=z re es ] rm a) =) = = — RO ey fais} No, ADAG 2 THE PROBLEM Investigate the probability density distribution of the ampli- tude of ocean ambient noise and ship noise; determine any differences in the distributions which might lead to the identification of ship noise masked by a high background- noise level. Also, determine, by standard statistical methods, whether the distributions are gaussian or non- gaussian. RESULTS 1. Ambient ocean noise was found to have a gaussian distribution of amplitudes (in the sense that the moments of the distribution satisfied specific tests) only when the am- bient noise was relatively clean, i.e., the noise did not contain high-level ship noise, biological noise, ice noise or any of the other extraneous noises discussed in the text. 2. The group of ship-noise samples recorded at close range contained a large number of samples that had a non-gaussian distribution. However the other types of ex- traneous noises were found to cause the same kind of de- viation from a gaussian distribution, so that it was not possible by these tests to distinguish between a sample with ship noise and a sample with the other types of ex- traneous noises (such as biological and ice noise), Ml TU iM MN 0301 0040503 RECOMMENDATIONS 1. Use the method of moments described here if better accuracy than that given by overlays is desired to estimate the moments and to determine whether a sample is gaussian or non-gausSian. 2. Inthe probability density analysis of a noise Sample, use a range of amplitudes covering at least +4 standard deviations; otherwise large errors in the estimates of the moments will frequently result. 3. In future applications of the PDA, have the output of the PDA in a digital form rather than a continuous curve so that the data will be available in a form more suitable for the calculation of the moments of the distribution. t) ADMINISTRATIVE INFORMATION Work was performed under SR 004 03 01, Task 8119 (NEL L2-4) by members of the Listening Division. The report covers work from January 1962 to June 1963 and was approved for publication 5 November 1964, The author wishes to express appreciation to the members of the Listening Division who contributed their time to perform much of the data processing; W. P. de la Houssaye who wrote the computer program; and Elaine Kyle who prepared the data for the computer. Thanks are also extended to Fred Dickson, who prepared the illustra- tions, and to G. M. Wenz, who made many helpful suggestions during the work phase and during the writing of the manuscript. CONTENTS INTRODUCTION... page 5 Ee Sie PRO GRAIE s7 Instrumentation... 7 Research Techniques... & Data Reduction Techniques...14 RESULTS... 30 CONCLUSIONS. . .37 PD of Ambient Ocean Noise...37 PD of Ship Noise...37 Comparison of Test Methods...37 RECOMMENDATIONS. ..38 RIE RIN CID 6 ost! APPENDIX A: DESCRIPTION OF THE PDA ANDITS OPERATION... 41742 APPENDIX B: DETERMINATION OF NUMBER OF DATA POINTS OF EACH SAMPLE... .43-44 TABLES Noise Samples Selected for Analysis, by Location... page Il Number of Curves Showing Significant Values of Skewness and Kurtosis... 31 Locations of Curves Showing Significant Values of Skewness and Kurtosis... 32 Locations of Curves Showing Skewness and Kurtosis at l Per Cent Probability Level... 33 List of Curves for Which Chi-Square Was Computed... .35 Curves Chosen by Normal Curve Overlay as Being Very Closely Gaussian...36 ILLUSTRATIONS Curve of probability density function of a gaussian random variable...page 6 Block diagram of Probability Density Analyzer system... 7 Selected probability density curves, compared with a MOA! CUIAVE>s 55 LAOS Examples of PD curves obtained by use of overlay method...15-17 Curve obtained with Probability Density Analyzer... 23 Normalized cumulative sums of tabulated values... 25 Curves with positive and negative skewness, computed with Edgeworth's series... 26 Curves with positive and negative kurtosis...27 Cumulative probability of noise samples shown in figures 3 and 4...28-29 Experimental PD curve of random noise showing skewness, calculated by Edgeworth's series...34 Theoretical and experimental PD curves of square wave and sine wave...41 Number of times random noise goes into an interval about x = 0 for PD of 0.4, vs. cutoff frequency of low-pass filter...45 INTRODUCTION The study reported here was undertaken to investigate the probability density distribution of the amplitudes of ocean ambient noise and ship noise with respect to various bandwidths in several frequency ranges. The question to be answered was whether ambient noise, without any ship noise or biological noises, can be considered gaussian, and whether the presence of ship noise significantly changes the probability of density distributions. A secondary objective was to investigate methods of data reduction of the probability density curves obtained with the B& K Probability Density Analyzer, using standard statistical tests. The probability density function, as treated throughout this report, may be defined as follows. 1iiaal plc) = Ax 0 ig | Be MN (1) N-o@ where x is a random variable, with its range of values divided into a large number of continuous intervals Ax. Measure its instantaneous value a great number of times /’. Let 7. be the number of measured values of x in the 7th interval (Ax). The above equation can be rewritten as Lim pod = Axao S%'*% I (2) Tae where A ts is the amount of time the signal spends in the interval Ax, and J is the total time of the sample. Equation 2 indicates more clearly how the B & K PDA measures the probability density function. A more detailed explanation can be found in reference 1. (See list of references at end of report. ) PROBABILITY DENSITY The function where X is the mean and go is the standard deviation, is illustrated in figure 1. 