33

n

=

81

d. f.

=

80

.050
t
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=

1.

99

=

2.

37

.025

30

70

80

Figure 28 Example of a two-tailed Gaussian filter operating on a set of

heterogeneous data to isolate, set aside, and eliminate outlying
data. The dark areas show the rejection; level, 0.05 (0.025 in
each tail). Crutcher (1966).

(a)
(b)

(c)

(d)

(e)

Here are the data in histogram form.

Here are the data with the theoretical fitting curve under
the assumption of normality and independence of data. The
rejection areas under each tail are shown beyond the 0.025
and 0.975 points. The observation at 75 is rejected.
Here is the theoretical fitting curve of the data which
passed the filter in (b). Again the rejection areas under
each tail beyond the 0.025 and 0.975 points are shown.
Data beyond 52 are rejected.

Here is the theoretical fitting curve of the data which
passed the filter in (c). Again the rejection areas under
each tail beyond the 0.025 and 0.975 points are shown.
Data beyond 47 are rejected.

Here is the theoretical fitting curve for the data which
passed the filter in (d). Although the rejection areas
are not shown, the singleton counts below 21 and above 25
would be rejected. This rejection is an unwanted rejection
but is a penalty which must be accepted at this point. The
next filtering step would result in no rejection.

l-ai

Mean =

28.

80

Variance =

139.

62

Standard deviation =

11.

81

n =

80

d. f. =

79

'.050 =

1.

99

70

80

30

20

10

(d)

Mean

=

27.

17

Variance

=

104.

06

Standard deviation

=

10.

20

n

=

75

d. f.
.050

74

1.99

60

— I —
70

80

30

20

10-

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20

Mean

=

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00

Variance

=

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52

Standard deviation

=

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23

n

=

64

d. f.

=

63

k.

— 1 —
30

.050

— I —
60

2.000

10

40

50

Figure 28 (continued)
142

SAMPLE = CANTON ISLAND

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c

+-> 3 3

•p-

ro to 01

o

S- (U •<-

Q.

4J S- 4-

</)

ro

3 <o c

+->

r^ •r—

ro

1 — lO

XJ

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cu

cu

(J in +J

1 —

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<a •»- +J

ro

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to

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x: .—

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CD

M6

STAMPEDE PASS, WASHINGTON

SURFACE TEMPERATURE (DEGREE C), U AND V WIND COMPONENTS (MPS)

DECEMBER 1966, 1967, 1968, 1969, 1970

HOURS 07 AND 13 LOCAL TIME (1ST 155 INPUTS ARE HR 07)

^REQUENI-Y UlblRlPUrUM ALONU AXIS 1

SACH X REPRESENTS — 1 SAMPuefSJ

CLASS Interval- o.32oS6

-<t. 80000!

mr

-4.<tB0C0l

zmr

-4.160001

TTT

-3.840001

rnr

-3.52000!

Ttr-

-3.20000!

or

-2.880001

crr

-2.56000 I

TTT-

-2.240001

— : bixxxxx

-1.920001

b!XXXXX

-1.600001

<iuixxyxxxxxxxxxxxxxxxxx

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24IXXXXXXXXXXXXXXXXXXXXXXXX
-0.9b000l

jaixxxxxyxxxxxxxxxxxxxxxvxxxxxxxxxxx

-0.640001

z^rrrrrrrrrnrrrrTTxruTTrrrr

-0.32000!

35!x»xxxxx»x>:xxxxxyxxxxxxxxxxx)'xxxxxxx

-0.000001

31 !XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.32000!

34IXXXXXXXVXXXXXXXXXXXXXXXXXXXXXXXXXX
0.6'»000l

26!XXXXXXXXXXXXXXXXXXXXXyXXXX

0.960001

24IXXXXXXXXXXXXXXXXXXXXXXXX
1.280001

19IXXXXXT,XX,«sXXXXXXXXXX
1.600001

VIXXXXXXXXX

1.92000!

8IXXXXXXXX

2.24000!

4IXXXX

2.56000!

ZTJOT

2.860001

m —

3.20000!

