NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS AN EVALUATION OF DISCRETIZED CONDITIONAL PROBABILITY AND LINEAR REGRESSION THRES- HOLD TECHNIQUES IN MODEL OUTPUT STATISTICS FORECASTING OF VISIBILITY OVER THE NORTH ATLANTIC OCEAN by Mark Diunizio September 1984 Thesis Advisor: Robert J . Renard Approved for public release; distribution unlimited T222051 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Wtimn Dmtm Enterad) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Sublllle) An Evaluation of Discretized Conditional Proba- bility and Linear Regression Threshold Techniques in Model Output Statistics Forecasting of Visi- bility over the North Atlantic Ocean 5. TYPE OF REPORT A PERIOD COVERED Master's Thesis September 1984 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORfs; Mark Diunizio 8. CONTRACT OR GRANT NUMBERf*; 9. PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, California 93943 10. PROGRAM ELEMENT, PROJECT, TASK AREA 4 WORK UNIT NUMBERS 11. CONTROLLING OFFICE NAME AND ADDRESS Naval Postgraduate School Monterey, California 93943 12. REPORT DATE September 1984 13. NUMBER OF PAGES 233 U. MONITORING AGENCY NAME 4 ADDRESSC// d///ofanf from Controtling Oltlc») 15. SECURITY CLASS, (ol this report) Unclassified 15«. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (ol this Report) Approved for public release; distribution unlimited 17. DISTRIBUTION STATEMENT (ol the abalracl entered In Block 20, II dllterent Irom Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse aide it neceasary and Identify by block number) Model Output Statistics, Visibility, North Atlantic Ocean Visibility, Marine Horizontal Visibility, Discretization, Conditional Probability, Physically Homogeneous Ocean Areas, Minimum Probable Error Threshold Models, Weather Forecasting, 20. ABSTRACT (Continue on reverse aide II neceaaary and Identity by block number) This report describes the application and evaluation of four primary statistical models in the forecasting of horizontal marine visibility over selected physically homogeneous areas of the North Atlantic Ocean. The main focus of this study is to propose an optimal model output statistics (MOS) approach to operationally forecast visibility at the 00-hour model initiali- zation time and the 24-hour and 48-hour model forecast DD , ]°Z^ 1473 EDITION OF 1 NOV 65 IS OBSOLETE S N 0102- LF-014- 6601 ]_ UNCLASSIFIED SECUBITY CLASSIFICATION OF THIS PAGE (Whan Data Bnlarad) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whun Dmlm Enffd) #19 - KEY WORDS - (CONTINUED) Maximum-Likelihood-of-Detection Threshold Models, Linear Regression, Natural Regression, Maximum Probability #20 - ABSTRACT - (CONTINUED) projections. The technique utilized involves the manipulation of observed visibility and Navy Operational Global Atmospheric Prediction System (NOGAPS) model output parameters. The models employ the statistical methodolo- gies of maximum conditional probability, natural regression and minimum probable error linear regression threshold techniques. Additionally, an evaluation of the 1983 predictive arrays/equations using 1984 NOGAPS data fields and a maximum-likelihood-of -detection threshold model were accomplished. Results show that two statistical approaches, namely a maximum conditional probability strategy uti- lizing linear regression equation predictors and the minimum probable error threshold models, produce the best results achieved in this study. S N 0102- LF. 014- 6601 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS P AG€.(Whmn Dalm Enlmrad) Approved for public release; distribution unlimited An Evaluation of Discretized Conditional Probability and Linear Regression Threshold Techniques in Model Output Statistics Forecasting of Visibility over the North Atlantic Ocean by Mark Diunizio Lieutenant, United States Navy B. S., United States Naval Academy, 1977 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN METEOROLOGY AND OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1984 ABSTRACT This report describes the application and evaluation of four primary statistical models in the forecasting of hori- zontal marine visibility over selected physically homogeneous areas of the North Atlantic Ocean. The main focus of this study is to propose an optimal model output statistics (MOS) approach to operationally forecast visibility at the 00-hour model initialization time and the 24-hour and 48-hour model forecast projections. The technique utilized involves the manipulation of observed visibility and Navy Operational Global Atmospheric Prediction System (NOGAPS) model output parameters. The models employ the statistical methodologies of maximum conditional probability, natural regression and minimum probable error linear regression threshold tech- niques. Additionally, an evaluation of the 1983 predictive arrays/equations using 1984 NOGAPS data fields and a maximum- likelihood-of-detection threshold model were accomplished. Results show that two statistical approaches, namely a maxi- mum conditional probability strategy utilizing linear regression equation predictors and the minimum probable error threshold models, produce the best results achieved in this study. TABLE OF CONTENTS I. INTRODUCTION AND BACKGROUND 13 II. OBJECTIVE AND APPROACH 18 III. DATA 20 A. VISIBILITY OBSERVATIONS AND SYNOPTIC CODES — 20 B. NORTH ATLANTIC OCEAN DATA 21 1. Area 21 2. Time Period 21 3. Synoptic Weather Reports 22 4. Predictor Parameters 23 C. DEPENDENT/INDEPENDENT DATA SETS 24 IV. PROCEDURES 26 A. TERMS AND SYMBOLS 26 B. COMPUTER PROGRAMS 28 C. MODELS 29 1. Preisendorfer PR Model 29 2. Preisendorfer PR+BMD Model 34 3. Equal Variance Threshold Model (EVAR) 34 4. Quadratic Threshold Model (QUAD) 36 5. Maximum-Likelihood-of-Detection Model 36 V. RESULTS 38 A. NORTH ATLANTIC OCEAN, AREA 2 39 1. Area 2, TAU-00 39 2. Area 2, TAU-24 43 3. Area 2, TAU-48 47 B, NORTH ATLANTIC OCEAN, AREA 3W 51 1. Area 3W, TAU-24 52 2. Area 3W, TAU-48 55 C. NORTH ATLANTIC OCEAN, AREA 4 60 1. Area 4, TAU-00 60 2. Area 4, TAU-24 64 3. Area 4, TAU-48 68 VI. CONCLUSIONS AND RECOMMENDATIONS 75 A. CONCLUSIONS 75 B. RECOMMENDATIONS 78 APPENDIX A: LINEAR REGRESSION AND THRESHOLD MODELS 83 APPENDIX B : NOGAPS PREDICTOR PARAMETERS AVAILABLE FOR NORTH ATLANTIC OCEAN EXPERIMENTS 99 APPENDIX C: SKILL AND THREAT SCORES, DEFINITIONS 107 APPENDIX D: BMDP LINEAR REGRESSION EQUATION PREDICTOR SETS, NORTH ATLANTIC OCEAN EXPERIMENTS (PR+BMD MODEL) 109 APPENDIX E. BMDP LINEAR REGRESSION EQUATION PREDICTOR SETS FOR TV70-STAGE THRESHOLD MODELS 113 APPENDIX F: TABLES — 118 APPENDIX G: FIGURES 127 LIST OF REFERENCES 228 INITIAL DISTRIBUTION LIST 231 LIST OF TABLES I. A suminary of 1200GMT observations, 15 May — 07 July 1983, North Atlantic Ocean homogeneous areas: TAU-00 118 II. A summary of 1200GMT observations, 15 May — 07 July 1983, North Atlantic Ocean homogeneous areas: TAU-24 forecast projection 119 III. A summary of 1200GMT observations, 15 May — 09 July 1983, North Atlantic Ocean homogeneous areas: TAU-48 forecast projection 120 IV. Number of observations of three visibility categories and 95% confidence intervals for the dependent and independent FATJUNE 19 83 data for the North Atlantic Ocean homogeneous areas 2 and 4, for TAU-00, TAU-24 and TAU-48 and area 3W for TAU-24 and TAU-48 121 V. Summary of the contingency table statistics results for all models used in the North Atlantic homogeneous areas 2, 3W and 4 for the evaluated TAU-00, TAU-24 and TAU-48 NOGAPS model output periods for FATJUNE 1983 123 VI. A summary of 1200GMT observations, 15 May — 23 June 1984, North Atlantic Ocean homogeneous areas: TAU-24 forecast projection 125 VII. Summary of the contingency table statistics results for all models used in the North Atlantic homogeneous areas 2 and 3W for the evaluated TAU-24 NOGAPS model forecast projection for 15 May--23 June 1984 126 LIST OF FIGURES 1. Proposed U.S. Navy Model Output Statistics (MOS) development schedule ■ 127 2. Physically homogeneous areas for the North Atlantic Ocean, May, June and July, from Lowe (1984b) 128 3. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 2, TAU-00 (PR model, MAXPROB II strategy) 129 4. Skill diagram and contingency table results for area 2, TAU-00 (PR model) for (a) the MAXPROB I Preisendorfer strategy, (b) the MAXPROB II Preisendorf er strategy and (c) the natural regression Preisendorfer strategy 130 5. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 2, TAU-00 (PR model, MAXPROB II strategy) 133 6. Same as Fig. 3, except for the (PR+BMD) model 134 7. Same as Fig. 4, except for the (PR+BMD) "model 135 8. Same as Fig. 5, except for the (PR+BMD) model 138 9. Contingency table results for the area 2, TAU-00 Equal Variance threshold model 139 10. Same as Fig. 9, except for the Quadratic threshold model 140 11. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 2, TAU-24 (PR model, MAXPROB II strategy) 141 12. Skill diagram and contingency table results for area 2, TAU-24 (PR model) for (a) the MAXPROB I Preisendorfer strategy, (b) the MAXPROB II Preisendorfer strategy and (c) the natural regression Preisendorfer strategy 142 13. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 2, TAU-24 (PR model, MAXPROB II strategy) 145 14. Same as Fig. 11, except for the (PR+BMD) model 146 15. Same as Fig. 12, except for the (PR+BMD) model 147 16. Same as Fig. 13, except for the (PR+BMD) model 150 17. Contingency table results for the area 2, TAU-24 Equal Variance threshold model 151 18. Same as Fig. 17, except for the Quadratic threshold model 152 19. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 2, TAU-48 (PR model, MAXPROB II strategy) 153 20. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 2, TAU-48 (PR model, MAXPROB II strategy) 154 21. Skill diagram and contingency table results for area 2, TAU-48 (PR model) for (a) the MAXPROB I Preisendorfer strategy, (b) the MAXPROB II Preisendorf er strategy and (c) the natural regression Preisendorfer strategy 155 22. Same as Fig. 19, except for the (PR+BMD) model 158 23. Same as Fig. 21, except for the (PR+BMD) model 159 24. Same as Fig. 20, except for the (PR+BMD) model 162 25. Contingency table results for the area 2, TAU-48 Equal Variance threshold model 163 26. Same as Fig. 25, except for the Quadratic threshold model 164 27. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 3W, TAU-24 (PR model, MAXPROB II strategy) — 165 28. Skill diagram and contingency table results for area 3W, TAU-24 (PR model) for (a) the MAXPROB I Preisendorfer strategy, (b) the MAXPROB II Preisendorfer strategy and (c) the natural regression Preisendorfer strategy 166 29. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 3W, TAU-24 (PR model, MAXPROB II strategy) 169 30. Same as Fig. 27, except for the (PR+BMD) model 170 31. Same as Fig. 28, except for the (PR+BMD) model 171 32. Same as Fig. 29, except for the (PR+BMD) model 174 33. Contingency table results for the area 3W , TAU-24 Equal Variance threshold model 175 34. Same as Fig. 33, except for the Quadratic threshold model 176 35. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 3W, TAU-48 (PR model, MAXPROB II strategy) ■ ■ 177 36. Skill diagram and contingency table results for area 3W, TAU-48 (PR model) for (a) the MAXPROB I Preisendorf er strategy, (b) the MAXPROB II Preisendorf er strategy and (c) the natural regression Preisendorf er strategy 178 37. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 3W, TAU-48 (PR model, MAXPROB II strategy) 181 38. Same as Fig. 35, except for the (PR+BMD) model 182 39. Same as Fig. 36, except for the (PR+BMD) model 183 40. Same as Fig. 37, except for the (PR+BMD) model 186 41. Contingency table results for the area 3W, TAU-48 Equal Variance threshold model 187 42. Same as Fig. 41, except for the Quadratic threshold model 188 43. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 4, TAU-00 (PR model, MAXPROB II strategy) 189 44. Skill diagram and contingency table results for area 4, TAU-00 (PR model) for (a) the MAXPROB I Preisendorf er strategy, (b) the MAXPROB II Preisendorf er strategy and (c) the natural regression Preisendorf er strategy 10 190 45. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 4, TAU-00 (PR model, MAXPROB II strategy) 193 46. Same as Fig. 43, except for the (PR+BMD) model 194 47. Same as Fig. 44, except for the (PR+BMD) model 195 48. Same as Fig. 45, except for the (PR+BMD) model 198 49. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 4, TAU-24 (PR model, MAXPROB II strategy) 199 50. Skill diagram and contingency table results for area 4, TAU-24 (PR model) for (a) the MAXPROB I Preisendorfer strategy, (b) the MAXPROB II Preisendorf er strategy and (c) the natural regression Preisendorfer strategy 200 51. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 4, TAU-24 (PR model, MAXPROB II strategy) 203 52. Same as Fig. 49, except for the (PR+BMD) model 204 53. Same as Fig. 50, except for the (PR+BMD) model 205 54. Same as Fig. 51, except for the (PR+BMD) model 208 55. Contingency table results for the area 4, TAU-24 Equal Variance threshold model 209 56. Relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for area 4, TAU-48 (PR model, MAXPROB II strategy) 210 57. Skill diagram and contingency table results for area 4, TAU-48 (PR model) for (a) the MAXPROB I Preisendorfer strategy, (b) the MAXPROB II Preisendorfer strategy and (c) the natural regression Preisendorfer strategy 211 58. Functional dependence, AO/Al statistics and 96%/05% confidence interval values for area 4, TAU-48 (PR model, MAXPROB II strategy) 214 59. Same as Fig. 56, except for the (PR+BMD) model 215 60. Same as Fig. 57, except for the (PR+BMD) model 216 11 61. Same as Fig. 58, except for the (PR+BMD) model 219 62. Contingency table results for the area 4, TAU-48 Equal Variance threshold model 220 63. Same as Fig. 62, except for the Quadratic threshold model • 221 64. Skill diagram and contingency table results for area 2, TAU-24, 1984 data (PR+BMD model) for the MAXPROB II Preisendorf er strategy 222 65. Graph of threat score versus threat frequency (unadjusted scores for a standardized two by two contingency table) for the minimum probable error threshold model and the maximum likelihood of detection threshold model. This graph reflects VISCAT I+II versus VISCAT III and VISCAT I versus VISCAT II separations 223 66. Same as Fig. 65, except a graph of false alarm rate versus threat frequency 224 67. Same as Fig. 65, except a graph of percent correct versus threat frequency 225 68. Contingency table results for the minimum proba- ble error threshold model (EVAR) for area 2, TAU-24. The contingency tables reflect VISCAT I+II versus VISCAT III and VISCAT I versus VISCAT II separations 226 69. Same as Fig. 68, except for the raaximum- likelihood-of-detection threshold model 227 12 I . INTRODUCTION AND BACKGROUND One of the most significant advances in objective weather prediction, since the introduction of numerical weather prediction in the 19 50's and satellite remote sensing capa- bilities in the 1960 's, has been the development of Model Output Statistics (MOS) weather forecasting method by Glahn and Lowry (1972). In general, this technique is the deter- mination of a statistical relationship between an operational weather element (predictand) , which may or may not be fore- cast by numerical methods, and numerical model output varia- bles (predictors), usually via linear regression methods. The resulting predictand/predictor regression equations provide the basis for generating a statistical weather pre- diction. The National Weather Service (NWS) has included MOS as an integral part of their weather forecasting operations since the early 1970 's. Currently, the NWS maintains MOS prediction equations for approximately 15 weather elements (e.g. ceiling, visibility, obstructions to vision, precipi- tation, etc.) at forecast times ranging from 6 to 48 hours. These forecasts are routinely provided to approximately 295 civilian and 190 military locations throughout the continental United States (CONUS) and Alaska [Glahn, 1983]. Based on the impressive results achieved with the NWS MOS program, the Department of Defense (DOD) , through the Air 13 Weather Service (AWS), implemented and operated a quasi- global version of the NWS MOS system at the Air Force Global Weather Center (AFGWC) , Offutt AFB , Nebraska [Best and Pryor, 1983] . The first operational forecasts obtained from the AWS MOS system were produced by AFGWC in December 19 80 and the system ran operationally for a period of approximately 18 months. Regions for which operational MOS forecasts were produced included Europe, Asis (including Korea and Japan), the South China Sea (including the Philippines and Taiwan) , the near and middle east and northern Africa. The AWS MOS program was terminated with the recent decision to replace the current hemispheric primitive equation (PE) model with a spectral global dynamic model [Klein, 1981]. Throughout its tenure as an operational forecast scheme, the AWS MOS system provided the U.S. Air Force with a rela- tively low cost, flexible and responsive prediction network. Further development of the AWS MOS system has been postponed until sufficient spectral model output is archived. The U.S. Navy, by virtue of its unique marine forecast- ing responsibilities, has a keen interest in applying MOS forecasting schemes to global oceanic regions. Through the research and development efforts of the Naval Environmental Prediction Research Facility (NEPRF) in Monterey, California, the Navy has sponsored a limited amount of research into naval applications of MOS. In particular, statistical studies have been done into forecasting Levante winds in Spain [Godfrey and 14 Lowe, 1979], ceiling and visibility prediction in the southern California (SOCAL) naval operating area [Lewit, 1980], marine fog and visibility predictability in the North Pacific Ocean [Renard et al., 1983 and Renard and Thompson, 1984]. Presently, a program is in operation which provides MOS forecasts for selected U.S. Navy and Marine Corps CONUS locations. These services, which are made available from NWS, are based on the National Meteorological Center (NMC) limited fine mesh model predictions. This MOS program was initiated on 10 November 1982 and provides forecasts for twelve weather parameters which include visibility, obstructions to visibility and cloud amount [Naval Environmental Prediction Research Facility, 1982] . The results of these limited studies along with the encouraging performances of both the NWS and AWS MOS programs and the implementation of the Navy Operational Global Atmos- pheric Prediction System (NOGAPS) dynamical primitive equation (PE) model at the Fleet Numerical Oceanography Center (FNOC) , in Monterey, California prompted the decision in September 1982 for the Navy to pursue its own MOS program. Fig. 1 is an overview of the currently proposed milestones for the Navy MOS program. The first operational weather parameter being investigated in this proposed ten-year Navy effort is horizontal visibility at sea, with the initial goal of this project being the investigation and development of statistical predictive schemes for forecasting horizontal visibility over the North Atlantic Ocean. 15 The impact of fog and other impediments to visibility on naval operations is well documented throughout maritime history. Records show countless catastrophes and accidents which were directly attributable to poor visibility at sea. For example, on 29 May 1914, the Canadian liner Empress of Ireland collided with the Norwegian vessel Storstad in dense fog on the Saint Lawrence River resulting in 1,024 fatalities and similarly, the legendary "North Sea haze" was a critical element in the World War I tactics employed at Jutland in 1916. Also, one of the most spectacular maritime disasters in the U.S. Navy's history took place on 9 September 1923 when seven Pacific fleet destroyers struck the rocks and ran aground in dense fog off of Point Arguello, California. Research into predicting marine visibility via traditional linear regression methodologies has taken place at the Naval Postgraduate School (NPS) since the early 1960 's. Generally, early visibility forecasting experiments identified potential physical air/ocean mechanisms [Schramn, 1966] and emphasized the inherent likelihood of human error in at-sea visibility observations [Nelson, 1972]. Later experimentation by Aldinger (1979), Yavorsky (1980) and Selsor (1980) concen- trated on various modifications to multiple linear regression schemes and the analysis of prediction skill measurements. This study presents a direct follow-on to the research presented by Karl (1984), in which statistical methodologies presented by Preisendorf er (1983a, b,c) and multiple linear 16 regression techniques presented by Lowe (1984a) were compared and contrasted. In Karl's preliminary study, Preisendorf er ' s three strategies, two based on maximum conditional proba- bility and one based on natural regression, as well as Lowe's linear regression threshold models were tested and applied to sets of FNOC model output parameters (MOPs) from both the North Pacific and North Atlantic Ocean areas. The North Atlantic Ocean study was separated into effective physically homogeneous areas [Lowe, 1984b]. Karl's study specifically dealt with an evaluation of the MOS scheme applied to oceanic regions for the TAU-00 model output during the period 15 May to 07 July 19 83. This study concerns itself with a continued evaluation and further refinement of statistical methods proposed by Preisendorf er as well as the linear regression threshold models presented by Lowe. With reference to Karl's study, other North Atlantic Ocean areas and model forecast projections (e.g. TAU-24 and TAU-48) are addressed. 17 I I . OBJECTIVES AND APPROACH The primary objectives of this study are to continue the previous NPS horizontal marine visibility prediction re- search initiated by Karl (1984) and to continue the search for an optimal Model Output Statistics (MOS) prediction scheme to operationally forecast coastal and open ocean visibility over the North Atlantic Ocean. The approach employed in meeting the stated objectives is listed below: A. Apply and evaluate the Preisendorf er maximum proba- bility and natural regression strategies (1983a, b,c) to addi- tional North Atlantic Ocean homogeneous areas [Lowe, 1983b] using May through July 198 3, NOGAPS predictand/predictor data . B. Expand the Model Output Predictor (MOP) data sets to include the NOGAPS model TAU-00, and the TAU-24 and TAU-48 prognostic times defined in Chapter III. C. Investigate specific two-stage, equal variance and quadratic multiple linear regression threshold models pro- posed by Lowe (1984a) for the oceanic areas and model output periods addressed in A. and B. above. D. Compare and contrast the individual results of the Preisendorf er statistical methodologies to those of the Lowe approach. E. Conduct a limited series of experiments in which a 1984 data set, 15 May to 23 June, is utilized as an 18 independence evaluation of the predictive models constructed with 19 8 3 NOGAPS data. F. Based on A. to E . above, present an interim recommenda- tion for an optimal statistical approach to forecast North Atlantic Ocean horizontal visibility as a function of pre- diction time and homogeneous area. 19 III. DATA A. VISIBILITY OBSERVATIONS AND SYNOPTIC CODES Horizontal visibility observations taken from seagoing platforms are reported as values of ten standardized World Meteorological Organization (WMO) synoptic weather codes. These codes range in value from 90, which corresponds to visibility less than 50 meters, to 99, which corresponds to visibility equal to or greater than 50 kilometers. Human observational error and inexactness in measuring visibility at sea necessitates a generalization of visibility classifi- cation for prediction purposes, as follows: Visibility Category Synoptic Code Visibility Range I 90-94 < 2 km II 95-96 ^ 2 km to < 10 km III 97-99 ^ 10 km The above scheme coincides with the classification scheme proposed by Karl (1984) and is based upon the below listed U.S. Navy operational criteria. 1. 10 km (5 n mi)--U.S. Navy aircraft carrier at-sea flight recovery operations change from visual (VFR) to controlled (IFR) approach guidelines [Department of the Navy, 1979] . 2. 2 km (1 n mi) — the sounding of reduced visibility signals for all vessels operating in international waters. 20 The term "reduced visibility" is not specifically defined in the International Regulations for Preventing Collisions at Sea, 1972. The distance of 1 n mi is generally considered to be the governing operational distance. B. NORTH ATLANTIC OCEAN DATA 1. Area The North Atlantic Ocean, from 0°-80° N latitude, was divided into homogeneous oceanic areas by Lowe (19 84b) using a statistical cluster analysis technique. The specific homogeneous areas evaluated in this study are identified as areas 2, 3W and 4 on Fig. 2. These areas were selected be- cause they individually represent a range of different rela- tive frequencies of poor visibility observations. Area 3W , which was used by Karl (1984) for his preliminary experimen- tation, represents an area of relatively frequent occurrence of poor visibility, while area 4 represents an area of rela- tively sparse occurrence of poor visibility and area 2 represents an intermediate case. 2. Time Period Data from mid-May 1983 to mid-July 1983 were combined to form a more extensive data set, hereafter referred to as FATJUNE 19 83. FATJUNE 19 83 was selected as the initial data set for statistical experimentation because of the high fre- quency of occurrence of poor visibility observations for many areas of the North Atlantic Ocean during this period. 