No description available.

The problem of estimating the parameters of several growth curves has been considered for the case where repeated correlated observations are taken on the same individual or population. These curves are the logistic, the Gompertz, the modified exponential, the ɵ-generalized logistic, and their modified forms with lower asymptotes different from zero. Three methods of estimation have been suggested and the mathematical procedure of each has been discussed. The different methods of estimation yield the vector equations for the estimators whose solutions require the inverse of the variance and covariance matrix. A procedure is given for obtaining the inverse of the type of covariance matrix used in our model. The procedure given holds good for all matrices of this type of any order and does not require the use of computers. The methods of estimation suggested are all of iterative type and require starting values of the parameters. A method for obtaining the starting values of the parameters has been given for each curve. The method for obtaining the starting values involves the estimation of the derivatives of the growth function w(t) or log w(t) with respect to t. The differentiation formulas for the estimation of these derivatives from the observed data, when the series of values may be given at equal or unequal intervals, have been obtained. The stochastic models for the logistic, the Gompertz, and the modified exponential laws of growth have been formulated as pure birth Markov processes. The solutions of the differential-difference equations describing the probability laws of the processes have been obtained by solving the partial differential equations for their generating functions. The properties of the processes have been studied by deriving the expressions for the means, variances and correlations. A method for obtaining the maximum likelihood estimators of the parameters involved has also been given in each case. The problem of distinguishing the different phases of growth has been attacked by deriving orthogonal expansions from the logistic, the Gompertz, and the exponential densities, in a manner similar to the way in which Gram (1879) and Charlier (1906) derived an orthogonal expansion from the normal density. The φ-generali zed Gompertz, the (φ,θ)-generalized logistic, and the (φ,θ)-generalized modified exponential densities have been obtained as generalizations of the Gompertz, the θ-generalized logistic, and the θ-generalized modified exponential respectively. The limiting cases of these densities have been found as φ or θ or both are allowed to go to infinity or zero. Lastly, the recurrence relation for the orthogonal polynomials qn (x) (leading coefficient one) of degree n associated n with the density function f(x) over the interval [ a, b] has been derived explicitly in terms of the moments of f(x). Further, an alternative proof has been given of the theorem that if f(x) is symmetrical about x = 0, then the polynomials qn(x) are even or odd functions according as n is even or odd. / Science, Faculty of / Mathematics, Department of / Graduate

This research study had shown various probable rational methods of quantifying subjective information in a probability distribution with particular reference to the evaluation of economic projects by computer simulation. Computer simulation to give all the possible outcomes of a capital project using the Monte Carlo technique (method of statistical trials) provides a strong practical appeal for the evaluation of a risky project. However, a practical problem in the application of computer simulation to the evaluation of capital expenditures is the numerical quantification of uncertainty in the input variables in a probability distribution. One serious shortcoming in the use of subjective probabilities is that subjective probability distributions are not in a reproducible or mathematical form. They do not, therefore, allow for validation of their general suitability in particular cases to characterize input variables by independent means. At the same time the practical derivation of subjective probability distributions is by no means considered an easy or exact task. The present study was an attempt to suggest a simplification to the problem of deriving a probability distribution by the usual method of direct listing of subjective probabilities. The study examined the possible applicability of four theoretical probability distributions (lognormal, Weibull, normal and triangular) to the evaluation of capital projects by computer simulation. Both theory and procedures were developed for employing the four theoretical probability distributions to quantify the probability of occurrence of input variables in a simulation model. The procedure established for fitting the lognormal probability function to three-level estimates of probabilistic information was the principal contribution from this study to research in the search for improved techniques for the analysis of risky projects. A priori considerations for studying the lognormal function were discussed. Procedures were also shown on how to apply the triangular probability function and the normal approximation to simulate the outcomes of a capital project. The technique of fitting the Weibull probability function to three-level estimates of forecasts was adopted from a paper by William D. Lamb. The four theoretical probability functions wore applied to a case problem which was analyzed using subjective probabilities by David B. Hertz and reported in the Harvard Business Review. The proposal considered was a $10/-million extension to a chemical processing plant for a medium-sized industrial chemical producer. The investigations of the present study disclosed that the log-normal function showed considerable promise as a suitable probability distribution to quantify the uncertainties surrounding project variables. The normal distribution was also found to hold promise of being an appropriate distribution to use in simulation studies. The Weibull probability function did not show up too favourably by the results obtained when it was applied to the case problem under study. The triangular probability function was found to be either an inexact or unsuitable approximation to use in simulation studies as shown by the results obtained on this case problem. Secondary investigations were conducted to test the sensitivity of Monte Carlo simulation outputs to (l) number of statistical trials; (2) assumptions made on tail probabilities and (3) errors in the three-level estimates. / Business, Sauder School of / Graduate

