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Using Modern Computing Tools to Fit the Pearson Type III Distribution to Aviation Loads Data

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Using Modern Computing Tools to Fit the Pearson Type III Distribution to Aviation Loads Data DOT/FAA/AR-03/62 Using Modern Computing Tools to Office of Aviation Research Fit the Pearson Type III Washington, D.C. 20591 Distribution to Aviation Loads Data September 2003 Final Report This document is available to the U.S. public through the National Technical Information Service (NTIS), Springfield, Virginia 22161. U.S. Department of Transportation Federal Aviation Administration NOTICE This document is disseminated under the sponsorship of the U.S. Department of Transportation in the interest of information exchange. The United States Government assumes no liability for the contents or use thereof. The United States Government does not endorse products or manufacturers. Trade or manufacturer's names appear herein solely because they are considered essential to the objective of this report. This document does not constitute FAA certification policy. Consult your local FAA aircraft certification office as to its use. This report is available at the Federal Aviation Administration William J. Hughes Technical Center's Full-Text Technical Reports page: actlibrary.tc.faa.gov in Adobe Acrobat portable document format (PDF). Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. DOT/FAA/AR-03/62 4. Title and Subtitle 5. Report Date USING MODERN COMPUTING TOOLS TO FIT THE PEARSON September 2003 TYPE III DISTRIBUTION TO AVIATION LOADS DATA 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. Peter Hovey* and Thomas DeFiore** 9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) *University of Dayton Research **FAA William J. Hughes Technical Center RPD-510 Institute Airport and Aircraft Safety R&D Division 11. Contract or Grant No. Structural Integrity Division Airworthiness Assurance Branch Grant 00-G-015 300 College Park Atlantic City International Airport, NJ 08405 Dayton, OH 45469-0120 12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered U.S. Department of Transportation Final Report Federal Aviation Administration Office of Aviation Research 14. Sponsoring Agency Code Washington, DC 20591 ANM-110 15. Supplementary Notes The FAA William J. Hughes Technical Center COTR was Thomas DeFiore. 16. Abstract The question of obtaining a relatively simple and at the same time accurate functional representation of asymmetric frequency distributions gives rise to one of the most important practical problems in mathematical statistics. The famous mathematician Karl Pearson solved this with a considerable degree of success in the development of his Type III frequency function. This function is a comparatively simple exponential expression, which is completely defined in terms of the mean, standard deviation, and the skewness of the distribution. The Type III function produces, with considerable accuracy, a large number of different distributions, both skew and symmetric, and reduces to the standard normal frequency function when skewness is zero. The Pearson Type III distribution is used by the U.S. Army Corps of Engineers in flood frequency analysis, by the National Oceanic and Atmospheric Administration in the analysis of precipitation data, and by the U.S. Navy in specifying airplane touchdown sink speed of non- carrier-based airplanes. A derivation of the Pearson Type III probability function, a table containing a complete set of Pearson Type III probability values for skewness values up to 1.1, an example describing how to use the tables, and a procedure and example for setting up an Excel spreadsheet and graphical applications are presented herein. 17. Key Words 18. Distribution Statement Pearson Type III distribution, Statistical analysis, This document is available to the public through the National Cumulative probability function, Skewness Technical Information Service (NTIS) Springfield, Virginia 22161. 19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price Unclassified Unclassified 83 Form DOT F1700.7 (8-72) Reproduction of completed page authorized PREFACE The Flight Systems Integrity Group of the Structural Integrity Division of the University of Dayton Research Institute (UDRI) performed this study under Federal Aviation Administration (FAA) Grant No. 00-G-015 entitled “Pearson Type III and the Gamma Distribution.” Mr. Thomas DeFiore of the FAA William J. Hughes Technical Center at Atlantic City International Airport, New Jersey, was the program manager for the FAA and also a major contributor for the preparation of the report. Dr. Peter Hovey was the Principal Investigator for UDRI. iii ACKNOWLEDGEMENT The authors would like to acknowledge the following for their assistance in the preparation of this