21 1200 GMT synoptic ship report data were used exclusively in this study. This time corresponds to general daylight condi- tions over the North Atlantic Ocean during FAT JUNE . In addition to FATJUNE 1983, a limited May 15 to June 23 1984 data set, possessing the same geographical coverage and day- light characteristics of FATJUNE 1983, was utilized in an independent test of the predictive arrays and equations generated in this study. For the purpose of this study, TAU-00 generally represents six-hour model forecast fields valid at 1200 GMT. Three specific fields, namely temperature, geopotential height and wind, are model initialization fields valid at 1200 GMT. TAU-24 and TAU-48 are defined as 24-hour and 48-hour model forecast fields, valid at 1200 GMT. TAU-00, TAU-24 and TAU-48 model output parameters (predictors) are employed in the 00, 24 and 48 hour forecast schemes, respect- fully. Summaries of the visibility frequencies for each visibility category, as a function of homogeneous area and prediction time, for FATJUNE 1983 and the 15 May to 23 June 1984 data set, are contained in Tables I through III and Table VI respectively. 3 . Synoptic Weather Reports All synoptic visibility observations (predictand data) for this study were provided by the Naval Oceanography Com- mand Detachment (NOCD) , Asheville, North Carolina which is co-located with the National Climatic Data Center (NCDC) . 22 The observations which contained systematic observer error or were obviously erroneous, as determined from the data quality indicators provided with the data, were deleted from the working data sets. 4 . Predictor Parameters Fifty TAU-00, fifty- four TAU-24 and fifty- four TAU-48 model output predictors (MOP's) were provided by the Fleet Numerical Oceanography Center (FNOC) , Monterey, California. These parameters are generated by their current operational atmospheric prediction model, the Navy Operational Global Atmospheric Prediction System (NOGAPS) . All MOP's were interpolated from model grid coordinates to synoptic ship report position using a linear interpolation scheme. In addition to the initial group of model output parameters, ten derived parameters representing calculated quantities, such as parameter gradients and products , were included as potential predictors. Of the entire group of potential predictor parameters, only forty TAU-00 and forty-seven TAU-24 and TAU-48 MOP's were actually used to develop the various Preisendorfer (1983a, b,c) and linear regression threshold models [Lowe, 19 8 3a] . The remainder of the NOGAPS model output parameters were dropped from consideration because 1) the MOP lacked a physical linkage to the visibility pre- dictand and/or 2) a lack of significant digits (lost during the transfer of FNOC data to the main computer center's mass storage system) rendered the particular MOP useless. 23 A list of all available TAU-00 , TAU-24 and TAU-48 MOP ' s are included in Appendix B. For each homogeneous area and model forecast projec- tion, a set of three linear regression equations, in addition to the aforementioned MOP ' s , were included as potential mop's for a separate evaluation of the Preisendorf er methodology (the PR+BMD model). These three predictor equations were obtained from two standardized linear regression software packages, namely P2R--stepwise regression and P9R — all possible subsets regression, as addressed in the BMDP Sta- tistical Software [University of California, 1983]. The P2R was initially employed in the evaluation of areas 2 and 4, TAU-00 data, while the P9R program was employed in the remainder of the cases studied. The change to the P9R program was initiated as a safeguard against any potential predictor selection bias incorporated in the P2R software. Specific details concerning these statistical software packages are addressed in Appendix A. C. DEPENDENT/INDEPENDENT DATA SETS Due to the limited amount of data available to this study for each of the North Atlantic Ocean homogeneous areas, it was necessary to withhold a significant amount of the observations from the developmental model to use as an independent data set. That amount was set as one»-third for the experiments reported here. This was accomplished by the 24 use of a counter and transfer statement in the computer programs which prevented every third observation from enter- ing the developmental computations. To ensure that the dependent and independent data were representative of the same population, a 9 5% confidence interval for proportions [Miller and Freund , 1977] was established from the entire data set, for each visibility category, and the dependent and independent data sets were constrained to have visibility frequencies within these established confidence intervals. Table IV summarizes the dependent and independent data for the North Atlantic Ocean data set. 25 IV. PROCEDURES A. TERMS AND SYMBOLS The terms and statistical symbols defined below will be used throughout the remainder of this report. The formal mathematical definitions are described in Karl (1984). 1. Maximum probability strategy--choosing forecast visibility category based upon the highest conditional probability of visibility within a predictor interval. a. MAXPROB I — designation of the maximum probability strategy in which ties of the highest conditional probabili- ties in a predictor interval are resolved by the generation of a random number. b. MAXPROB II--designation of the maximum probability strategy in which ties of the highest conditional probabili- ties in a predictor interval are resolved by assigning the lowest visibility category, of those tied, as the forecast category . 2. Natural regression strategy--choosing forecast visi- bility categories based upon the statistical average of the conditional probabilities of visibility within a predictor interval . 3. AO — the probability of a zero-class visibility cate- gory forecast error (e.g., if visiblity category I is forecast and observed) . 26 4. Al — the probability of a one-class visibility category forecast error (e.g., if visibility category I is forecast and category II is observed) . 5. CE — class error parameter defined as A0+2A1, used as the primary aid in identifying the first predictor. 6. PP — the potential predictability of visibility by any given predictor. 7. Functional dependence. This is a measure of the stochastic dependence of one predictor upon another. Func- tional dependence is the probability that one of the predic- tors will change when the other does. High functional dependence values between one already selected predictor and another potential predictor, indicates that little addi- tional information beyond the selected predictor is possible. Conversely, a low functional dependence value between the same two predictors, indicates that each predictor possesses a high degree of linearly uncorrelated information concerning the predictand. Functional dependence range is 0.0 to 1.0 (1.0 = highest functional dependence). The specific deriva- tion and mathematical description of the concept of "func- tional dependence" is discussed in greater depth by Preisendorfer (1983c). 8. Root-sum-squared functional dependence. The functional dependence of a predictor on all predictors already included in the developmental model. It is equal to the square-root 27 of the sum of the squares of the individual functional dependence values. 9. TSl — threat score for visibility category I, as computed from a contingency table (see Appendix C) . 10. ATSl--adjusted threat score for visibility category I which removes the influence of the data set category frequency (see Appendix C) . 11. AAO--ad justed AO . A contingency table statistic which removes the influence of the most frequent visibility category in a set of data (similar to a normalized value) (see Appendix C) . B. COMPUTER PROGRAMS Four computer programs were developed to test the pro- posed Preisendorf er (1983a, b,c) methodology. The programs are on file in the Department of Meteorology, Naval Post- graduate School, Monterey, California, 93943. 1. A program to compute AO , Al , CE and PP for all predic- tors, all strategies (MAXPROB I, MAXPROB II and natural regression) for a particular number of equally populous predictor intervals. Statistics for the three strategies are based upon the same predictor (s) rather than the best predictor (s) for each strategy. 2. A program to compute functional dependence values for all predictors, on a given predictor, for a given number of equally populous predictor intervals and to compute the 28 associated 96% critical confidence interval value, referred to as functional dependence (9 6) in this study, by Monte Carlo means . 3. A program to construct contingency tables and to compute skill and threat scores, for both the dependent and Independent data sets. 4. A program to generate 100 random data sets, from the marginal probabilities of the predictor (s) in the develop- mental model, and to compute upper and lower 5% critical confidence interval values for AO and Al to be used for testing the significance of the results from each of the Preisendorf er mcdels against chance. These confidence interval values are calculated via Monte Carlo means. C . MODELS 1. Preisendorf er PR Model This model represents the first of two different applications of the basic Preisendorf er methodology [Preisendorf er , 1983a, b,c]. Karl (1984), in his preliminary research, provides a rigorous interpretation and results associated with this statistical forecasting methodology. Karl's study provides the necessary background for the con- tinued investigation and evaluation of this model and readers interested in specific details are advised to consult this document. The PR model utilizes the working set of NOGAPS model output parameters (MOP's) and derived parameters 29 (Appendix B) as potential predictors in constructing a developmental model, based upon the dependent data set, which provides the structure by which the independent data set is tested and evaluated. In general, these potential predictors have their range of values partitioned into discretized equally populous predictor intervals ("cells") and conditional probabilities of the predictand are calcu- lated according to the three modified visibility categories (VISCAT) I, II and III. Three separate strategies of deter- mining the specific VISCAT which is to be identified with each predictor value, are proposed. These strategies, two based upon maximum probability and the third based on a natural regression approach, are addressed as MAXPROB I, MAXPROB II and natural regression in the remaining portions of this study. Initial evaluation of this model involves varying the equally populous predictor intervals from sizes of four through ten, and selecting an optimal first predictor which provides one of the following requirements in the designated order : a. the lowest CE value of all the potential predictors b. the highest PP value of all the potential predictors Once a first predictor is identified for each of the four through ten equally populous predictor intervals, corresponding VISCAT I, II and III threat and AO skill scores (Appendix E) are calculated for both the dependent 30 and independent data sets. The practice of selecting an optimal equally populous predictor interval from the eligible grouping sizes of four through ten, was proposed by Karl (1984) as a practical procedure which would permit the realization of peak skill scores as well as maintain asso- ciated computer storage requirements at a manageable level. An unfortunate consequence of this range of potential group- ing sizes is that certain statistical calculations associated with equally populous predictor intervals of eight, nine and ten are terminated before completion due to a two mega- byte storage ceiling at the NPS W.R. Church Computer Center. When considering potential predictor intervals, the size of the interval is of obvious importance, with lower values being the most desirable. The criterion for determining the optimal equally populous predictor interval is to select the smallest interval value which maximizes the dependent data set adjusted AO and independent adjusted VISCAT I threat score. For this study, this interval value was fixed for all ensuing aspects of the model evaluation. In practice, the selection of equally populous predictor intervals was based upon the initial adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the MAXPROB II strategy. The MAXPROB II scores were routinely found to be the highest for each case evaluated, at this early stage in the evaluation process, and therefore used as the basis for grouping selection. As the equally populous 31 grouping interval remains constant throughout the Preisen- dorfer models, the MAXPROB I and natural regression strate- gies practically play no role in the predictor selection process. Once the first predictor and its associated equally populous predictor interval have been identified, a func- tional dependence test of the first predictor against those remaining potential predictors is run. The second, third and all subsequent predictors are selected only if both of the following criteria are met: a. subsequent predictors must increase AO over the AO value attained at the preceding level, and b. the selected predictor must have the lowest functional dependence and root-sum- square functional dependence of all the remaining potential predictors. After each predictor selection stage has been com- pleted, significance tests are run upon the developmental model to determine if the results are suitably significant as compared to random chance. This testing is accomplished via Monte Carlo testing methods using the conditional probabilities of the selected predictors and assuming equal probability of occurrence for the three modified visibility categories. Functional dependence/root-sum-square functional dependence, AO , and Al statistics are calculated for each of 100 randomly generated data sets. For the developmental model to yield results which are significant at the speci- fied confidence interval values, each one of the following criteria must be met: 32 a. AO must be equal to or greater than A0(96) b. Al must be equal to or less than Al(05) c. the functional dependence value for a selected predictor must be less than functional dependence (96) As with the process of selecting equally populous predictor intervals, the AO , A0(96), Al and Al(05) statistics (Appendix G) , reflect scores for the MAXPROB II strategy. The AO statistics routinely were found to be the highest for this strategy and thus were used as the basis for ensuring the aforementioned predictor selection criteria were met. However, the MAXPROB I strategy often' produced AO values identical to MAXPROB II. The natural regression strategy regularly lagged the two maximum probability strategies in AO and Al scores and consequently played no real role in the prediction selection process. Specific trends in AO/Al scores can be seen in Appendix G. From a practical standpoint, the model development continues until computer storage limitations preclude further addition of predictors. This generally occurred at the fifth predictor level. Once the developmental model is completed, contingency tables of forecast visibility category versus observed visi- bility category are constructed for both the dependent and independent data sets, and threat and skill scores are computed and compared. 33 2. Preisendorfer PR+BMD Model This model is still the PR model described above. Now, sets of three linear regression equations (Appendix D) are added to the list of potential NOGAPS model output and derived predictor parameters. The inherent difference of these predictors is evidenced in both the predictor selec- tion process as well as in the resulting skill and threat scores, as will be demonstrated in Chapter V. 3. Equal Variance Threshold Model (EVAR) This model represents the first of two threshold models, developed by Lowe (1984a), which were evaluated in this study. The model uses an algorithm which requires the assumption that the variances of two normally distributed populations which are to be separated by a threshold value are equal, while their means are unequal. A detailed dis- cussion of the theoretical background of this scheme is addressed in Appendix A. . " A two-stage separation scheme was used to effectively divide the visibility categories (VISCAT) I, II and III into a first-stage VISCAT I versus a combined VISCAT II plus VISCAT III separation, and subsequently VISCAT II versus VISCAT III separation for each homogeneous area and model output time. This separation was accomplished by setting all VISCAT I observations equal to an arbitrary integer value of zero and the combined VISCAT II plus VISCAT III observations equal to an arbitrary integer value of one and generating 34 a linear regression equation to suitably describe the resulting two distributions. This linear regression equation was then used in the graphical plotting program BMDP5D, from the BMDP Statistical Software [University of California, 1983] , to generate a set of histograms describing the first stage separation. Included with the graphical histogram output is a listing of the individual frequency of observa- tion (P) , mean (y) and standard deviation (a) of each of the specified visibility distributions. These statistics are incorporated into the equal variance threshold algorithm and a corresponding threshold value is calculated. Following the first-stage threshold calculation, a second linear regression equation is generated, based upon only those VISCAT II plus VISCAT III observations which exceed the previously calculated threshold value. This effectively eliminates any VISCAT II plus VISCAT III obser- vations less than the threshold value (i.e., those observa- tions contained in the tail of the distribution) , from being included in the second-stage regression. The previous proce- dure of generating corresponding histograms and statistics is repeated, based upon all VISCAT II observations being assigned an arbitrary integer value of zero and all VISCAT III observations being assigned an integer value of one. A second-stage equal variance threshold value is then calcu- lated which separates VISCAT II from VISCAT III. 35 with the two-stage separation complete, the indepen- dent data set is processed through the governing equations and thresholds to obtain a set of observed visibility value results versus calculated "forecast" visibility value re- sults. These results, in contingency table format for each evaluated case, are presented in Chapter V and Appendix G. 4. Quadratic Threshold Model (QUAD) This model represents the second of two threshold models, developed by Lowe (1984a), which were evaluated in this study. The model uses an algorithm which requires the assumption that both the variances and the means of two normally distributed populations, which are to be separated by a threshold value, are equal. Similar to the EVAR model, a detailed discussion of the theoretical background of this scheme is addressed in Appendix A. The general two-stage separation procedure employed with this model is identical to that described for the EVAR model in IV. C. above. The only difference between the QUAD and EVAR model is the algorithm, based upon a solution to a quadratic equation in this model, used to calculate the appropriate threshold values. 5 . Maximum- Likelihood- of -Detect ion Model The maximum-likelihood-of-detection criteria (MLDC) is an additional threshold technique which is included in this study as a possible alternative to the aforementioned EVAR and QUAD minimum probable error threshold models. The 36 MLDC involves calculating a threshold value based upon the assumptions that the population frequencies and variances of two normally distributed samples which are to be separated are identical. This technique is particularly well suited for cases where the threat frequency (i.e., number of threatening events divided by the total number of threat and non-threat events) approaches very small values (e.g., statistical rare events) . Unlike the EVAR and QUAD models, the two-stage separation employed with this technique utilizes a first- stage VISCAT I+II versus VISCAT III followed by a second- stage VISCAT I versus VISCAT II separation. In calculating the specific threshold values, the lowest frequency visibility category (usually the VISCAT I threat category) is assigned an arbitrary integer value of one. The remaining larger visibility category/ies are assigned the arbitrary integer value of zero. Proceeding in the same manner as described with the EVAR and QUAD models, population means are calcu- lated for each separation stage. The threshold value is simply the mid-point between the two population means. A detailed discussion of the theoretical background of this scheme is addressed in Appendix A. 37 V. RESULTS The general procedures outlined in Chapter I were fol- lowed in evaluating the statistical scoring techniques for the oceanic homogeneous areas 2, 3W and 4. Certain slight modifications were required to handle the relatively low frequency of visibility category I, in area 4 for the TAU- 00, TAU-24 and TAU-48 model output data sets. Fig. 2 displays the individual oceanic homogeneous areas for FATJUNE 1983. Tables I through III identify the frequency of occur- rence of visibility categories I, II and III at TAU-00, TAU-24 and TAU-4 8 for each of the evaluated homogeneous areas. In discussing the results of this study, specific comment is focused upon the optimal model for each case as well as any significant finding observed by the author. Certain characteristics of the evaluated cases are repetitious and are considered adequately described by their associated figures. Consequently, the entire assemblage of figures in Appendix G are not individually addressed. These figures are nevertheless considered noteworthy, as they document the performance of each tested model in this study, and are included as a matter of record. The following presentation of the results of this experimentation are arranged accord- ing to the specific oceanic homogeneous area and model output period. 38 In general/ four models are evaluated for each of the predefined homogeneous areas/model forecast projections. The four models are: the Preisendorf er methodology utilizing NOGAPS model output predictors and a limited number of derived predictors (PR) , the Preisendorf er methodology uti- lizing both NOGAPS model output predictors, derived predic- tors and linear regression equation predictors (PR+BMD) , an equal variance linear regression threshold model (EVAR) and a quadratic linear regression threshold model (QUAD) . A. NORTH ATLANTIC OCEAN, AREA 2 Area 2 encompasses a geographic region that extends from the southeastern tip of Newfoundland, across the North Atlantic Ocean to the eastern coast of England, north to the Five Fingers of Iceland and back to the Canadian coast north of Newfoundland. Fig. 2 gives the pictorial repre- sentation of the area. 1. Area 2, TAU-00 Fig. 3 shows the relationship of equally populous grouping size to the adjusted P:0 (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. For this case, a grouping size of eight was se- lected. Results of the individual MAXPROB I, MAXPROB II and Natural Regression strategies are shown in Figs. 4a though 4c. The MAXPROB II strategy (Fig. 4b) produced the largest overall independent data VISCAT I adjusted threat 39 score, namely 0.23 (unadjusted, 0.30). This peak threat score occurs with the inclusion of the first predictor, E850, and declines marginally with the addition of the re- maining four predictors. Of the three strategies, the natural regression strategy (Fig. 4c), yields the poorest overall threat scores with its peak threat score occurring with the addition of the fourth predictor. The predictors selected for this case are E850, ENTR, DVDP , UlOOO, and STRTH. The associated functional dependence and AO/Al sta- tistics and 96%/05% confidence interval values for these predictors are shown in Fig. 5. The trend of functional dependence versus its 96% confidence interval shows that the specific functional dependence values associated with the chosen predictors never falls within the 96% confidence interval. At the first predictor level, for example, the functional dependence of ENTR upon E850 has a value of 0.1146 as compared to a 96% confidence interval value of 0.1039. This infers that the corresponding scores (i.e., threat scores, AO and Al) are not statistically significant at the preselected 96% confidence interval level. Fig. 6 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. For this case an equally populous grouping size of seven was selected. Results of the three individual 40 Preisendorf er strategies, along with the corresponding con- tingency tables, can be seen in Figs. 7a through 7c. As with the PR model, a maximum independent VISCAT I threat score was obtained with the MAXPROB II strategy using the first predictor selected, namely the linear regression equa- tion predictor BMDl (Appendix D) . The overall independent adjusted VISCAT I threat score achieved with this model is 0.29 (unadjusted, 0.36), which is .06 greater than that for the PR model. The natural regression strategy (Fig. 7c) provides the poorest resultant threat scores and these reach their peak with the inclusion of the fifth predictor. The predictors selected for this case are BMDl, ENTR, DVDP , PS and PBLD. The functional dependence, Al/AO statistics and 9 6%/ 05% confidence interval values for this model can be seen in Fig. 8. As with the PR model, the specific functional dependence values associated with the selected predictors never fall below the calculated 96% functional dependence confidence interval. Figs. 9 and 10 show the contingency tables results for the EVAR and QUAD threshold models. For each of these models the independent adjusted VISCAT I threat scores have identical values of 0.32 (unadjusted, 0.38). The two-stage linear regression sequence employed for both of these threshold models yields very similar basic statistics. For the EVAR model, a threshold value of 41 0.648497 was calculated for the first-stage VISCAT I versus VISCAT II+III separation. This threshold was based upon a VISCAT I sample size of 190 observations, a mean of 0.659 and standard deviation of 0.205 and a combined VISCAT II+III sample size of 1722 observations, a mean of 0.927 and standard deviation of 0.122. The second-stage VISCAT II versus VISCAT III separation was based on a calculated threshold of 0.580128. Associated with this threshold value were 311 VISCAT II observations with a mean of 0.708 and standard deviation of 0.142 and 1473 VISCAT III observations with a mean of 0.850 and standard deviation of 0.131. For the QUAD model, a threshold value of 0.642104 was calculated for the VISCAT I versus VISCAT II+III first- stage separation, based upon the sample addressed above. A second-stage quadratic threshold separating the VISCAT II and VISCAT III samples was calculated to be 0.580569. This VISCAT II sample contained 358 observations with a mean of 0.643 and standard deviation of 0.142 while the VISCAT III sample contained 1402 observations with a mean of 0.846 and standard deviation of 0.140. While no significant difference appears to exist between the results of the two threshold models, the QUAD model yields a slightly higher AO and slightly lower Al values for both the dependent and independent data sets. Table V shows a synopsis of the key statistical results for this case. The best models, as determined by 42 independent adjusted VISCAT I threat scores, are the two threshold models. Of these two models, the QUAD model achieves the highest adjusted AO , namely 3.16% (unadjusted, 80.73%) . 