0.5 0.4 0.3 0.2 0.1 0 — =3 = =I 0 1 2 3 AMPLITUDE IN STANDARD DEVIATION UNITS IER OUrG Iho CUumDIe) Of. Te DROVE Diet CY Qe TUS tit Ye faerie Chas tion of a@ gaussian random variable (normalized to Uri C eM O@elOn ING Cire NCA DCM MCILLTADICY Ite TEST PROGRAM Instrumentation The equipment used for the investigation is described below and illustrated in figure 2. An Ampex Model 350 was used as the record and play- back recorder. This model has a good low-frequency response to below 20 c/s. The filters following the recorder were an Allison Laboratories Model 2-A (used mainly as a low-pass filter) and a B & K Band Pass Filter Set, Type 1611. A McIntosh amplifier, Model MC30, was used to raise the signal level to 1 volt rms or greater. The B & K Probability Density Analyzer, Model 160 (to be referred to as the PDA) was the main piece of equip- ment and has been primarily designed to obtain the prob- ability density curves of disturbances that are essentially random in character. A brief description of the PDA and its use in this investigation is given in Appendix A. A com- plete and detailed description of the PDA can be obtained from the instruction manual. ? ALLISON LABS FILTER McINTOSH B&K AMPEX 350 MODEL 2-A Heenan PROBABILITY DENSITY VARI PLOTTER PLAYBACK UNIT OR MODEL MC30 ANALYZER XY RECORDER B &K, TYPE 1611 MODEL 160 COUNTER Figure 2. Block diagram of Probability Density Analysis system. An XY recorder by Electronic Associates, Inc., was used to record the analog 4 and Y outputs of the PDA. A cathode ray oscilloscope monitored the signal out- put of the filter. The counter used responded to frequencies of at least 10 Mc/s for use with the PDA. The counter can be used in place of an XY recorder and, in fact, is essential if measurements are to be made at low probability densities. Research Techniques Data which had been recorded for previous ambient- noise studies were available for this study. These samples had been recorded on 10-inch reels of g-inch tape, at 32 inches per second, and were from three locations. Two groups had been made in shallow water -- one, about 2 miles from the western side of an island off the coast of Southern California, and the other in the Bering Straits. These con- sisted of short ambient-noise samples recorded at regular intervals throughout the day, so that one reel covered data for one day. The third location represented was in deep water in the North Pacific between Hawaii and Alaska; most of these samples were of longer duration than the other two groups, but covered only a few days. Samples of ship noise were desired, so that their probability density curves might be compared with those of "clean'' ambient noise. Recordings were made of ships entering San Diego Harbor, with the sampling made at ap- proximately the closest point of approach. These included Navy surface ships, submarines (surfaced), and commercial ships. Several factors were considered in choosing the data samples to be used in this study. 1. ''Clean'' ambient noise was used to determine whether the distributions of the amplitudes were gaussian or near-gaussian according to certain tests which will be discussed later. Ambient noise was judged to be ''clean'' when it was free from ship noise, biological noises, or any man-made sounds when the sample was monitored. A band- pass filter and oscilloscope were used to determine whether 60-c/s hum or any other single frequency components were present in the noise sample. 2. All noise samples should be stationary for their entire length. When the sample is ambient ocean noise, this condition will not in general be true. For a noise sample to be stationary it is necessary for the sample parameters, the means and the variances, to remain un- changed as measured from samples taken at different times. It is possible that no significant difference in the sample parameters will be found if the time between samples is short enough. In a previous study* it was concluded that ocean noise is a Slowly varying, not a stationary, process. This conclusion was based on a comparison of samples that were 3 or more minutes apart. However, no significant difference was found among the values of some other samples which were only 3 minutes or less apart. Thus it appears reasonable to assume that ocean noise is stationary during a short interval of time (less than 3 minutes). 3. The PDA requires a noise sample of about 30 minutes duration for a complete automatic analysis of the amplitudes from -3.00 to +3. 00 standard deviations. The need for a long noise sample that is stationary can be satisfied by recording a short noise sample on mag- netic tape and then making a loop of the tape. A loop length was selected according to the following requirements. a. The loop should be short enough so that the noise could be considered stationary and so that the entire loop could be analyzed for each amplitude interval. The PDA (in the particular position used) requires 30 seconds to sweep a range of amplitudes equal to the window width, which is 0.1 times the rms value of the input signal. A sample length of 7 seconds met all the above requirements 10 and this gives a loop size of 52.5 inches, which was conveniently handled. b. The recorded noise on the loop should be con- tinuous, i.e., there should be no blank intervals on the loop, since a blank interval would