"(a)

Frequency disd^ihution AiaNG axIs 2

'EACH X fttPkESENTS 1 SahPlE(S)

CLASS INTERVAL" 0.21000

-3.360001

2ixx
-3.150001

i 1 XX
-2.940001

0!
-2.730001

1 Ix

-2.520001

2iyy

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1 IX
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(51
-1.89000!

2ixx

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Ol
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llx

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8IXXXXXXXX
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11 IXXXXXJ<XXXXX
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0.000001

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0.21000!

32IXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.42000!

32 1XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
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7IXXXXXXX
1.47000!

2! XX
1.68000!

61 (b)
1.89000!

Figure 31 Distribution of wind and temperature standardized components
along the three principal axes of the Stampede Pass, Easton,
Washington, December 1968-1970 data.

147

FreOUeNCY DISTRIBUTIflN AlOnG AxH i

EACH X Represents — i samPle(S>

CLASS INTERVAL* 0.32000

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rrr

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or

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mr

-2.560001

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0.320001

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0.6'tOOO I

0.96000

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1.28000

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xxxxxxxx
xxxxx

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3.200001

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TX-

Figure 31 (continued)
148

ALBROOK AFG CANAL ZONE RADIO!
(PRESSURE AT SUHFACei, TEMPERATURE, OEW POIN
AT SURFACE. 850Me. 70CMB. WOMB
JULY 12Z 1961 THRU 1970

I AND V WIND COMP

CLASS IKiTERViL- O.*loflfl

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i()\xxKn».t>xtx,.xjxxx>iX^xxt.ttxtrit

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3.280001

3.69J00I

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1

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1

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Figure 32 Distribution of wind, temperature, height, and dew point
standardized components along the 20 principal axes of
the Balboa, C.Z., July data. Four levels are involved,
surface, 850- , 700- , and 500-mb.

149

.^

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(ii

Figure 32 (continued)
/ 150

1

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Figure 32 (continued)
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Figure 32 (continued)
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154

10. PREDICTION

In the multimodal multivariate case, prediction first should be to the modes
or clusters. Once a mixed multivariate data set has been separated into its
various homogeneous parts, prediction within the cluster can be made. The
characteristics of the group are known and the appropriate regression equation
for each homogeneous group can be developed.

In a time series which may be composed of two or more periodic or aperiodic
parts, it will be necessary to determine which components are active at the
moment. Prediction is then again straightforward.

The problem still remains as to the deterministic regime operating at the
moment. This may require prediction into the appropriate cluster and then
prediction within the cluster.

The problem of prediction is an important one considered to be beyond the
scope of the present paper. It is one to which the authors intend to return
and to provide the necessary procedures to forecast to the cluster then within
the cluster. Briefly, this has been touched on in section 3 on Discriminant
Techniques.

155

SUMMARY

Clustering techniques to separate mixed distributions of meteorological data
are presented. The techniques and data sets are restricted, to multimodal
multivariate distributions. Dichotomous or polychotomous and non-normal dis-
tributions have not been discussed.

The major element considered is wind. Three specific examples include the
situations of the land and sea breeze, the quasi-biennial oscillation, and
winds in a pass. Three other examples are continental tropospheric winds, a
location on the ocean, and a tropical troposphere. One of the examples does
include temperature; another includes temperatures, dew points, and heights
of pressure surfaces; another includes pressure, temperature, and dew point;
while another includes heights of the pressure surfaces and temperatures.

The electronic computer program available to make the separation of mixtures
into their homogeneous parts for weather data effectively does the job.

The technique will be useful in establishing weather or climate groups which
can be set aside for study and for use in guidance and forecasting.

156

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165

AUTHOR INDEX

Alt, F. B., 140

Anderson, Edgar, 29

Anderson, T. W., 15, 16, 18, 19

Andrews, D. F. , 22

Angel 1, J. K. , 33

Arnett, J. S., 41

Aversen, J. N. , 9

Bailey, Daniel E., 15, 19

Baker, Frank B., 20

Banarjee, K. S. , 18

Barnard, M. M. , 10

Bartlett, M. S., 15, 16, 17

Berlage, H. P., 33

Boehm, Albert, 21

Bose, R. C, 2, 17

Box, G. E. P., 22

Brooks, C. E. P., 32

Brown, George W. , 10

Buchhorn, J. , 27

Burrau, C. , 23

Carr, R. N., 18

Carruthers, N. , 32

Cattell, Raymond B., 15

Charlier, C. V. L., 23

Clayton, H. H. , 33

Clutter, Jerome, L., 23, 33

Cochran, William G., 6, 81

Cohen, A. C. , 23, 138, 154

Cox, G. M., 29

Crutcher, Harold L., 10, 12, 13=
14, 23, 32, 33, 41, 57, 137,
139, 140, 141, 144, 145, 146