report: (1) University of Dayton Research Institute’s John Rustenburg for his technical review and final assembly into the report, (2) Federal Aviation Administration (FAA) William J. Hughes Technical Center’s Carman Munafo for the preparation of the proofs presented in Appendices B, C, and D, and (3) FAA William J. Hughes Technical Center’s Richard Micklos for his final technical edit and suggestion for the insertion for sample problems. iv TABLE OF CONTENTS Page EXECUTIVE SUMMARY ix 1. INTRODUCTION 1 2. DISCUSSION 1 3. EXAMPLES 3 3.1 Percentile of Distribution Table Look-Up 3 3.2 Generating Tables 4 3.3 Generating Exceedance Curves 5 4. CONCLUDING REMARKS 9 5. REFERENCES 9 APPENDICES A—Cumulative Probability of the Standardized Pearson Type III Distribution B—Derivation of the Pearson Type III Probability Density Function C—Deriving the Three-Parameter Gamma Probability Density Function D—Moments of the Pearson Type III Probability Density Function v LIST OF ILLUSTRATIONS Figure Page 1 Histogram of Wind Speed Parallel to the Runway at Touchdown 6 2 Histogram of Wind Speed Perpendicular to the Runway at Touchdown 6 3 Pearson Type III Probability Plot of Wind Speed Parallel to the Runway 7 4 Pearson Type III Probability Plot of Wind Speed Perpendicular to the Runway 7 5 Exceedance Probability for Wind Speed Parallel to the Runway 8 6 Exceedance Probability for Wind Speed Perpendicular to the Runway 8 LIST OF TABLES Table Page 1 Example Using the Excel Inverse Gamma Distribution Function to Calculate Pearson Type III Variates 4 vi LIST OF SYMBOLS AND ABBREVIATIONS a,c0, c1, c2 Constants in the Pearson differential equation f(x) Probability density function F-1(x) Inverse of the Cumulative Gamma Distribution Function F(x) Cumulative Gamma Distribution Function κ skewness K, m Constants in the Pearson Type III Density Function P(x) Pearson Type III Density Function P Percentile Pe Exceedance Probability x Sample Mean XP Standardized Pearson Type III Percentile s Sample Standard Deviation α Parameter of the Gamma Distribution β Scale Parameter of the Gamma Distribution Γ Gamma Distribution Function µ mean of distribution σ2 variance of distribution vii/viii EXECUTIVE SUMMARY The question of obtaining a relatively simple and at the same time accurate functional representation of asymmetric frequency distributions gives rise to one of the most important practical problems in mathematical statistics. The famous mathematician Karl Pearson solved this with a considerable degree of success in the development of his Type III frequency function. This function is comparatively simple exponential expression, which is completely defined in terms of the mean, standard deviation, and the skewness of the distribution. The Type III function produces, with considerable accuracy, a large number of different distributions, both skew and symmetric, and reduces to the standard normal frequency function when skewness is zero. The Pearson Type III distribution is used by the U.S. Army Corps of Engineers in flood frequency analysis, by the National Oceanic and Atmospheric Administration in the analysis of precipitation data, and by the U.S. Navy in specifying airplane touchdown sink speed of non- carrier-based airplanes. A derivation of the Pearson Type III probability function, a table containing a complete set of Pearson Type III probability values for skewness values up to 1.1, an example describing how to use the tables, and a procedure and example for setting up an Excel spreadsheet and graphical applications are presented herein. ix/x 1. INTRODUCTION. The Pearson Type III distribution is used by the U.S. Army Corps of Engineers in flood frequency analysis [1 and 2], the National Oceanic and Atmospheric Administration in the analysis of precipitation data [3], and by the U.S. Navy in specifying airplane touchdown sink speed of non-carrier-based airplanes [4]. The Federal Aviation Administration has been conducting video landing parameter surveys at high activity airports. Some of the parameters measured include sink speed, ground speed at touchdown, roll, pitch, and yaw rates. For almost every civil airplane model type for which substantial data has been collected, the most important measured parameter, sink speed appears to have a substantial degree of asymmetry, i.e., skewness to the positive side. Engineers need a simple statistical tool to analyze, present, and extrapolate frequency distributions for this type of skewed data in which the parameters of the distribution have a clear physical meaning. The three-parameter (mean, standard deviation, and skewness) Pearson Type III distribution appears to fulfill this need. The Pearson Type III distribution is better known today as the Gamma distribution. This report describes the relation between the parameters of the Pearson Type
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