2. Area 2, TAU-24 Fig. 11 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. For this model, adjusted dependent AO values of -0.03 and adjusted independent threat score of -.01 were obtained for grouping sizes four through nine. At the grouping size of ten, a jump in scores was realized and thus ten is identified as the only possible selection. An associated difficulty in utilizing a grouping size of eight, nine or ten, is that local computer storage resources are limited to two megabytes. This decreases the usual five predictor array to only four predictors as witnessed in this case. The results of the three Preisendorf er strategies are shown in Figs. 12a through 12c. For this model, the MAXPROB I and MAXPROB II strategies yield identical maximum independent adjusted VISCAT I threat scores of 0.21 (unadjusted, 0.27). For each of the maximum probability strategies, an initial threat score of 0.19 (unadjusted, 0.25) was achieved with the first predictor, E850, solely. The slight increase to the overall peak threat score was obtained with the inclusion of the second predictor, ENTR, with subsequent independent 43 VISCAT I threat scores decreasing at the third and fourth predictor levels. Of the three strategies, natural regression (Fig. 12c) yielded the poorest overall threat score and per- cent correct values. These relative peak scores for the natural regression strategy occur with the inclusion of the fourth and final predictor. The predictors selected for this model were: E850, ENTR, DVDP and DIV9 25. The associated functional dependence, AO/Al statis- tics and 96%/05% confidence intervals for this model are shown in Fig. 13. For this case, the third and fourth pre- dictors' root-sum-square functional dependence values exceed the associated 96% confidence interval values, indicating significant statistical interdependence of these predictors at this confidence interval level. Fig. 14 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. The dramatic increase in independent threat score at grouping size of seven identifies it as the optimal selection. The results of the three Preisendorf er strategies are shown in Figs. 15a through 15c. For this model, the MAXPROB I and MAXPROB II strategies yield identical maximum independent adjusted VISCAT I threat scores of 0.26 (unad- justed, 0.32) . This peak score was achieved with the inclu- sion of the first predictor. In this case, the first selected predictor is the second generated linear regression equation 44 predictor, BMD2 (Appendix D) . Following the initial threat score maxima, the scores decreased with the addition of the subsequent four predictors. While some fluctuation in the threat score trend was observed with the MAXPROB II strategy, independent VISCAT I threat scores never surpassed their initial maximum value. Of the three strategies, natural regression (Fig. 15c) provides the poorest overall indepen- dent VISCAT I threat score of 0.21 (unadjusted, 0.28). This score was achieved with the addition of the fifth and final predictor. The predictors selected for this model were: BMD2, VRT925, ENTR, UlOOO and RH. The associated functional dependence, AO/Al statis- tics and 96%/05% confidence intervals are shown in Fig. 16. For this model, a comparison of functional dependence and functional dependence 96% confidence interval values indi- cates that the final three predictors have root-sum-square functional dependence values which are too large to ensure significant statistical independence at the 96% confidence interval level. Figs. 17 and 18 show the contingency tables and associated statistics for the EVAR and QUAD threshold models. For each of the models, the independent adjusted VISCAT I threat scores have identical values, namely 0.29 (unadjusted, 0.24). The two-stage linear regression sequence employed for both of these models yields fairly similar statistical results. For the EVAR model, a threshold value of 0.674932 45 was calculated for the first-stage VISCAT I versus VISCAT II+III separation based upon a VISCAT I sample size of 180, a mean of 0.682 and a standard deviation of 0.227 and a VISCAT II+III sample size of 1580, a mean of 0.938 and a standard deviation of 0.109. The second-stage VISCAT II versus VISCAT III separation was based upon a calculated threshold value of 0.601717. Associated with this threshold were 300 VISCAT II observations with a mean of 0.733 and standard deviation of 0.149 and 1339 VISCAT III observations with a mean of 0.857 and standard deviation of 0.121. For the QUAD model, a threshold value of 0.675210 was calculated for the first-stage VISCAT I versus VISCAT II+III separation based upon the sample statistics addressed above. The second-stage threshold separating the VISCAT II and VISCAT III samples was calculated to be 0.617455. The VISCAT II sample contained 300 observations with a mean of 0.739 and a standard deviation of 0.125. The VISCAT III sample contained 1339 observations with a mean of 0.885 and standard deviation of 0.118. While the VISCAT I threat scores for both the dependent and independent data sets are identical for the two models , differences in other statistics are apparent. The EVAR model (Fig. 17), for example, has the higher independent adjusted AO scores, namely 2.96% (unadjusted, 81.34%), as compared to scores of -63.31% (unadjusted, 68.60%) for the QUAD model (Fig. 18) . 46 In general, for area 2, TAU-24, the threshold models again provide the highest independent VISCAT I threat scores (Table V). Of the two threshold models, the EVAR model has a slight edge in AO scores. 3. Area 2, TAU-4 8 Fig. 19 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. For this model, the initial peak values of dependent AO and independent VISCAT I threat score at the grouping size of four did not sufficiently ascertain four as the optimal grouping selection. For this grouping size, the second selected predictor ENTR had a functional dependence of 0.2952 as compared to the calculated functional dependence 96% confidence interval value of 0.1932. The large dis- parity between the two functional dependence values indicates a significant statistical correlation between E850 and ENTR at a grouping size of four and thus grouping size four was dropped from consideration. The selected grouping size of nine, which unfortunately carries with it the requirement of a very large computer storage forecast array at the fifth predictor level, had a functional dependence value 0.09 30 as compared to a functional dependence 9 6% confidence interval value of 0.0970 and thus was selected as the optimal grouping size. The associated functional dependence, AO/Al statistics and 96% confidence intervals are shown in Fig. 20. The first 47 three predictors selected have functional dependence values sufficiently low enough to ensure no significant predictor interdependence . The results of the three Preisendorf er strategies are shown in Figs. 21a through 21c. The maximum independent VISCAT I threat score achieved for the three strategies was 0.17 (unadjusted, 0.26) and was obtained with the MAXPROB II strategy with the addition of the fifth predictor. It should be noted that the independent adjusted VISCAT I threat scores achieved by both the MAXPROB I and MAXPROB II strate- gies reached near peak values of 0.16 (unadjusted, 0.24) with the addition of the second predictor, thus greatly mini- mizing the size of the associated forecast array. Of the three strategies, natural regression (Fig. 21c) yielded the poorest overall adjusted independent VISCAT I threat score, namely 0.09 (unadjusted, 0.18). This score was achieved with the inclusion of the fourth predictor in the forecast array. The predictors selected for this model were E850, ENTR, DVDP, DRAG and DIV9 25. Fig. 22 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. For this model a grouping size of nine was selected. The results of the MAXPROB I, MAXPROB II and natural regression strategies are shown in Figs. 23a through 23c. For this model, MAXPROB I and MAXPROB II provide 48 identical maximum independent adjusted VISCAT I threat scores of 0.31 (unadjusted, 0.37). These scores were achieved with the inclusion of the second linear regression equation pre- dictor BMD2 (Appendix D) . For each of these strategies, the independent VISCAT I threat scores decrease with the addi- tion of the second and subsequent predictors. While a slight upward progression is noticed with the MAXPROB II strategy, the peak score observed at the first predictor level is never surpassed. Of the three Preisendorf er strategies, natural regression (Fig. 23c), yields the poorest overall independent VISCAT I threat score, namely 0.18 (unadjusted, 0.26). This score occurs with the inclusion of the fifth predictor and culminates in a slow increase in threat score as each predictor is sequentially added to the forecast array, The predictors selected for this model were BMD2, VRT925, ENTR, U500 and DRAG. Fig. 24 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for the selected predictors. For this model, the second and third predictors' functional dependence values fall below the 96% confidence interval and thus are not significantly inter- dependent upon one another. This trend changes with the fourth and fifth predictors which have functional dependence values greater than the calculated 96% confidence interval values . Figs. 25 and 26 show the contingency table results for the EVAR and QUAD threshold models. For each of these 49 models, the independent adjusted VISCAT I threat scores have identical values of 0.21 (unadjusted, 0.29). The two-stage linear regression sequence used to separate the three visibility categories yield very similar results for the two threshold models. For the EVAR model, a threshold value of 0.652554 was calculated for the first- stage VISCAT I versus VISCAT II+III sample separation. This threshold value is based upon a VISCAT I sample size of 182 observations with a mean of 0.686 and a standard deviation of 0.267 and a combined VISCAT II+III sample of 1670 observations with an associated mean of 0.930 and standard deviation of 0.106. The second stage VISCAT II versus VISCAT III regression separation yielded a threshold value of 0.572257 based upon 355 VISCAT II observations, with a mean of 0.711 and standard deviation 0.135, and 1408 VISCAT III observations with a_ mean of 0.834 and a standard deviation of 0.130. , " For the QUAD model, a very similar threshold value of 0.652554 was calculated for the first-stage VISCAT I versus VISCAT II+III separation based upon the sample first- stage statistics addressed above. A second-stage threshold value of 0.564579 was calculated based upon 330 VISCAT II observations with a mean of 0.724 and standard deviation of 0.128, and 1407 VISCAT III observations with a mean of 0.833 and a standard deviation of 0.127. 50 In general, the results of these two threshold models are nearly identical. The EVAR model shows a very slight advantage in adjusted independent AO scores, namely 7.07% (unadusted, 80.11%) as compared to 5.05% (unadjusted, 79.68%) for the QUAD model. Similarly, the EVAR model yielded a slightly higher independent adjusted threat score for VISCAT I combined with VISCAT II of 0.02 (unadjusted, 0.23) versus an adjusted score of 0.01 (unadjusted, 0.22) for the QUAD model . For Area 2, TAU-48 the PR+BMD model provides the highest overall independent VISCAT I threat score (Table V) . The difference between the independent adjusted VISCAT I threat scores for the PR+BMD model and the two threshold models is minimal, namely 0.02, while the PR model is 0.14 lower . B. NORTH ATLANTIC OCEAN, AREA 3W Area 3W was the North Atlantic homogeneous area selected by Karl (1984) for his initial TAU-00 MOS experimentation. This area borders the United State's eastern seaboard from the vicinity of Cape Charles, Virginia to the southeastern tip of Newfoundland. The area encompasses a large portion of the Georges Banks region and extends to approximately 45° W longitude. The specific detail and proximity of this area can be seen in Fig. 2. Area 3W constitutes the homogeneous area with the highest relative frequency of VISCAT I observations with approximately 51 19% of the total number of visibility observations being less than 2 kilometers in the TAU-00, TAU-24 and TAU-48 periods. The TAU-24 and TAU-48 prognostic periods will be addressed in this document. The reader is advised to consult Karl (1984) for detailed information concerning area 3W , TAU-00. 1. Area 3W, TAU-24 Fig. 27 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. For this case a grouping size of six was selected. Results of the three Preisendorf er strategies are shown in Figs. 28a through 28c. The MAXPROB II strategy achieves a slightly higher independent adjusted VISCAT I threat score of 0.21 (unadjusted, 0.36) as compared to a score of 0.20 (unadjusted, 0.35) for the MAXPROB I method. For each of these strategies, the maximum threat score is reached with the inclusion of the fifth and final predictor in the fore- cast array. The general trend of these two strategies is nearly identical and show an initial rise in threat score at the first predictor level, a slight decrease with the addition of the second and third predictors and a secondary rise at the fourth and fifth predictor levels. The poorest results for this case were achieved with the natural regres- sion strategy (Fig. 28c) , for which an independent adjusted VISCAT I threat score of 0.16 (unadjusted, 0.32) was achieved This score was similarly reached with the addition of the 52 fifth and final predictor. The predictors selected for this model were DTDP, SHWRS , ENTR, UlOOO and DUDP . Fig. 29 shows the functional dependence, AO/Al statistics and 96%/05% confidence intervals for this model. For this case only the second predictor has a functional dependence value which falls below the corresponding 9 6% confidence interval and thus meets the requisite conditions regarding predictor interdependence. Consequently, the greatest independent threat score achieved, which coinci- dently meets the functional dependence criteria, occurs with the MAXPROB II strategy at the inclusion of the second pre- dictor. The threat score achieved in this particular instance has a value of 0.13 (unadjusted, 0.30). Fig. 30 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. For this case a grouping size of five was selected. Results of the MAXPROB I, MAXPROB II and natural regression strategies are shown in Figs. 31a through 31c. For this model, the two maximum probability strategies pro- vide identical peak independent adjusted VISCAT I threat scores of 0.28 (unadjusted, 0.42) at the first predictor level. For both of these strategies, the addition of subse- quent predictors produces a steady drop off in threat score values. The poorest overall results for this case are achieved with the natural regression strategy (Fig. 31c). 53 This method yields an independent adjusted VISCAT I threat score of 0.17 (unadjusted, 0.33) which was obtained with the addition of the fifth and final predictor. The predictors selected for this model were BMDl, D500, DVDP , ENTR and U850. Fig. 32 shows the associated functional dependence, Al/AO statistics and 96%/05% confidence interval values for the predictors chosen for this model. The functional depen- dence versus the 96% confidence interval follows a peculiar trend where the second predictor is significantly dependent upon the first predictor but the third and fourth predictors are conversely sufficiently uncorrelated with the prior predictors to ensure no significant functional dependence. The final predictor returns to being functionally dependent upon the previous predictors. This trend indicates that the relative contribution of the second and subsequent predictors is statistically hot significant at the preselected 96% confidence interval level. Figs. 33 and 34 show the contingency table results for the EVAR and QUAD threshold models. The results of these models are very similar with the EVAR model yielding an independent adjusted VISCAT I threat score of 0.17 (unad- justed, 0.33) as compared to a corresponding threat score of 0.16 (unadjusted, 0.32) for the QUAD model. For the EVAR model, a first-stage threshold value of 0.561855 was calculated based upon 270 VISCAT I observations 54 with a mean of 0.590 and standard deviation of 0.203 and 1145 VISCAT II+III observations with a mean of 0.861 and standard deviation of 0.168. The second-stage VISCAT II versus VISCAT III separation was based upon a calculated threshold value of 0.542363. Associated with this threshold were 299 VISCAT II observations with a mean of 0.647 and standard deviation of 0.146 and 938 VISCAT III observations with a mean of 0.794 and standard deviation of 0.153. For the QUAD model, a similar threshold value of 0.5559971 was calculated based upon the first-stage regres- sion separation listed above. A second-stage threshold value of 0.540874, separating VISCAT II from VISCAT III, was calculated based upon 30 5 VISCAT II observations with a mean of 0.639 and standard deviation of 0.157 and 940 VISCAT III observations with a mean of 0.793 and standard deviation of 0.154. In general, the PR+BMD model produced the best overall results for this case, followed by the PR model and lastly the two threshold models (Table V) . The independent adjusted AO score of the PR+BMD model, which corresponds to the maxi- mum independent adjusted VISCAT I threat score, is similarly a maximum value for this case, namely 21.68% (unadjusted, 74.96%) . 2. Area 3W, TAU-4 8 Fig. 35 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the 55 adjusted VISCAT I threat score (independent data) for the PR model. For this model, an equally populous grouping size of six was selected. The results of the three Preisendorf er strategies are shown in Figs. 36a through 36c. For this case, the MAXPROB II strategy achieves the highest indepen- dent adjusted VISCAT I threat score of 0.18 (unadjusted, 0.33) as compared to 0.17 (unadjusted, 0.32) for the MAXPROB I strategy and 0.12 (unadjusted, 0.22) for natural regression, The maximum score for each of the three methods was achieved with the addition of the fifth and final predictor. The statistical score trends for the two maximum probability strategies are very similar and reach identical near peak independent VISCAT I threat scores of 0.15 (unadjusted, 0.30) at the first predictor level. This is particularly note- worthy when considering that the computer forecast array size may be of significant operational concern. The poorest strategy for this case is natural regression. The predictors selected for this case are DTDP, SHWRS , ENTR, U850 and DIV925. Fig. 37 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. In this case, only the second predictor strictly meets the requisite functional dependence criteria ensuring no significant dependence of one predictor upon another. The MAXPROB II independent adjusted AO score, which corresponds to the peak independent VISCAT I threat score for this case. 56 is -2.47% (unadjusted, 66.49%) as compared to AO scores of 4.94% (unadjusted, 68.91%) for MAXPROB I and -16.87% (unad- justed, 61.78%) for natural regression. Fig. 38 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. For this case a grouping size of five was selected. The results of the three Preisendorf er strategies are shown in Figs. 39a through 39c. For this case, the MAXPROB II strategy provides the highest independent ad- justed VISCAT I threat score of 0.30 (unadjusted, 0.43). This peak score slightly surpasses the score of 0.29 (unadjusted, 0.42) achieved by the MAXPROB I method. The trends for these two strategies are nearly identical, showing only a slight oscillation in independent threat scores as predictors are added. The peak score achieved by the MAXPROB II scheme is at the fifth predictor level while the peak value for MAXPROB I is obtained with the inclusion of the first predictor. It should be noted that the results at the first predictor level for the two maximum probability strategies are identical. A forecast array predicated upon a one predictor versus five predictor array size requires four orders of magnitude less computer storage resources and is therefore a desirable characteristic for an operational forecast system. Additionally, the independent adjusted AO scores, achieved by both schemes, have identical maximum 57 values of 19.25% (unadjusted, 73.76%) at the first predictor level as compared to a maximum value of 12.76% (unadjusted, 71.47%) for the natural regression strategy at the fifth predictor level. The poorest strategy for this case is natural regression (Fig. 39c). The independent VISCAT I threat scores for this scheme initially yield very low threat score values at the first and second predictor levels with a subsequent rapid rise at the third, fourth and fifth predic- tor levels. This rapid rise however produces a threat score value of only 0.19 (unadjusted, 0.34) and a corresponding AO of -1.65% (unadjusted, 66.76%) at the fifth and final predictor level. The predictors selected for this model are BMD2, UlOOO, ENTR, DVDP and EAIR. Fig. 40 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. For this case, three of the five selected predictors do not meet the 96% confidence interval criteria for functional independence. This further justifies the use of a single predictor forecast array for possible operational use. Figs. 41 and 42 show the contingency table results for the EVAR and QUAD threshold models. The results of these two models are very similar with the EVAR model showing a slight advantage in independent adjusted VISCAT I threat score of 0.15 (unadjusted, 0.33) versus 0.14 (unadjusted, 0.31) for the QUAD model. Similarly, the EVAR model achieves a slightly higher independent adjusted AO of 13.17% 58 (unadjusted, 71.60%) versus 12.76% (unadjusted, 71.47%) for the QUAD model. For the EVAR model, a first-stage regression thres- hold value of 0.577452 was calculated based upon a VISCAT I sample size of 290 observations with a mean of 0.620 and standard deviation of 0.211 and a combined VISCAT II+III sample size of 1197 observations with a mean of 0.860 and standard deviation of 0.153. A second-stage threshold value of 0.548587 separating VISCAT II and VISCAT III was calcu- lated based upon a VISCAT II sample size of 328 with a mean of 0.654 and standard deviation of 0.142 and 971 VISCAT III observations with a mean of 0.777 and standard deviation of 0.136. The first-stage threshold value of 0.572592 for the QUAD model was generated with the above VISCAT I versus VISCAT II+III sample statistics. A second-stage threshold value of 0.548717 was based upon 333 VISCAT II observations with a mean of 0.649 and standard deviation of 0.138. In general, the model which produces the highest independent VISCAT I threat score for this case is the PR+BMD model while the highest independent AO score is achieved with the EVAR threshold model (Table V) . The rela- tively large independent threat score dominates the scores however, and therefore the PR+BMD model is determined to be the optimal model in this case. 59 C. NORTH ATLANTIC OCEAN, AREA 4 Area 4 was selected for evaluation because of its rela- tively low frequency (approximately 3% of the total) of VISCAT I observations. It was hoped that this area would statistically represent a region where there was an insuffi- cient number of VISCAT I observations to allow for study of a forecast region where results were anticipated to be poor, yet enough VISCAT I observations to avoid any "rare event" statistical entanglements. This area encompasses a broad region of the North Atlantic Ocean which is generally to the south of area 2 and east and southeast of area 3W. Area 4's southern border reaches to the northeastern tip of Portugal and extends northward through the English Channel to encompass the southern portion of the North Sea. 1. Area 4, TAU-OQ Fig. 43 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. Several unique characteristics were encountered for this case which had not been previously been observed. The previously observed variation of dependent AO and inde- pendent threat scores, associated with the sequential varia- tion in grouping size from four through ten, was not initially achieved. For this case, non-zero values of dependent AO and independent VISCAT I threat score were only achieved 60 after three iterations of the predictor selection procedure. The grouping size of four was deleted from