Davies, Owen L. , 18

Dempster, A. P. ,18

Dickey, Woodrow W. , 118

Draper, N. R., 18

Duda, Richard 0., 19

Ebdon, R. A., 32, 33, 76

Essenwanger, Oskar, 23

Falls, L. W., 23, 138, 140, 154

Finney, D. J., 18

Fisher, R. A., 10, 15, 17, 19, 26, 29

Fix, E., 10

Friedman, H. P., 2

Fruchter, B. , 15

Gal ton, Francis, 6, 7

Girschik, M. A. , 16

Good, I . J. , 3

Graybill, Franklin A., 21, 22

Graystone, P., 32

Gupta, S. S., 9

Hald, A., 23

Hall, K., 27

Harman, Harry H. , 15

Hart, Peter E., 19

Hartigan, J. A., 19

Hartley, H. 0., 23, 33

Heastie, H., 32

Hodges, J. I., 10

Hoerl, A. E. , 18

Hotelling, Harold, 15, 16, 17

Hsu, P. L., 17

Hubert, Lawrence J., 20

Kendall, M. G., 15, 18

Kennard, R. W. , 18

Korshover, J. , 33

166

Kullback, Solomon, 16

Landsberg, H. E. , 33

Lawley, D. N., 15

Lee, Alice, 5, 7

Ludlum, David, 118

MacQueen, J., 28, 30

McCabe, G. P., Jr. , 9

McCreary, F. E. , Jr. , 32

Mahalanobis, P. C, 15, 17

Martin, W. P., 29

Maxwell, A. E., 15

Miller, Robert G., 10

Moore, P. G. , 22

Mulaik, Stanley A,, 15, 16

Myers, Raymond H. , 18

Newell, Reginald E., 32, 33

Pearson, Karl, 6, 7, 10, 16, 23

Phillips, Earl L., 117

Rao, C. R., 10

Reed, R. J., 32, 33

Rogers, D. G., 32, 33

Roy, S. N., 2, 17

Rubin, H., 15

Rubin, J., 2

Schneider-Carius, K. , 23

Smith, B. Babington, 15

Smith, C. A. B., 10

Sneath, P. H. A., 15, 19

Snedecor, George W., 6, 81

Sobel, M., 9

Sokal, Robert R., 15, 19

Spearman, C. , 15

Stephenson, P. M. , 32

Stromgren, B, , 23

Stuart, A., 18

Student (W. S. Gosset), 17

Tatsuoka, M. M., 10, 17

Thurstone, L. L., 15

Tidwell, P. W., 22

Tryon, Robert C, 15, 19

Tucker, G. B., 32

Tukey, John W., 22

Veryard, R. G. , 32, 33, 76

Wagner, A. C, 41

Ward, Joe H., Jr., 27

Wicksell, S. D.. 23

Wilks, S. S., 17

Wing, Robert N., 118

Wolfe, John H., 20, 26, 28, 29,

30, 31, 35, 36, 41, 42, 111,

121, 129, 134, 153

'J'U.S. GOVERNMENT PRINTING OFFICEi 1977-240-848/117

167

(Continued from inside front cover)

EDS 16 NGSDC 1 - Data Description and Quality Assessment of Ionospheric Electron Density Profiles for
ARPA Modeling Project. Raymond 0. Conkright, in press, 1976.

EDS 17 GATE Convection Subprogram Data Center: Analysis of Ship Surface Meteorological Data Obtained
During GATE Intercomparison Periods. Fredric A. Godshall, Ward R. Seauin, and Paul Sabol,
October 1976.

EDS 18 GATE Convection Subprogram Data Center: Shipboard Precipitation Data. Ward R. Seguin and Paul
Sabol , November 1976.

PENN STATE UNIVERSITY LIBRARIES

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NOAA— S/T 76-1912