UNCERTAINTY ANALYSIS WITH HIGH DIMENSIONAL DEPENDENCE MODELLING

By KENNETH LEE

CONTENTS INTRODUCTION WAGS AND BOGSATS UNCERTAINTY ANALYSIS AND DECISION SUPPORT: A RECENT EXAMPLE OUTLINE OF THE BOOK ASSESSING UNCERTAINTY ON MODEL INPUT STRUCTURED EXPERT JUDGMENT IN OUTLINE ASSESSING DISTRIBUTIONS OF CONTINUOUS UNIVARIATE UNCERTAIN QUANTITIES ASSESSING DEPENDENCIES UNICORN UNICORN PROJECTS BIVARIATE DEPENDENCE MEASURES OF DEPENDENCE PRODUCT MOMENT CORRELATION RANK CORRELATION KENDALL’S TAU PARTIAL, CONDITIONAL AND MULTIPLE CORRELATIONS COPULAE FR´ECHET COPULA DIAGONAL BAND COPULA GENERALIZED DIAGONAL BAND COPULA ELLIPTICAL COPULA ARCHIMEDEAN COPULAE MINIMUM INFORMATION COPULA COMPARISON OF COPULAE BIVARIATE NORMAL DISTRIBUTION BASIC PROPERTIES MULTIVARIATE EXTENSIONS MULTIVARIATE DEPENDENCE MEASURES MULTIVARIATE COPULAE MULTIVARIATE NORMAL DISTRIBUTION CONCLUSIONS EXERCISES 3 10 SUPPLEMENT HIGH-DIMENSIONAL DEPENDENCE MODELLING JOINT NORMAL TRANSFORM DEPENDENCE TREES TREES DEPENDENCE TREES WITH COPULAE EXAMPLE: INVESTMENT DEPENDENCE VINES VINES BIVARIATE- AND COPULA-VINE SPECI?CATIONS EXAMPLE: INVESTMENT CONTINUED PARTIAL CORRELATION VINES NORMAL VINES RELATIONSHIP BETWEEN CONDITIONAL RANK AND PARTIAL CORRELA- TIONS ON A REGULAR VINE VINES AND POSITIVE DE?NITENESS CHECKING POSITIVE DE?NITENESS REPAIRING VIOLATIONS OF POSITIVE DE?NITENESS THE COMPLETION PROBLEM SUPPLEMENT PROOFS RESULTS FOR SECTION 4 4 6 EXAMPLE OF FOURVARIATE CORRELATION MATRICES RESULTS FOR SECTION 4 5 2 OTHER GRAPHICAL MODELS BAYESIAN BELIEF NETS DISCRETE BBN’S CONTINUOUS BBN’S INDEPENDENCE GRAPHS MODEL INFERENCE INFERENCE FOR BBN’S INFERENCE FOR INDEPENDENCE GRAPHS INFERENCE FOR VINES SAMPLING METHODS (PSEUDO-) RANDOM SAMPLING REDUCED VARIANCE SAMPLING QUASI-RANDOM SAMPLING STRATI?ED SAMPLING LATIN HYPERCUBE SAMPLING SAMPLING TREES, VINES AND CONTINUOUS BBN’S SAMPLING A TREE SAMPLING A REGULAR VINE DENSITY APPROACH TO SAMPLING REGULAR VINE SAMPLING A CONTINUOUS BBN EXERCISE VISUALIZATION A SIMPLE PROBLEM TORNADO GRAPHS RADAR GRAPHS SCATTER PLOTS, MATRIX AND OVERLAY SCATTER PLOTS COBWEB PLOTS COBWEB PLOTS LOCAL SENSITIVITY: DIKE RING RELIABILITY RADAR PLOTS FOR IMPORTANCE; INTERNAL DOSIMETRY 7 10 UNICORN PROJECTS 7 11 EXERCISES PROBABILISTIC SENSITIVITY MEASURES SCREENING TECHNIQUES MORRIS’ METHOD DESIGN OF EXPERIMENTS GLOBAL