change the average rms value of the recorded noise. A typical analysis procedure was as follows. A portion of data was selected for analysis from the recorded data available. The noise was re-recorded on a loop. The loop was played back at 7% ips and the analysis proceeded as indicated by the diagram in figure 2. The filter was set to the desired bandwidth, and the noise was amplified to 1 volt rms or greater. The PDA was carefully calibrated and adjusted just before each analysis. Its input level of noise was adjusted to 1 volt rms by its potentiometer, thus normalizing its output. Probability density of the amplitudes was recorded on the Y scale of the Y recorder and the amplitude around which the probability density was measured was on the X scale. Scale factors were selected to give a deflection of 4 inches on the / scale for a probability density range of 0 to 0.4, and a deflection of 1 inch per standard deviation of amplitude on the X scale. The automatic sweep time of the PDA was set at X = -3.00 standard deviations, and would automatically sweep through to 4 = +3.00 standard deviations, based on a 1-volt rms input. Total running time was about 30 minutes. This procedure was repeated for each band- width on every loop analyzed. Table 1 lists the number of samples analyzed from each location, the total number of probability density curves obtained from the samples, and the filter used to analyze these curves. When the Allison Laboratories filter was used, the system cutoff frequency at the low end was about 20 c/s and the upper cutoff frequency was determined by the filter which was set at 2500, 1500, 1200, 600, 300, or 150 c/s. The B & K filter was used in both the octave and third-octave positions for center band frequencies of 100, 200, 400, 800, and 1600 c/s. TABLE 1. NOISE SAMPLES SELECTED FOR ANALYSES, BY LOCATION. (FOR THE BANDWIDTHS USED, SEE ABOVE) NUMBER OF NOISE SAMPLES NUMBER OF P D CURVES OBTAINED LOCATION FILTER USED FOR ANALY SIS OF DATA SOUTHERN CALIFORNIA 8 SAMPLES WITH ALLISON LABS FILTER; 1 SAMPLE WITH ALLISON LABS AND B &K BERING STRAITS ALLISON LABS NORTH PACIFIC SAN DIEGO (SHIP NOISE IN HARBOR) ALLISON LABS Actual probability density curves of ambient noise are shown in figures 3 and 4. The large fluctuations in some of the traces are caused by substantial variations in the level of the noise sample. Since some of the curves appeared to be closely gaussian, the methods used to measure the parameters of the distribution included over- lays, calculated moments, and cumulative probability graphs. Tests of significance and the chi-square ''good- ness of fit'' tests were used to determine what values of skewness and kurtosis were improbable at a 5 or 1 per cent probability level. 11 12 PROBABILITY DENSITY 3 =2 DATA TAKEN IN SHALLOW WATER (So. Calif. ) ‘ DATA TAKEN IN BERING STRAITS ell 0 1 2 3 AMPLITUDE IN STANDARD DEVIATION UNITS Figure 3. Examples of some PD curves taken shallow water, compared with a@ normal curve. in PROBABILITY DENSITY Figure 4. =2 DATA TAKEN IN NO. PACIFIC a a DATA TAKEN IN SAN DIEGO HARBOR (Ship Noise) =I 0 1 2 3 AMPLITUDE IN STANDARD DEVIATION UNITS Examples of some PD curves taken in both shallow and deep water, compared with a normal Curve. 13 14 Data Reduction Techniques OVERLAY METHOD Since it was expected that the probability density curves obtained with the PDA would have a gaussian or nearly gaussian distribution, an overlay with a gaussian curve was used. The curve had parameters of a mean equal to zero and a standard deviation equal to one. Figure 5 illustrates the use of this method with two curves, one judged to be gaussian and the other non-gaussian. Some probability density curves obtained with the PDA were judged to be very nearly gaussian. One disadvantage of the overlay method is that de- cisions about how well a particular curve compares with the overlay are purely subjective. Skewness and kurtosis can be detected, but the magnitudes of these moments cannot be estimated with accuracy. An extension of the overlay method which will allow estimates of skewness and kurtosis 15} CESCiEMHSNC) |S, The extension is an overlay with several curves in- stead of just one. Each curve has a different set of values for skewness and kurtosis. The curves are positioned over the actual probability density curve and the parameters are estimated by interpolation between the two closest curves. The curves of the overlay can be computed with the use of Edgeworth's series approximation for nearly gaussian distributions.* The first four terms of this series are Oe) * gq. g, Gea) = h(x) — Feo) + FA (xc) + h.© (x) 1 6! where f(x) is the normalized gaussian distribution, nh” (cc) is the nth derivative of h(x), Gg. is the standardized skew- ness, and 95 is the standardized kurtosis. (3) PROBABILITY DENSITY a— NON-GAUSSIAN =3 ae =I 0 1 2 3 4 AMPLITUDE IN STANDARD DEVIATION UNITS IDA Owe Be Examples of two PD curves which were determined to be gaussian or non-gaussiany using @ normal curve as an overlay. 