consideration in the third iteration because the associated AO value, achieved at that predictor level, did not exceed the previous AO value at the second predictor level. For this case, the independent VISCAT I threat scores maintained indentically low values, while a relative peak in AO was achieved at a grouping size of eight. For this reason, eight was selected as the optimal grouping size for this model. Figs. 44a through 44c represent the results of the three Preisendorf er strategies. For each of the schemes, the independent VISCAT I threat scores at the first three predictor levels reveal the near-zero scores encountered in the grouping size selection process. The highest independent adjusted VISCAT I threat score, namely 0.08 (unadjusted, 0.11) is achieved with the MAXPROB II strategy at the fifth and final predictor level. For this model, the MAXPROB I and natural regression strategies yield only slightly inferior, identical independent adjusted threat scores of 0.04 (inad- justed, 0.07) which are achieved at the fifth predictor level. The MAXPROB I strategy yields the highest independent adjusted AO score of -15.77% (unadjusted, 82.45%) as compared to scores of -28.63% (unadjusted, 80.50%) for natural regression and -34.85% (unadjusted, 79.56%) for the MAXPROB II strategy. The predictors selected for this model are V500, DVDP, STRTTH, E500 and ENTR. 61 Fig. 45 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. In this particular case, only the third predictor displays a functional dependence value less than the 9 6% confidence interval value. This renders the threat scores achieved by this model, beyond the first predictor level, statistically not significant, if strict adherence to the basic functional dependence criteria is followed. Fig. 46 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. For this case a grouping size of nine was selected. The results of the three Preisendorfer strategies are shown in Figs. 47a through 47c. Generally, the results for this model differ very little from the previously dis- cussed PR model. This case reflects the first and only occurrence where the Preisendorfer methodology coupled with linear regression equation predictors (PR+BMD model) did not yield superior results to the PR model. The trends for these three strategies are generally quite similar. The MAXPROB II scheme provides the highest independent adjusted VISCAT I threat score of 0.09 (unadjusted, 0.11), as com- pared to scores of 0.08 (unadjusted, 0.11) for MAXPROB I and 0.07 (unadjusted, 0.10) for natural regression. For each of these three strategies, the maximum independent VISCAT I threat score was achieved with the inclusion of the fifth 62 and final predictor. The independent AO scores associated with the peak threat scores are near their lowest values at the fifth predictor level with the MAXPROB I scheme yielding the highest relative independent adjusted AO of -19.09% (unadjusted, 81.95%) followed by natural regression with a score of -26.56% (unadjusted, 80.82%) and lastly MAXPROB II with a score of -39.42% (unadjusted, 78.87%). The predictors selected for this model are BMD2, DUDP, ENTR, DEDP and UIOOO. Fig. 48 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. Generally, the relative difference between the func- tional dependence and 96% functional dependence confidence interval values is much less severe than with the previously discussed model. While only the third predictor's functional dependence value meets the 96% confidence interval criteria for significance, the other predictors are only marginally insignificant. The application of the EVAR and QUAD threshold models to this case presented results which had not been previously encountered. The first-stage VISCAT I versus VISCAT II+III separation calculation results in a QUAD threshold value which is imaginary and an unrealistic EVAR threshold value of 209.588882. These thresholds were calculated based upon a VISCAT I sample size of 85 observations with a mean of -1.012 and standard deviation of 6.280 and a combined VISCAT 63 II+III sample size of 3096 observations with a mean of -1.864 and standard deviation of 7.092. These results are linked to the preponderance of VISCAT III observations in this area coupled with the fact that these employed threshold models are designed to provide for a minimum error when separating samples. These results indicate that a forecast model predicated upon the dependent data set employed in this case would strictly forecast VISCAT III. 2. Area 4, TAU-2 4 Fig. 49 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. This case required three iterations of the four through ten grouping size calculations before any non-zero dependent AO or independent VISCAT I threat score values were achieved. Additionally, for the grouping size of four, no increase in AO was observed at the second predictor level and therefore was deleted from consideration. A grouping size of five was ultimately selected for this model. Figs. 50a through 50c represent the results of the three Preisendorf er strategies. Generally, the independent VISCAT I threat scores yielded for these schemes are poor with the highest independent adjusted VISCAT I threat score of 0.05 (unadjusted, 0.07) being achieved by the MAXPROB II strategy at the fifth predictor level followed by MAXPROB I with a score of 0.02 (unadjusted, 0.05) and natural regression 64 with 0.01 (unadjusted, 0.04). The AO scores corresponding to these values provide for a slightly different scoring hierarchy. The highest independent adjusted AO score, namely 0.94% (unadjusted, 85.64%), is attained by the MAXPROB I strategy as compared to scores of -30.05% (unad- justed, 81.34%) for natural regression and -37.56% (unad- justed, 80.05%) for MAXPROB II. The predictors selected for this model are VRT9 25, DTDP, ENTR, V8 50 and DVRTDP . Fig. 51 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. For this case, each predictor following VRT925 proved to be significantly functionally dependent on its predecessors and therefore only a single predictor forecast array is justifiable for this model. Fig. 52 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. As in the previous case, three iterations of dependent AO and independent VISCAT I calculations were required before any non-zero scores were achieved. Addi- tionally, in this case, the grouping sizes of four and five were deleted from consideration as they did not provide an increase of AO at the second predictor levels. The grouping size ultimately selected for this model was nine. Figs. 53a through 53c show the results of the three Preisendorf er strategies for this model. The scores for 65 this model, as in the previously described case, are quite poor and show very little improvement over the PR model. The highest independent adjusted VISCAT I threat score, namely 0.06 (unadjusted, 0.09), was achieved by the MAXPROB II strategy followed by scores of 0.05 (unadjusted, 0.07) for the MAXPROB I strategy and 0.0 3 (unadjusted, 0.0 6) for natural regression. The corresponding independent adjusted AO scores show a maximum score of -19.25% (unadjusted, 82.71%) for the MAXPROB I strategy followed by scores of -30.05% (unadjusted, 81.14%) for natural regression and -39.91% (unadjusted, 79.71%) for the MAXPROB II strategy. Fig. 54 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. For this case, the relative magnitude of the differ- ence between the actual functional dependence and its 9 6% confidence interval value is quite small. It is only at the second predictor level, that the calculated values do not exceed the corresponding 96% confidence interval value. Fig. 55 shows the contingency table results for the EVAR threshold model. The QUAD model provided an imaginary threshold value at the second regression stage and therefore did not allow completion of the entire separation sequence. This represents the only occurrence where a valid equal variance threshold was calculated but a corresponding quadratic threshold proved to be imaginary. 66 The results of the EVAR model were in keeping with those of the previously described PR and PR+BMD models. An independent adjusted VISCAT I threat score of 0.0 5 (unad- justed, 0.07), was achieved with a corresponding independent adjusted AO value of -13.15% (unadjusted, 83.59%). The first-stage regression separation for this model was based upon a calculated threshold value of 0.908275. Associated with this threshold were 449 VISCAT I observations with a mean of 0.953 and standard deviation of 0.030 and 2489 VISCAT II+III observations with a mean of 0.976 and standard deviation of 0.027. The second-stage VISCAT II versus VISCAT III separation was based upon a calculated threshold value of 0.683569. Associated with this threshold is a VISCAT II sample size of 69 observations with a mean of 0.831 and standard deviation of 0.066 and 887 VISCAT III observations with a mean of 0.912 and standard deviation of 0.078. For the QUAD model, an initial first-stage threshold value of 0.908275 was successfully calculated with the sample statistics addressed above. The second-stage regres- sion attempt was based upon a VISCAT II sample size of 65 observations with a mean of 0.829 and standard deviation of 0.067 and VISCAT III sample size of 853 observations with a mean of 0.905 and standard deviation of 0.079. These sample statistics produced an imaginary threshold value. In general Area 4, TAU-24 is characterized by very poor independent VISCAT I threat scores. This indicates 67 that there is very little skill in forecasting visibility conditions of less than or equal to 2 kilometers in this area. The evaluated models show little variation in scores with the best relative model for this area and forecast projection being the PR+BMD model (Table V) . 3. Area 4, TAU-4 8 Fig. 56 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR model. As in the TAU-00 and TAU-24 forecast projections for this area, the calculation and evaluation of dependent AO and independent VISCAT I threat scores had to be run through three iterations before any non-zero statistics were obtained. The grouping sizes of four and five were deleted from con- sideration because the addition of predictors at those grouping sizes did not provide for any increase in AO scores. Based on an evaluation of the results as shown on Fig. 56, a group- ing size of seven was selected. Figs. 57a through 57c show the results of the three Preisendorf er strategies for this model. In general, the near-zero statistical scores encountered in the grouping selection process, can be seen through the third predictor level, along with a noticeable increase in scores at the fourth and fifth predictor level. The MAXPROB I strategy yields the highest independent adjusted VISCAT I threat score, namely 0.18 (unadjusted, 0.20), for this model 68 followed by a natural regression score of 0.16 (unadjusted, 0.19) and lastly by MAXPROB II with a score of 0.13 (unad- justed, 0.16). This is the first and only encountered case where the natural regression strategy effectively achieved a maximum independent VISCAT I threat score which is higher than either of the two maximum probability strategies. The independent AO scores associated with these peak independent VISCAT I threat scores, adhere to this same scoring sequence, with MAXPROB I achieving a value of -14.94% (unadjusted, 81.86%) followed by natural regression with a score of -27.80% (unadjusted, 79.83%) and MAXPROB II with an indepen- dent adjusted AO of -43.98% (unadjusted, 77.78%). Fig. 58 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. In this case, only the third predictor's functional dependence value falls below the associated 96% confidence interval value. The predictors selected for this model are VRT925, DVRTDP, ENTR, DUDP and RH. Fig. 59 shows the relationship of equally populous grouping size to the adjusted AO (dependent data) and the adjusted VISCAT I threat score (independent data) for the PR+BMD model. As in the previous area 4 cases, three com- plete iterations of the four through ten grouping size calcu- lations had to be performed before any non-zero dependent AO or independent VISCAT I values were achieved. The grouping size ultimately selected for this model was nine. 69 Figs. 60a through 60c represent the results of the three Preisendorfer strategies. A unique result of this model is that for the first time, the PR+BMD model did not achieve independent VISCAT I threat scores which exceed those achieved by the PR model. The peak independent adjusted VISCAT I threat score of 0.17 (unadjusted, 0.20) is achieved by the MAXPROB II strategy at the third predictor level. The predictors selected for this model are BMDl, DDVDP, DUDP, ENTR and PRECIP. Fig. 61 shows the functional dependence, AO/Al sta- tistics and 96%/05% confidence interval values for this model. Only the second predictor sufficiently meets the 9 6% confidence interval significance criteria. Based upon a strict adherence to the preselected 96% confidence interval significance requirements, these functional dependence values provide cause for uncertainty in the representative- ness of the scores achieved after the second predictor level. Figs. 62 and 63 show the contingency table results for the EVAR and QUAD threshold models. The results of these models are very similar and generally quite poor. The QUAD model achieves the highest relative independent adjusted VISCAT I threat score of 0.01 (unadjusted, 0.04) as compared to a score of -0.01 (unadjusted, 0.02) for the EVAR model. The QUAD model similarly achieves the highest independent adjusted AO score of -2.07% (unadjusted, 83.89%) versus a score of -7.88% (unadjusted, 82.97%) for the EVAR model. 70 For the EVAR model, a first-stage regression thres- hold value of 0.847203 was calculated based upon a VISCAT I sample size of 109 observations with a mean of 0.918 and standard deviation of 0.052 and 2947 VISCAT II+III observa- tions with a mean of 0.967 and standard deviation of 0.037. The second-stage VISCAT II versus VISCAT III separation was based upon a calculated threshold value of 0.629338. Asso- ciated with this threshold is 495 VISCAT II observations with mean of 0.861 and standard deviation of 0.103. For the QUAD model, a first-stage separation thres- hold value of 0.847203 was calculated upon the associated sample statistics addressed above. A second-stage threshold value of 0.613739 was calculated based upon 481 VISCAT II observations with a mean of 0.770 and standard deviation of 0.089 and 2522 VISCAT III observations with a mean of 0.862 and standard deviation of 0.100. The overall results associated with the area 4, TAU- 48 case are particularly unique. The independent adjusted TAU-48 VISCAT I threat score represents the highest area 4 independent VISCAT I threat score (by a minimum of 0.09) achieved, as compared to TAU-0 0 or TAU-24. The maximum independent VISCAT I threat score is achieved by the PR model. Following the completion of the testing and evaluation of the FATJUNE 19 83 data set, a series of preliminary experi- ments were performed with the May 15 to June 23 1984 data 71 set for the TAU-24 model forecast projection. These experi- ments consisted of evaluating the TAU-24, 1983 forecast arrays and equations (generated with FATJUNE 19 83 data) with training and testing cases of TAU-24, 1984 data. In performing this evaluation, the 1984 data set was divided into "dependent" and "independent" portions. This data separation is a function of the specific mechanics of the computer programs utilized in this study and is not associated with the generation of additional forecast arrays or equations Two homogeneous areas were evaluated, namely area 2 and area 3W. This essentially provided an independent verification of the utility of the 1983 forecast arrays and equations in predicting observed 1984 visibility in these areas. In general, the skill and contingency table results for these experiments compare very favorably to those achieved with the FATJUNE 19 83 data. A summary of the results of each of the evaluated models is provided in Table VII. For area 2, a peak independent adjusted VISCAT I threat score (1984 data), namely 0.27 (0.33 unadjusted), was achieved with each of the two threshold models . This compares to a peak independent adjusted VISCAT I threat score (1983 data) of 0.29 (0.34 unadjusted) achieved by the same two models. For area 3W, a peak independent adjusted threat score (1984 data) of 0.28 (unadjusted, 0.42) similarly compares to a peak independent adjusted threat score (1983) of 0.13 (unadjusted, 0.36). 72 The overall results of the 1984 data experiments can be seen in Table VII and are represented by Fig. 64 which illustrates the results of the PR+BMD model, MAXPROB II strategy, for area 2, TAU-24. A review of the results associated with area 4 for TAU-00, TAU-24 and TAU-48 indicates that none of the models evaluated achieved very encouraging skill and threat scores. Consequently, the maximum-likelihood-of -detection criteria (MLDC) was proposed as an alternative technique to increase threat scores in area 4. A series of experiments involving an arbitrarily selected population of two hundred normally distributed events, partitioned into eight separate threat/non- threat samples, were performed to demonstrate the theoretical utility of the MLDC at low threat frequencies. Threshold values were calculated, for various threat frequencies, using the EVAR minimum probable error and MLDC techniques and two by two contingency tables were constructed to tabulate the asso- ciated threat score, percent correct and false alarm rate results. Fig. 65 shows the resulting plot of threat score versus threat frequency which illustrates the amount of in- crease in threat score associated with the MLDC model. Asso- ciated with these higher threat scores are correspondingly higher "costs," namely higher false alarm rates, illustrated in Fig. 66, and lower percent correct scores, illustrated in Fig. 67. 73 A set of two experiments was performed, utilizing FATJUNE 1983, TAU-24 data and the two-stage separation scheme outlined in Chapter IV (MLDC model) , to evaluate the relative performance of the MLDC and EVAR models on area 4. In general, the results of these two experiments were consis- tent with the results predicted by the aforementioned theoretical experiments. The most obvious area of agreement is the significantly lower independent adjusted VISCAT I and VISCAT II threat scores (both are considered threatening events in this study), namely 0.01 (unadjusted, 0.04) and -0.14 (unadjusted, 0.00), achieved by the EVAR model. Fig. 68, as compared to the corresponding scores of 0.04 (unad- justed, 0.07) and 0.03 (unadjusted, 0.15) achieved by the MLDC model. Fig. 69. 74 VI. CONCLUSIONS AND RECOMMENDATIONS A. CONCLUSIONS The primary objective of this study was to expand upon the initial research and experimentation presented by Karl (1984) and to propose a viable statistical forecasting scheme suitable for eventual employment in an operational U.S. Navy marine visibility MOS forecasting system. In general, while the results of linear regression and the evaluated Preisendorf er models are roughly comparable, it has been shown that two specific statistical approaches, namely the PR+BMD model's MAXPROB II strategy and the linear regression models, yield the best results (as measured by independent VISCAT I threat score) achieved in this study. The PR+BMD model achieved the best results for six of the eight evaluated cases: area 2, TAU-48; area 3W, TAU-24 and TAU-48; and area 4, TAU-00, TAU-24 and TAU-48. The nearly identical results of both the equal variance and quadratic linear regression threshold models provided the best skill and threat scores for area 2, TAU-00 and TAU-24. A common characteristic of each of the evaluated cases is that the predictability of visibility category II is relatively very poor and nearly always poorer than that for visibility categories I or III. This pattern affirms the findings of similar Pacific Ocean visibility studies [Renard and Thompson, 1984] as well as 75 t±iose documented by Karl (19 84) and further supports Karl's recommendation to change from a three-category to a two- category visibility forecasting scheme. An evaluation of the overall results of this study shows that no real connection between individual model/strategy and either the homogeneous oceanic area (2, 3W and 4) or model output time (TAU-00, TAU-24 and TAU-48) can be made. The linear regression threshold models performed best for area 2, the intermediate poor visibility oceanic area, while the Preisendorf er methodology incorporating linear regression equation predictors proved the best in the evaluated homogene- ous areas with the greatest and lowest relative concentration of poor visibility observations. The trend of visibility category I skill and threat scores, for each homogeneous area and model output time, seems to contradict the preliminary supposition that peak skill scores would be associated with the area containing the greatest frequency of poor visiblity observations and the TAU-00 model output time. This result is most apparent with area 4, where threat scores increase with increasing model forecast projections until they achieve values at TAU-48, which are nearly identical to those for the other two homogeneous areas. This type of trend in skill and threat scores most likely reflects the overall strength of the statistical relationships for the predictand/predictors involved irrespective of the frequency of specific visibility observations . 76 In several cases, the maximum independent visiblity category I threat score achieved by the PR+BMD model was reached at the first predictor level. In several additional cases, threat score values which were only marginally lower than peak value, were similarly achieved at the first pre- dictor level. Forecasting arrays involving only one predic- tor drastically reduce required computer storage and consequently such arrays are a desirable attribute to any operational MOS forecasting system. A MOS-type forecasting system predicated upon such a small number of predictors would prove extremely beneficial in an independent single station forecasting scenario such as that experienced by an aircraft carrier based U.S. Navy Oceanography Officer. The concept and practical employment of functional dependence, associated with the Preisendorf er methodology, provides a greater restriction on the statistical significance of the skill and threat score results achieved in this study, as compared to that which was previously experienced by Karl (19 84). It was shown that the calculated functional dependence values for each respective predictor or group of predictors often exceeded the associated 96% confidence interval value at the first or second predictor level and rarely met the requirements for significance for an entire array of selected predictors. This restriction further indicates that any operational forecasting scheme should most likely be composed of only a minimal number of select predictors . 77 The difference between the equal variance and quadratic threshold models was shown to be very minimal. The two-stage visibility category separation approach is designed to handle cases with distinct separability between categories while providing for minimal error in the calculated threshold values. This condition was not met in the area 4, TAU-00 and TAU-24 cases and subsequently lead to unrealistic thres- hold values. The preliminary independent evaluation of the 15 May to 23 June 1984 data set provided a crucial test and verifica- tion of the utility of the forecast arrays and equations presented in this study. The introduction and initial evaluation of the maximum- likelihood-of-detection threshold model offers another technique to the pool of visibility prediction schemes. This method appears to be most beneficial in areas of low threat frequency. B. RECOMMENDATIONS The following recommendations are offered to future researchers : 1. Remove the MAXPROB I and natural regression strategies of the Preisendorf er methodology from further consideration in the forecasting of marine horizontal visibility. 2. Delete one of the two threshold models evaluated in this study, and investigate additional thresholding 78 techniques based on the Beta distribution and the maximum- likelihood-of-detection criteria . 3. Revise the current three-category visibility scheme to a two-category scheme where visibility categories I and II are combined. This should be particularly beneficial in those homogeneous areas with extremely low frequencies of visibility category I and II observations. 4. Expand the initial set of potential predictors to include air-sea temperature differences as well as additional derived predictors such as the advections and gradients of temperature, vorticity and moisture in order to more fully simulate the physical processes associated with poor marine visibility. Additionally, include TAU-00 and TAU-24 model output parameter fields as potential predictors in future evaluations of TAU-24 and TAU-48 MOS forecasts. 5. Evaluate OOOOGMT data sets to determine the effect of nighttime conditions on both visibility observations and NOGAPS model output parameters. 6. Investigate new procedures to determine the number of equally populous predictor intervals. The following procedure [Preisendorf er , 1984] is proposed: a. To establish the number of equally populous predic- tor intervals for any predictor, consider a bivariate predictand/predictor [Preisendorf er , 1983a]. Start with m = 1 and find the potential predictability (PP) for the resultant plot, call it "PP(1)." In general, PP(m) is the 79 PP for the general case of m. Successively, find PP(m) for m = 1,2,..., and continue to subidivde the predictor range as long as PP increases: PP(m) < PP(m+l). Stop at PP(m) if PP(m+l) £ PP(m) or if PP(m+l) < PP(m+ll96), where the later is defined by Preisendorf er (1983a) and denoted by "PP(96)." This last condition avoids sparse bivariate data plots, caused by too large an m. It was Karl's 1984 experi- ence that five to eight equally populous predictor intervals are sufficient for all predictors. Hence m, for each pre- dictor, is expected to be in this neighborhood. b. Order the set of available predictors in descending value of potential predictability (PP) . Break ties with AO (PP and AO are defined by Preisendorf er (1983a)). AO is the actual skill, after the prediction has been made. c. The first predictor is that with the greatest PP. Compute associated AO and Al. Call them AO ^ and Al ^ . 