This work concerns the evaluation of statistical disclosure control risk by adopting the position of the data intruder. The underlying assertion is that risk metrics should be based on the actual inferences that an intruder can make. Ideally metrics would also take into account how sensitive the inferences would be, but that is subjective. A parallel theme is that of the knowledgeable data intruder; an intruder who has the technical skills to maximally exploit the information contained in released data. This also raises the issue of computational costs and the benefits of using good algorithms. A metric for attribution risk in tabular data is presented. It addresses the issue that most measures for tabular data are based on the risk of identification. The metric can also take into account assumed levels of intruder knowledge regarding the population, and it can be applied to both exact and perturbed collections of tables. An improved implementation of the Key Variable Mapping System (Elliot, et al., 2010) is presented. The problem is more precisely defined in terms of categorical variables rather than responses to survey questions. This allows much more efficient algorithms to be developed, leading to significant performance increases. The advantages and disadvantages of alternative matching strategies are investigated. Some are shown to dominate others. The costs of searching for a match are also considered, providing insight into how a knowledgeable intruder might tailor a strategy to balance the probability of a correct match and the time and effort required to find a match. A novel approach to model determination in decomposable graphical models is described. It offers purely computational advantages over existing schemes, but allows data sets to be more thoroughly checked for disclosure risk. It is shown that a Bayesian strategy for matching between a sample and a population offers much higher probabilities of a correct match than traditional strategies would suggest. The Special Uniques Detection Algorithm (Elliot et al., 2002) (Manning et al., 2008), for identifying risky sample counts of 1, is compared against Bayesian (using Markov Chain Monte Carlo and simulated annealing) alternatives. It is shown that the alternatives are better at identifying risky sample uniques, and can do so with reduced computational costs.

One technique that is commonly used to replace the costly daily sampling of sedimentconcentration in assessing sediment discharge is the "rating curve" technique. This technique relies on the form of the relationship between sediment concentration and water discharge to estimate long-term sediment loads. In this study, a regression/time-series approach to modelling the relationship between sediment concentration and water discharge is developed. The model comprises of a linear regression of the natural logarithm of sediment concentration on the natural logarithm of water discharge and an autoregressive time-series of order one or two for the errors of the regression equation. The main inferences from the model are the estimation of annual sediment loads and the calculation of their standard errors. Bias correction factors for the bias resulted from the inverse transformation of the natural logarithm of sediment concentration are studied. The accuracy of the load estimates is checked by comparing them to figures published by Water Survey of Canada. / Science, Faculty of / Statistics, Department of / Graduate

I study a simple model of a turbulent dynamo. Using a combination of analytic solutions and scaling arguments I derive a set of governing equations that describe the evolution of the magnetic field. This simplified system predicts behaviour which is qualitatively similar to that seen in the Earth's magnetic field and in numerical simulations. In particular, the model predicts multiple steady-field states, and that in certain cases the larger field states are more stable than their weaker counterparts. This provides a possible explanation for the observed periods of high stability in the geomagnetic field as well as for some of the observed behaviour in numerical simulations. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