15 16 Estimates of the skewness and kurtosis can be found with the above method; but it does not give any indication of whether these estimates are significantly different from the expected values, if the sample is taken from a gaussian distribution. Using the previous overlay, a method can be developed so that a sample can be accepted or rejected at any desired level of probability. Basically the method is to have two of the curves on the overlay plotted so that they will represent the maximum deviations allowed in the par- ticular parameter of a sample with (J) points. The method will be developed for kurtosis, but a similar method can be used for skewness. The variance of kurtosis is given by* weir.) = 24/N (4) for large 7. This holds for a sample taken from a normal parent population, The standard deviation of kurtosis is (24/) 2; if the kurtosis is distributed normally, then from the ratio of a particular value of kurtosis (9, ‘) and the standard deviation we can obtain the probability of getting a value of kurtosis as large or larger thang, ‘, The ratio is / Pz Same (6) The probability of getting a value of kurtosis as large as or larger than g, “is given by the amount of area under a normal curve outside the -f and +f standard deviations. A value of 8 = 1.96 corresponds to a probability level of PROBABILITY DENSITY Figure 6. Overlay indicating g,‘ of +9.50 and of =0.5@- 5 per cent, or 1/20th the total area. A ratio as large as 1.96 may be considered sufficiently improbable and hence Io ‘ can be assumed to result from a non-gaussian distribution. The sample would therefore be rejected as coming from a gaussian distribution. The value of 9, ‘ therefore depends Gin Wl, Gi,” FF We 96(24/W”)&. Edgeworth's series would then be used to compute two curves, one with -g, ’ (for negative kurtosis) and one with +g, ‘ (for positive kurtosis). These curves would represent the limits, at a 5 per cent probability level, within which a sample of V points would be considered as coming from a gaussian distribution. Figure 6 shows two curves as they would appear in the overlay. These two curves are the limits for a sample =6) =e =I 0 1 2 3 4 AMPLITUDE IN STANDARD DEVIATION UNITS A curve having a value of kurtosis as IBPOG OF LOrPEe it than these values will be non-gaussian at a@ 5 per cent level for a sample of 370 points OTs, equivalently, a pandwidth of about 55 c/s. 7 18 of bandwidth about 55 c/s, with V given by the equation / = Grad I? where i is the bandwidth. The equation is obtained from Appendix B, using a time constant 7 = 2.3 seconds. The overlay method was not used extensively because of the complexity that comes from considering different values of / and also different combinations of skewness and kurtosis in the same sample. A method using computed moments of the curves is described next; it was felt that this method would yield accurate values of the mean, standard deviation, skewness, and kurtosis. METHOD OF MOMENTS The method of moments is basically a general method of forming estimates of the parameters of a distribution by means of a set of measured sample values. The first few moments of the actual distribution are calculated and these are used as estimates of the moments of the parent population. On the basis of these moments a suitable theoretical dis- tribution curve is selected. For any particular distribution curve the moments are functions of the parameters of that curve. The parameters are determined and tests of sig- nificance are made on the skewness and kurtosis. The moments about the origin are defined as” ae ir 7 = > Pole, U where p;(x) is the probability that a value selected at ran- dom from the population will lie in the tth class. The variate x with which we are concerned may be discrete or continuous. The moment (6) ( ) is defined as the mean value of x, m, Tes oe Another more important set of moments is obtained by changing the origin to the arithmetic mean. Equation 8 defines the moments about the mean. r m= Gc) (x, - 2)" (8) Bey eel)\es tb For computing purposes, the relations between the mand the m_’ are convenient. Expressing the lites in terms of the ™_ ‘ we have the relations m, =m,’ (9a) iis = ida = (ae De (9b) Mz, = M,' - 3Mgm,’ + 2(m, *)? (9c) i, = iD,” 2 Bil “Wa” =P Oils Wi, 2 Sita, (9d) Grouping errors are negligible, so Sheppard's corrections are not applied. — These moments can be expressed in standard units by the use of a standardized variable 2, by dividing the variable x by oe the standard deviation. id (x-x) Zz (10) s 5g The standardized moments are defined by the equations ur Ol = —=—, lormir = i, wy Go eine! 4 (11) iF if s x 19 20 The first four standardized moments are Gi. = O (12a) Qe = I (12b) m Chg =. (12c) s 3 x m Oy Aas (12d) Ss & x The third moment, @,, is a measure of the skewness of the distribution. A positive value indicates a distribution with a longer positive tail than a negative tail. The fourth standardized moment, @~,, is a measure of the kurtosis of the distribution. In some cases it is a measure of the ''peakedness"' of the distribution, though it is now understood that the length and size of the tails are very important in this measurement. For a normal curve the values of ~, anda@, will be O and 3, respectively. We redefine the skewness and kur- tosis as Oe = ANG (13a) 9, = % - 3 (13b) so that g. is 0 for a normal curve. It is not very likely that the third and fourth moments of a random sample will be zero. Depending on the distri- bution and on the actual sample values, the third and fourth moments will have some value different from zero. To de- termine whether this difference is significant, it is neces- sary to use the variances of the third and fourth moments. * var(g,) = 6¥(W-1)(W-2)~ 1(V-1)~*(W-3)7 + (14a) var(g,) = 24N(W-1)? (W-3)7+(W-2)7 *(W-3)~ * @-5)"* (14b) For large JV use, var(g,) = 6/W (15a) var(g,) = 24/M (1 5b) The hypothesis to be tested is that the data sample is taken from a gaussian distribution. To test the hypothesis compare g, to (6 /N) 2 and G, WO (24/ I) (see ref. 5), then g, if > 1.96 reject the hypothesis at the 5 per cent level Al. (6 /i)® > 2.57 reject the hypothesis at the 1 per cent level. Similarly, for OE 9. if > 1.96 reject the hypothesis at the 5 per cent level (24/1)2 > 2.57 reject the hypothesis at the 1 per cent ewe. CHI-SQUARE ''GOODNESS OF FIT" TEST The y* test will be applied to the hypothesis that a sample of individuals forms a random sample from a population with a given probability distribution. The param- eters of a distribution are known and are not estimated from the sample itself. Later a modification will be given for the situation where the parameters are estimated from the sample. 