7. Associated with recommendation 6. above, improve the predictor selection procedure as follows : a. Suppose k-1 predictors have been chosen, let them be X, ,...,X,_,. Let Y be a new predictor candidate. Admit Y as the kth predictor if the three following conditions are satisfied: (1) Functional dependence (YJX.) < functional dependence (y|x.;05) for i = l,...,k-l (i.e., the functional dependence of X. and Y is not significantly large for each 80 i = l,...,k-l. Find functional dependence (yIx.) and functional dependence (y|x.;05) as described by Preisendorf er (1983c)). (2) AO^^) > A0(^-^) and Al^^^^ > Al^^'^^ (3) AO^^^ > A0(96) and Al^^^ > Al(05) All three conditions must hold for admittance of Y to the predictor set. b. A less stringent predictor selection process would be to form functional dependence (Y.lx.)/ where X., i = l,...,k-l are the selected predictors, and the Y., j = l,...,q are the as-yet unselected predictors. Here (k-l)+q = p, the original number of potential predictors. Next, form |min|max functional dependence (Y. |x.) | ]. This J i -" ^ fixes that Y. for which functional dependence (Y. Ix.) is the least possible of the maximum functional dependence values over the present X set. This makes the best out of the worst case of functional dependence (i.e., select the Y. farthest from the set of X.). c. Continue to repeat step 7a. above until all potential predictors are used up (the critical values of A0(96) and Al(05) are as defined by Preisendorf er (1983b). Another reason for stopping may be that allotted CPU time is used up before the predictors. 8. Investigate a further and more complete verification of the forecast arrays and equations presented in this study 81 utilizing available 1984 data sets. Specifically, utilize the 1984 data set as an entire independent test case without first removing a portion of the data for use as a dependent forecast array /equation training set. Additionally, gener- ate an additional set of forecast arrays and equations based on combined 1983 and 1984 FATJUNE data and evaluate the statistical stability of the equations as different years of data are merged into a larger data base. 82 APPENDIX A LINEAR REGRESSION AND THRESHOLD MODELS A. LINEAR REGRESSION The linear regression techniques used in this study expand upon and slightly modify those first presented by Karl (1984). In this study, two separate least-squares, multiple linear regression software programs; referred to as the BMDP2R — Stepwise Regression and BMDP9R — All Possible Subsets Regression computer programs in the BMDP Statistical Software [University of California, 1983], were used. The independent variable selection procedure employed in the BMDP2R program is referred to as a forward, step-wise selection process where predictors are selected from a large group of available potential independent variables based upon the highest correlation with the dependent predictand (visibility in this study) . This correlation is calculated based upon certain F-to-enter and F-to-remove limits, where F is a ratio which tests the significance of the coefficients of the predictors in the regression equation. The regression model fitted to the data is y = a + b,x, + b„x„ + . . . + b x + e ^ 112 2 n n where : 83 y = the dependent variable (predictand) which can be either a continuous function or a discrete value X, / . . . ,x = the independent variables (predictors) b, , . . . ,b = the regression coefficients In ^ a = the intercept p = the number of independent variables £ = the error with mean zero The predicted value y, and the general form of the resulting equation, is y = a + b,x, + b_x^ + ... + b x -^ 112 2 n n The step-wise selection of predictors continues until there are no predictors remaining which meet the requisite F-to- enter criteria. The regression equation generated by the BMDP2R program is outputted at each regression step where variables are selected as independent predictors, along with its corresponding R value (the correlation of dependent 2 variable y with the predicted value y) and R value. The resulting equation sets are reviewed, and that equation con- 2 taining only those predictors which increased R by at least 0.01 are retained for application. The procedure employed with the BMDP9R program varies from that of the BMDP2R, in that a "best" possible subset, derived independently of variable or variable sequence, is calculated from the group of potential predictors. Once this 84 t "best" subset is identified, a linear regression equation is fitted to the data, based only upon those selected predic- tors, in a fashion identical to that for the BMDP2R program. The "best" possible subset is calculated by a Furnville- Wilson algorithm which provides the user with a variety of subordinate subsets in addition to the "best" subset. Three criteria are available to define the "best" possible subset as a function of independent variables (predictors) and a 2 dependent variable (predictand) : the sample R , the adjusted 2 R and Mallow's Cp . For this study, the Mallow's Cp criteria is defined as: Cp = RSS/S - (N - 2P' ) where: RSS = the residual sum of the squares for the new subset being tested S = the residual mean square based on the linear regression using all independent variables P' = the number of variables in each subset N = the total number of cases For this method, "best" is defined as the smallest Cp value. Independent variable selection for the BMDP9R program begins with a general screening of the entire set of potential predictors. Variables which are identified as redundant. 85 linear combinations of other variables, with respect to the predictand, in this general screening are deleted from further consideration. The t statistics for the coefficients which minimize the Cp value for each reviewed subset identifies the "best" subset. The number of predictors assigned to each subset can be predefined and for this study each subset equation was required to have six predictors. The role of regression, once appropriate predictor varia- bles have been selected, is simply that of dimension reduction (representing a multivariate structure by a univariate proxy which constitutes a classif icatory or predictive index). This proxy takes the form of a polynomial, linear in its coefficients, of the components of the multivariate structure. The problem now becomes one of determining the form of the state conditional distributions (one for each group of interest; e.g., one, two and three for visibility categories I, II and III, as used in this study) . Once an appropriate form has been selected, it remains, then, to determine the parameters of the class conditional distribu- tions (e.g., means and variances) and then apply an appro- priate decision criterion or threshold model. B. THRESHOLDS [Lowe, 19 84a] 1 . Notation E = an event; this is an indicator variable which when E = 1, the threatening event occurs, and when E = 0, the non- threatening event occurs. 86 C = the classification of an unknown event which when C = 1, the event is classified as a threat, and when C = 0, the event is classi- fied as a non-threat. P[E=1] E unconditional probability of occurrence of threat . P[E=0] E unconditional probability of occurrence of non- threat . Error of the 1st kind (false alarm) [C=lnE=0]. Error of the 2nd kind (miss) [C = 0 n E = 1] . P[C=lnE=0] = joint probability of an error of the 1st kind . P[C=OnE=l] E joint probability of an error of the 2nd kind. P[C=1|e=0] = class conditional probability of misclassi- fying a non-threat. P[C=0|E=1] e class conditional probability of misclassi- fying a threat.. P[C=lnE=0] = P[C=1Ie=0] P[E=0]. P[C=OnE=l] = P[C=0|E=1] P[E=0]. z = a value of the predictive index (equivalent to y , above) . Z = range of the predictive index on the real line For a dichotomous problem, Z is into two parts Z , Z , C = 0 if z t Z 1 if z e Z The decision regions are mutually exclusive and exhaustive (i.e., Z nZ =0 and Z = Z uZ.,). 87 Thresholds = boundary (s) between decision regions. P(z|e=0) e class conditional density of z given that E = 0. p(z|E=l) = class conditional density of z given that E = 1. A(z) = p(z|e=1)/p(z|e=0) = the maximum likelihood ratio (i.e., the ratio of class conditional densities) . p = p{[C=lnE=0] u [C=OnE=l]} = the total probability of error. 2 . Minimum Probability of Error Criterion p = probability of an incorrect classification. p = p[C=l|E=0] p[E=0] + p[C=0|E=l] p[E=l] where p[E=l] + p[E=0] = 1. NOte that the events E = 1 and E = 0 are mutually exclusive and exhaustive. The objec- tive is to select decision regions (thresholds) so as to minimize p . ^e p[C=0|E=l] = / p(z|E=l)dz = the probability of ZeZQ misclassif ying E = 1. p[C=OlE=l] = / p(z|E=l)dz + / p(z|E=l)dz ZeZ_^ ZeZ, - / p(z |E = l)dz p[C=0|E=l] = 1 - / p(z|E=l)dz these are zeZ, substituted into the expression p(C=llE=0] = / p(z|E=0)dz for p ZeZ, ^ 88 then, p = p[E=0] / p(z|E=0)dz + p[E=l)[l - / p(z|E=l)dz] ^ ZeZ Z£Z and algebraic rearrangement yields. p = p[E =1] - / {p[E=0] p(z|E=0) - p[E=l) p(z|E=l) dz} ® ZeZj^ In order to minimize p , Z, (the decision region for C = 1) el will include all those values of z for which the integrand in the expression for p will be negative. The decision regions can be symbolically represented as follows: Zq = {z: p[E=0] p(z|E=0) - p[E=l] p(z|E=l) > 0} Z, = {z: p[E=0] p(z|E=0) - p[E=l] p(z|E=l) < 0} An alternative representation is given by, Zq = {z: p[E=0] p(z|E=0) > p[E=l] p(z|E=l)} = (z: p[E = 0]/p[E = 1] > p(z|E = l)/p(z|E =0) } Likewise , Z^ = {z: p[E = 0]/p[E = 1] < p(z |E = l)/p(z|E = 0) } 89 These statements can be combined to give, c=l p(z|E =l)/p(z|E =0) = A(z) ^ p[E =0]/p[E =1] c=0 Thresholds are the value (s) of z for which A(z) = p[E =0]/p[E =1] This equation can be solved for z either analytically or numerically depending on the forms of the density functions 3 . Threshold Cases In order to examplify the model, the assumption is made that the class conditional distributions are Gaussian. There are essentially three distinct cases that can arise. a. Case I: Equal variances; different means (Referred to as the equal variance model (EVAR) in the text) p(z|E=l) = k exp{ (-1/2) (z -y^)^/a^} p(z|E=0) = k exp{(-l/2) (z -Uq)^/o^} where: ,„ ^-1/2 -1 k = (2it) ' a 2 2 c=l exp{ (-1/2) (z -y, ) /a } p^ A(z) = ^ J- -^ exp{(-l/2) (z -Mq) Va"^} ^^q ^1 90 where A is the likelihood ratio and p = p[E =0] and p = p[E =1] . Thus, the threshold value is ;* = (yQ+u^)/2 + o In {p^/p^} / {^-^ - \1q) Classification index (z) The position of the threshold depends on the relative values of p, and p^ . The threshold moves toward the group with the smallest p.. If p, = p- the threshold will be the value of z where the densities intersect (i.e., where the densities are equal) . b. Case II: Equal means; different variances A(z) 2 2 c=l a^expi {-1/2) {z - u^) /o^} ^ p 0 o^exp{{-l/2) {z -Uq) /o^} ^^Q ^1 with the threshold Z* = ± 0 2 2 . 2 2. (o^-Qq) In Pl^O n 1/2 91 Note that in this situation there are two thresholds. The group having the smaller variance will lie between the two thresholds . E= 1 Classification index (z) The thresholds shown are typical of a situation where p-, < p Note that these thresholds lie between the two intersections of the densities. If the inequality of prior probabilities were reversed, the thresholds would lie outside of the region between the two density intersections. Further note that the decision region for the group having the lesser variance lies between the thresholds. c. Case III: General Solution (Referred to as the Quadratic Model (QUAD) in the text) 0 p(z|E=l) = k/o^ exp{ (-1/2) (z - p^)^/o^} p(z|E=0) = k/o^ exp{(-l/2) (z - p )^/a^} 0 92 z - y 0x2 z - y A(z) = exp{l/2 I (— ^)'- - ( •^0 i)2 c=l c=0 -1/2 where k = (27t) . Algebraic manipulation produces 2 2 2 2 2 2 2 2 2 2 2 + [ (o-^Pq - a^y^) - 20^0-^ In (PqO^/p-^Oq) ] c=l > < c=0 which is recognizable as a quadratic equation in z where 2 1/2 z* = -b ± (b - 4ac) /^/2a a = 2 2 ^1 - ^0 b = 2(aQy^ - al^Q 2 2 2 2 2 2 c = (0;l^o ~ ^o'^i^ ~ ^^1^0 "^^ ^Pq'^i^Pi^o 93 _ r, ^=' r\ ^' >> c o Q E = 0 1 \ \ "^ ^^ ^2 , Classification index (z) The remarks given for the figures in cases I and II are also applicable here. More often than not, only one of a pair of thresholds induced by differing variances will be of real interest. If the variances of the two groups are radically different, then both members of the threshold pair become important. 4 . Th e-Maximum- Like lihood-of -Detect ion Criteria For this specific model the following background is provided : event space: 2' mutually exclusive populations 7T-, 7T, forecast decision space: 2 possible forecasts ^0' ^1 d„ is a correct forecast if tt actually occurs d-j is a correct forecast if tt., actually occurs Problem: select the decision rule d(z) which maps the observation space Z into some forecast space in some optimal manner. 94 Z may be an observed variable or it may be an univariate index derived from a number of variables. For this two decision problem, Z is partitioned into two parts, Z^ and Z, . d(z) = d^ if z e Zq d(z) = d, if z fc Z, where Z^ n Z, = 0 and Z^ u Z^ = Z 0 1 0 1 The maximum-likelihood-of-detection criteria repre- sents the simplest decision model. The basic involves selecting the forecast (decision) corresponding to the obser- vation (signal) which is the most likely symptom of the event subsequently observed. Consider the following example: problem: diagnose disease A or disease B. The observed symptoms occur with probability 0.75 for A and 0.1 for B. By the maximum-likelihood-of-detection criteria (MLDC) , diagnose disease A because A is the most likely cause of the observed symptoms (if there is no more information) . But if we know that A is rare and B is common, the above decision may not be optimal and MLDC may not be appropriate. MLDC requires only that we know the event conditional probability density functions of the observations. That is: 95 p(z I tTq) and p ( z | tt , ) decision rule: d(z) d^ if p(z|7T^) > pCzItIq) d^ if p(z|tt^) < p(z|7Tq) In the following development the Gaussian density is used to exemplify the model. -o "" z P(z|tt ) = l//2^ exp{-l/2( ^) ^ } 0 z - z 1. 2 P(z|tt^) = l//27TOj_ exp{-l/2( -) ^ ] P(Z I 7T ) definition: likelihood ratio A(z) = ^ P(zhQ) for convention sake we assume z, > z , 2.2 ^1 > ^0 96 2 ^ 2 ^1 ' ^0 * note the class having the largest variance has a bifurcated decision region. In the case where the variances are equal, the situation simplifies considerably. 2 — — 2—2—2 2o (2^ - Zq) - a (Zj_ - z ) > 0 0 2z ^ (^1+^0^ < 2 z 97 ^0 = °1 ^0 "- /^i It is obvious that z* is simply the average of the means of the class-conditional distributions and is found at the intersections of the two density curves. In the foregoing, normal class conditional distribu- tions were assumed. This was done because the Gaussian form admits of a rather clean analytical solution. However, the general concept of the minimum probable error decision criteria may be applied to any form of density function. Indeed, the density function of one group need not even be the same form as that for another group (one might be exponen- tial and the other Gaussian) . The difficulty with most non- Gaussian forms is that they seldom admit of closed analytical forms and require numerical means in determination of thresholds . 98 APPENDIX B NQGAPS PREDICTOR PARAMETERS AVAILABLE FOR NORTH ATLANTIC OCEAN EXPERIMENTS I. Area: Entire North Atlantic Ocean and Mediterranean Sea Model output time: 1200GMT (TAU-0 0) 15 May — 7 July 1983 A. Model output Descriptive name of parameter parameter DIOOO 1000 mb geopotential height D925 925 mb geopotential height D850 850 mb geopotential height D700 * 700 mb geopotential height D500 500 mb geopotential height D400 * 400 mb geopotential height D300 * * 300 mb geopotential height D250 * 250 mb geopotential height TAIR Surface air temperature TIOOO 1000 mb temperature T925 925 mb temperature T700 * 700 mb temperature T500 500 mb temperature T400 * 400 mb temperature T300 * 300 mb temperature T250 * 250 mb temperature EAIR Surface vapor pressure ElOOO 1000 mb vapor pressure E925 925 mb vapor pressure E850 850 mb vapor pressure E700 * 700 mb vapor pressure E500 500 mb vapor pressure UBLW Boundary layer zonal wind component 99 UIOOO U925 U850 U700 * U500 U400 U300 U250 VBLW VIOOO V9 2 5 V850 V700 V500 V400 V300 V250 * VOR9 25 ** VOR50 0 ** PS SMF PBLD STRTFQ STRTTH SHF ENTRN DRAG * * 1000 mb zonal wind component 925 mb zonal wind component 850 mb zonal wind component 700 mb zonal wind component 500 mb zonal wind component 400 mb zonal wind component 300 mb zonal wind component 250 mb zonal wind component Boundary layer meridional wind component 1000 mb meridional wind component 925 mb meridional wind component 850 mb meridional wind component 700 mb meridional wind component 500 mb meridional wind component 400 mb meridional wind component 300 mb meridional wind component 250 mb meridional wind component 925 mb vorticity 500 mb vorticity Surface pressure Surface moisture flux Planetary boundary-layer depth Percent stratus frequency Stratus thickness Surface heat flux Entrainment at top of marine boundary- layer Drag coefficient (Cp^) B. Derived parameters DTDP DEDP DUDP Vertical gradient of temperature (1000-925 mbs) Vertical gradient of vapor pressure (1000-850 mbs) Vertical gradient of zonal wind (1000-850 mbs) 100 DVDP RH TV DDVDP DVRTDP ESUM EPRD Vertical gradient of meridional wind (1000-850 mbs) Surface relative humidity Virtual temperature Vertical gradient of geopotential height (1000-850 mbs) Vertical gradient of vorticity (500-925 mbs) Sum of vapor pressures (1000-850 mbs) Product of vapor pressures (1000-850 mbs) II. Area: Entire North Atlantic Ocean and Mediterranean Sea Model forecast projection: 1200GMT (TAU-24) 15 May — 7 July 1983 A. Model output parameter DIOOO D925 D850 D700 D500 D400 D300 D250 TAIR TIOOO T925 T700 T500 T400 T300 T250 EAIR Descriptive name of parameter 1000 mb geopotential height 925 mb geopotential height 800 mb geopotential height 700 mb geopotential height 500 mb geopotential height 400 mb geopotential height 300 mb geopotential height 250 mb geopotential height Surface air temperature 1000 mb temperature 925 mb temperature 700 mb temperature 500 mb temperature 400 mb temperature 300 mb temperature 250 mb temperature Surface vapor pressure 101 ElOOO E925 E850 E700 * E500 UBLW UIOOO U9 2 5 U700 * U500 U400 * U300 * U250 * VBLW VIOOO V9 2 5 V850 V700 V500 V400 V300 V250 VOR9 2 5 VOR500 PS SMF PBLD STRTFQ STRTTH SHE ENTRN DRAG PRECIP 1000 mb vapor pressure 925 mb vapor pressure 850 mb vapor pressure 700 mb vapor pressure 500 mb vapor pressure Boundary layer zonal wind component 1000 mb zonal wind component 925 mb zonal wind component 700 mb zonal wind component 500 mb zonal wind component 400 mb zonal wind component 300 mb zonal wind component 250 mb zonal wind component Boundary layer meridional wind component 1000 mb meridional wind component 925 mb meridional wind component 850 mb meridional wind component 700 mb meridional wind component 500 mb meridional wind component 400 mb meridional wind component 300 mb meridional wind component 250 mb meridional wind component 925 mb vorticity 500 mb vorticity Surface pressure Surface moisture flux Planetary boundary-layer depth Percent stratus frequency Stratus thickness Surface heat flux Entrainment at top of marine boundary- layer Drag coefficient (C^^) Total amount (mm. ) of model precipitation in the last six hours 102 SHWRS Total amount (mm. ) of model precipita- tion associated with cumulus convection in the last six hours INSTAB Boundary layer inversion instability DIV925 925 mb Divergence B. Derived parameters DTDP Vertical gradient of temperature (1000-925 mbs) DEDP Vertical gradient of vapor pressure (1000-850 mbs) DUDP Vertical gradient of zonal wind (1000-850 mbs) DVDP Vertical gradient of meridional wind (1000-850 mbs) RH Surface relative humidity TV Virtual temperature DDVDP Vertical gradient of geopotential height (1000-850 mbs) DVRTDP Vertical gradient of vorticity (500-925 mbs) ESUM Sum of vapor pressures (1000-850 mbs) EPRD Product of vapor pressures (1000-850 mbs) III. Area: Entire North Atlantic Ocean and Mediterranean Sea Model forecast projection: 1200GMT (TAU-48) 15 May — 9 July 1983 A. Model output Descriptive name of parameter parameter DlOOO 1000 mb geopotential height D925 925 mb geopotential height D850 850 mb geopotential height D700 * 700 mb geopotential height D500 500 mb geopotential height D400 * 400 mb geopotential height 103 D300 * D250 * TAIR TIOOO T925 T700 * T500 T400 T300 T250 EAIR ElOOO E925 E850 E700 * E500 UBLW UIOOO U925 U850 U700 * U500 U400 U300 U250 VBLW VIOOO V9 2 5 V850 V700 V500 V400 V300 V250 300 mb geopotential height 250 mb geopotential height Surface air temperature 1000 mb temperature 925 mb temperature 7 00 mb temperature 500 mb temperature 400 mb temperature 300 mb temperature 250 mb temperature Surface vapor pressure 1000 mb vapor pressure 92 5 mb vapor pressure 850 mb vapor pressure 700 mb vapor pressure 500 mb vapor pressure Boundary layer zonal wind component 1000 mb zonal wind component 925 mb zonal wind component 850 mb zonal wind component 700 mb zonal wind component 500 mb zonal wind component 400 mb zonal wind component 300 mb zonal wind component 250 mb zonal wind component Boundary layer meridional wind component 1000 mb meridional wind component 925 mb meridional wind component 850 mb meridional wind component 700 mb m.eridional wind component 500 mb meridional wind component 400 mb meridional wind component 300 mb meridional wind component 250 mb meridional wind component 104 VOR 925 VOR500 PS SMF PBLD STRTFQ STRTTH SHF ENTRN DRAG PRECIP SHWRS INSTAB DIV9 2 5 925 mb vorticity 500 mb vorticity Surface pressure Surface moisture flux Planetary boundary-layer depth Percent stratus frequency Stratus thickness Surface heat flux Entrainment at top of marine boundary- layer Drag coefficient (C^^) Total amount (mm. ) of model precipitation in the last six hours Total amount (mm. ) of model precipitation associated with cumulus convection in the last six hours Boundary layer inversion instability 925 mb Divergence B. Derived parameters DTDP DEDP DUDP DVDP RH TV DDVDP DVRTDP ESUM EPRD Vertical gradient of temperature (1000-925 mbs) Vertical, gradient of vapor pressure (1000-850 mbs) Vertical gradient of zonal wind (1000-850 mbs) Vertical gradient of meridional wind (1000-850 mbs) Surface relative humidity Virtual temperature Vertical gradient of geopotential height (1000-850 mbs) Vertical gradient of vorticity (500-925 mbs) Sum of vapor pressures (1000-850 mbs) Product of vapor pressures (1000-850 mbs) 105 * Parameters which were not used due to their being considered as physically unimportant in forecasting marine visibility. ** Parameters which were not used due to loss of significant digits during transfer from tape to mass storage. 106 APPENDIX C SKILL AND THREAT SCORES, DEFINITIONS [Karl, 1984] :;3 R S T < o LU 2 U V W O ^ 1 X Y Z 1 2 3 OBSERVED Total = R+S+T+U+V+W+X+Y+Z PI = (R+U+X) /Total P3 = (T+W+Z ) /Total P2 = (S+V+Y)/Total PN = greatest of Pi, P2 or P3 Raw Scores AO = % correct = (X+V+T) /Total Al = one-class error (U+S+Y+W) /Total TSl = Threat score for visibility category I = X/(R+U+X+Y+Z) TS2 = Threat score for visibility category II = V/(S+V+Y+U+W) TS12 = Threat score for visibility categories I and II (X+V)/(Total-T) 107 TS12 is designed to represent the skill of forecasting visibility categories I and II as separate categories, rather than their skill as a combined category, which would be (U+V+X+Y) /(Total-T) . Adjusted scores AAO = (A0-PN)/(1-PN) ATSl = (TS1-P1)/(1-P1) ATS2 = (TS2-P2)/(1-P2) ATS12 = (TS12-(P1+P2) )/(l-(Pl+P2) ) 108 APPENDIX D BMDP LINEAR REGRESSION EQUATION PREDICTOR SETS, NORTH ATLANTIC OCEAN EXPERIMENTS (PR+BMD MODEL) I. Area 2, TAU-00 (BMDP P2R) BMDl = 2.842 - 0.21767*E850 + 0 . 837882E-05*D500 + 0.03293*SHF + 7.057*DTDP + 0.05872*ESUM ** BMD2 = -10.4469 + 0.11854*EAIR + 0.10124*SMF - 0.07409*T925 - 0.16481*E925 BMD3 = 3.47713 - 0.22482*EM + 45 . 06116*DEDP + 0.00521*EPRD II. Area 2, TAU-24 (BMDP P9R) BMDl = -20.9733 - 0.20905*E850 - 0 . 078694*T925 + 0.0533674*SHF - 0 . 0316725*INSTAB + 0.0862939*TV + 0 . 0862983*ESUM BMD2 = 2.68106 + 0.0356103*TM + 0 . 53048E-04*V500 - 0.141302*E925 + 22.0764*DEDP + 0.0125618*DDVDP + 0 . 00563327*EPRD ** BMD3 = -35.2882 + 0.0381891*PS + 0 . 0273575*T500 + 0.00449538*PBLD - 0 . 00625203*STRTFQ - 0.0083686*STRTTH + 0 . 00272894*DTDP III. Area 2, TAU-48 (BMDP P9R) BMDl = -37.7157 - 0 . 147084*E500 - 0 . 0897567*T925 - 0.128407*E925 + 0.022881*SHF + 0 . 00860574*RH + 0.145537*TV 109 BMD2 = 1.85487 + 0.0777253*TM - 0 . 0266753*E850 - 0.0000390116*U500 - 0 . 0000366663*V500 + 0 .0240246*DDVDP + 0 . 105648*ESUM ** BMD3 = -13.9637 + 0.0160572*PS + 0 . 00308705*PBLD - 0.0031323*STRTFQ - 0 . 00 84644 3 *DTDP + 25.6871*DEDP - 0 . 0029 6342*EPRD IV. Area 3W, TAU-24 (BMDP P2R) BMDl = 2.673 - 0.09363*E850 - 0.05101*T925 - 0.20451*E925 + 0.0305*SHF + 0.15111*ESUM ** BMD2 = 1.15536 + 0.16326*EAIR + 0.01509*SMF + 0 .13014E-04*DM + 8.08795*DEDP + 0.02091*DDVDP - 0.00788*EPRD BMD3 = -18.55031 + 0.02089*PS - 0 . 30643E-03*VBLW - 0.01151*STRTFQ + 0 . 02772*STRTTH - 0.05736*DUDP V. Area 3W, TAU-4 8 (BMDP P9R) . • BDMl = 1.92874 - 0 . 0719 817*T925 - 0 . 201663*E9 25 + 0.0376905*SHF + 7.66796*DEDP + 0 . 182705*ESUM - 0.00585094*EPRD BMD2 = -33.2574 - 0.1459*E850 - 0 . 