A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, for the degree of Master of Science. Johannesburg, 1980 / Monthly mean s e a - l e v e l p r e s s u r e ove r th e o c ea nic areas a d j a c e n t t o t h e Kepublic o f South A f r i c a i s an aly s e d . R ela t i o n s h i p s between t h e oceanic p r e s s u r e and r a i n f a l l over t h i s p a r t o f t h e c o n t i n e n t a re d i s c u s s e d . P r i n c i p a l compon e n ts a n a l y s i s is used t o d e r i v e u n c o r r e l a t e d f u n c t i o n s of th e o r i g i n a l p r e s s u r e v a r i a b l e s . Three major p r e s s u r e f i e l d s were i d e n t i f i e d , termed a g e n e r a l , a l o n g i t u d i n a l and a l a t i t u d i n a l p r e s s u r e f i e l d . The r e l a t i o n s h i p s between p r e s s u r e and r a i n f a l l a re a s s e s s e d by r e g r e s s i n g monthly r a i n f a l l on t h e p r i n c i p a l comnonent s c o r e s . R a i n f a l l in w in te r maxima a r e a s appears t o be d i r e c t l y r e l a t e d t o oceanic s e a - l e v e l p r e s s u r e s i t u a t i o n s , whereas the r e s t o f th e country shows an o u t - o f - s e a s o n r e l a t i o n s h i p between r a i n f a l l and p r e s s u r e over n o n - c o n t i n e n t a l a r e a s .

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / by Patrick Jaillet. / M.S.

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yes / Highlights: Non-additive probabilities and Choquet integrals in a classical context. The use of Choquet integrals in a quantum context. Approximation of partially known density matrices with Choquet integrals.

This dissertation is a study of certain aspects of restricted ranking, a method intended for use by a panel of m judges evaluating the relative merits of N subjects, candidates tor scholarships, awards, etc. Each judge divides the N subjects into R classes, so that n₁ individuals receive a grade i (i = 1, 2, R; Σnᵢ = N) where the R numbers nᵢ are close to N/R (nᵢ = N/R when N is divisible by R) and are preassigned and the same for all judges. When this method is used, all subjects are treated alike, the grading system is the same for all judges and the grades of each judge are given equal weight. Equally important, the meaning of a particular grade is clear to each judge and the same for each judge. Under the null hypothesis that all nR = N subjects are of equal merit, tests of significance are developed to determine whether (1) a particular individual is superior or inferior to the rest of the subjects; (2) two particular subjects are of equal merit; (3) the individuals with the highest and lowest scores are respectively superior and interior to the rest of the subjects and (4) the nR subjects form a homogeneous group. The critical values of the test statistics for (1), (2) and (3) are tabled for small to moderate values of m, an approximation based on the asymptotic normality of the appropriate test statistic proving suitable for large m. The test of homogeneity (4) employs a sum of squares of subjects’ scores which is shown to be asymptotically distributed for m→∞ as chi-square with nR-1 degrees of freedom. For the special case of complete ranking (R=N), this statistic is identical to one proposed by Friedman (1937) form rankings. The behavior of two of these tests is theoretically investigated for the non-null case of nR-1 subjects having equal merit and one "outlying" subject whose merit exceeds the others. The assumption is made that each judge j assigns a grade to every subject i on the basis of a "subjective random variable" xᵢⱼ with mean equal to the "true" merit of subject i and that the distribution of xᵢⱼ is the same for all j. The probability, P(δ), that subject #1 with true mean differing from the others by an amount δ would receive a significantly high score according to the test for outliers is obtained and presented graphically as a function of for xᵢⱼ distributed as (1/2) sech² (x-δ) and also as N ( δ, 1). Using a result due to Hannan (1956), an expression for the asymptotic relative efficiency of the chi-squared homogeneity test for restricted vs. complete ranking for the aforementioned non-null case is obtained and values of this A.R.E. for 2 ≤n≤10 and. 2≤R≤8 are tabled. This A.R.E. is found to be at least 0.9 for all cases where n≤10 and R≥4. A further comparison of the performances of restricted (R) and complete (C) ranking is made by way of some simulation studies performed on a high speed digital computer tor the non- null ease where xᵢⱼ is normally distributed with unit variance and a mean δ₁ having as many as three different possible values. The complete and restricted ranks assigned by the jth judge to the ith subject are assigned on the basis of the value of xᵢⱼ obtained by experimental sampling using a random normal number generator in the computer program. A group of Nₛ subjects with the highest rank sums for (R) and for (C) are then selected in each study. The observed difference in true means between selected and remaining groups is then used as a measure of goodness of the two selection procedures. The results of these studies are presented graphically, displaying a very close agreement between (R) and (C) in all instances. / Ph. D.