21 22 The quantity* is a measure of the deviation of the sample from the ex- pectation, where /’, is the number of observed frequencies in the 7th interval, and Vp, is the number of expected fre- quencies in the 7th interval as predicted by the theoretical distribution. Karl Pearson proved that the above quantity, in the limit, is the ordinary y* distribution which is now tabulated in most statistics books. The y* computed with equation 16 is compared with Ss the 5 per cent point for (4-1) degrees of freedom from a val distribution table. The tabulated value of xy” at the 5 per cent probability level with 29 degrees of freedom is 42.6. Now, if y* , as calculated by equation 16, is greater than 42.6, then the hypothesis is rejected by this test; that is, the sample is non-gaussian. The application of the ee test to the data was as fol- lows (fig. 7). Let f(x) represent a probability density curve obtained from the PDA. Divide the curve into 30 TMCIAVEMNS Iieon 59 = —3,0 vow = 16 W5 ILS é. be the mid- point of Ax. , one of the intervals, and INE 5) the value of the probability density at €,. The area under the curve is then estimated by A, ‘, where A ae = F(E,)Ax. Let A. - Zt | ~(z)dz, where ~(z) is the theoretical probability zZ, t-1 * See ref. 5, pp. 197-200. (16) F(x) PEG EG Ho Curve obtained with one PDA (jo = 9¢(ee)) BSc MEER sured probability density at &. (fF(E;)). The area within the rectangle (A,’) is an estimate of the area under the curve for the interval f(é,) Ax =x -x ; z Cel i-1 f(€;) : MEASURED PD AT &, p = f(x) : CURVE OBTAINED WITH PDA density distribution with which the experimental curve is being compared. Also let Ny = NA’, and Be = NA ,. (n,-e,¥ ee a epee rar (17) Ss : D L y glen coals 6 22 (18) , t U WV is estimated according to the method described in Appendix B. 23 24 The only modification needed when the parameters are estimated from the sample itself is a reduction in the number of degrees of freedom by one for each parameter that is estimated from the sample. For example, ifa gaussian distribution is assumed and the mean and standard deviation are estimated from the sample, then the number of degrees of freedom are (4-1)-2, where % is the number of groups. CUMULATIVE PROBABILITY PLOTS The use of cumulative probability paper was also in- vestigated. On this type of plot a gaussian distribution is represented by a straight line. The data are normalized and plotted, and deviations from a gaussian distribution are seen as departures from a straight line. The data are the p,(€,) used for the method of moments where é, is the mid- (DOMAI, Oi AIF, = 83 he ae The points plotted are the normalized cumulative sums, i.e., the first point is DB, We) op, (é,) the second point is 9,3)? 2. (E) rere 210, (3) INNS GAINS VISSC! WS weOna oe = —So00 tos = +3. 00 So that a gaussian curve resembles figure 8. The curve de- viates from a straight line at the ends because of the small amount of area (0.27 per cent) outside three standard de- viations. This curve should be used for comparison with the data instead of the straight line. 99.99 NORMALIZED CUMULATIVE AREA UNDER PD CURVE 0.1 0.01 -4 =3 =2 =I 0 ] 2 3 AMPLITUDE IN STANDARD DEVIATION UNITS Figure 8. Normalized cumulative sums of tabulated values of the area of a normal ChrOe fOr OBHUGS Of 36 EeWeen S3I6OO wpe #3200 standard deviations. 25 The type of deviations that would result for curves with skewness and kurtosis are shown in figures 9 and 10. The distribution curves for certain amounts of skewness and kurtosis are computed using Edgeworth's first four 99.99 99.9 NEGATIVE SKEWNESS NORMALIZED CUMULATIVE AREA UNDER PD CURVE -4 =) = =| 0 ] 2 3 4 AMPLITUDE IN STANDARD DEVIATION UNITS Figure 9. Curves illustrating positive and negative skewness, computed using HEdgeworth’s series. 26 terms, equation 4. The f(x) were computed using a mean of zero and a standard deviation of one. For skewness, Che +0.24, g = O were used. For kurtosis, g, = 0, g, = +0.48 were used. 99.99 NORMALIZED CUMULATIVE AREA UNDER PD CURVE a Oo POSITIVE 2 KURTOSIS 0.1 —— NEGATIVE KURTOSIS 0.01 A -4 =) 2 -1 0 i 2 3 4 AMPLITUDE IN STANDARD DEVIATION UNITS Figure 10. Curves illustrating positive and negative kurtosis, computed using Hdgeworth's series. 28 Several cumulative probability examples are shown in figures 11 and 12. 99.99 99.9 NORMALIZED CUMULATIVE AREA UNDER PD CURVE pS oO 0.1 ~ ee) ae) ell AMPLITUDE IN STANDARD DEVIATION UNITS SO, CALIFORNIA BERING STRAITS | pes 0 1 2 3 HEB OB PE Ito Cumulative probability of noise samples shown 1 fF UGBrEG Bo 99.99 NORMALIZED CUMULATIVE AREA UNDER PD CURVE 10 SHIP NOISE ~J 2 4 NO. PACIFIC -4 =e) =2 -1 0 1 2 3 4 AMPLITUDE IN STANDARD DEVIATION UNITS Figure 12. Cumulative probability of noise samples shown in figure 4. 