000205441*V925 + 0.325802*SHWRS + 0.0168064*RH + 0.124434*TV + 0.0 24 72 7 7*DDVDP ** BMD3 = -10.1316 + 0.0126085*PS - 0 . 00032403*VM + 0.000112099*U500 - 0 . 00880168*STRTFQ + 0.0159356*STRTTH - 0 . 00174911*DRAG 110 VI. Area 4, TAU-00 (BMDP P2R) BMDl = 2.86828 + 0 . 4877E-04*U500 - 0.64632E-04*V500 - 0 . 30475E-02*STRTFQ BMD2 = 2.70156 + 0 . 14946E-04*DM - 0 . 00904*STRTTH + 0.01888*SHF + 4.09377*DTDP ** BMD3 = 2.84881 + 0 . 24549E-03*VBLW - 0.10113*E£50 - 0.25666E-3*V850 + 0.03273*ESUM VII. Area 4, TAU-24 (BMDP P9R) BMDl = 3.00017 + 0.0773367*TM - 0 . 0000464491*V500 - 0.103205*T925 - 3 . 76267*VRT925 - 0.000477853*DTDP + 9.51302*DEDP BMD2 = 3.05949 - 0 . 088302*E500 + 0 . 00372204*PBLD - 0.00492842*STRTFQ + 0 . 0154289*SHF - 1.89105*VRT500 + 0 . 0143745*DVDP ** BMD3 = -24.6366 + 0 . 00275966*PS - 0 . 0549077*E850 + 0.195977*T500 + 0 . 0140852*INSTAB + 0.0886378*TV + 0 . 0231264*DDVDP VIII. Area 4, TAU-48 (BMDP P9R) BDMl = -27.0959 - 0 . 0732462*E850 + 0.01616*T500 - 0.117787*T925 + 0 . 287098*SHWRS - 4.13253*VRT925 + 0.111855*TV ** BMD2 = -3.1619 + 0 . 00678538*PS - 0 . 0850887*E500 - 0.0000502297*U500 - 0 . 0000396501*V500 - 0.00389051*STRTFQ - 0 . 00917554*RH 111 I BMD3 = 2.08319 + 0.0771067*TM - 0 . 0282767*E925 ■ ** - 1.5829*VRT925 - 0 . 00114379*DTDP + 13.2073*DEDP + 0 . 0289832*DDVDP Equation selected as predictors in the PR+BMD model i 112 APPENDIX E BMDP LINEAR REGRESSION EQUATION PREDICTOR SETS FOR TWO-STAGE THRESHOLD MODELS I. Area 2, TAU-00 Threshold Equations (BMDP P2R) a. VISCAT I vs. VISCAT II+III separation VIS = 0.87475 - 0.11042*E850 + 0.01173*SHF + 2.61984*DTDP + 0.03863*ESUM b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = 1.03838 + 0.03668*EAIR - 0.10423*E850 + 0.01560*SHF c. VISCAT II vs. VISCAT III separation (QUAD model) VIS = 1.03730 + 0.03727*EAIR - 0.10505*E850 + 0.01552*SHF II. Area 2, TAU-24 Threshold Equations (BMDP P9R) a. VISCAT I vs. VISCAT II+III separation VIS = 0.792157 - 0.00404602*SMF + 0.0271941*TM - 0.0767329*E850 - 0 . 0385021*T925 - 0.0656522*E925 + 0 . 0683159*ESUM b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = -13.9955 - 0.0612856*E850 + 0.0478582*SHF - 0 . 0303761 *INSTAB - 1.585*VRT925 + 0.0539318*TV + 0.0102962*DDVDP 113 c. VISCAT II vs. VISCAT III separation (QUAD model) VIS = -13.1384 - 0.107716*E850 + 0.00002923*V500 + 0.0122899*SHF + 0.0498382*TV + 0.0105859*DDVDP + 0 . 0232983*ESUM III. Area 2, TAU-48 Threshold Equations (BMDP P9R) a. VISCAT I vs. VISCAT II+III separation VIS = -18.0327 - 0.0569882*E850 - 0.104458*T925 - 0 . 00116092*PBLD + 0.0957819*PRECIP - 0.0705180*TV - 0.0118901*DVDP b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = -14.4404 - 0.0670910*E850 - 0.11931*E500 + 0.0157788*SHF - 0 . 023835*DUDP + 0.0551551*TV + 0.00761201*DDVDP c. VISCAT II vs. VISCAT III separation (QUAD model) VIS = -56.9191 - 0.168467*TM - 0 . 116816*E500 + 0.0201013*SHF + 0.000599864*DTDP + 0.210713*TV - 0.0487371*ESUM IV. Area 3W, TAU-24 Threshold Equations (BMDP P2R) a. VISCAT I vs. VISCAT II+III separation VIS = 0.88319 - 0.04039*E850 - 0.03385*T925 - 0.11313*E925 - 0.02692*DVDP + 0.0843*ESUM 114 b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = 0.59606 + 0.000012576*DM + 0.0151*INSTAB + 10.23334*DEDP - 0.0019117*EPRD c. VISCAT II vs. VISCAT III separation (QUAD model) VIS = 0.73427 - 0.12863*E850 + 0 . 0015056*PBLD + 0.02073*SHF + 0.04270*ESUM V. Area 3W, TAU-48 Threshold Equations (BMDP P9R) a. VISCAT I vs. VISCAT II+III separation VIS = 0.424612 - 0.0000997789*VM + 0.0000403063*U500 - 0 . 0360747*T925 - 0.132972*E925 + 0 . 114795*ESUM - 0.00286176*EPRD b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = -7.19287 - 0.000788627*PS - 0.0685555*E850 - 0 . 05799 88*E500 + 0.0222827*SHF + 0 . 195715*SHWRS + 0.0208835*ESUM c. VISCAT II vs. VISCAT III separation (QUAD model) VIS = -7.28759 - 0 . 000797890*PS - 0.701888*E850 - 0 . 0547155*E500 + 0.0226117*SHF + 0 . 202564*SHWRS + 0.0212283*ESUM VI. Area 4, TAU-00 Threshold Equations (BMDP P2R) a. VISCAT I vs. VISCAT II+III separation VIS = 0.98483 - 0.75281E-03*STRTFQ + 0.70578*DTDP 115 VII. Area 4, TAU-24 Threshold Equations (BMDP P9R) a. VISCAT I vs. VISCAT II+III separation VIS = -3.02386 - 0.200326*T925 - 0.000654475*STRTFQ - 0 . 000295813 *DRAG - 0.466227*VRT925 - 0 . 0019 2255*RH + 0.0153547*TV b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = 1.19369 - 0.0709575*E500 - 0.000617273*V500 + 0 . 19 3681*SHWRS - 0.000551506*DRAG - 2 . 96306*VRT925 - 0.0180048*DUDP c. VISCAT II vs. VISCAT III separation (QUAD model) VIS = 1.36198 - 0.0737982*E500 - 0.000061033*V500 + 0 . 185475*SHWRS - 3.05142*VRT925 - 0 . 0184565*DUDP - 0.00185381*RH VIII. Area 4, TAU-48 Threshold Equations (BMDP P9R) , ■ a. VISCAT I vs. VISCAT II+III separation VIS = -8.24765 - 0 . 0085996*T500 - 0.0333863*T925 - 0 . 0177121*E925 - 0.00059629*STRTFQ + 0 . 000128675*DTDP + 0.0343295*TV b. VISCAT II vs. VISCAT III separation (EVAR model) VIS = 1.20788 - 0.026601*E850 - 0.0000310346*U500 - 0 . 0000480308*V500 + 0.203575*SHWRS - 3 . 2719 *VRT92 5 + 4.10651*DEDP 116 VISCAT II vs. VISCAT III separation (QUAD model) VIS = -12.5384 - 0.0397823*E850 - 0.0000388604*V500 - 0 . 043269 *T925 + 0.201266*SHWRS - 3 . 46254*VRT925 + 0.0500493*TV 117 APPENDIX F TABLES o\o o\o o\o o\o o\o o\o 0V3 o\o o\o ^ <-{ •^ CM 00 -D CN ^ ro 00 o a^ CN o> LTl (Ti CM a> i^ ^-' r- — ' C\] --^ LO — ' fH -^ rH — ' X) ^ 00 •^ rH ■ — - '^ ■ — - LD — r-H ^ rH 'S' ^-' r- in 0V3 o o o\o o\o o\o o\o o\o o\o dvO OP ^ CO (T> ^ CO 00 ■^ in rH in O • 00 • • o • KO . rH O CO CO in in rf rH CO o CM O CM • (Ti * • • r- r-\ •-i r-i rH CM a\ rH o <-{ CM •^ CM >^ CO ■^ O CM CM CM CM r-{ '^ H CM • O • • kO CTi • • • • • O in cn CTi •^ a^ rH in CM CM o O •-{ O O <-\ rH ■^ (N ^ 00 <-{ ^ CO CO rH --' in r~ 00 <-\ ^ r~ •^ CM o\o (N r— I a\ CO 1^ — ' CO o o nH 00 00 o r^ 0\° o iH 00 00 UD un 1^ . in in CN 00 O CM rH in c3^ CO — ' o\o iH in ro en CN ^^ in o\o o\o o\o o\o o\o ^-^ ^.^ , , .—^ CN tH CTi ^X) cr> o\o o\o o\o 0\° • t in CTi in in M "^ rH rH CM in rH • • • CN .H 00 rH CM ^ rH -^ o eg .— 1 ^-' CTi -^ 00 00 ^-■ tH ~-' CO "-^ ^ -^ H ^ H — ' , . ,_^ ^^ o\o ,_^ ^_^ , — . , . , — . o\o o\o o\o ■^ o\o o\o o\o o\o o\o in H O • x) in CT\ r^ n r^ ro r^ M3 r^ t~^ U3 •^ 00 a^ cTi ■^ o^ v^ lCO 0 v^ 00 x: fO CTi w i '"' g U} ;3 >, fo W rH QJ P iH < f^ td H 89 Q > o\o o\o o\o c\o o\o ^^ , , ^-^ , , CM r^ 00 CM in o\o o\o o\o o\o • • • • • ^ VO r- in in — rH ^' f-\ ^^ CO --^ CTi --' <-{ — ' r-\ ■—' CM —- Eh ,_^ o\o 6 o\o O CM &f^ CM • O O O 'sD CO ,-\ Q > ^ —' <-{ -^ r-\ 5 CO a> in CO P in ^ EH CM CM in 00 CM o 00 o in U3 00 CO CO o\o o\o f^ o\o •^ 00 • O Xi r^ rH <-K CM o CM 00 r-\ CM r^ in CM o o r-\ '^^ H CM as CO r^ \D CO o ■^ o r- rH r-\ in 00 <-\ H CO r^ ^ vD (^ M r- rH ^ >vD rvj • • • • LD M O M (N I-H rH a\ rH CM ro 00 vo iH "^ CN CN .-) --^ tH --' CTi I-H ^— iH — ' TT ^-' in ^' I-H ^-' <-\ ^^ in H J 2 H H > ^-^ ..^ . , (X> . . , — , , — , ,—^ ^^ dP OP o\o 00 LD •^ CN CN • • • CN r~« • • • • • (N lO rH a^ a\ ro <-\ LO CS] O ■^ rH r-\ O in — G\ ^~' ■^ - — rH -^ "=r — ' vo ■ — :-\ — ' ro - — ' ^ •~^ dp dP 0\° o\o dP o\o o\o OP o\o in 00 CN o ^ ro CTi CTi o ^ • o • • ^ • t-i • a\ • ro • • • in (J^ ^ r-- O r-{ CO "^ ro v£> in r^ <-{ * r^ (y\ [^ O KD in 00 O G\ r~ G\ r^ cTi r-\ a^ r-\ —^ •-{ -^ in ■^ <-^ -^ CN ^- CN ^' r-i — ' ■ca' ^- 00 -^ &1 <«5 CfP o\o o\o oV> , , ^-s ,— ^ ^-. *i) '^ '^ in ro o\o o\o o\o o\o • • • • rH CO •^ O r- rf O CN r- VD CN 'JD ro • CN • • • \0 r-i ro rH ro (N 00 rH O rH o -^ ro ^ Csj -^ r-- CN CN ■— ' CM —^ s ^ o rH in CSJ 00 00 CTi CM CN O CTi CM in OS o CO rH O in CM o\o CO A ■^ tf,o 00 o o o o ^ ro CN CM in CM O ro rsi CM CO 00 in in CM CO CO CTi 00 CM in 00 CM CM CTi in CO CM in CO o rH CO CTi CN in CM rH in CN 00 CM in 00 rH 00 00 rH in o CO o rH CM O CO in o in CM CO 00 r-\ CN W ro ^ ■^ in VO [^ 00 120 TABLE IV. Number of observations of three visibility categories and 95% confidence intervals for the dependent and independent FATJUNE 19 83 data for the North Atlantic Ocean homogeneous areas 2 and 4, for TAU-00, TAU-24 and TAU-48 and area 3W for TAU-24 and TAU-4 8 TAU 00 Area 2^ Total VISCAT I VISCAT II VISCAT III DEP 1912 190 (.099) 214 (.112) 1508 (.789) IND 955 87 (.091) 103 (.108) 765 (.801) 95% C.I. (.086-. 107) ( .099-. 122) ( .778-. 808) Area J_ Total VISCAT I VISCAT II VISCAT III DEP 3181 85 (.027) 400 (.126) 2696 (.848) IND 1590 44 (.028) 197 (.124) 1349 (.848 95% C.I. (.022-. 032) (.116-. 135) (.838-. 858) TAU 24 Area J_ Total VISCAT I VISCAT II VISCAT III DEP 1760 180 (.102) 206 (.117) 1374 (.781) IND 879 71 (.084) 98 (.111) 710 (.808) 95% C.I. (.081-. 106) (.103-. 127) (.774-. 805) Area 3W Total VISCAT I VISCAT II VISCAT III DEP 1415 270 (.191) 173 (.122) 972 (.687) IND 707 137 (.194) 89 (.126) 481 (.680) 95% C.I. (.173-. 205) ( .109-. 137) ( .665-. 705) Area 4 Total VISCAT I VISCAT II VISCAT III DEP 2938 81 (.028) 368 (.125) 2489 (.847) IND 1469 38 (.026) 175 (.119) 1256 (.855) 95% C.I. (.022-. 032) (.114-. 133) (.839-. 860) 121 TABLE IV (CONTINUED) TAU 4 8 Area 2 Total VISCAT I VISCAT II VISCAT III DEP 1852 182 (.098) 230 (.124) 1440 (.778) IND 925 91 (.098) 107 (.116) 727 (.786) 95% C.I. (.087-. 109) (.109-. 133) (.765-. 796) Area 3W Total VISCAT I VISCAT II VISCAT III DEP 1487 290 (.195) 186 (.125) 1011 (.680) IND 743 132 (.178) 111 (.149) 500 (.673) 95% C.I. (.173-. 205) (.119-. 147) (.658-. 697) Area 4 Total VISCAT I VISCAT II VISCAT III DEP 3056 109 (.036) 406 (.133) 2541 (.831) IND 1527 45 (.029) 196 (.128) 1286 (.842) 95% C.I. (.028-. 039) (.122-. 141) (.824-. 846) 122 Q W H H 00 en to ■=3^ -P 1 nH CO D P P < w 0 Eh (U 0) ^ c T3 (D C w U^ (13 u 0 ■H 6 •^ 4-1 0 CN w x: 1 •H D -P 0 < ro fO •H H 00 -p -P CTi w c o r-\ fd o 0) rH o W iH +J 1 2 ^ <: D D (d <: 1^ +j x: Eh Eh +J < >i M 13 fa 0 0 Q) c s -P Sh OJ fO 0 &^ QJ P M-l c x: <-\ •H 4-> td W +j > T! c c (D 0 0 •H -H u 0) Sh T! ^ 0) Q) Q) -P a. x: cn -U P 5h -p 0 p ^-^ w M-l a 0 -H -p OJ ■<^ p >iT3 0 S-i 0 T3 fO e r- M P fd 0) g 0) Xi P ^ S 0 CO +j PO e Q rH S CO H Eh Q o Q — 2: o H < O^ CO W Eh Q < W CO Q Eh cu o W O w Q O O '^ 03 00 ro m n P-) en KD m ^ in r-\ X) CTi ID in CO o CO 00 C?i o o t~- r~ 00 CO Q + « > Q D a in CM CO •^ 00 r- r- CM CO o 00 CO I in 00 00 CO CM CO 00 en in CO us O 00 + > Q < D O CM CM CO 00 o ^ Q PQ + PU Oi CO CO CTi in 00 00 CN CO CO CO rH nH CM 00 00 in O CO 00 ^ O CO CO 00 r^ r- <: > 00 ^x> Q < D O >i) CM CO CN CO •^ CO CO r-- CO ^ r-H r^ 00 CT\ CM CO [^ CTi CO r^ ^ CM ^ CM 1 ^ CM 00 1 1 rH CM a^ O iH 00 CN •^ o r- V£) o CO <£> CO CO 'vD •^ a> o '* rH o 1— 1 00 r- ^ rH nH 00 00 00 UD ^ r^ r^ r^ iH r~ KD UD 00 o CO CM CM CM CM CM 00 CO CM CM 00 •^ 00 r- •^ CO CO 00 r^ r- r-- m r^ in CM CO CM in KD in r~- in in + o o Q CQ Di < > in CO ■^ CM CM CO CO Q a. Oi w o H CQ < Eh D < Eh O O o o CN CM fd (U u CM CN CO 123 Q cn 2 Eh M < in >vO LD o o o rO CM CN 00 O n CO r- r-l <-i H E-i o o o r^ vo v£) ro ro ro CO m M CM fo ^ n 00 o o CN CM CM "^ O O Q D H EH O U Q o Q ^ 2 O M X) [^ VO ■^ CO CO r^ r^ in o o •^ [^ r-\ r^ ^ CO o 'sD r- r- Ln ^ (T\ CM '^r r- r^ r~- en 00 r^ r^ CTi ON 00 CO w Q O Q Q Q Q s s S S P3 « Q PQ Pi Q PQ rt Q CQ Pi Q + < <: + < < + < < + < < Di; cc; > D X « > D Di cc; > 3 p:; C^ > D D^ a, u o (h &. w O CU cu w a 0^ eu w O D < Eh CM 00 CO 00 q; u < CN 5 124 en CN I I >i s o •H 4J O (D •n O U a, tn :=! o (U c en fd o u en c o •H -p > U (U w Xi o s o x; o M-l c 03 -^ U O 0 ■H -P c fd Eh M S +J o o x: CN +J o M-l S o en ■H C -H C >i"=3' O 5-1 00 x; td CT^ CO 6 w D dJ (d w C 0) <: i-D (d J H M 00 o o I— I CO *> CO (AO rH 00 O rH CM ^^ 0\° o o CO I— I o I— I CO CO ^ CN CTl rH 00 ^-' ro iH c\o o\o M r-- ■^ • • r^ 00 CM 00 ro r^ r-- '^ rH ^' a> — ' CN n CN r^ tH o\o O ro n CO r^ r- r- 00 KD m n f^ PO m c^ w MH d a & o Q < ro CO o o^ in 00 CN CTl ro Cvj o 00 t^ x> CO 00 r- • • • • • • • • r- 00 00 a\ 00 CN CN CT\ 00 r^ r^ ^ VD r~ r^ I CN (N 0) CN 126 APPENDIX G FIGURES OPERRTIONRL U.S. NRVY MODEL OUTPUT STRTISTICS (MOS DEVELOPMENT SCHEDULE ^^> ^%^ VISIBILITY CEILING CLOUD RMOUNT OBSTR TO VIS SIG WAVE HT WIND WAVES OCEAN SWELL SEC WIND SEC TEMP SPC DEWPOINT PRECIP OCCUR PRECIP RMT TRW OCCUR NORTH RTLRNTIC MEDITERRRNERN NORTH PRO I PIC INDIAN SOUTH RTLRNTIC SOUTH PRCIPIC 1987 1988 1990 1991 1988 1989 1989 1990 1991 1992 Figure 1. Proposed U.S. Navy Model Output Statistics (MOS) development schedule 127 c n 0) o o u •H -U C ^^ ra XI •-{ "^ ■p CO < .H x: - — -p S-l (U 0 5 2 0 hJ 0) x: e 4J 0 u S-i M-l 0 U-( «. >i w r-{ fd 0 Cl) 1^ >-i fl TS c w (13 Id 0 OJ a; C c :3 QJ f^ D^ 0 «. 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Q_ Ti 1 • • UJ rj n n n a III 111 y z Q a a * X • LJ CD z 8'0 9-0 V'O Z'O [ lUOSIA 38035 l^iimi ? oy asisnroy g-o- -P c Q) Ti C o Q) >i 0) •. U £ -H (N c P cu e u ft3 cn ■H 0 0) C X -P ^^ ■H fd u fO +J e -H Q C tj UJ M 0 0) (D > 0 u x; s-i S. 4-4 -p a UJ OJ t/l to x: x: -p O -P H -p cn -H •H ^ a 5 -H w • l+-t 0) >it:! ^^ cn 0) (U Q} -p x: QJ -P 03 -P rH ffl •H Xi S-l O -P rd -P 0 n3 +J w >iM fd 0) u H > c cn cu 0) CQ 0) -H CPO ^ x: c ec; 0 0 •H cu U fO -p X cn 0 g a; u u If) 0 0) 0 u T3 x; x: w c -p -p rO -p M 0 rd e 0 -P 0) n3 U-l 5-1 S-i tn s: cn .— ^ T3 -P n3 -H c •H 0) 0 M 'd X! D^ 0 cn E-i rH s (U < t-{ u u ■H Cii >-l CO ^ C^ 0 M (^ ^ u > • •^ CNJ o r- 00 O h o d • CM 11 c« o ^ tfl l/l (/I < < h- t- >— 6« If) iri 00 II o < < 00 (d o OJ CN CO 00 d O o a ^ •^ CM CM CN ISV03dOd < < z UJ a z UJ & UJ Q O) •^ o CO CO CO o o in in isvoadod CO LJ ID a o LJ f-i cc: o o CD LD q; f— 1 rr f- CO (-. tij ""^ rK a: •— . • o z o , n CC X f— . 1 in n cc CL I— o (M 03 CiJ til o cc -L . t—\ rn a_ Q_ a (T en [■1 ■ • LJ LJ (.1 cc 1 n (1. f 1 n h 1 rr o LJ L.1 Z 2 Od cc 1— CJ CJ —^ *"• n 1 1 1 I Q n • (_> CiJ a LO O"! 9-Q 9-Q VO Z'O [ lyasiA 38005 i\nm\. 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Q > Q o o o D I cn >1 D^ cu 0) 4-1 u (13 c M q; +J T3 w •H 4-J M c H 0 u CQ o o\o Cii m cu o X g >x> fO c .H 0 (d •H > -p M u QJ c -P D C P4 •H in -H En 133 • o t t rsi \ \ -01 cc \ \ LJ \ \ or 0= \ ^^ \ LO 1 t-cn i \ — CO C£) 0 tro 'Z. a =3 LJ D_ 01 >• \ M ZD t- LJ \ a oce \ Iri cc 1 (EO \ CD 0 1 cs LJ \ -tv z L_ 21 z iJi— "-«, \ qT o LJ UQCC --, 1 ZD X cs2;y ''-- \ 0 2: 0 CO LJ tJ ce LJ !-• --- CD ■— -■ 0 \ ' ".fc (— • , 1 ^"• ^ LJ — ■ "-♦ -<0 _J CD CD LJ 0 3 • cn Q_ X o: cc ' ■sz 1 1 ■M. • ■ \ -Ul 1 — ^ ^ . ^ \ i — — t — r -» S-Q >-0 £-0 Z-Q fO Q- 0 ro- 20- £'0 - I 3d03S lyayHi -qni Tay /oy •d3Q Toy M T3 ^3 H (U CD +J 4J a U Ul 0 Q; P 0) oi 1—1 ■r-^ U c^ (U TJ 0 X CO (T3 U g CO (U (d x: 4-1 •. 5 -P (13 M OJ 0) C 0 i^ T3 (U -p ^ 0 > +j e (U 0) CO N H Q -H 2 M-l W E-i CQ 0 < + 1710 cc; 0) c w a, N •H M — •H a> CO P 0 0 T3 0 cn S-^ 0) 1 c Cri4-i D •H W < O. cn p E-< P p -1-1 0 0 TD •. u iH (13 (N cn P a 0) (d 03 0 x: 0) a-u S-i q; (T3 CO >iT3 (T3 ^ c S-i u c— 1 (t3 0 rd 4-1 CO P — •H cr 03 ,— .. 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N. o o O h- M " w* (H o 1/) m 1/1 4 K* »- r— < < < < # 1^ ^ CO CO 0) CO 00 CO 00 o o CD It '"0- [ lyosiA 3yo3s ibnym ? oy aiisnrau 9-0- >> u ^ c cN oj e 1 ty 0 DCS < -H -H £-• -U X c (d - 0 e ^ CN 0 -P (U 14-1 (T3 QJ X: -H (1) x; 4-> u-i S-i E-" fo x: OJ +J £ >-l . -H +J Q 0 >i 5 UJ 4-1 en -P > 0) T3 (T3 tr en -U 0) UJ +J fl +-I tJ 10 .-1 !-i (T3 (D ffi 3 -P -H > 0 en w 0 i 0) U cn U 5-1 en C -P d) rH QJ (« CP (t3 en 0) C 5-1 0 M •H a x; x: +j -p 4-) -P , C fO 0 c 0 M U -P OJ E-t Td x: en (< C -P T3 U 03 C W rH 5-1 0 M (1) g 0 a> > 03 u-i en dJ 5-1 (D -P r-t CH'- S-l c fO M 5-1 0) 5-1 -H (1) 0 Ti 0 TJ T3 0 C -P 0 0) U .H E (u a-H .— ( iH 0) TD ■H cc: X! Td (u ^ Ot 03 C 5-1 CO ~-' 4J -H Oi • 0 CM ^^ 0) M p tT> •H Em 144 in o < CO CD CO CO CO Cvj to < T— d ^ Q U. O d -* CO CO T- d CO CO CD d in 00 00 T- d o o a. o to 00 LU Z L'J Q. Q > in c:) > Q lO o fX. \04 o\o — 1^ (Ti ^ CN T3 1 C D n3 < Eh W 0 - •H rsi -P CO ro •H 0) -P 5^ rO rU -P CO S-i 0 rH U-l < --> \ W >1 O (U D^ < 3 QJ rH +J - (T3 03 0) > M U -P C rH W CD (T3 ^ > M C M H CU QJ a-p DQ QJ C O ^ -H « 0^ rH 0) X c c i 0 (U •H TS •. -P -H rH U U-i 0) d c X) =1 0 0 P-i u e 00 0) M •H 145 LJ M Q_ a a o LJ LJ en az a: OJ en E— LJ LJ O CO CD O q; Q_ X az - H- f-CE (E O CJ LO cn~ — > > LJ Oct: CEO CJ (— (T) aZ Z LJf-- UiCHTL C^ZLJ tZji^ct: ^Q.31 UJM a OQ LJZ H- — in o • — ) — ) QQ (EtE X • ~» ♦ Q- O cs i'O £'Q Z-Q I 3yo3S lys^Hi ro Q'Q ro- 20- •QNI TQy /OH •d3a Toy £•0- H T3 T3 H QJ QJ •P -P PQ O cn O Q) D (U cc; rH •r- > S-J c^ 0) T3 0 X cn fd 0 to $ to OJ (d x: -p - 5 -P (d M 0) 0) c C ^ Ti OJ +j x: 0 > +j e 0) OJ to N M Q ■H s <4-i CO Eh m 0 < + Cr>U Di OJ C w CU c^ •H M ■ — ■ •H Q.> CO 3 ■^ 0 T3 CN tT> >H OJ 1 C Cn+J z> •H U) < a W D H a D ■r— 1 0 0 T3 •. M rH 03 CN en P Qu (U ra (d 0 JZ 0) 0,4-1 u iT3 (d -— 1 C u u 1— 1 fO 0 ft3 U-l to D , ^ •H CT fd ,-v x: 0) +J 03 -p fO -p iw Tl fd ^ 0 T3 o -P Em a c -p •H cu c x: T3 cu • w C T3 ^^ c i o a (u Cn •H OJ a, 0) 4-> ^3 q; +J (U ^-' T3 fd r-H C u 0) 3 H p « < — ' to U •H fa 146 < Q Z UJ O z lU Q. lU Q Z (^ N co cvj « (0 r- o CM 1 d o 1 d II ^ (N O i/i Vl 1/1 < 1- h- >- < < < < CO 2/3 U5 CO o o C\J o 00 ■ ^ d o d o ^. 1/1 U) 1/) < < K- y- t- 00 o < 00 N. O < o T-* d CM d to i- < in CO CO r- T- q d o I II 1/1 < o o 1/1 I— < CO CM T— CO CD O CD 00 O O CM CD CM O ID iSVOBdOd < < Q Z LU O z UJ c UJ Q o CM o o 0) (0 o CO 0) CO o •r-. iswoaaoj CO ZD a CO u. [jj o M •3 ?^ -J o q: CO " cn 5; a E— I o a cr UJ o; Ct: ^ Q_ I L_ CM I- LiJ CD X 1 1 1 i t ■■\ K t X - ♦ 'f^ \ - „ ' 1 \ '; LEGEND - DEP. flO - DEP. THR. I - INDEP. RO - INDEP. THR. - a * X • \ 1 1 H , , 1 I _ 80 g-Q VQ 2-0 ro z-Q- va- [ lyosiA 3800S iuimi ? oy a3isnrQy 1 D >i e < U ■H ir> C X (U 03 - cn e 4J oj C tn •H 4-1 U 0 nj U x: 0) -p x: S-i dJ •H -P 0 x: 5 u-i E-i -P T! fd w 0) o +J . +J 'd > -H >i fC cu D iTi ■H > UJ CO 0) U Q) 10 dJ -p 0 •H CD s-i fd w r O 5-1 w U OJ 4J (13 03 -H W i2 U) 0) rd H Q) M -p U 0 CQ 0 u >iO U cn u cc; W C CI. -p OJ X OJ 03 cn (U 0 S-i •H x: x: -l-J 0) -p +j c r 0 +J 0 M u ■p 5-1 B T! 0 cn < C ^ X! u ro c CO i-l ^^ 0 M CU e -H a> > (T3 0) CO Q) J-1 T3 0) -P rH Cr> 0 s^ c ta g S-I 0) V^ •H o XS 0 'a Q u C -P s QJ U i-l CQ 0) Oi-H iH + -H QJ Ti -H Di XJ T3 0) ^i 0. 03 C S^ 01 -- -P •H Q. nJ in H 0) M p CJ^ ■H U, 147 ^ N. CO CVJ CO CD ^~ o 1 CVJ CD o 1 o ^* l/l (/I < «J ►- ►- t- CO < t- < Q z lU Q Z tu a lU Q z CD (D o 0) CO o o CVJ (0 CM o in en o o en Q (jj Or: Q_ LJ I lSV03«0d < Q z OJ o z LU c LU Q CO LJ [ — I LJ M CD tn CD Z! "^ ct: o o m Q LJ a: c^ ■ N. T- o 1~ CVJ o 00 6 1 d II ;: c^ (/I 1/) < t- 1— < < < < ^ a? CO CVJ N. "t O CO o 00 T- o o d N. T— „ i 3 D 0 6 < C -H Eh OJ X CP 03 «. c e -p CM •H CO fd c x; -H 0) 0 -1-1 u-i u u re +J >i-l-J TD IZ nH Cn 03 OJ UJ P (U -H > 1/1 CO -P 0 OJ 01 -i to ^ -P CO o Q) CO 03 03 i-t X! H CO o; (d H OJ M +J U 0 CQ 0 U >iO o CO u cc CO c Oh -P (U X 0) 03 en <; CO 0) C 2 0 ^^ •H ^ jr +J 0) -P +J , c j:: 0 -POM u -p ^ Eh T3 0 CO < C 4h T3 CJ n3 c cn i-H ^ 0 M (U E r-i a> > (T3 0) CO OJ U T3 (U -P -H Cn O 5-1 C (13 6 S^ 0) 5h •H 0 T3 0 Ti Q O C -P r-i oQ (u a-H iH + rH QJ TD •H a XI T) QJ -i^i CU 03 C Vj C/1 — -P -H Q^ XI iTi -H 0) M p D^ •H tu, 148 ^ -* "* ,_ CO CM O if} CM 1 d o CD CD II ft ^ CI O m in trt < t- ►- 1- < 4 < 4 c^ ^ -* T- O) lO CO CM ou Tf CVJ , d lO 0) o JD ^ ^_ M o ^ I/) CO in < «l h- >- ►- < a z lU Q Z Ui a. UJ Q Z CO LJ ID a LJ o CvJ r- CD C» T- CM (0 CO CM 00 CM CM CM CM CM isvoaaoj < z lU o z UJ c m o CD " o % CD Ci: oj CO f_ or en O ^ f— I I O oo I — I I — I UJ LJ ^ cc ^ Q_ 1 L_ <~vj LJ CD - — q: > o r (—1 q; en f— V CJ X r, 1 CL H- . • rn a. Q. Ti 1 ■ • LJ LJ Q. a. C.3 a liJ Ill 1' :^ a a D * X • c^ - (0 Ci T- CO "* 00 <0 r»- K o o o h- „ II ^• M o in cn m < 1- h- ^ < 4 < 4 # CO o 0) CO CO (0 0) lO 00 d o o 0) "*' CN " II ^ (N o in in in 4 4 h- h- t- in CO 0) T— O 03 CO CM - CO CM CO lO 9-0 lyosi 1 A 3yoGS iy3yHi Q-0 J-Q- ►•0- ? oy QGisnray 9-Q- >1 o c (U cr> c '^ -H CM -P g 1 C P P 0 e < o ■H ^ X Q) fd -x: e ^ (N Eh 4-1 d) U-l fd x: •H 0) • 4-> <+-( v^ >i fd cnx; QJ (D -p ja >-i +J •H 4-1 0 fd 5 M-l M 4-1 4J T3 fd cn 03 QJ 4-) 4-> "X^ rH C fd QJ 0 0 •H > W -H u QJ 0) W 0 •H M W w x: QJ w u 0) !h td fd rH cri X! QJ w 0) (T3 !-i 0) M 4-) >-H 0 M 0 o >i fd 0 cn U 5-1 w C P 4-J (1) -U 0) fd CT" (d cn QJ c c 0 S-l •H x: x: +J QJ -p 4-1 c x: 0 +J 0 M u -p u H T3 0 CO < C U-i T! u fd c; CO i-H , — . 0 M QJ e -H Q.> > (d 0) en QJ M T3 0) +J iH cn 0 S-i c fd e S-t QJ S-l -H 0 T3 0 T3 Q 0 C 4-1 s QJ U -H CQ Q) a ■H M + rH 0) T3 •H Cci Xi TD QJ ^ 0^ fd c U cn ^ -P •H a t o in rH CL 1- z UJ o o o X rtO in o \ o\o a\ -^ CNl T! 1 C P (T3 < en t/) u - -H CS -P 03 tC ,— ^ ■H 0) >i +J >-l cn td d 0) +J +J C/5 V-l rd 0 U 1— 1 <+-! ■P < w \ w O Q) H < 3 H -H - fd CQ q; > O u Oi C rH eu 0) (T5 X 13 > i 0) cu a-p •. 0) C rH t3 -H cu T3 rH (U 0 d u e d c 0 0 Q ■H T3 S -P -H CQ U M-l + C C ttJ P 0 04 t4 U »^ iH 0) M 3 Dv •H Ci4 150 DEPENDENT DATA < O 2 UJ q: O 80 172 1278 25 23 61 75 11 35 1 2 3 OBSERVED AO=78.18%AAo = 0.52% Ai 15.28% Tsi= 0.33 ATS1. 0.26 TS2=0,08 ATS2=-0.04 TS12=0.20 ATS12=_Q 02 INDEPENDENT DATA 31 74 668 7 14 27 33 10 15 1 2 3 OBSERVED AO'81.34%*Ao= 2.96% Ai=13.42% Tsi= 0.34 ATsu 0.29 TS2= 0. 1 1 ATS2 -0.01 TS12= 0.22 ATS12= 0.04 Figure 17. Contingency table results for the area 2, TAU-24 equal variance threshold model 151 DEPENDENT DATA V) < U 2 lU cr O 33 115 1053 72 80 286 75 11 35 2 3 OBSERVED Ao=68.64%AAo = -43 01% Ai= 27.50% TS1-0.33 ATsi=0.26 ^22=0.14 ATS2=0.03 Tsi2= 0.22 ATsi;>= 0.0 INDEPENDENT DATA 19 57 539 19 31 156 33 10 15 1 2 3 OBSERVED Ao-68.60%AAo= -63.31% Ai = 27.53% Tsi = 0.34 ATsi= 0.29 TS2= 0.1 1 ATS2= 0.6 TS12= 0. 19 ATS12- 0.0 Figure 18. Contingency table results for the area 2, TAU-24 quadratic threshold model 152 2 Q- o CD O o CJ cr LJ 21 LJ CJ CD CD O CC Q_ X CE _ >_< i— f-cr tE CJ CJ cn cn — — >• > LJ o ai dZ o o e-co az z tJf-H Ljcjir Ci3 Z UJ UJLJCC ^JCLX UJI— a □ a M z i-^ — to =) • —1 —> QQ (Tcr X • £•0 VO C'Q fO I 3yoos iu3yHi 10 •QNI Q-Q ray /oy ■d3a Z'O- TQy £0- 'O TJ 0) cu -p -p w H u P cu H cu ■r-i U <-\ 13 0 CQ cu fO U O cn cn « OJ O, cn x: +) X (d g 5 0 i^ cu 4-1 s: V c -p <-{ •H cu cu G N H ■TD •H 0 4-1 cn Eh e O < tnU p:^ CD C w CU N •H H ■ — • •H a> cn p 00 0 T! ^ CT" >-i cu 1 C Cn-P D •H cn < a cn a Eh p P -m 0 0 'n V u iH fO CN cn p a (U (T3 fd 0 x: cu a-p M (U fd cn >inc! (d rH C S-l u rH n3 0 tC 4-1 cn p ^ -H CP (T3 .— , s: CU -P rC -p (fl +J U-i 'O fd SH 0 T( 0 4-1 t^ a c -P ■H cu c X: T3 cu • cn c TS ^-^ c cu C >i 0 CL, CU cn •H cu a (U -P TD cu -P rd — t: fd iH c u (U o •H -P cr: < --^ cn cr* Q < CVJ > Q OP in o cc; \a^ o\o — ' >J3 ON 00 •^ 'U 1 c :3 fo i O 0) Cn < a 0) 1— 1 4-1 - (T3 (T3 (U > M O 4-> C -H cn QJ n3 T! > M C S-l M (U (U a-p CQ OJ G O T! -H a cu iH OJ X 03 CJ C C s 0 (U •H T3 ^ 4J -H rH U 4-1 0 c c T3 P 0 0 t. u e o (N f? -rt ^ in T- CN o ' . 00 ■ d o o O ^ in i/l (/I < < K h- t- < 2 lU Q Z UJ a lU Q Z 00 0) CN 8 d n in iri d 9 V* M o t/) to in < 1- f- 1— < < < <. 6^ O o ^ r^ h- CN lO n CM o r- ^ CN d It o o (N " II w- (N »- o 10 (n O^ < < t- (- t— to n 00 CN 0) o o T— 00 in 04 CO to LJ ID a o LJ t— 1 CD O O CO L_ O CH LJ CD isvoaaoj ui -: Q:: cE o ^ I — I LJ LJ o; Q_ I < Q 2 LU O Z lU c. UJ n CO 01 CO O s CN ■«t 00 CO 00 n (N - isvoaaoj — ^ o X n ce en M ?• o X ci: ^ ■ • rn a. Q- U 1 • • M CJ 1 n n I 1 a r>i [;1 7 ^ e_3 C_3 ■"• a * X • CO Cd o o LJ 5Z CO 1 Q) D ja < 03 &-I -p Ti «. >i e c rg 0 p o c s u n3 QJ ■H (U CU Cn X en V-l C 03 03 •H -P e 0) x: u c OJ 4J 0 0 jn U-l 0 +J -P 03 en 0) x; -U s: +J T3 rH H •H OJ P ^ > W Q) QJ • T3 •H M >i oj x: Cn-P U -l 0 M -P -p tn 0 en en U >i 03 en O H c w ■P (U OQ OJ (0 cr>o M a c Cc: 0 M •H CU u x: 4-1 X en +j 0 g (D M 0 en (U 0 H T^ s: x: < c -p -p u (T3 (^ rH >^ 0 M a; e 0 -p > > fO U-l Q) >^ en -P <-{ Cn ,-». TJ c fO iH c (U U •H 0) 0 TD 0 TD Xi CL C +J 0 en 0) u iH e (U a H rH ^1 cu T3 •H 0^ v-i T! 0) ^ Cl^ 0 c ^4 W ^-' u ■H a 9'0 9-Q »-0 2-0 Q-Q J'O- VO- [ lyosiA 3yoos ly^yHi i oy asisnray u •H 155 h- h- (O II o < < CO oi o fe5 o (/I < CO CN d ;; S o < Z UJ Q Z lU a. Ui Q Z CD LJ ID a -J- CD LJ ° [ — I u M CD 0^ ^ CD 1 — I z CJ g en CD en z 617 00 CO CN in O CN OJ in CJ) 00 iswoaaoj < z tu o z UJ C. m Q (n Q- LJ ct; X rr •. ( ) i: O t- LT) m r ) QJ (M (D f 1 (T or LJ UJ rt- o [K cc o Q- 1 L_ 00 n LJ ID CK LJ CD in 00 o < in (O CD CJ) o < 00 CN CO o <3) 00 O 00 CN 00 OO CD' to 00 o CI (/I 5? CO 00 t ^ CO CN CO CN ■^- CJ) iSV03dOd _ — o o cc cr H- -?• O -L fl 1 (T s- * rn U- Q_ ti 1 • t L.1 UJ Q. a. ,^ a Cil Ill :^ :?■ □ a a « X • en o 00 0) •^ rH 1 X! o (d < +J E^ >i - U e ^ CN C a -p (D s M-l (13 cn •H •H 0) c X m J-l -H (T3 fO 4-1 E Q) c x: V4 0 OJ -p 0 U x: m 4J -p 0) (0 Q en x: x; UJ -p B ■p T3 > 1—1 •H (U a p 5 > UJ en • QJ 1/1 0) >i'ri ■H ffi >-i en (U x: O (D -P u QJ +J (C (0 .H (0 •H XI M u QJ fO -P 0 M ■P CO en 0 en U >iM (13 en O M c en +J OJ CQ cu (C CnO v^ QJ c c^ 0 U ■H Oj o x: +J X en +J c < 0 s CD M u en QJ 0 E-i TI x: x: < C 4J -p U 03 CO -H u 0 M QJ e 0 -p > > n3 M-i QJ 5-1 tn -P rH en-- ■n c (T3 rH c QJ U •H Q) 0 T3 0 T3 T) a c -P 0 en QJ U -H g (U a-H i-H S-l QJ 13 ■H Cc: j-i T3 Q) A:: cu 0 C S-i U) ^ u •H a 8'0 9-Q VO 2 0 [ lyasiA 3^035 ly^yHi Q-O Z'Q- >•[>- ? oy QGisnrob 9-0- Q) 3 •H 156 < o z UJ D z lU Q. Ol Q Z 2f3 0> CO •^ CO 0) o '". Oi o o o CO 1 o 1 1 u II ^ (N o (fl 1/) U) < H ►- »- < < - h- 00 o to CO o (D 0) ?3 r^ 00 CO O CM I — I LJ CjJ CC Q_ I CO o i LJ CO iSV03dOd < z UJ o Z UJ o. UJ o en LJ :z) a LJ o , , CD o ^ tn g CD 0:1 O O _ cc • c ■) X n ce a_ t- "?■ o J. tii 1 u CO c •^ (U e 1 en P D c g <: •H •H Eh -p X c (d x: •. 0 e -p rsj u 0) M p n3 (U x: 0 , 5 U-l CTi +j 0) ID fd Q en -P 0) UJ -M fd +j T3 > r— 1 5-1 fd 0) cc a -p •H > UJ I/) w en u i 0) u tn u M en c 4-1 OJ M 0) fd en fd tn (U c U 0 >-i •H 0 x: x: ■P -p 4J -p c fd 0 c 0 H u OJ -p Eh T3 x: en < C -p T! U fd c CO i-H V-l 0 M CD e 0 a> > fd M-l en (D M (U -P rH D^ .--, M c fd 1— 1 ^ QJ >H •H OJ 0 T3 0 '13 0 u C -P 0) U rH e (U a-H l-\ rH QJ 'Ti •H « XI T3 Q) ^ (Ih fd C U w ^-' -p •H a O'l 8"0 9"Q >□ Z'O [ lyosiA 3yoos iBsyHi ? Oti Q3isnrQy o rH 0) en ■H 157 • \ 0 01 L \ -0> az -•--, \ CJ ---- \ a: *■*■* \ _ "'---^ \ " T Ul 1 \ *■** "■■^ -co CS CO CEO z — > 1 LJ Q- 01 >• 1 fvl 13 f- LJ ' a oce f LO ry 1 (EO , CS UJ 0 i-cr) : : t C3 -c^ Z L_ 2: ZLJf— " I qZ O LJ X ; 0 2: 0 0 CO LjLJ cc: -"LJf- I I CS — Q 1 LJ ' ' C2CD ; • - i -U1 S-D 1 1 1 I ' ► 'Q fO 2-Q ro Q- 0 !