This thesis contains a review of existing statistical techniques for the prediction of extreme waves for coastal and offshore installation design. A description of the four most widely used probability distributions is given, together with a detailed discussion of the methods commonly used for the estimation of their parameters. Although several of these techniques have been in use for several years, it has never been satisfactorily shown which are capable of yielding the most reliable predictions. The main purpose of this thesis is to suggest a practical method of solving this problem and achieving the best estimate. The basic theory for the prediction of extreme values was described in detail by Gumbel (1958) who concentrated largely on the double exponential distribution which is named after him. An order to evaluate the quality of fit between this law and the data, Gumbel derived expressions which enabled one to plot confidence intervals to enclose the data. The method described in this thesis in partly an extension of Gumbel's work, and similar confidence interval methods are given for the remaining distributions, thus permitting direct comparisons to be drawn between their performances. The outcome of this is that the most reliable model of the data may be chosen, and hence the best prediction made. The method also contains a curvature test which has been devised to facilitate computation and lead more directly to the end result. The particular form of the wave data, which is quite different from wind records, is also taken into consideration and a working definition of the sample tail is suggested. / Applied Science, Faculty of / Civil Engineering, Department of / Unknown

Various aspects of duality in a series of Abelian lattice models defined on topologically non-trivial lattices are investigated. The dual theories on non-trivial spaces are found to contain extra topological degrees of freedom in addition to the usual local ones. By exploiting this fact, it is possible to introduce topological modes in the defining partition function such that the dual model contains a reduced set of topological degrees of freedom. Such a mechanism leads to the possibility of constructing self-dual lattice models even when the naive theory fails to be self-dual. After writing the model in field-strength formalism the topological modes are identified as being responsible for the quantization of global charges. Using duality, correlators in particular dimensions are explicitly constructed, and the topological modes are shown to lead to inequivalent sectors of the theory much like the inequivalent ^-sectors in non-Abelian gauge theories. Furthermore, duality is applied to the study of finite-temperature compact U(l), and previously unknown source terms, which arise in the dual Coulomb gas representation and consequently in the associated Sine-Gordon model, are identified. Finally, the topological modes are demonstrated to be responsible for the maintenance of target-space duality in lattice regulated bosonic string theory and automatically lead to the suppression of vortex configurations which would otherwise destroy the duality. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In this thesis several sampling methods for Statistical Learning with imbalanced data have been implemented and evaluated with a new metric, imbalanced accuracy. Several modifications and new algorithms have been proposed for intelligent sampling: Border links, Clean Border Undersampling, One-Sided Undersampling Modified, DBSCAN Undersampling, Class Adjusted Jittering, Hierarchical Cluster Based Oversampling, DBSCAN Oversampling, Fitted Distribution Oversampling, Random Linear Combinations Oversampling, Center Repulsion Oversampling. A set of requirements on a satisfactory performance metric for imbalanced learning have been formulated and a new metric for evaluating classification performance has been developed accordingly. The new metric is based on a combination of the worst class accuracy and geometric mean. In the testing framework nonparametric Friedman's test and post hoc Nemenyi’s test have been used to assess the performance of classifiers, sampling algorithms, combinations of classifiers and sampling algorithms on several data sets. A new approach of detecting algorithms with dominating and dominated performance has been proposed with a new way of visualizing the results in a network. From experiments on simulated and several real data sets we conclude that: i) different classifiers are not equally sensitive to sampling algorithms, ii) sampling algorithms have different performance within specific classifiers, iii) oversampling algorithms perform better than undersampling algorithms, iv) Random Oversampling and Random Undersampling outperform many well-known sampling algorithms, v) our proposed algorithms Hierarchical Cluster Based Oversampling, DBSCAN Oversampling with FDO, and Class Adjusted Jittering perform much better than other algorithms, vi) a few good combinations of a classifier and sampling algorithm may boost classification performance, while a few bad combinations may spoil the performance, but the majority of combinations are not significantly different in performance.

In this paper we lay the foundation of the concentration measurement for statistical tables with more than two columns. A concentration function and a coefficient of concentration are defined which can be used in a similar way as the Lorenz diagram and the Gini coefficient in case of tables with two columns. For computational purposes we derive an explicit formula and give an algorithm. The mathematics behind our approach is formally equivalent to the statistical theory of the comparison of experiments. / Series: Forschungsberichte / Institut für Statistik