29 30 RESULTS An overlay of a normal curve was constructed and used on the probability density curves obtained with the 4 recorder. This method was found to have a limited useful- ness since it would not yield results with the desired amount of accuracy in all cases. Cases where it could be used are for curves which do not deviate very much from the overlay. As the deviations become greater it becomes more difficult to estimate the parameters accurately. It was decided not to use an overlay with more than one curve (as described in "Data Reduction Techniques'') because, even though the curves were normalized to a standard deviation of 1, the error in setting the input to a value of 1 volt rms can be as mauchyashOsperIcentn although in most cases it was less than 5 per cent. Such error makes it difficult to estimate the skewness and especially the kurtosis, since the overlay will not fit if the standard deviation is other than 1. With the method of moments it is possible to get estimates of the mean, the standard deviation, the skewness, and the kurtosis. The limited range of amplitudes analyzed (which was thought to be sufficient before the data reduction) causes an error in the calculated moments for those samples which have amplitudes extending beyond three standard de- viations. The error is more evident in the higher (3rd and 4th) moments, because of the higher powers of x used in the calculations of these moments. A correction can be applied to the computed skewness and kurtosis. The correction takes into account the ''ignored'’ amplitudes, i.e., it com- pares the computed moment with the moment of a truncated normal distribution. Actually only the moment of kurtosis was corrected, since skewness was not found very fre- quently among the noise samples. The hypothesis to be tested is that the noise samples are taken from noise with a gaussian distribution of ampli- tudes. The test used is the same one described on page 21 and it is applied to both the skewness and the kurtosis. The hypothesis is then rejected by the test if both moments of the sample are significant at the 5 per cent level, and also if either moment is significant at the 1 per cent level. Table 2 lists the sample for which the 3rd and/or 4th moments were found to be significantly different from the TABLE 2. NUMBER OF CURVES SHOWING SIGNIFICANT VALUES OF SKEWNESS AND KURTOSIS. g, (SKEWNESS) | g, (KURTOSIS) SAMPLE 13* aan ee BIOLOGICAL NOISE (eS | ae x CLEAN NOISE 20 || | |X | UOC ICAISNOISE 2ST | aaa | ee | XT | | OWE EREOREIUNIEFANKS B x CLEAN NOISE 277 ao X X SHIP NOISE 287 X xX CLEAN NOISE 29% (ios Sapeetl eemiete lias Gan he cara | MOSTLY CLEAN NOISE 42 apes SHIP NOISE i a ICE NOISE SL es a CINE 52m ee eX a CEANOISE 607 X X 60 C/S aa ee cs 69 eat ee es | 84* pa ee a ee 100 isc eal aa ALL THE FOLLOWING ARE SHIP NOISE SAMPLES — Ww No} *Hypothesis rejected for that sample. 31 32 expected values. An asterisk by the sample number indicates that the hypothesis was rejected by that particular sample according to the conditions of the previous paragraph. Comments on the type of noise (clean, biological, ship noise, etc.) inthe sample are also given. The number of noise samples that were found to be significant are listed by lo- cation in table 3. Note that the shallow-water locations have many more non-gaussian samples than the deep-water North Pacific location. TABLE 3. LOCATIONS OF CURVES SHOWING SIGNIFICANT VALUES OF SKEWNESS AND KURTOSIS AT 1 AND 5 PER CENT PROBABILITY LEVELS. NO. OF LOCATION SAMPLES SHALLOW, SO. CALIF. SHALLOW, ALASKA NORTH PACIFIC SAN DIEGO (SHIP NOISE IN HARBOR) If the hypothesis that the noise samples are taken from a gaussian distribution is true, then it is expected that one sample in 20 may have significant parameters at the 5 per cent level. If it is not true, it is expected that more than one sample in 20 may have significant parameters. Table 3 shows that the values of skewness for the first three locations are weil within the expected number, except perhaps for the Southern California location, where two samples out of 29 were found with significant amounts of skewness. However, since this location had considerable biological and other non-gaussian type, noises, a greater number of rejected samples is to be expected. The table also shows that for kurtosis the expected number (one out of 20) was exceeded for both of the shallow-water locations but was not exceeded for the deep-water (North Pacific) location. For the ship-noise samples, the number of ex- pected significant values of skewness and kurtosis are ex- ceeded at all levels. Of interest are the number of sig- nificant values of skewness in the ship-noise data, since there were not many of these for the other three groups. For all groups the data indicate that kurtosis is a sensitive indicator for the presence of ship noise, biological noise, ice noise, and in general any type of noise which has a non- gaussian distribution. Table 4 indicates the bandwidth of the noise samples which had significant values of skewness and kurtosis at the 1 per cent level. For the most part the bandwidths involved are the larger ones. TABLE 4. NOISE SAMPLES FROM FOUR LOCATIONS, SHOWING SIGNIFICANT MOMENTS OF SKEWNESS AND KURTOSIS AT 1 PER CENT PROBABILITY LEVEL. SOUTHERN CALIFORNIA BERING STRAITS BANDWIDTH BANDWIDTH | g, (1%) | g, (1%) BROADBAND ] 20-1500 20-1200 2 a 20-600 0 2 NORTH PACIFIC ) SAN DIEGO (SHIP BANDWIDTH g, (1%) NOISE IN HARBOR) 1 OCTAVE BANDWIDTH 1600 c.f. 20-1200 1/3 OCTAVE 1600 c.f. 600-1200 33 34 An Edgeworth series was fitted to an experimental curve using the parameters calculated by this method, to insure that they were good estimates to the extent that they provided a good fit to the data. Figure 13 is the trace of an experimental curve of random noise which shows con- siderable skewness. The fit of an Edgeworth series ap- proximation