A theorem by Shah and Khatri [A.M.S. (1961) 321883-887] is extended to give the distribution of Q/X² , where Q is a positive definite quadratic form involving non-central normal variates and X² is an independently distributed chi-square variate. Conditions are given under which the distribution of this ratio reduces to that of non-central F. Kramer’s method for analysis of variance of a two-way classification with disproportionate subclass numbers is reviewed and shown to satisfy these conditions. Various functional forms of the non-centrality parameter for evaluating the power of his method are given. Additional algebraic and numerical results are obtained to compare the power of Kramer's method and the method of fitting constants (least squares) outlined by Yates [J.A.S.A. (1934) 24151-66]. A mnemonic rule, based on 310 randomly generated two-way classifications, is given for discriminating against use of Kramer’s method in situations where his method potentially may be very deficient in power as compared to the method of fitting constants. / Ph. D.

Though some of the discrete distributions, for example the binomial, hypergeometric, Poisson, are well tabulated, often statisticians use the percentage points of approximating continuous distributions when analysing discrete data. In this thesis, the exact relationships between certain discrete and continuous distributions are established, and these relationships are used for setting confidence limits and significance testing of hypotheses. In Chapter 1, statements of all distributions and mathematical functions used in this thesis are made, and also some approximations are mentioned without proofs. In Chapter 2, exact relationships between discrete distributions (the binomial, negative-binomial, and Poisson) and continuous distributions (the F and χ²) are proved. In Chapter 3, use is made of the approximate and exact relationships between discrete and continuous distributions, for setting confidence limits on the parameters of the discrete distributions. Chapter 4 consists of the approximate and exact significance testing of hypotheses by using the approximate and exact relationships, given in Chapter 2. In Chapter 5, two-sample, exact and approximate, significance tests of hypotheses on the Poisson distribution are performed, in the case of fixed number of events experimentation and fixed time experimentation. / M.S.

A brief introduction to the concepts and terminology of spectral analysis and a review of the standard methods for cross-spectral estimation, based on discrete time history data, are incorporated in Chapter 1. Co-spectral and quadrature-spectral estimators which are characterized by non-negative spectral windows are developed in Chapter 2. While the spectral windows for the co-spectral estimators are non-negative for all relevant values of the assignable constants, certain restrictions on these constants are necessary to assure the non-negativity of the quadrature-spectral window. The properties of these estimators are considered in detail. In Chapter 3 randomized co-spectral and quadrature spectral estimators are presented. These estimators depend on the random selection of sets of time differences, as opposed to the systematic evaluation of all possible time differences for the standard estimators. By suitable choices of probability distributions for the time differences and of weight functions, the expectations of the randomized estimators can be made equivalent to the expectations of the standard estimators or the estimators of Chapter 2. Since the randomized estimator is much simpler to use than the standard estimator, these estimators are compared in terms of their variances, given that they have equal expectations. The choice of probability distributions to yield minimum variance, given that the expectation is specified, is considered. Extremely simple co-spectral and quadrature-spectral estimators, for the case where the coefficients of the Fourier series expansions of realizations of the processes over a finite time interval can be obtained by means of suitable analog equipment, are developed in Chapter 4. The expectations, variances and covariances of these estimators are derived. / Ph. D.