* 1 ro- z-Q- CO - ^ I 3^035 iy3yHl -QNl TQd /oy •d3a TOy Ti M TJ 0) OJ +J CQ -U tn 0 u 3 (U cc; 0) •r-i >-i CL, <-{ TJ C OJ a 0 ■-Si tn CO 2 0) en x: -P 0 fd 4-> fd •-{ 5 dj (U 0 u T3 (1) ■p x: 0 c ■p E •H U PC (U C W cu N •H M — ' ■H a> en a CO 0 T3 '^r Cn U 0) 1 C tj^-p 3 ■H en < a en D Eh a P -r-i 0 0 T3 » !-l i-i 03 CM cn P a 0) fd (d 0^ 0) a+j 5-1 0) fd en >iT3 fd -H c S-l u -i 0 TD 0 -U t, a c +j •H 0) c x: TD OJ • en c T3 ,— « c a; c; >i 0 a oj Cn ■H 0) a 0) -P T3 QJ -P fd — T3 (d rH G v^ 0) 0 -H -p cc: < en CN CN 0) M p en -H p^ 158 en 00 in CO CJ CO d CD d r^ d 1 ti ^ n o - h- < < < < ^ o CM CO o C» 00 CM 6 o d II ^ CO < < K K < Q Z UJ Q Z LU a UJ Q Z CD LJ a o LJ C£l O O CD 00 o O O) O (D to CO O in in ISV03dOJ < Q Z m Q Z lU c LU Q in CO o oo 00 0) o CN CO o CO i SW 3 3d O J o o ID CD or tr O ^ ^-^ I 5 £ LJ ^ (H a: Q_ I L. 5; LJ CQ 21 CM Q r: CD tn a: o ~ ~ a: ■ 0 r 0 or a f— -^ 0 r Ti 1 CL h- • ■ r"i 0- Q- Ti 1 • • ClJ Ul n n n Q (i! (ii i* 2 t_) CJ •~* •"^ a * X • 1 a'Q g-Q vo z'o [ lyOSIA 3^035 lb3yHi o LJ QQ Z3 Q-O Z-Q- ^-0- ? oy aaisnray g-Q- CO 1 e D >. e <: U -H Eh C X 0) rd ^ D^ e -p rN4 c to ■H 0) ^ ^ +J x: -H OJ C +J M-J U 0 03 u j:: 0) Q +J x: UJ !-^ 0) -H -P > 0 x: 5 a. M-l H -p Lii t! fO to to 0) 0 -P . +J 'TI rH >i n3 QJ a Cr>-H > Ui OJ U CD 0) -P 0 -H Sh fd to x: M to u (U -P fd (d rH to X3 to QJ (T3 M CD M -P U 0 moo >iO u to 0 K to C Oh -P > H ■H 0 TS 0 X3 Q U C -P 1— 1 CQ a Q,-H <-\ + rH QJ T3 •H cc; X! 'd Q) ^ 0. 03 G Sh CO ^ -p -H a • (0 m CN Q) U 0 tT> •H Cn 159 # T" 00 CJ 00 CO '" o in o o o N^ 1 " ^ n o m !/> U1 < 1- [» 1- < < < < # ^ CJ fs. N. CO OJ in CO o CV4 o ^ O o o 00 y- , H 1 ^ (N o ^ yl 0^ (A < < »- h- ►- r- CM 6 q d CM CD 1 1 II ^ r( M o m in W <« t— h- h- l 3 D U 6 < C H Eh QJ X Cn (U - C 6 -p (N -H en -P (U U 03 C s: •H CD 0 4J m U U 03 x: 0) OJ -p jn u x: •H -P 0 Eh 5 m -P T3 03 CO • OJ 4-) >,-lJ 13 rH CTl 03 CU D (D •H > en -P U CU 0) 03 0 •H u u w x: +J to 0 0) en 03 03 <-{ n H CO q; 03 M 0) S-( -P S-l 0 CQ 0 u >iO U en O O:^ W C P^ -P (U X QJ 03 a>< cn OJ c 2 0 u •H x: x: +J OJ +j -p C j:: 0 +J 0 M u -p u Eh Ti 0 en < C M-l TJ u 03 c CO r-) ,-~, 0 M (U e -H a> > 03 0) en 0) S-i TD (D -p rH Cn 0 M c 03 e >^ OJ 5-1 •H 0 T3 0 t3 Q U c -P s o O ■-I PQ 0) D. ■H r-l + rH 0) T3 ■H CC i2 TD QJ ^ Ol 03 c; !-i en ^ -p ■H a • X! fO CN (U U S Cn •H li^ 160 < Z LU Q Z lU a. o z CD # o CM O CM 1 II O < < ?^ ^ t«* CD CM Tf "* 0) h- 1— O < « no '^ o o o O o 1 ,, M (N ^ (N in Ul m t- V- ►— < < T- o 1- in 00 CD cj 6 I/) irt i/i 4 < in 00 CO N. 6 c> -^ II H o 00 CN ID 00 in CM r^ CO in CO CO (D CM CM 00 o CD i SVD 3a Od < Q z lU Q Z m c UJ O 00 CN CM O ^ (D 0) O o CM CO O in T— isvoaaoj CO LJ LJ o UJ N cn ^ r o % CJ o on o o oo 1—1 or I — I u LJ cc Q^ a: ^ ~ r) T r~i ftr a. (- T' O X fil a: H- - ■ rn 0- Q- fil • ■ OJ LJ n n D Q ii 1 III ■^ Z a CI a * X • g-Q 9-0 VO Z'O [ lyosiA 3yo3S lyayHi ? oy QGisnroy ro- 00 1 D T) < cu Eh > QJ 'T3 0) - j:: , 0 OJ (t! cn W ^ 0) cn 0 5-1 -P fd u O (t3 W U-^ M W -P 0) 4-> w cn 5-1 03 4J 0 QJ M C U 5-1 :3 0 cn x: cn -H 4J oj cn 0) 5-) cn cn H OJ 0 QJ 5-< j:: Eh ■H CP-P < i3 Q) U (« 5-1 0 W -P -P M rH > >i fd cn CJ in T3 P rH C P c c OJ QJ -P 0 (U > tJl fO QaTD 0) c c cn c -H •H O (U +j oj 5-1 a, 5^ c x: s-i Q) O O -P 0 T3 -P u u c u 5-1 -H •H T) 0 (U TS C >+^ M g (U IT3 X! P S-l — 03 g a g ^ P -H (T3 CU X sz 5-1 t3 >i ra 4-) CP 0 U g mh fO g C ■H •H q; 0) U-l TD Q cnx: S C -P 0) rH CQ -H £ rH + -p x; -p •H tti C -P ^i CU 0 -H -p W — u 5 fC3 • u n CN Q) M D Cr> •H fa 161 ID O < CO CO 34.7% CVJ 00 CD < cvi c\J 00 0) CO Cvj CO 0> o < q 06 CO CO 6 CO CO o < 00 10 CO c» CD 0) Q u. CO 00 CO 0 d CO CO CO 6 0) CVJ T- CO Oi d z LU 0 0 10 3 C3 < q: Q OJ 0 C M 0) H T3 •H PQ m 0 c »: 0 P^ 0 X o\o 2 in 0 ^ \M o\o 0) VO T3 c^ 0 e T3 C Q fO S PQ u: + 0 K •H &^ -P — (J) •H 00 -p •^ (T3 1 4-) 3 tn (0 , — . C i-H >i 0 rd CP •H > (U ■P U -p 0 (U 03 C -P U D C -P P_, -H W ■^ CN OJ ^ P Cn •H Ci. 162 DEPENDENT DATA < O 2 UJ a O 89 197 1351 28 21 57 65 12 32 1 2 3 OBSERVED INDEPENDENT DATA Ao = 77.59%AAo=-0.73% Ai = 15.87% Tsi = 0.29 ATsi= 0.21 TS2 = 0.07 ATS2 = -0.07 TS12=0.17 ATS12= -0.07 40 88 685 10 15 24 41 4 18 AO-80.1 1%AAO=7.07% Ai=13.62% TS1' 0.36 AT SI- 0.29 TS2=0.11 ATS2 -0.01 TS12= 0.23 ATS12= 0.02 1 2 3 OBSERVED Figure 25. Contingency table results for the area 2, TAU-48 equal variance threshold model 163 DEPENDENT DATA < U 2 c: O 101 197 1348 16 21 60 65 12 32 1 2 3 OBSERVED Ao=77.43%AAo = -1.46% Ai = 15.39% Tsi=0.29 ATsi= 0.21 TS2 = 0.07 ATS2=-0.06 TS12= 0.17 ATS12= -0,07 INDEPENDENT DATA 44 91 684 6 12 25 41 4 18 1 2 3 OBSERVED Ao.79.68%AAO=5.05% Ai'13.62% Tsi = 0.36 ATsi= 0.29 T52= 0.09 ATS2' -0.03 TS12 = 0.22 ATS12-0.01 Figure 26. Contingency table results for the area 2, TAU-48 quadratic threshold model 164 CD Q_ ZD O C3 O O U -J LJ in az LJ a: az az E— CjJ LJ o en CD o ccz Q_ X or , . I_l 1— t-o: cro cj on en — — > > LJ OCC (T O O s-cn qz ZLJH- uQcc SZQJ LJLJQC IjO. X -'liJf- a aa txJ z h^ — en 3 • -) -) QQ ceo: X • O cs s-0 ► •0 £-0 Z-Q ro Q-o fO- Z'O I IHODS lySiJHl •ONI Toy /oy •d3a TQy 13 cu ^ -p H QJ en H -p P ^ tn p CN 0 T3 1 Cn >-i OJ D C cn-P < ■H m &H a m 0 p P -r-i - 0 0 T! s u M fO ro en P a OJ rO n3 0 x: i 0 a q; cn •rH 0) a 0) -P TD QJ -P fd — TD rc f— ( c S-l QJ O •H 4-) « < ^ cn • CN OJ M p en •H P4 165 Tt # o ""t O a> CM o o 1 - ►- 1— < < < in CO T- co CO 00 cvi O o d < Q D Z LU a. ill o z CO LJ a o t CO CO 00 CO lO CN -^ CO T— (O isvoaaoj < a z LU O z lU a Ui !o C3) CO CM ?j in - cs ^ CN iS»33HOd (jj CD ID 1 — I z O 3 CJ g CO CD en en X o t— I ' <^ J. I I ro O tn Ct; (T D- I r '*" Li_ CM ° ^ q; LJ QD — — n X [— t ct- a: i— ';^ o i [. 1 a: i- ' • f n Q. o. fi 1 • e LJ LJ 1 a. a. L.) L") Ill (>i ^ 2 t_) LJ ^~^ ■"^ a ♦ X • o I— Q o-i •0 9-0 »'Q 20 Q-0 2'Q- i'Q- [ lyosiA 3yoas ly^yHi ? oy aaisnray g-Q- >l u e C P (U g tT>-H C X - -H fd S +J g ^ ro C 4-) 0 0) U-l tc u x: ■H dJ 4-> CW 5-1 0) <^ s: sz 1T( (d w Cn 0) +J (U 4-) n M -P 03 OJ a fd -H > 0) M o CU 0) -u 0 ■H >-^ w w x: tn u QJ M (C (d iH X! OQ W CU fd O -i 0 CU 0 0 >iX u en U < CO c s -p CU 0) (d en OJ en CU c x: 0 u •H +j x: x: -P 4-> -p C S-J 0 0 0 M U M-l +J E-« -ri -- CO < C .H T3 u (d CU c en rH 'O 0 M CU e 0 a> > rd E CO (U U 0) +J rH cpa sh c (d Oi M CU >-l ■H — 0 13 0 ^ o C -p ■^ CU u rH - ►- GO CO CD r- ^ CN CN o CM 00 1^ iSV03dOJ < Q z UJ o z UJ c UJ Q CJ b UJ f-l CD CO cs cn -r tn o o Q_ Ll- o UJ CQ ' 5 ZD I— — ^ o X ct (X. (— -^ o X cr l— • • cs Q_ Q_ r. 1 > f LJ CJ a. n LJ □ Ill III -^ Z a a ■"* D * X • 00 CO C3) r- h^ 00 r^ 00 in o o O 00 ,1 - O) CN 00 CO in CN CO C4 in to CN isvoauoj CQ fl-l g-'o 9-'o vb Z'O Q'O Z'Q- »'0- [ lyosiA 3yo3S lyayHi ? ou aGisnray "* CM (U 1 -H D i2 < fd E^ +J - >i S u e x: ro C :3 +J (U e ^ fd cn H -H (D C X M-l S-i -H fd fd -P e OJ C x: >^ 0 0) -P 0 u x: U-l +j -p (D Id w x; x: +J EH -P 13 ^ •H 0) :3 5 > cn • Q) Q) >iT3 -H J-i cn QJ x; 0) 4-) U 0) 4J (d td iH fd •H X) u U 0) fd -P 0 u +j tn tn 0 cn u >iH fd tn U H C to -P (U CQ 0) rd cnO U 0) C Di 0 M •H a^ u x: AJ X cn 4-1 0 s (U H u cn (D O E-t T! x: x: < C -P •P u (d Oi tH IH 0 M 0) g 0 -p > > fd ^ cu M cn -P -H cn-- T! C fd -H C (1) J-l •H OJ 0 T3 0 T! n a c -p 0 cn 0) u -1 e 0) a, •H rH ^ 0) T3 •H D:^ >-l T3 Q) >i 0^ 0 c S-l W -' U -H Q. XI 00 CM Q) U P cn •H fa 167 < Q Z UJ Q Z ui a. tu o z a ?^5 h- CO CM (0 CO o CO T" o o 1 o o 1 ^ r< ^- O Ul lA UI < K h- h- < < < < # ^ 0) CO o CvJ CO CD CO co (0 (0 CVJ 6 o n d 1 y ^ fN CD «- UI (/> tn < < ►- >- k— c^ •* CM CM CO 00 lO 1^ o o O o 0) II ^ M o l/l UI UI < t- ►— f~ < < < < ^ CO ^ (0 Tt T' CO CO q 00 d (0 d 00 d 0) 0) o to n oi g ^ ^ 00 ^ 00 CO in isvoaaoj < Q Z UJ G Z lii UJ a 1 o 00 - o in T— O - 00 CO CM iSV03aOd LJ ° fe r-j □ C3 o^ - -:^ CD ■^- cs o I — I z o ^ 51 3 I — I ce cc: CO cs t- cn cn o o Q LJ Qi Q_ L_ O Q:: LJ CD **N \ \ \ ^'^^^N ii:^ "^ • ' ■ ... ^^v^ "^, ^ > \ / T^ -^ /' LEGEND - DEP. flO - DEP. THR. I - INDEP. P.C. - INDEP. THR. r / / '' ^ > tL - □ ♦ X • 1'/ 1 1 1 1 1 w 1 OJ CD OT fl-Q 9-Q »'Q 2'0 I lyosiA 3^005 lysyHi ? Ob Q3isnray ro- >i 'T U CN c 1 0) e Z) cn d < c 6 H •H -H -U X •. C ra 2 0 g x: 00 U 4-> QJ 4-1 fO oj j:; -H OJ x: -p u-i J^ E-i fU x; cu +j x: u • -H -P 0 >i 5 ^+-1 cn 4-) tn CO U QJ (U 0 -H S-i c CO x: 0 CO U Q) •H 03 03 n-t cn XI cn CO QJ fd QJ QJ V4 4-1 Sh S-I 0 cn 0 O >i (U 0 CO u U cn c 4-> QJ rH 0) 03 cn 03 tn QJ C S^ 0 !^ •H P x; x: 4J 4-1 4-) 4-) C 03 0 COM U 4-) Q) E-i T! ^ cn < C 4-1 X! u 03 C CQ ^4 O M e 0 Q.> n3 M-i tn >-i QJ 4J cn ^ s^ c (T3 M >-i QJ •H QJ 0 T3 Ti 13 o C 0 QJ iH e QJ a .H r-H QJ •H Oi Xl T3 ^ fc 03 C U^ — 4-1 -H O 00 (N 0) u 0 cn Q) 168 o < cvj T- co CO a') CVJ CO d CJ c» d < CVJ cvi a? CVJ T- CO d in cj <0 0) o < CO CO cvi CO in o d CO CJ o < CO cvi CNJ CO 2P d . CO 00 in d 0) Q CD CO CVJ 6 CVJ O) T- d CO CO CVJ d T- o CO d CO o> Q LL o If) CO d CO d CJ o o CsJ d CO CVJ d 0) cn eg d O •4— o -a 0) Q. a. Q 1- Q CO cc X CO 2 LU o o o 3 Q. Q Q >. en (U 4-1 Q) fC U S^ C +J QJ en T3 •H M U-l H C 0 pa u o cc; o\o cu in X o < \2 o\o VO ^ OV rH 0) TS Ti c 0 fd e CO D:^ u O. •H — ' -p en ■^ •H fN 4-) 1 fa D 4J < to ^ M ^ < S \ro o < fO 0) •. >-i Q) ^ U c Sh 0) 0 T3 M-H C (D cn a (D Q) a TD 1— 1 fd .— 1 > fd c rH 0 fd •H > +J S-i u Q) c -U p C Ph -H cn CM (U U Cr> •H 169 LJ M CO C3 O CC CD CD o a LJ LJ en cc cc CjJ SI o CO CD O CC Q_ X cr — . f— (—a: cr CJ CJ ui in — —■ r> > LJ O CC tr o o H-CO Q^ 2: CJ s— LJOOZ CS Z M ujCJQ: ^d-x cut-. o Q Q CJ Z f— — - CD r) • -) —) CDCl aza: X # ■■■ > s-0 ► •0 CO Z-a 10 Q-Q I 3y00S lUSiiHl "ON I TQU /Od fO- J'O- •d3a TQy £0- T3 TD CD PQ CD +J O +J W « U D 0) Oi 4J •H (D Q M-l NHS •H a U-l W Eh + 0 < a CPU Oi CD C CO — N •H M •H a> -^ CO P 04 0 n 1 cn 5q (D D c D^ -M < •H 0) E-i a CO a p D -t-i - 0 0 ^ s >-i rH (0 n cn P Oi CD 03 fd 0 x: CD CU-P u CD (0 CO >iTD (d M C S-i u •-1 03 0 fd 4-1 CO P -- ■H cr rO ^ x; (D -P fO +j (T3 -P I+-I T3 n3 >H 0 TD 0 -P ^ a c +j •H (D c x: T3 CD • en C T! .^-H C CD C >1 0 a CD tj^ •H dJ d, CD -P T3 CD -P fd -"13 fd .-1 C u CD O -H -p X < -- to • O n cu u P en •H Cm 170 00 CM II o < CO ■ o < ^ o CO y— i g Eh O -H C >< ^ GJ ro 2 Cn e -M ro c CO •H 0) >-i fd -M x: -H 0) C 4-) IH U 0 ro o x: cu +J j:: ^ 0) -H +J 0^5 M-i Eh 4-1 Q XI (T3 UJ to QJ > 4-1 . 4-1 T3 S -H >i 03 QJ UJ P Cn-H > Ul CO 0) O (D CD OJ 4J 0 -H O >-i 03 CO ^ ^4 en u 0) +J 03 03 rH W X! U) CD 03 M CU M 4J >-i 0 CQ 0 U >iO o to 0 a; CO CO, 4J OJ X OJ 03 cn<; CO (u C S 0 )H •H x: x: 4-1 0) 4-1 4-1 c ^ 0 4J 0 M U 4J !h t^ T! 0 CO < C UH T) U 03 C CO M ^ 0 M 0) g rH a> > 03 OJ CO OJ >H T3 0) 4J 1— 1 Cn 0 >H C 03 g M (D Sh •H 0 T3 0 n a o c -p X 0) o -H CQ QJ a-H rH + rH 0) T3 •H ff; X! X! OJ -i^ Oi 03 C M c/2 — +j -H a 0 9"Q VQ Z'D 0*0 Z"Q- »"0- [ lyosiA 3yo3S luayHi ? oy aiisnray m u cn ■H 171 00 00 "t o a> CM 6 o CM O 1 1 M ^ p« O (fl 1/1 I/) < t- ^ 1- < < < 4 ^ ^ CM r- 0) ir> ""t o CO "* CJ C5 o o h- T~ *! r« ^ ,» (N O ^ to lA in < < h- ►- >- z LU Q Z UJ a UJ Q Z a 2 01 01 5 o CM o CO in o 5 > UJ O O iSW03dOd < a c. UJ Q O O CD CO "> Q CD CO o ^ o 3 (-—1 ro Q cr LiJ LJ or a: O 0:1 UJ CD lO CM CM II O < < CO N. to o < •^ O T— CO d d 1 ^ N (rt U1 H t- < < ^ CO •«t CM -* o CM o o CM O d CO CO d < >- »- o o 00 (0 in CM o CO C4 o T— o CO ID n (N - to o :3 9-0 VQ Z-'O Q"0 2"Q- ►■'>- [ lyosiA 3^035 itiimi 1 OB a2i9nray 9-0- ■^ CN 1 e D >i 3 < u s Eh C -H Q) X - Cn ra see -p n -H CO -U OJ s^ ra c x: H q; 0 -p U-l u o (0 x: 0) 0) -M x: s-j x: -H -p 0 Eh 5 M-l -p t3 fd W • dJ 4J >i-P Tl iH cn fO Q) P (U -H > CO -P u (D CD fd 0 •H 5^ s-l w x: +J CO u 0) CO td fd iH i2 M CO Q) rO M OJ u -1-1 S-l 0 m 0 u >.0 O CO O Pi CO C CU -p (U X 0) fd cn<; CO Q) C S 0 U •H x: x: 4J (U JJ -P c x: 0 -P 0 M U 4J >-i Eh TD 0 w < C ^ 13 u (d c en rH -^ 0 M OJ e nH a> > fO 0) CO s OJ u rH CQ 0) a •H n-l + rH o T3 ■H Oi X) 13 0) ^ CU fd c S-l cn — -p ■H Cu 0) •H 172 ■* o lO o q 6 1 o o 6 (N r~ p* o in lO (/I < >- ►- ►- < < < < ^ ^ CO o CO CO CO r^ 00 CO CM d o 1 cJ 1 ^ ^ tn - »- H < z Q z UJ a UJ Q Z CO LJ o 00 CO CN ID o CO CO CN CO ■^ CO CO o ID iSV03dOJ < Z lU o Z UJ 0. LU n r T u. LJ o n 1— 1 CjJ LO M en I — 1 ) Li) to z: r") nr f— 1 ^ H- ctr: n 1,1 f ) ^ CJ (_r IT n U) 1 > a en ^ LJ O LC LO n tn f- QC 7- •v i ) Q E-* 1 1- m f 1 3 f ) a) a: ( 1 tE CD LJ UJ CI rr az Q_ I CJ LJ (Y L_ Q- CJ Qi LJ OD in d (0 II o < < CO o 00 CO CO o CO lO d d d It II (M ^ e< tn ifl d d .^_ II II M crt (/I I/) 00 o 05 5 CO o CJ) CM 't CN CO (O in o CN iSV03MOj ^_^ QC . c> L n ce CL t- ■?• o J- r, 1 a. H- • • ro a. Q- ■ ■ I.J UJ 1 a a ( 1 n [ii 111 7' ■z. LJ o •"* ""' a * X • Q) x: ■p 1 -C D -p < •H H 5 0) •. 0) T3 x: [2 x: cu ■p m Eh (d -p (13 •H (d (U • o v^ >i 0 T3 fd Cn w CD QJ cn > i^ 4J fd 0) 0 n3 •H u-i U cn x: -P QJ U w W >^ fd -p 0 <-\ C U 0) p 0 cn ^-1 w ■H 0 Q) cn (D u >-l cn o cn Q) u s: 4-> <-{ cn-M fd X! (U Q) 03 M 0 u -M -p x: i-H 4-) >i (T3 cn U S-i T3 M C P C OJ -P 0 E^ en fd a< c c cn CJ •H QJ CO -IJ OJ u M c x; u > rH 0 4J 0 (U u u -P > U c (U T3 0 OJ Q) <-\ C M-l -H T3 fd 12 C u --^ fd 0) 0 e M -p a-p (d a; QJ u >-i ^ >iT! ■H en 0 U C T! (d g c •H Q) •H 0) U 13 Q cn e a s c P rH m ■H e x: M + 4J •H -p -H a C X y-i M di 0 fd •H CO ^-^ u e m 9'D g-Q VO Z'O 0"0 J'Q- ^"0- [ lyosiA 3^035 lyayHi ? oy aaisnray o iH n Q) cn •H 173 o < co oo CO CO 0) CO CVJ 00 < OJ cvj q o d CO lO <0 0) o < ca CO (0 00 CO d o < 00 o CVJ cvj 00 CVJ T— c» Q CVi CO 00 6 CM o o CVJ d CVJ CO CVJ d CO o CO d CO a li. 00 d 0) 00 ^— CV4 d 00 CD d CO 0) o CO d CO CO d o o 0) Q. Q in o o Q Q > Q 2 o 00 0) U M C M OJ Ti CQ •H O M-i 0:5 C Cm 0 x u < o\o lD ■* O rH \ 0) o\o T3 v£) 0 en e T! Q c s fT3 CQ + u) cd u cu •H — ' -1-1 CO -^ •H (N 4J 1 (T3 3 ■p < W Eh tH - < 2 \ro O < 03 (U " V^ 0) (13 U C V^ Q) 0 TD ^ C 0) cn CL OJ (U p n r-i fO -H > 03 ,-^ C M >1 0 03 CP •H > CD -P S-l 4-) U 0) 03 C +J U D C -P t^ -H W (N ro (U 5-1 a Cr> ■H IJ^ 174 DEPENDENT DATA < UJ O 105 120 878 42 32 60 123 21 34 1 2 3 OBSERVED INDEPENDENT DATA 1 2 3 OBSERVED AO=73.00%AAo=13.77% Ai= 17.17% Tsi = 0.38 ATsi= 0.23 TS2= 0.12 ATS2=-0.01 TS12= 0.29 ATS12= -0.04 58 62 431 25 17 35 54 10 15 Ao-71.00% AAo= 9.29% Ai=18.67% Tsi= 0.33 ATsi= 0.17 TS2= 0.1 1 ATS2 = -0.01 TS12«0.26 ATS12= -0.09 Figure 33. Contingency table results for the area 3W, TAU-24 equal variance threshold model 175 DEPENDENT DATA < U 2 UJ ex O 98 121 886 54 32 54 118 20 32 Ao=73.22%AAo=14.45% Ai = 17.60% Tsi=0.37 ATsi = 0.22 TS2= 0.1 1 ATS2= -0.01 TS12= 0.28 ATSl2=-0.04 1 2 3 OBSERVED INDEPENDENT DATA 54 62 436 31 17 31 52 10 14 1 2 3 OBSERVED Ao-71.43%AAO = 10.62% Ai = 18.95% Tsi=o.32 ATS1-0.16 TS2=0.1 1 ATS2= -0.02 TS12=0.25 ATS12=-0.10 Figure 34. Contingency table results for the area 3W, TAU-24 quadratic threshold model 176 CO f ■■■■ LJ M UJ Ul 1 C3 Q- n CD E- cc C3 O LJ SI a 3= O 1— CJ 1 ^ tij LJ CD a n-- Q_ X (— f-CC (r CJ CJ LD en ~ — . >• > LJ oce cc o CJ (- en ca^ ZLJi— uCJo: SZLJ LJCJCC ^Q_I Lje-' Q aa LJ z S-— . en ZJ • -) -) QC3 an en X • s-0 — I 1 1 — ► ■Q CO Z-Q I imOZ lB3iJHl — r- ro o 'QNI ray /ou fO- •d3a z-o- TQU £■0- 0) TD 4-> M 0) W H -P 3 U ■(-. 0) CQ 0 T! J^ O -H fd 0 Cri OJ O Oi cn OJ 0) X x: -U 2 en 03 nj 5 0 q; - ■P U --i X x: Q) •H q; +j x3 en N 0 •H M g 4-1 W Eh 0 < « CnCJ Oi 0) C CO — N •H M •H ai> cx) en p '^j" 0 T3 1 cn S-i (U D c tn-P < •H en Bh a en D p p -r-i - 0 0 T3 [2 M M fC ro tP P a OJ nj 03 0 r OJ a.+j s-i Q) fO en >i^ 03 •-i c u o ■H to 0 fC M-l en p — •H cr n3 ^ x: 0) -P rn -p (T3 -P M-i X! fd >-i 0 T) 0 -P Cm a, c +J •H 0) c x; T3 0) • en C T3 ^^ C 0) c >i 0 a dJ cn •H (D a Q) -P T3 (U +J 03 -- T3 03 ^ c S-i a) o -H -P p:; < — en • in m 0) M P cn •H fa 177 ^ lO V.V ■* r^ o , 0) ^. o O 1 "* o 1 N ^ « 1/1 t/1 < < h- ►- h- o 0) II o < 4 CO CO a o < N- h- K. N. (0 O o '1 o • w~ n 1/1 to U1 t- h- r- < < < ^ CVi o (.0 CO CO t^ ' 00 CO o o cJ < Q t- 2 UJ Q Z UJ a. lU O z CD LJ a CJ t- LJ ° f — I LJ M CD tn 1 — I z CO 13 00 CO O CO 00 in CD o CM h- 00 in iswoadoj < t- < a z UJ O z lU c m a 8 r- CN in CN 5 CO co o CO iSV33«Oj cn en X or: g o I , to O a: UJ ^ LJ DO — , — o X (V cr s— -^ o X Ti 1 (E !— < ri O- a. r,i • • Cil CxJ 1 a. 11 1 1 n (ii (.1 i' z LJ LJ '"' ■"^ a * X • LJ CO sz o-i S^O g'Q V'Q Z'O Q-Q Z^O- ^'O- [ lyosiA 3yoas lu^ym ? ob a^isnray 9-0- co I < Eh m u 3 03 (U -H C •H -P C o +J p w U 0) rH X5 03 -tJ +j m •H U-l CD +J T5 0) > •H x: u (T3 U H c (U CQ C Di Ol X U (U o ^^ CO O cn o rd cn cn +J 0) (d -p c 0 o C -P (T3 M E O CP — (d rH ■H 0) O u x: cn -P cn o -p o -p cn -P C OJ 0 T3 a c cn (U >-i O o S-l o +J u T3 0) C ^ ■H D< (d ro (U M •H 178 cvi I M O < < CO • o < 00 00 00 o o CO m CO ^ d d O t- < 00 CVJ o CSj M o < < CO CO Oi o < CO o 01 in l/l K t— h- < < < ^ CJ ^ t*J 00 h-' CO • (S CO o O ,1 ^ (N «- Ul 10 1/1 < Q Z LU Q Z UJ a LU Q z CD LJ O o CO ifi in ifi CO CN lO CD CM isvoadoj < < Q Z lU O Z OI c ai a o CM CO 00 in CM CD (y> 00 00 CM isvoaaoj o O Ul h 1 oo E— 1 OJ CD M > CD Qi Q- CJ ai cn ^> CD o = E— ' 1 CJ ^ I 1 CO Q az LJ w O LJ CQ — r — — n ce m a: (r f— -;^ o JZ Fi 1 CI. h^ • ■ en u. Q_ • « M LJ 1 n n ( 1 Q (ii Ul z -i: CJ i_j ^^ a * X • — r 9-0 — r ct: o Z-Q [ lyGSiA 3yoos ib^aHi ? oy a3i9nray 9-0- 00 I D < E-i o fd o g c :3 0) g ftj Cn-H C X •H ra +j e c O 0) u x; 4-1 0) w x: x: -P Eh -u ■H -H 0) • V^ Cn (1) dJ 4-) -U (13 fC -H !-l U -M O W W U) >i H (T3 U M c to Q) CQ Q) tji O i-i C 0:^ O ■H d^ U c -l X3 fO x: -p m •H (U x: 4-) 4J (d > •H x: 0 (d (D s^ 0 u 4-1 (d cu x: 4J > 4-1 rH C QJ Sh 13 O C 4-1 (D O a,-H QJ T3 Ti QJ C S^ •H Oj QJ U a cn P4 179 00 I II o < i ^ CO CO N. r- 00 CVJ W t- < CM O d 1 < U5 < CM CM CO I o < W5 0) lO lO o CD o CD o UJ I/) O O isvoauod z UJ a z lij LU a O) CO - in CO CO CO o in in in CO CM isvoawoj CO LJ ZD a 31 '^ LJ ° f— I LJ M ^ CD tn o CJ _ CO ci) LJ r, LTJ cn f- Or: i o CJ 3 I I ro Q cc LJ ^ o- , UJ — ~ o X l— 1 (ir a. e-^ ■?• o -L r.l ct 1— • • r1 00 u '^ C 1 0) e 3 !T> 3 < c g E-i -H -H -P X - C (T3 s 0 e r ro U P QJ •H to 0) r H 0) x: +j 4-1 M £h (d x: 0) ■p £ >-l • -H -P 0 >i 5 14-^ !T> +J cn cn u 0) (U 0 ■H V4 C cn x: 0 w o QJ -H (0 CO r^ cn j3 cn cn QJ (0 0) Q) >-i +J >-l S-l 0 cr^ 0 U >i -i ■H a x; x: -p -P -P 4J c to 0 C 0 M U -P 0) Eh x3 ^ cn < C -P T3 U fO C cn ^ !^ 0 M Q) e 0 a> > (0 1+-* cn 0) s-i a; -P .H cn^ )-i c (0 --H >-l (U U •H (U 0 TD 0 T3 T3 U c +J 0 0) u -H e ro (U }^ 3 tn •H Ci^ 180 in o < CO C\J in CO CO d C\J d C\J in d < c\i in cvi CD cvj r- 00 00 00 d 0) o < T- 00 CO cvj C33 C\j in 00 d CO o < o CJ) CD CO CD (0 0) CO 00 CO d 0) D CO 6 CO CSJ d CD O CD CsJ d o CO d CO 0) «-* Q u. CO T- d in o d CD in CO CM d o d o o CO d o o Q. a. Q 1- Q 03 en 2 LU o in 00 3 in C\J c» > Q Q) U c Q) 13 ■H M m M G 0 CQ U O Pi 0\0 CLi LT) X O < \S o\o 1^ ^ 03 ^-^ C -H >i 0 (ti Cn ■H > q; 4J h -p U 0) (T3 c +J M P c 4-1 Ua -h cn r^ ro OJ M 3 tn •H Cm 181 LJ N C3 Q_ O cr CD CJ -J LJ en CO cr az LjJ 51 O CO CD O ce Q_ X or „ I — 1 ^- s-a: cc o CJ en CD — — > > LJ O (K -. in ZD • -1 -) QQ azcc X • ^•0 I 3yoos iy3iJHi fo ray /oy •d3a O'Q Toy T! (U +J 0) W > P OJ •H •r-1 S-i U-i T! 0 fO 0 M-l CO 0 0) - x: -u rH 0) -P (13 Q) N 0) Ti ■H 0 u 0 W 4J ^ e ^ Cn (U Q C N M s •H •H CQ Q. W E-i + P < cc: 0 CnU ex >-j c en ^— Cn •H M a> CO (d p "H* 0 T3 1 (U 5-1 0) D en Cr«4J <: fd w Eh u W D P -r^ «, U) 0 T3 2 •H M rd ro x: P +j a oj (d 0 x: 0) >-t a-u u 0 a D-, >i'^ rH C u rH (d 0 • (T5 4-1 ^-^ p — >i CP (d ^-^ Cn QJ -U fd (U 03 -(-) -U 14-1 TD (d fd 0 T3 S-l -(J V a c +J U) Ti •H 0) c QJ x: T3 QJ H 4J en c T3 M O C (U C OJ 0 a (U CQ r-\ •H (U ao (U 4-) T3 0) a: cn (T3 -' 13 fi^ 1— 1 C X w (U O ■H iCjJ fd Pi < "''—*' ^ 5 CO m cu u p en •H U. 182 in en o # 00 in o lO 0) ^- h* CJ o o 0) o 1 1 T- M M H ^. «N o tfl (/I Ul < »- h- t- < < < < ^ c^ CJ) CVJ CD CM o CO -* o o C> h- ^m n ^- o ^m 1/1 Ul 10 < < 1- ►- K CO T- CNi (I o < < in 00 "^ s. o < 00 CM ^ 5 Q) O I T- CM CM O I CM CO O 6 o < z UJ o z UJ a. UJ o z CO LJ ID CD LJ ° t — 1 LJ M CD (?) Z CD 1 — 1 z en cB g o CO CO s o o CO lO o 5 > IX UJ m O isvoaaod < z UJ o z UJ c. UJ a o 0) o r^ CN CO o in T— o CO iSV03HOd D > Q en Qc: Q_ O LJ 1 X 2 CJ LJ *"* *"* a « X • Ift QC O i- O Qi CO Q-Q 9'0 VQ 2-0 : lyDSiA 3yoos iu3yHi Q-O J-0- VO- ? OB a3isnrau 9-0- CX3 ^ 1 fO ID < > tH 0 c •- 0) s ^ e +J n c d CO •H e u fO -U •H •H QJ c X M-l S-< 0 ftj rd u e OJ J-i 0) Q) -p 0 x: x: m Eh -p -p a (T3 UJ CO x: > -P • -p T3 oe iH >i-H Q) LU a O^ 5 > iO m CO u cc; c Oi CO -P 0) X (D (d en -i OJ c ^ 0 M •H u x: -P 0) CO -p c x: 0 -p QJ M u CO Jh 0 Eh T3 0 x: < C m -p U fd CO rH . — , 0 M 0) E -H 4-) > > (d 0) QJ i-l Ti CO -p ^ Cri 0 TD c fO e c Q) >-i •H 0 T! 0 T3 Q a C -P s CO QJ O rH CQ 0) a-H nH + S-l QJ T! •rH a u T3 Q) ^ ex 0 C U 01 ^ o •H CLi • o^ m Q) U a cr> •H Cm 183 CO o CO o C4 O cvi d 6 1 O T- ,, M ^ d o 1/1 00 1/1 < h- ^— ►- < < < 4 ^ ^ CO N- o CO T- Tt 0) Tj- ^ T-^ h". 6 o CD h". r- n (N ,j ^■ ^a o ^ LO i/> (/I < «J 1- h- H < Q Z UJ Q Z ID a. lU Q Z cn ID a CJ UJ CD CO CM in CN CN CO (D CN CO CM CO CM a isvoauoj z UJ o z UJ c UJ Q Q >• Q o ^» CD on ^ ai cn O ^ E-' I Q LJ UJ r. oD a; *"■ UJ DD (r o -, — o X q; cr H- T o J. cr t-- ■ • a_ Q_ fl 1 ■ • LJ M 1 n n n a OJ (ii -^ z o t_j •""* *"* Q ♦ X • ^ CO 0> CM CO U5 in CD d O d M " ^ M O Ul l/l l/l < ►- H ^ < < «1 < CM 0) o (0 CM s. OJ 00 in o o CD " „ *- (N o ^ 1/) l/l l/l < «» f- h- K CM 01 CO CO in CM CO cn CM CN CN CD r- CN iswoaaod g'o 9-'0 >0 20 Q"0 Z'Q- ►'0- [ lyosiA 3yoos iu3yni ? oy a3isnray 9-0- 00 1 E 3 >1 p < u e tH C •H cu X V cn fd s c e ^ ro -H 4-) -P (U M-l rC C x: •H q; 0 -p 1+4 U 0 03 iH 0) 0) -p ^ 5-t s: •H 4J 0 Eh ^ U-l 4J T3 n3 cn . 0) ■M >i4-) Xi rH cn (T3 0) ;3 OJ •H > to 4-> O CU 0) (T3 0 -H !-i M cn £ 4-1 cn U (U cn fO (T3 <-{ n H cn (U fa M OJ >-i -p M 0 CQ 0 u >iO u cn o cc; cn C 0^ 4-» 0) X OJ (C3 CTi< tfl 0) c s 0 M •H j:: x: -P (U 4-1 4-1 c r 0 -P 0 M 0 4-1 u Eh nd 0 cn < C U-i T! u (13 C CO .H 0 M OJ e ^ a> > fO 0) CO (U U TD CD 4-1 rH cn 0 U C (c e U (U 5h •H 0 T3 0 T3 Q U c 4-> S (U o <-\ m 0) Q. •H M + r-H 0) T! •H Di i: TD QJ M cu fO d M w -- 4J •H Qu XI CTi PO (D M p cn •H Pm 184 in CO O) o in < 1- < < ^ ^ (0 0) CO (0 1^ o • ^ o ^ U1 4 < ^- CO o ^ o CD 1 1 II N in 1/1 l/l < Q Z lU Q Z UJ a LU Q Z cn LJ CD "^ fe LJ ° E— t UJ M CD cn 1 — I z cn o 5 CO to to CN 00 (O o If) i n3 ■rs fd cn •H QJ (U U > u +J 0 cu 0 (d cn •H m >-i m -C -p (d u CO w fd -U cn <-\ c 0) cu ^ 0 M 5-1 w ■H 0 0 OJ cn u 0 S-l cn 0) tn cn 0) ^ cu -p iH cn cn (d XI 0) 0 (U fd S-< -C u +j rH -p -p >i rd o u U -p M c P CD -P ■cn tH D^ fd Ti < C c G U •H 0 CO +J OJ OiM c x: cn > iH 0 -p QJ CU o ^ -p > u M c CU T3 0 0 cu rH c 14-1 u X! (d C u ^-^ 0) CU 0 e rH rH a-p (d (U XI cu U S-l Ti (d X! -H cn 0 -p c T3 (d e ■H CU •H >i Sh Td Q u e Q. s c p i-i CQ OJ E x: rH + cn H -P •H « C X U-l ^ O^ H (d H cn -P e ^ u CTi m (U U 0 cn •H fe 185 o < CO co 0 If) CO CO cvi CO CVJ 06 < cvi CO cvl T- cvi d 0) 10 to a> ^^ o < CO a 00 CO CO CO d o < 00 CVJ 10 ■ d 00 d CO Q LL Oi ■f- d 10 T- €VJ d CO CVj d CO CO d CO 0) Q T- 6 Oi 0 r^ CO C\J d CVJ CO 0 CO d CO CO d o o •5 w Q. CVJ Q m 0 0 0 a 1- z UJ Q > Q DC < QJ 0 C 0) T! •H U-J G 0 U o\o m * 0 .H \ QJ o\° t3 UD 0 OV g ti Q C X (d m + to tf U 0^ •H -' 4-» Ui CO •H "<;r -P 1 (d D -P < M Eh rH •» < s \rn 0 < (0 0) - M CD (T3 U , — , C S-i >1 0) 0 en TD 4-1 QJ C +J 0) W rO a 0) U oj 0 -P T3 i-H w fl -H > H It M C rH 0 (T3 m •H > 0 4J M rt U QJ a, C -P X i • 0 ■'S' 0) ^1 3 en •H h 186 DEPENDENT DATA 3 1- 125 125 920 < O 2 50 28 51 IT O 1 115 33 40 1 2 3 OBSERVED Ao=71.49%AAo=10.92% Ai = 17.42% Tsi=0.32 ATS1- 0.15 TS2=0.10 ATS2=-0.03 TS12=0.25 ATS12=_0.10 INDEPENDENT DATA 57 73 461 21 17 20 54 21 19 1 2 3 OBSERVED Ao-71.60% AAo=13.17% Ai = 18.17% TS1=0.31 ATS1.0.17 ^S2 = 0.1 1 ATS2--0.04 TS12=0.25 ATS12=-0.11 Figure 41. Contingency table results for the area 3W , TAU-48 equal variance threshold model 187 DEPENDENT DATA 1/; < O 2 124 124 918 1 55 30 54 111 32 39 1 2 3 OBSERVED INDEPENDENT DATA 1 2 3 OBSERVED ' AO = 71.22%AAo- 10.08% Ai=17.82% Tsi= 0.31 iTsi=0.14 TS2-0.10 ATS2^-0.03 TS12=0.25 ATS12=-0.11 54 74 459 24 18 22 54 19 19 Ao.71.47%AAO=12.76% Ai=18.71% Tsi=0.32 ATS1-0.17 TS2= 0.1 1 ATS2= -0.04 TS12-0.25 ATS12--0.11 Figure 42, Contingency table results for the area 3W, TAU-48 quadratic threshold model 188 u o ■"T" -Ol cc CJ UJ cc __ t— t , 1 01 1 (— ca CD crcj ; -oo ■z. O ocn en ^— •— ' =) — . >• r. 1 Q_ cr > M n H- UJ o a cc (T) q: 1 (E o C3 LJ CJ 1— CD ' CS Li_ 2= ZUJf-i Q_ a QJ ujao: =) X OJZOJ O o o CD ljujq: LiJf-r a CS [—1 ,__, • ■ \ CJ LJ '— ' □ C3 LJZ ) -to ; CD tn M a 13 • cn Q. X tree cr 1 1 2= X • -U) _ S-Q k-0 I 3 1 1 1 1 1 1 £-Q J-Q to Q-Q fO- Z'Q- £'0 yoDS iu3yHi 'ONI Toy /ou •d3a Toy -* TJ 73 Q) OJ -P 4J O W H QJ D QJ H M •n 1 ^-1 QJ T3 0 CQ U) (T3 u o m vc W 0) cu fT3 x: 4-) X 5 -P (T3 (D g +J 0 M x; -p x; «. Cji -p r-l •H OJ QJ QJ N M T3 •H 0 >+-i W < 6 0 CnU « 0) C en a, N •H M - — ■H a> W D o 0 -tJ o Cn u OJ 1 C tn4-i D •H to < a en P H p D •i—i 0 0 xi «. Sh -H (T3 •^ en a a OJ (13 fd 0 j:: QJ a+j >-i QJ fd w >,T) (d rH c U u .-1 (T3 0 td M-i 0) p , — , •H cr (T3 — ^ x: 0) 4J (T3 -p (0 -P 4-1 'TJ n3 u 0 TJ 0 +J fc a C -P -H 0) C x: Td Q) • w c T3 .-^ c QJ G >^ 0 a QJ cr> •H QJ O* QJ -P ■D QJ -p (T3 ^ t3 fd rH C i-i 0) o H +J cc; < ^~^ w ro •<3' 0) V^ D Di •H fc 189 ^ N N Tf CO o lO o o o ^- o o 1 1 » (N ^ — < < < 4 ?^ ^ in CVJ N- If) ^ Tj- CJ o ' . Cvj 10 o o o 00 T- n CN u ^ CN o «» 1/1 t/» cn <* < t- t- ►- < Q t- Z liJ Q Z ai a lU Q z CO ID a ^ eg CO 0) CO CO o 6 o n " w* rt o If) 1/1 (/I < f— h* ^ < < < < ^ CO c^ r- q <3) lO CO CVJ 6 00 C) CO CD CD CN 00 00 in CM r- «* CO T- "^ iSV03dOJ < Q 2 a z UJ c m Q 0) 00 CN CO T— CO CO 1^ co CO o ^ 0) CN (O CjJ o CD ct: o CJ en en " en Sc Ct:: cc o ^ CJ ^ 5 £ {jj q: Ct: "^ L. § I— LJ CD - o ct X o a: CE e- T' CJ -L r,i a. t— • • rn CL Q_ lil • • LJ U 1 a. a. t:i CI [i] li) i- 2 CJ CJ *"• ^^ □ * X • [ lyosiA 3yoos iy3iiHi ? Ob aaisnroy -1 — cc CJ o o 1 1 rt: B^ (U x: - >i +j '<3' U C 0) -p fO dJ ^ 03 (U Cn-P M c n3 n -H s: (U -p -p > S-l C -H Q) 0 0 ^ •H 4-1 u x: TD o t/) 0) 0) 03 ■P -C -P nH E-< fT3 QJ P -H M (/] u 0 0) • 0 O >-i >i cn en cr> en dJ QJ (T3 4-> .-1 +J 03 X! fd w CU fd s^ (u S-l -P -P J-i x: en 0 -p >i u u M tn M c (U CQ CJ £h cnO CO < c « 0 U •H a, x: cn -p x: +j M c < > -^ 0 S 0 GJ U -P •P > CU C (D "O x: fi (U <-{ c -p "o n 03 c C >-i u o 0) 0 e 0 a a-p (13 m W Q) U >-l 0) 10 -H CP-- s-l C T3 03 .-( 5-1 •H OJ •H 0) 0 5-1 73 T3 U e a 0 p nH e 0) e x: ^ M •H 4-1 •H PC XI X 4-1 Ai Pi 03 03 -H cn -- -P e ^ n3 ■^ •^ 0) M p cr> •H fo 190 HZ 00 CO I II o < < iq oj ^r o < 00 o CO o O) CNl 1^ o o CD <-> — II " M ^ (^ r- trt U1 U1 < < < - u^ !^ # CM h- ^_ o 00 t^ 00 CO o 00 o 6 o n w- M o in 10 ifl < t- h- h- < < «< < ?