to the experimental curve is seen to be very good, 0.5 f(x) 0.4 EXPERIMENTAL PD 0.3 0.2 PROBABILITY DENSITY THEORETICAL NORMAL Ts 0.1 AMPLITUDE IN STANDARD DEVIATION UNITS Figure 13. Experimental PD curve of random noise showing considerable skewness. The points shown are calculated using Edgeworth's series with the moments calculated from the curve itself by the method of moments described in the teste A theoretical normal curve is shown for comparison. The chi-square test was performed on a few selected curves and the results are shown in table 5. An X marks those noise samples for which y*__ was equal to or greater than 49.6, which is the tabulated value of y* for 29 degrees of freedom at a 1 per cent probability level. The value of x” _ is given for the samples for which the hypothesis was not rejected by the test. The figure also shows whether SAMPLE | x2 NO. SS ake B x (5%) 14 ied ae 1S al 7 || Ge [SPA Acl aes i [OE EG (Gt eS Le = aS TABLE 5, CURVES FOR WHICH X?_ eel Gael WAS COMPUTED, COMPARED 37 = WITH RESULTS BY METHOD OF eS | MOMENTS, THE x's INDICATE 42 See X(5%) THOSE CURVES WHICH REJECTED LI) Ces (eee THE HYPOTHESIS. (oe Es (Glee eas Se er a Ss ees 1a |e ae (ee Se ee B4 oe bs xc ey) 147 reese X(5%) 150 X(5%) the 3rd and 4th moments were Significant at the 5 per cent or 1 per cent level. Generally the results of the chi-square test agree with the results of the method of moments. Only one hypothesis was tested and this was that the noise samples were taken from a gaussian distribution with the mean equal to zero and the standard deviation equal to one. A better hypothesis is to assume that the distribution is gaussian with a mean x = m and standard deviation Ss = $ , where ™ and ou are estimated from the sample itself. Table 6 shows parameters for several noise samples which were selected by the overlay normal curve as very closely approximating a gaussian curve. The four computed moments of each sample are given (the mean, the standard deviation, the skewness, and the kurtosis). The ratio of the skewness and kurtosis to the square root of their variances are given. The computed value of X" iy is given for comparison. All three methods agree that these samples 35 TABLE 6. NOISE SAMPLES CHOSEN, BY VARIOUS METHODS, AS BEING VERY CLOSELY GAUSSIAN. 9 G2 COMPUTED ene SAMPLE STD. DEV. NO. 1 2 apa hy aaa (SKEWNESS) |(KURTOSIS)| (var g,)% | (var g,)” 0 97868 L 0. 33 L 0.39 . 26 se 0.05984 | 0.99511 | -0.01290 | 0,014 0. 06427 0.98920 | -0.01077 0. 030 I 0. 98493 Syoye |e |e ie |e |S Is fe bean dl Ke on es 10 | ON | INO JIN [00] OS JU] in 1 Oo Oo Ine} 1 co wi S S) oS No) ; 0. 02767 0. 04498 0.97240 | -0.01331 0.019 0.3 a oO — are very closely gaussian. The largest value of chi-square occurs for sample no, 112, which has avery good shape; however it does have a mean which is different from zero, thus causing the large \* .: Sample no, 68 also has a large y° for the same reason, Cumulative probability graphs of the noise samples can reveal if the curve has skewness or kurtosis and can also reveal other deviations. However this method was not used beyond plotting a few curves. The graphs would per- haps require an overlay to estimate skewness and kurtosis but, as before, the sample distribution needs to be stan- dardized for each sample before the overlay can be applied. Analysis with the cumulative probability method was not performed, as it was felt that it would not provide much more information than the method of moments and the chi- Square test, and the estimates would not be as accurate as those obtained by the method of moments. This method does, however, give a better indication of whether a noise sample is gaussian than does the overlay method on the XY probability density curves. CONCLUSIONS PD of Ambient Ocean Noise The results obtained from the analysis of ambient ocean-noise samples from the three locations indicate that the hypothesis (that is, the assumption that the ocean samples are taken from a gaussian noise distribution) is not rejected when using ''clean'' ambient noise. The hypothesis is rejected for ''contaminated'' ambient ocean noise. The contamination may be ship noise, biological noise, noise from ice (in polar regions), etc. Thus it has been shown that ambient ocean noise, under certain con- ditions, can be assumed to have a gaussian distribution of amplitudes. PD of Ship Noise The same hypothesis used for the ambient noise samples was used to test the ship samples. The results indicate that ship-noise samples are not gaussian, since six out of nine samples tested rejected the hypothesis that the samples were taken from a gaussian distribution. It was not found possible to distinguish between the types of contamination of the ambient noise; that is, there were no obvious differences in the probability density distributions for ambient noise contaminated by ship noise and that con- taminated by other sources. Comparison of Test Methods The method of moments was found to be the most suitable, of the four methods of data reduction, for providing the most accurate and useful estimates of the parameters. However, when using this method the tails of the distribution should not be left out of the calculations, since their con- tribution at the higher moments is very significant. The 37 38 chi-square test, although it does not provide estimates of the parameters of the distribution, does provide a good in- dication of the ''fit'' of the sample distribution to the theo- retical distribution. The chi-square method can be used to give an independent check on the method of moments. RECOMMENDATIONS 1. Use the method of moments described here if better accuracy than that given by overlays is desired to estimate the moments and to determine whether a sample is gaussian or non-gaussian. 