SENSITIVITY MEASURES CORRELATION RATIO SOBOL INDICES LOCAL SENSITIVITY MEASURES FIRST ORDER RELIABILITY METHOD LOCAL PROBABILISTIC SENSITIVITY MEASURE COMPUTING ?E(X|GO) ?GO PROBABILISTIC INVERSION EXISTING ALGORITHMS FOR PROBABILISTIC INVERSION CONDITIONAL SAMPLING PARFUM HORA-YOUNG AND PREJUDICE ALGORITHMS ITERATIVE ALGORITHMS ITERATIVE PROPORTIONAL ?TTING ITERATIVE PARFUM SAMPLE RE-WEIGHTING NOTATION OPTIMIZATION APPROACHES IPF AND PARFUM FOR SAMPLE RE-WEIGHTING PROBABILISTIC INVERSION APPLICATIONS DISPERSION COEF?CIENTS CHICKEN PROCESSING LINE CONVOLUTION CONSTRAINTS WITH PRESCRIBED MARGINS IPF AND PARFUM 10 UNCERTAINTY AND THE UN COMPENSATION COMMISSION 10 1 INTRODUCTION 10 2 CLAIMS BASED ON UNCERTAINTY 10 3 WHO PAYS FOR UNCERTAINTY WAGS AND BOGSATS ‘WHETHER TRUE OR NOT [IT] IS AT LEAST PROBABLE; AND HE WHO TELLS NOTHING EXCEEDING THE BOUNDS OF PROBABILITY HAS A RIGHT TO DEMAND THAT THEY SHOULD BELIEVE HIM WHO CANNOT CONTRADICT HIM’ SAMUEL JOHNSON, AUTHOR OF THE ?RST ENGLISH DICTIONARY, WROTE THIS IN 1735 HE IS REFERRING TO THE JESUIT PRIEST JERONIMO LOBO’S ACCOUNT OF THE UNICORNS HE SAW DURING HIS VISIT TO ABYSSINIA IN THE 17TH CENTURY (SHEPARD (1930) P 200) JOHNSON COULD HAVE BEEN THE APOLOGIST FOR MUCH OF WHAT PASSED AS DECISION SUPPORT IN THE PERIOD AFTER WORLD WAR II, WHEN THINK TANKS, FORECASTERS AND EXPERT JUDGMENT BURST UPON THE SCIENTI?C STAGE MOST SALIENT IN THIS GENRE IS THE BOOK THE YEAR 2000 (KAHN AND WIENER (1967)) IN WHICH THE AUTHORS PUBLISHED 25 ‘EVEN MONEY BETS’ PREDICTING FEATURES OF THE YEAR 2000, INCLUDING INTERPLANETARY ENGINEER- ING AND CONVERSION OF HUMANS TO ?UID BREATHERS ESSENTIALLY, THESE ARE STATEMENTS WITHOUT PEDIGREE OR WARRANT, WHOSE CREDIBILITY RESTS ON SHIFTING THE BURDEN OF PROOF THEIR CAVALIER ATTITUDE TOWARD UNCERTAINTY IN QUANTITATIVE DECISION SUPPORT IS REP- RESENTATIVE OF THE PERIOD READERS INTERESTED IN HOW MANY OF THESE EVEN MONEY BETS THE AUTHORS HAVE WON, AND IN OTHER EXAMPLES FROM THIS PERIOD, ARE REFERRED TO (COOKE (1991), CHAPTER 1) QUANTITATIVE MODELS PERVADE ALL ASPECTS OF DECISION MAKING, FROM FAILURE PROB- ABILITIES OF UNLAUNCHED ROCKETS, RISKS OF NUCLEAR REACTORS AND EFFECTS OF POLLUTANTS ON HEALTH AND THE ENVIRONMENT TO CONSEQUENCES OF ECONOMIC POLICIES SUCH QUAN- TITATIVE MODELS GENERALLY REQUIRE VALUES FOR PARAMETERS THAT CANNOT BE MEASURED OR