(5 CO 2^ f^ CO q oi 6 CO CO CD CO 6 " It ,. (N o lO 1/1 1/1 < < t- 1- t- < a z ai a. UJ Q 2 CD LJ ID a 00 o CN in CM CO CO 00 CO CN ^ en o UJ > cr m 10 o o isvoaaoj < Q Z LU Q Z LU C UJ Q T— CM (0 CM 5 ^ 00 in 00 CO t^ - CM CM 00 iSvoaaoj X 03 ri L. LJ CD cc 1-' F— 1 LJ z M LJ rn CD V z: fO ►— * ^ in ct: n UJ f ) 33 ^ CJ O U) CS 1 i— m en I^ K ^ m n C ) ( ) rtr > (/) a. Q Q^ rr ^ ( ) i: O E-' 1 in r ) > ■«" tn f 1 a: cc LU UJ or a: i O e x: - c p 4J ^ 0) e U-i cr^ -H •H Q) c X m M -H nj fd +J e 0) c x: U 0 0) 4-1 0 u x: ^ 4-1 4-1 w • QJ 0) >iTi •H )>4 en 0) j: 0) +J U 0) 4-1 rd (T3 rH rtJ •H i2 u u 0) fO -U 0 5-1 4-1 CO en 0 en U >lM (0 tn U H c to 4J 0) CQ 0) m cno J-i (D C D:^ 0 5-1 ■H Oi 0 x: 4J X en 4J 0 S OJ M u en Q) 0 ir< T) x: x: < C 4-> 4-1 u n3 CO iH i-( 0 M Q) E 0 4-1 > > ra 4-4 QJ M en 4-1 M en-. 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Q > X h- f- □c 1- co 0 0 10 UJ 1- Z UJ o\o LO 0 Di \.a^ o\o ^^ V£) CTi 0 0 T! 1 C D td -l 0 <-{ U-l <: \ w ^~. 0 cu >i <: p tJi r— 1 QJ - (13 +J QJ > fd U u C -H U) 0) fd T3 > M C M H 0) 0) a-u CQ (U c 0 13 -H « cu 1— 1 0) X (d u < c c s 0 0) •H 13 •. -P -H rH U M-l 0) c c 13 :=! 0 0 Cu u e • in ^ QJ M P CP •H fa 193 \ '* o •f \ \ -o> cr LJ \ * UJ a: cr \ \ »— t , , \ 1 CO 1 f- \ '^ (-cc y * -CD CD a cr c_3 \ :z o 3 CJ en on-- LJ Q_ cc > 1 M 3 E- UJ » a oq; * tn az I OIO ^ CD LJ o t- tn : [ t Li- ZI zc-Ji- \ ; CL O LJ ijoai \ 1 rj X fiZtiJ \ . O o CO LEI DEPE THR \l a: c^ (— . \ 1 CJ , ^ acD c • - S-i c T! 0 ftJ U M^ U) 0 0) x; -p •. (D -P rrj -H N 0) q; -H 0 u 13 cn -P £ 0 -P e en OJ c N M Q •H •H s Q. 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OB anisnrau 9-0- ■^ CM 1 D T) < OJ H > 0) xi (U -^ a; -H ^ t^ 4J ^ d o fO -H (li 0) • u M >, 0 (U (d iT> en M (U W 0 Q S^ -U fO u UJ 0 fT3 W > U-l >-| w cr 4J QJ 4-) UJ if) m u ^ LO CQ O 4-1 0 QJ M C U M :3 0 en r; 0) -H 4-1 dj en OJ V4 en tn M (D 0 0) U x: ir* rH Cr>4-1 < XI (D U 03 J-l 0 CO 4J 4J M rH > >i (li cn U S-i Id 4-1 -H C D C G Q) CU 4J 0 0) > D^ (d CLtJ 0) C C LO C -H ■H (U CD 4J (u j-i a >-i c x: !-i 0) 0 0 4-10^4-1 u u c u >-l -H -H Ti 0 CU 'O C U-J rH g Q) fd Xi 0 u -- d g a e rH 4J -H fd Qj X x: M T! >, fd 4-) cp 0 u e ^ fd e c; -H ■H OJ 0) ^ T) Q Cnx; S C 4J OJ rH CQ -H J^ rH + 4J x: -p •H a c 4J ^i CU 0 -H +J en — u 5 fd u ro in 0) j-i a Cr> -H t. 207 o r- < CO CO CD CO in CVJ < in cvi csi T- (0 CO CO 0> *-» o < CO 00 CO in CO CvJ o < 2P CO CO CO CO CD CO CVJ 0) CNJ 06 0> Q CO CO CO 0 c> 6 CO in 6 in 0 CJ a CO 0) >»» Q li. 0) CJ> CO 0 6 in in 6 00 Oi T- 0 T— CVJ 6 o o ■5 Q. C\J Q m 0 0 0 IT- LU Q. Q 3 Q in CVJ 0) > Q OJ u c q; TD •H M-l C 0 0 o\o lD 0 ^ ^-H o\o QJ ^ XJ (T\ 0 e TD C Q 03 S CQ en + U 0^ ■H &, -u — en •H vr -p CN fT3 1 4J :d en < H r-l < - \^ 0 < (K Q) - S^ 0) fd 0 ^-^ C i-i >i 0) 0 tji T^ M-l 0) C 4-1 (1) cn 03 a 0) U 0) D 4-1 TD -H cn ■TJ -H > M 03 M C -H 0 fd CQ •H > 0 4-1 U Di U Q) O, C -P X H^ C < Cl, -H s ■^ LO QJ V-i a !T> ■H |i( 208 DEPENDENT DATA < O 65 299 2386 5 37 71 11 32 32 1 2 3 OBSERVED INDEPENDENT DATA 31 145 1207 2 16 34 5 14 15 A0^82.85% AAo = -12.25% Ai.13.85% Tsi=0.08 ATsi=0.05 TS2=0.08 ATS2 = -0.05 TS12=0.08 ATSl2=-0.09 AO>83.59% AAo=-13.15% Ai=13.27% Tsi=0.07 ATsi=0.05 TS2=0.08 ATS2 = -0.05 TS12-0.08 ATS12=-0.08 1 2 3 OBSERVED Figure 55. Contingency table results for the area 4, TAU-24 equal variance threshold model 209 \ f - o ■«r \ ^ -OT en LJ \ 1 LJ N \ i ^^ CO 1 V 4 -CO ca 00 or o ■z. 3 c_j en in —. — ' > \ i r. 1 Q_ CE > \ I M r) E- UJ \ [ o a q; \I to fV^ 1 (TO Y C3 LJ i- en L CD qz T L- ■sz ZLJM u Q- O QJ X Mate CSZOJ \\ O a to ljIjJCc: LJl— \ \ tt: 1 — ■ Q \ \ (— ' ,__, \ * ■— ' LJ z X • -to -J CD en u O :3 • en cc: X —1 — ) cr en CE 1 1 z: X • -Ln s-o I >0 1 ■ T ■ ■ ■ t 7 £-0 j-0 10 0-0 r 1 0- E'O- fO - ^ I 3 yoos lya^Hi -qni Toy /ou "d3a tqu T3 T3 QJ 0 -P 4-> O W M QJ D QJ H 1— 1 •n S-j QJ TD 0 m cn (d U o CO a w . > -M M QJ 0) cu tn N M T! •H 0 "4-1 CO Eh e 0 < CPU ce; QJ c w CU N •H M ■ — •H a> to p CO 0 T) •^ Cn J-i 0) 1 C Cn-P ID •H CO < a CO D H p D -r-i 0 0 T! •. ^ rH fO "^r cn p a 0) 03 03 0 x: 0) a+J S^ QJ 03 CO >,T3 03 rH c U U t-\ 03 0 03 M-l CO p — •H cr (T3 ,— , x: 1 0 a 0) cn •H 0) a QJ +J TD (U -P 03 — TD 03 rH c U 0) o ■H -P a < "^ CO • U3 LD 0) M p Cr> •H Cn 210 o < i 111 ^ ^ a CD o « X • LJ m ■Q 9-0 Va 2'0 [ lyosiA 3yoos ly^yni O'O 2"Q- ►"0- ? oy QGisnray 00 ^ 0) 1 rH D i2 i e ^ 'g' U :3 -P c e m td OJ -H H QJ Cn X M-I ^-1 C rd d -H e 4^ O +J n3 en 0) j:: -P ^ -M 'T! rH H -H tu :3 5 > en i oj x: en -P u (U QJ ra n3 rH 4-1 -H X) (T3 U 0) (T3 V-l 0 S-I -P -P cn 0 en CO u >i (0 en U M C en -P (U pq Qj fd tT^O u 0) C P^ 0 IH •H p^ u x: -p X en +j C < 0 S QJ M u en 0) 0 E- Ti x: x: < C -P -P U (T3 cn rH >-i 0 M 0) B 0 V > > fd M-i 0) s-i en -P rH CP^-T! c tC rH C (U U •H OJ 0 T3 0 T3 TD a c 4J 0 en 0) u rH e OJ a •H rH >^ Q) n ■H C^ u T3 d) ^ O^ 0 c U W — ' u •H Qa • fd r-- LO 0) >-i :3 en ■H fa 211 ^ 00 CO CO o o 1 6 o o - o q d 00 II o < CO CD (d o> o 00 6 (/I < 1 5 CNi CO 1/1 CO 00 < Q 2 lU Q Z OJ Q. UJ Q Z LJ a CD cn o ^ 't CM CO 00 CN ID CN If) CN CD ■<* or LJ CD ISVDBdOJ < Z Ui O 2 UJ c UJ o If) tD CM CN CD ■<* h- CO CO CO CD ■^ o O iSV33«Od CD a CO CD cn 'Z > § CO Q- O n o a (E cc: cn ^ — o (—1 ce a: t- "P- o X T i— ■ . rn a. Q- * ■ t.j OJ 1 n n. in tJ (ij i>) w z L-l t_i ^^ *"* o ♦ X • O'l 80 9"0 »'0 Z'Q 0^0 Z'O- >"0- [ lyosiA 3yo3S ly^yHi ? ob a^isnray 9-0- 00 •^ E 1 e < ■H E-t X >i fT3 - U E x: '^ c +-I 0) (U M-i (0 cnx: •H 0) c -p iw 5-1 -H nj 4J x: 0) c +j x: V4 0 •H -P 0 0 5 4-1 +J OJ T! n3 cn s: OJ +J Eh 4J t3 M (T3 CU a •H > U) ' 0 CU i 0 •H M cn cn x: OJ cn u (U -P rO fCj i-H (d -Q u cn 0) fd +J QJ S-l 4-) W u 0 0 o >.M o cn U M cn C -p 0) CQ (U fO cr^O cn QJ c c^ 0 5-1 •H a, x: X +J X 4J +J c < 0 s 0 M u 4-> (1) E-" TJ x: CO < C -P TI u rO C C/2 rH S-i 0 H 0) e 0 a> > rd U-i cn (U >-i (U -p ,— 1 tj^^ M c (C 1— 1 ^-( QJ u •H 0) 0 Ti 0 Tl T3 u C -p 0 (U o -H E 0) a •H rH 1— i CU TD ■H ec; X! TD (U ^ 0. fO G V-i w -^ 4J •H a ^ r^ in » <0 T- r- , 0) N- d CD '1 a ,1 ^ *- o ^ to 1/1 lA < < h- h- >- # o (0 CO d a CO d d N- ,r (N ^. (N o in 1/1 l/l < »— 1- ^- < < < •* ^ •^ ^ CO o (6 CO o 0) CO ri o> CO d d " ^ (N o CO 1/1 i/i < < »— 1- t- < a z O z UJ a 111 Q z CO ZD a M CD ■^ IT) 00 T— 00 CO CO in in in 00 CN iswoaaod o o en tn q; en ^ en tE o ^ o ^ 5 £ ct: "^ Q_ I o g I— LJ CD < z o z UJ C. LU Q CN in CN o CO CN CO CO - T- CN 00 isvoaaod Q-I T 8°0 1 '- n X r~i ct. a: I- (.D J. r, 1 (L. (- • a. a. r. 1 • ■ LJ LJ a. n CJ Q Ill III i; Z a o *"■ o * X • g-Q VO ZQ [ Ltiosw 3yo3s iy3yHi Q-0 z'o- yo- ? OB a3isnrad >i O 00 C rr QJ 6 1 en 3 DCS 0 >i ^ S M-l tji 4-> LU QJ TO fO l/l Ul 4-1 (U m 4-1 fd 4-1 X! O rH ^ ft3 dJ 13 4-1 -H > CO W U OJ 0) 0 -H V4 c w x: 0 en u QJ -H fd (d rH cn xj to w oj fd -i 4J Vh S-l 0 en 0 u >i 0) u w u 5-1 en C 4J 0) >H dJ n3 CT' fd en 0) C S-i 0 5-1 ■H D x x; 4J 4-1 4-1 4J c (d 0 C 0 M U 4J (D Fh T3 x: w < C 4J TD U (d C CO rH 5-1 0 M QJ e 0 a> > td ^4-1 en 0) 5-1 0) -P rH en-- 5q C fd rH s-i 0) 5^ •H 0) 0 T3 0 TJ 'd U C 4-1 0 QJ U rH g dj a-H M rH 0) 73 •H CC; ia TD Q) ^ (1h td c ^ CO ^ 4J -H a u r- in Q) u p en ■H IJH 213 o < Cvj q CO 31.5% T- CM < CO CO CO CO CM # -^^ t^ CO CM CO Oi o < CD CO CO d CO d in CO C£> CO o < co CO CVJ CO CO CO CO CO d 0) CO d 05 Q LL CM CM d CO d CO 0 CM d CO in CM d CO 0) «_« Q 0) CO 0 6 CM in d in CO T- d CO CM d 0 0) CO CM d o o ■5 0 in C\J 0) 1- > Q. Q > Q DC h- Z LU Q- Q D Q >i en 0) OJ -P 0 (T3 c v^ QJ 4-1 T) in •H m M c M 0 u m 0 o\o « LO 0, 0 X 0^0 g V£) (T\ •. nH T3 0) C 13 (T3 0 S cn u d:; •H Oh -P *^ W ■H CO -P •^ (0 1 +J 3 w <: Eh .— t < - \"^ 0 < (d 0) •. M OJ (T3 u c U (1) 0 -T! y-i C dJ cn a 0) (D a -T! -H (t3 r-\ > (T3 C rH 0 (T3 •H > -U u u (U c 4-> a G P-, -H in u 214 % X • _o ■ Y\ LJ Q_ az \ "v M ID t- tj \ ^\ —* a occ \ ^ V cn q; a: o \ ^ o V CD C3 t-cn > LJ aZ 1 — 1 Li- zr ^LJ!- , Q_ □ LJ X / O z CJ / a: o (j~i CD E— . CJ LJ ■— ' a a LJ z t— * •—• t [ - \^ cr cc \ CE 1 1 t 21 X • J k • -m ^•0 1 1 1 fQ z'o ro — 1 1 ■ 0-0 ro- Z'O- £'C - I 3iJ03S iy3yHl -QNI " rob /oy "d^a Toy T3 (U -P CO D (U ■r-i U n 0 (13 U 4-1 cn O 0) x; 4-1 - oj +J (13 .H N 0) 0) -H 0 >-l X! cn -p x: 0 ■p e cn Q) c N M Q -H ■H SO. cn E-i CQ :3 < -H 0 !TiU ff; i-l c cn 0^ cn •H H ^ a.> 03 D 00 0 T3 "^ Q) s-i (U 1 cn Cn-P D (13 cn <; u cn p &-I D -r-i cn 0 "0 --H rH (13 -^ J^ D -p a, oj fT3 0 x: Q) V4 a-P J-i 0 (U &-. >iTD rH C i-l fH (13 0 • (T3 H-i ^ D — >. CT ro ,-^ cr> CU +J (T3 (U T3 (d -P +J QJ U-i T3 (13 (T3 -P 0 T3 J-i U -P -P Q) a C -P cn rH •H QJ C q; x; T3 0) M tn cn c X! H C QJ C cn 0 CU (U CQ fa •H Q) CUO ^ +j Ti oj 0^; (13 -^ TD D-i Q) -H C X C 0) o -H <; •H Oi < — s C o^ Ln dJ S-i P D^ 215 «i8 CM to I II o < < CO 0) CO "' T ^ o < in o o o d o d ,, ■« n ^ CN ifl U1 (/I >- h- h- < < < # in lO 00 CO CM r- d d d ^ a CM CO 00 CJ) CO CO CO o 0) d o d " V M o (/I 1/1 1/1 < h- h- ^ 4 < < < # (0 CO 00 CO 00 o > 0) o CJ) 00 CM o o o ^" II r* " ,1 ^ fN v> o ^ Ul 1/1 U1 < < »- t— H- < Q Q Z a. lU Q Z CO ZD a LiJ o o ^ CJ o -^ o (/) S; Ci; cr O ^ I — I (jj LJ CE tt: <^ Cl_ 1 L_ h; (— a: UJ CD c:5 in ? 00 ^ CN CN isvoaaod < Q z LU O z LU C (U Q CJ) CO in CN - CO - o CO (O isvoaaod > Q •— • Q „ . ^ cc ,_i ■ ( 1 X a (— 1 (K cr. 1— i: T" O X m fi 1 cc t— • rq Q_ CL LO M • ■ UJ LJ ce n 11 n a o iij LlI ^ z i— o a u a • X • 90 9'0 VO Z'O [ lyosiA 3yo3S iB3yni O'O J-Q- VO- ? OH Qsisnrau 00 1 1 o >1 < u Eh C Q) 0) x; - cn 4-J -T C •H dJ -p CO -p x: 03 0) C -P Sh 0 n fO U £ Q) -p > V4 (U -H i to dJ 13 Cn-H 5-1 W 0) U 0 (U -P 0 u ^ fo cn Vi M w 0) -P (C +J rH en fd x: w OJ (C M (U u -P u sz CQ 0 4-) >iO O u IX in M C 0^ 0) X Q) Eh fTi<; w < c s 0 u ■H x: en -P 0) +J 1— 1 c ^ > rH 0 -P 0 OJ U -P ■P > Sh C 0) Ti 0 W 0) rH C U-i TJ rQ ra c c u ^ 0 -i T3 OJ T3 -H en 0 Sh C T3 (0 e M •H (U •H 0 >-l T3 Q U E a. S a -H CQ (D e ^ M + -H •H 4J •H ce; X! X 4H ^ eu 1X3 rt3 -H W -' -P e ^ (13 o MD q; u a en -H P4 216 ^ CO (0 1^ lO CVJ n 00 CM 1 r- d d d ^ f>< o 0) i/i Ifl < ►- h- ►- 4 < < < # ^ o 00 o s- a> in CD o d d K Y« It a CN • ^ M o ^ m 1/) trt < < t- H- ►- < Q I- Z UJ Q Z UJ a UJ Q Z CO LJ ZD a in 7— CN en 00 CN in in CO O CN CO r~ CiJ isvoaaod < Z UJ O z UJ c UJ Q O fe o ^ E— I LJ M UJ ct: CD cn Cl. Z ai cc Q -z. (-) ID o CJ •. en 1 Q en — ^ > en m o > or: rr CJ z: _■ M Q CJ I — 1 ••(>- ? oy a3isnrab CO 1 >l D O < c Eh 0) 'cr -H 4J QJ -p IT3 c x: ra (D 0 -P V^ U t3 (T3 x: a 0) -P > 5-1 ^ -H (U 0 Eh 5 -H U-i s^ ncJ O W • 0) n3 4-1 >i+J r-i Cri (T3 Q) d -l 4J !^ j:: CQ 0 4J >iO U U K W M C Oi (L) X (U Eh Cn<(; CD < C S 0 CJ •H x: CO 4J (U 4J M ex: > rH 0 4J 0 ^ c 0) T3 0 en 0) 1— 1 C MH Ti T! (d C C !-l -- 0 CU 0 E ,-1 Q. a.4J fd (U eo (U u >H TD (U TS •H cn 0 !-i C T3 03 6 ^ -H (U ■H 0 v^ ^ Q u e a s ^ i-H CQ Q) e t3 rH + rH -H M ■H a X! X ■H ^ Oi 03 03 -C CO -- -p e -p n o iX> 0) u a D^ ■H P-i 217 < < Z UJ Q Z XI Q. Ui Q Z Cvj CM I H o < CO (0 d 00 o < '^ o o d o o ,, ^ - t~ K 00 CN in CN CO o CO 1^ CO r- o in n (N - iSV03HOd S^-0 5S'Q 5£'0 SrO SD'O- S2'0- [ lyosiA 3yoos iBsyHi ? oy a3isnray 00 1 4J 3 ■H < ^ £h QJ TD «■ x: 0) ■^ Eh 4J fd 0) 03 •H > OJ • 0 QJ S-i >i 0 ■H fd cn cn x: CU cn u S-l 4-) fd fd 0 fd M-l 5-1 cn OJ 4-) 0) ^ cn tn S-i 0 -P 0 o M c 0 cn D 0 cn cn -H 4J CD cn Q) fd >-l cn cn 4-1 X5 0) 1-3 S-l 0 M 4-1 l-l 4-1 tH l>i fd cn < 0 U T3 u iH c 13 c a: cu QJ 4-> 0 M > cn fd a> Q) c C cn i-i ■H 0) 4-> 4-1 Q) i-i C u C x: ^4 QJ 0 0 ■p 0 T^ 4-) 0 u C o u 0) •H XJ 0 0) axi c 4-4 rH 0) (U fd XI T3 u ^^ fd C cu e rH 4J •H (d dJ x: M T3 >. e ■p cn 0 u D 4-1 fd a G e ■H •H OJ •H 4-1 T) Q cn X s C fd (D M CQ •H e -c M + 4J 4-) •H a c Q) ^ a. 0 x; 4-1 cn '^'"^ u 4J fd u o o 0) M 3 cn Em 218 o r- < O ir> CO CD CO 2^ 00 si CVJ CO s." < CO CO r- CO CO CD T- lO 0) o < CO CVJ CVJ CD in CO CvJ o < CO 00 2P CVJ CO 00 CD 00 CO (J) -* 00 D U. CVJ 00 o 6 o CO 6 in CVJ CD T- d O) o u. CO 0) o d o CVJ CD o CO lO T- d in CVJ 00 T- d o CVJ d o o Q. Q CD a. Q > Q Q Q. Q Q Z LU CL o LU q: CL 0) 0 c QJ T! •H M-l c 0 a o\o lD o * \-l o\o Q) ^ nj CTi 0 e 'a C Q fT3 S CQ tfl + U (^ •H li. -p — m •H OD 4J ^ ra 1 -U ID en < Fh M < - ^-^ o < fO 0) - )-i CU tT3 0 ^-» C )^ >1 0) 0 en T! M-( QJ C 4-J 0) U) 03 a 0) u (u a -p T3 -H en fd M > M ra M C iH 0 (T3 CQ -H > o 4J V^ « U 0) a^ C -P X 3 c < Ci-l -H CD 5-1 a 219 DEPENDENT DATA to < U 2 92 353 2468 14 53 73 3 0 0 1 2 3 OBSERVED INDEPENDENT DATA 39 175 1245 5 21 40 1 0 1 Ao = 82.59% AAo = -3.30% Ai- 12.73% Tsi = 0.03 ATsi=-0.01 TS2= 0.11 ATS2- -0.03 TS12=0.10 ATS12= -0.09 Ao.82.97%AAo = -7.88% 4, .14.41% Tsi= 0.02 ATsi = -0.01 TS2=0.09 ATS2=-0.05 TS12=0.08 ATSl2=-0.09 1 2 3 OBSERVED Figure 62. Contingency table results for the area 4, TAU-48 equal variance threshold model 220 DEPENDENT DATA U1 < O 2 LLI a. O 96 368 2511 8 37 30 5 1 0 1 2 3 OBSERVED Ao = 83.54% AAo= 2.33% A,=13.32% TS1=0.05 ATS1= 0.01 TS2=0.08 ATS2- -0.06 TS12=0.08 ATS12=_Q .j "I INDEPENDENT DATA 42 181 1266 1 13 18 2 2 2 1 2 3 OBSERVED AO'83.89% AAo= -2.07% Ai=13.23% TS1 = 0.04 ATS1- 0.01 TS2 = 0.06 ATS2= -0.08 TS12=0.06 ATS12^-0.12 Figure 63. Contingency table results for the area 4, TAU-48 quadratic threshold model 221 !P 0) h- lO o CO (0 1 6 d 1 o ^. (N o «i l/l U1 < t— >- ^— < < «t < ^ CD ^" Tj- 05 CVi (0 CO d O d d ^1 — *» f< „ ^ o ^ l/> l/l 10 < < ^- ►- *- oo T- O) CVi CM m CO • CO T^ o o (0 O II ^ (S o i/i U1 (/I < >- ^ ^. 4 < «< < 5P 2P 0) co> CO O CO CO lO h- o o d d 00 N- „ (N " ^ M o I/) 1/1 in <« < >- 1- >- Q Z UJ o z LU a o z CO a CO c:^ CO (0 «o •* c:) 5 CO 1— in 00 iSW03dOd < a z m Q Z UJ c LU n o CJ) CD CO CO 5 O CO CM CM o 00 X ( » LJ LJ f—i U) f-) cs Z ^ k — t Q- Ql O r ) ct: CJ LU Ul 1 _; ce LD lJ-> a: IT :> - cr •. — • o ' <-) cc X f— ' SI n a; a. 1— o cn LJ cn nz CD in LJ O- Q- Q cn o UJ n n UJ n LJ Q fi 1 M Ul LJ -i Z rr ' o 1-1 CJ ■"* ■"• Cl. 'D Q 1 ' 1 1 a-j CC D • « • O LU LJ Q-l g-o vo 20 lyjsiA 3^03^^ lyayHi Q'O Z'Q- ►'0- ? oy aaibnrau "^ CN 1 4-) D •H < 0) 5 Eh s: -p £h TD (d •. 0) CN +J T! t (d 0) (T3 >fH > (U CP u 0) ^ XI fd 0) 0 M +j x: 4-) -p H >, cn < 0 S^ TD u c o c CO (U M-l 0 M i-t cn a> cu C — cn > •H iH CD 4-> a; +j a U c r-^ C T3 U OJ 0 0 0 ^ U u e o c 0 0) 4-1 T! Q OJ a 0 C 2 ,— 1 (U •H (13 CQ XI T! T! + rd C (U e a 4-) ■H !-l (T3 CU a ^ -- >i e tJ^ u =J x: (d (T3 c e 4-) ■H 4J (U ■H M T! (d cn X P T3 c fd 0 rH •H e M-l iH -^ -p H 00 c (U OJ ^ C7\ 0 x: x: cn -H 0 4-> 4-J ■^ >X) (U U 0 cn •H Cn 222 en s: LJ zn CJ en a -J a ni en LJ Or: a 2: a en q; az Q_ z; a u ■z. 0 (-H 0 LJ h-i LJ Q Lt: OL_ QCO Qd LJQ a CD 2^ LIO LJ -JOI C3 QD-- LJ CC _] _1 03 LJ 0 ji ct: — Q ) z: jz ZDZD t: tz 1—1 t— 1 z: X "CC rrsr II II > » o LJ ID a LJ CC LJ 9*0 S'Q VD £-0 2-Q 3>joas iu3yHi ro 0"0 03 0) -(J ^ W OJ OJ W 3 rH x: ■H •i-ii2 -p -C TD (0 Eh 03 4-1 t3 C c 13 >i (13 • 13 — 0 rH c C rH 0) fd >i OJ 0) ^ 0 CnT) 0 H C C 0 e H CD -H e H D +J XI cr c T3 H Eh 0) 0 r— 1 0 < S-i u 0 x: U U-l x: w cn en 0 U) (U H c +J 5 0) M > 0 fd -M S-I s: ■H 0) x: 4-) U) +J M >1+J a 03 x: XI c 03 S-l -I-) u 0 !-l 03 0 0 H QJ Q. W 15 S-l +J > 0) P -u S-l u CO w QJ 0) H U Xi -U H H q; 0) 0 0) + H > N 1— 1 Ti H •rH XI t ^ 0) T3 03 (T3 0 > -M 4-1 e s: W n3 cn 0 •H cn D dJ e H -P 03 S-I 03 ■H 0) u S-l x: C ^ (U 0) V u •H H H > 0 e H M-l 14-1 M-l 1 0) H 0 0) e S-l w x: p Eh x: in U3 CD U D cn 223 CD IxJ 51 [jj X o en a _] a CO q: X E-- U- Q a CO I — I az Q_ a u 2 O 1— 1 s— • o LJ t— LJ O ce OL_ CCCD en L-ia Q o a* LJO LJ .J>3Z CD CQ^ LJ a: _i _) CDLJ CD^ Qi" Q 1 zr 5z 33 r: z= 1— 1 •— 1 Z X — az sz sz II II > • 0£*Q SZ'Q 02"0 SI'Q Ql'O 3iyy wyyny 3snoj SQ-Q 00 -Q 0) X! 'O +J en (U +j -p S^ u M w 0 cu H D m r-{ •n >4-l EH HD ,.^ 1 CU < 03 (1) E 5-1 u C ^ a C/^ D i3 e. s: M ^ ta -H o +j X (T3 >i (13 5-1 en U > E CP D c O en 0) C 0) en ^ p CD x: ■H cu cr cn-p s: > dj c Eh u •H T3 H 4-1 -p c C (T3 • H 4J 0 -H < n3 u rH OJ u OJ 0) 13 en >- o U 0 T3 0 M x: :5 0 E > LJ ■p -p e 3 a "0 T3 LJ U) >iTi rH C L_ D XJ .— 1 0 fO en o X !— u 0 x: en M 01 LJ OJ ^ en cu M Qi > -p vj ra u u tT3 T3 OJ 0) en rH C -p a (T3 (;3 Q) cu en -p ^ TD 5-1 0) w XI 1 cu w td U-t > M (T3 i3 0 03 0 1 M cn M-l >-l S-l T3 H c 0 a 0 + 0 U-l 4H 0 M •H 0 E X -P en 3 •H Eh (d x; QJ e rH < 5-1 a S-i •H (U u (d (t! 0 c M en a 5-1 u ■H •H M cu o tn E rH > en VC >^ (U S-l a CP 224 tn LJ H LJ HZ CJ en a _i a iz CO or X i— CD a en en az Q_ 51 a CJ Z o h- ' o CJ t-' LJ □ cc OL_ CCD CC Lja o o z LJO LiJ _) t: CD QD" (. 1 CC-J _J (XILJ Oi^ Q^ — Q ) z: x; 33 r: 2= 1—1 1— — < Z X -^az 5= ZI X • Q'OQI Q"S6 Q-06 Q-S9 0) CU G s: 1 Eh 0 -p •H < -H "0 •-{ U -P Q) s^ QJ c/J (d 4-) 0 ^ H M W 4-1 •H > fd D iH Q. •1—1 , — , I UJ QJ T3 CD e 4-1 OJ ra r-i m O c Si e QJ H U3 :3 (K ■H rH H a ' — ' +J X <4-l QJ E-i >i >i e V^ < u U U c C QJ x: cn Q) 0) x: a,M tn 0 tji 4-1 fd > a cr c u 0) ■H T! tJi 01 u 4-1 c P M-l C 0 fd W t/J •H S-i -U U iH x: QJ ^ (0 QJ E-i > a QJ 0 T3 V-. U 5 0 H CJ x: 4J e • ^ -p -H t^ LJ 3 >iX! QJ < O w -Q M -ri u ^^ D 0 0 cn ° L- w 0 x: e H cn >-l 5 w > QJ 4-1 QJ T3 UJ > U -H XJ T3 x: 0 C h- +J Q) 4-1 x: (d fM U N cn CD 0) ■H S-l QJ M M T3 0 U M J-l U J-l -C M 0 (tl i-l 4-1 u QJ Eh — +J IT3 QJ 0 u O c 4-1 i-H •H cn OJ W J3 4J M u (T3 u > u fU ^ Q) Q) 0 4-1 W o a M 5-1 QJ 0 o 0 axi w 4h U-4 . S-i 0 g 4-1 QJ w p 0 > s: QJ e , CU S-i •H 13 M rd 0 c 0 M ^ U •H 0 + o U) e x: M r^ 'sD 0) >-l p D^ •H pL, 225 DEPENDENT DATA < 35 133 1662 0 0 0 46 235 827 1 2 3 OBSERVED AO = 58.13%AA0^0.01% Ai= 12.53% Tsi = 0.04 ATsi=0.01 TS2= 0.0 ATS2- -0.14 TS12=0.05 ATSl2=-0.04 INDEPENDENT DATA 16 58 857 0 0 0 22 117 399 1 2 3 OBSERVED Ao-59.84% AAo=o.01% A1= 11.91% TS1=0.04 ATS1--0.01 TS2=0.0 ATS2=_0 14 TS12-0.04 ATS12= 0.0 Figure 68. Contingency table results for the minimum probable error threshold model (EVAR) for area 4, TAU-24. The contingency tables reflect VISCAT I+II versus VISCAT III and VISCAT I versus VISCAT II separations 226 DEPENDENT DATA to < lU cr O 35 133 1662 22 134 659 24 101 168 1 2 3 OBSERVED Ao=61.95% AAo= 0.01% Ai = 40.78% Tsi=0.07 ATsi=0.04 T^S2 = 0.13 AT^S2^ 0.0 TS12= 0.05 ATS12=0.01 INDEPENDENT DATA 16 58 857 11 75 319 11 42 80 1 2 3 OBSERVED AO-64.19% AAO=0.01% Ai= 29.27% Tsi= 0.07 ATS1-0.04 TS2 = 0.15 ATS2= 0.03 TS12=0.14 ATS12= 0.04 Figure 69. Contingency table results for the maximum- likelihood- of - detection threshold model for area 4, TAU-24. The contingency tables reflect VISCAT H-II versus VISCAT III and VISCAT I versus VISCAT II separations 227 LIST OF REFERENCES Aldinger, W.T., 19 79: Experiments on Estimating Open Ocean Visibilities Using Model Output Statistics > M.S. Thesis (R.J. Renard, advisor), Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, 81 pp. Best, D.L.,'and Pryor, S.P., 1983: Air Weather Service Model Output Statistics System Project Report. AFGWC/ PR-83/001, United States Air Force Air Weather Service (MAC), Air Force Global Weather Central, Offutt AFB , NE, 85pp. Department of the Navy, 1979: CV NATOPS MANUAL, Office of the Chief of Naval Operations, Washington, D.C., 156 pp. ■Glahn, H.R., 1983: MOS Support for Military Locations From the Techniques Development Laboratory. TDL Office Note 83-9, National Weather Service, NOAA, U.S. Department of Commerce, 10 pp. , and Lowry, D.A., 1972: The Use of Model Output Statistics (MOS) in Objective Weather Forecasting. J. Appl. Meteor. , 11, pp. 1203-1211. Godfrey, R.S. and P.R. Lowe, 1979: An Application of Model Output Statistics to Forecasting the Occurrence of the Levante Wind. Preprints, Sixth Conference on Probability and Statistics in Atmospheric Sciences, Banaff, Alta., Amer. Meteor. Soc . , Boston, MA, pp. 83-86. Karl, M.L., 1984: Experiments in Forecasting Atmospheric Marine Horizontal Visibility using Model Output Statis- tics with Conditional Probabilities of Discretized Parameters . M.Sc Thesis (R.J. Renard, advisor), Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, 165 pp. Klein, W.H., 1981: Design of a MOS System for the Navy. Final Report. NEPRF Contract No. N00228-80-C-LV22 , Intercon Weather Consultants, Inc., Camp Springs, MD. Koziara, M.C., R.J. Renard and W.J. Thompson, 1983: Estimating Marine Fog Probability Using a Model Output Statistics Scheme. Monthly Weather Review. Ill, pp. 2333-2340. 228 Lewit, H.L., 1980: SOCAL MOS Ceiling/Visibility Forecast Algorithms. Task 3 Final Report: Forecast Algorithm Development, and Task IV, Section I: Test of Forecast Algorithms Employing Dependent Data. NEPRD Contract No. N00228-78-C-3289 , Ocean Data Systems, Inc., Monterey, CA. Lowe, P., 1984a: The Use of Decision Theory for Determining Thresholds for Categorical Forecasts, unpublished manu- script. Navy Environmental Prediction Research Facility, Monterey, CA. , 1984b: The Use of Multi-Variate Statistics for Defining Homogeneous Atmospheric Regions Over the North Atlantic Ocean, unpublished manuscript. Navy Environmental Prediction Research Facility, Monterey, CA. , 1984c: The Use of Adjusted Scores and Significance Tests in the Verification of Categorical Forecasts, unpublished manuscript. Navy Environmental Prediction Research Facility, Monterey, CA. Miller, I., and Freund , J.E., 1977: Probability and Statis- tics for Engineers , 2nd. Ed., Prentice-Hall, Inc., Engle- wood Cliffs, NJ, 528 pp. Naval Environmental Prediction Research Facility, Monterey, CA, 19 82: MOS Forecasts for U.S. Navy and Marine Corps CONUS Locations — User's Manual, NAVENPREDRSHFAC Document No. 7W0 513-Um-0 7 , Commander, Naval Oceanography Command, Bay St. Louis, MS, 18 pp. Nelson, T.S., 19 72: Numerical-Statistical Prediction of Visibility at Sea. M.S. Thesis (R.J. Renard, advisor), Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, 3 3 pp. Preisendorf er , R.W., 1983a: Proposed Studies of Some Basic Marine Atmospheric Visibility Prediction Schemes Using Model Output Statistics, unpublished manuscript. Depart- ment of Meteorology, Naval Postgraduate School, Monterey, CA, 2 8 pp. , 19 83b: Maximum-Probability and Natural-Regression Prediction Strategies, unpublished manuscript. Department of Meteorology, Naval Postgraduate School, Monterey, CA, 10 pp. , 1983c: Tests for Functional Dependence of Predic- tors, unpublished manuscript. Department of Meteorology, Naval Postgraduate School, Monterey, CA, 5 pp . 229 1984: Update of MAXPROB MOS Prediction Method, unpublished manuscript, Department of Meteorology, Naval Postgraduate School, Monterey, CA, 3 pp. Renard, R.J. and W.T. Thompson, 1984: Estimating Visibility Over the North Pacific Ocean Using Model Output Statistics National Weather Digest, Vol. 9, No. 2, pp. 18-25. Schramm, W.G., 1966: Analysis and Prediction of Visibility at Sea. M.S. Thesis (R.J. Renard, advisor), Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, 22 4 pp. Selsor, H.D., 1980: Further Experiments Using a Model Output Statistics Method in Estimating Open Ocean Visibility. M.S. Thesis (R.J. Renard, advisor), Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, 121 pp. University of California, 1983: BMDP Statistical Software, 1983 Edition, Department of Biomathematics , University of California at Los Angeles, University of California Press, 726 pp. Yavorsky, P.G., 1980: Experiments Concerning Categoriecal Forecasts of Open-Ocean Visibility Using Model Output Statistics . M.S. Thesis (R.J. Renard, advisor), Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, 87 pp. 230 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Technical Information Center 2 Cameron Station Alexandria, VA 22314 2. Library, Code 0142 2 Naval Postgraduate School Monterey, CA 93943 3. Meteorology Reference Center, Code 63 1 Department of Meteorology Naval Postgraduate School Monterey, CA 9 39 43 4. Professor Robert J. Renard, Code 63Rd 6 Chairman, Department of Meteorology Naval Postgraduate School Monterey, CA 9 3943 5. Chairman (Code 68Mr) 1 Department of Oceanography Naval Postgraduate School Monterey, CA 93943 6. Dr. Rudolph W. Preisendorf er 1 NOAA/PMEL/R/E/PM Bin 015700 Bldg. 3 7600 Sand Point Way, N.E. Seattle, WA 98115-0070 7 . Mr . Paul Lowe 1 Naval Environmental Prediction Research Facility Monterey, CA 93940 8. Dr. Robert Godfrey 1 Naval Environmental Prediction Research Facility Monterey, CA 9 39 40 9. Lt. Mark Diunizio 1 10855 Charbono Point San Diego, CA 92131 10. Director 1 Naval Oceanography Division Naval Observatory 34th and Massachusetts Avenue NW Washington, DC 20390 231 11. Coiranander Naval Oceanography Command NSTL Station Bay St. Louis, MS 39522 12. Commanding Officer Naval Oceanographic Office NSTL Station Bay St. Louis, MS 39522 13. Commanding Officer Fleet Numerical Oceanography Center Monterey, CA 93940 14. Commanding Officer Naval Ocean Research and Development Activity NSTL Station Bay St. Louis, MS 39522 15. Commanding Officer Naval Environmental Prediction Research Facility Monterey, CA 9 39 40 16. Chairman, Oceanography Department U.S. Naval Academy . Annapolis, MD 21402 17. Chief of Naval Research 800 N. Quincy Street Arlington, VA 22217 18. Office of Naval Research (Code 480) Naval Ocean Research and Development Activity NSTL Station Bay St. Louis, MS 39522 19. Commander (Air-370) Naval Air Systems Command Washington, D.C. 20360 20. Chief, Ocean Services Division National Oceanic and Atmospheric Administration 8060 Thirteenth Street Silver Spring, MD 20910 21. Dr. Alan Weinstein Leader, Code 422 Ocean Sciences Division Office of Naval Research Arlington, VA 22217 232 J I 22. LCdr. Michael L. Karl USS Peleliu (LHA-5) FPO San Francisco, CA 96624 24. LCdr. Mike Wooster SMC #1136 Naval Postgraduate School Monterey, CA 93943 25. LCdr. Kris Elias SMC #1542 Naval Postgraduate School Monterey, CA 9 3943 26. Mr. Gil Ross, Met 09 Meteorological Office Bracknell, Berkshire England 27. Chief, Technical Procedures Branch Meteorological Services Division National Oceanic and Atmospheric Administration National Weather Service Silver Spring, MD 20910 28. Chief, Technical Services Division United States Air Force Air Weather Service (MAC) Air Force Global Weather Central Offutt AFB, NB 68113 233 3 3 7 5 Thesis D59T2 c.l ' The sis -B59T2 c.l 2iuiG2 Diimizio An evaluation of discretized condition- al probability and linear regression threshold techniques in model output sta- tistics forecasting of visibility over the North Atlantic Ocean. 210162 Diunizio An evaluation of discretized condition- al probability and linear regression threshold techniques in model output sta- tistics forecasting of visibility over the North Atlantic Ocean.