2. Inthe probability density analysis of a noise sample, use a range of amplitudes covering at least +4 standard deviations; otherwise large errors in the estimates of the moments will frequently result. 3. In future applications of the PDA, have the output of the PDA in a digital form rather than a continuous curve, so that the data will be in a form more Suitable for the calculation of the moments of distribution. REFERENCES 1. Bolt, Beranek and Newman, Inc., Report 895, Probability Density Analyzer, Summary Report, by H. L. Fox, September 1963 2. Bell Telephone Laboratories Contract Nonr-2461(00); Technical Report 10, A Study of the Bivariate Distribution Function of the Ocean Noise, by A. H. Green, p. II-2 - II-3 13 April 1962 3. Cramér, H., Mathematical Methods of Statistics, p. 229-230, Princeton University Press, 1946 4. Kenney, J. F. and Keeping, E. S., Mathematics of SUeASinKes, ZACleCle s W745 i065 Weoule)O, Wein INOsimeeiacl, Ii do. Kenney, J. F. and Keeping, E. S., Mathematics of Statistics, 3ded., v. 1, p. 90-103, 116, Van Nostrand, 1954 REVERSE SIDE BLANK 3 39 Al Pail) jaw rcs zi! t oH > Le = n D Ss = a a > : > = H ES — A. SQUARE |: = B. SINE co WAVE | ee WAVE as co S ae oO 0.225 ; T =I I | -] 0 l =f) -] 0 1 AMPLITUDE IN STANDARD DEVIATION UNITS WE phrE Ai c Theoretical and experimental probability density curves of the square wave and the sine wave. 41 42 The PDA has one input and several types of outputs. The input level to the PDA is controlled by a precision 10- turn potentiometer and the level can be in the range from 1 volt to 50 volts rms. The frequency range is from dc to 10 ke/s and this range should never be exceeded. The in- put level is set with the aid of a true-rms voltmeter. There are two analog outputs. The X (amplitude) output is from -10 to +10 volts and corresponds to +5 to -95 times the rms input level. The Y (probability density) out- put is from 0 to about 7.5 volts and corresponds to a PD from 0 to 10. However a meter on the instrument gives a reading of PD on either of two, scales, from 0 to 0.4, or 0to 1.0. A PD greater than one can be measured using the digital outputs, of which there are two: al-Mc/s and a 10-Mc/s output. Both digital outputs are on when the in- put signal is inside the amplitude interval Av, at any selected amplitude. The PD is obtained by counting the number of cycles per second from either digital output. The counting interval can be varied as the occasion demands. Another output available on the PDA is the pulse out- put. This particular output gives one pulse each time the signal goes into the amplitude interval, or two pulses for signals exceeding the set-in level (one pulse when signal passes through on the way up and another pulse when it is on the way down). Some of the important controls on the PDA are the in- put potentiometer (to determine and set input level), the ''X"' 10-turn calibrated potentiometer (to select the amplitude around which the probability density is to be measured), and the output-damping-sweep time control (to select the averaging time and sweep speed). Other controls deal with the calibration of the PDA and are better described in the PDA manual.* APPENDIX B: DETERMINATION OF NUMBER OF DATA POINTS OF EACH SAMPLE The application of the y* test to our type of data was desired as a means of testing the ''goodness of fit" of the data to certain theoretical distributions and, in particular, the fit to the gaussian distribution. There is some difficulty in applying this test directly because /, the total number of points in our sample, is not known. However it is possible to obtain an estimate of 7 which can be used. The estimate of V is also used in determining whether the moments of the distribution, that are calculated by the method of moments, are significant at a given probability level. The estimate of / is found as follows. From reference 1 use eh (3/8n)2 on FT wGey Ax oa where I. is the input bandwidth of the signal, 7 is the sampling interval or atime constant which gives an equivalent averaging, and w(x) is the value of probability density ob- tained from the PDA. The o_* is the normalized variance of the estimate of the probability density, and is not to be confused with the variance of the distribution of amplitudes of the input signal. In reference 1, V is given as itor pee (B-2) Here J represents the total number of times that the input signal enters the amplitude interval Ax at the (x) of interest. After combining equations B-1 and B-2 we have 2 f_T w(x) = 2.97 w (x) (B-3) (3/87)= ; 43 44 The total number of points in the entire sample is what is needed and / represents only that number in the interval Ax. However, we have normalized data and therefore Dw(x)Ax = 1 ane le Oo Vn = DVAx% = eee (3/87)2 Calculations performed with this equation agree well with some data taken using the pulse output of the PDA. Figure Bl, curve B, shows the results of equation B-3 for T = 1 second, and w(x) = w(0) = 0.4, or Nie eel ell heey O The theoretical curve is compared with actual measurements taken from the PDA pulse output using a counter with a 1-second sampling interval. The slight difference between the two is due to an error in the actual cutoff frequency of the filter. The effective bandwidth of the filter is a little larger, about 1.12 times the set value. 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