Subir Kumar Bhandari

Investigation of the global dynamics of cellular automata using Boolean derivatives

Global dynamics of a non-linear Cellular Automaton (CA), is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In this paper, efforts have been made to systematize non-linear CA evolutions in the light of Boolean derivatives and Jacobian matrices. A few new theorems on Hamming Distance between Boolean functions as well as on Jacobian matrices of cellular automata are proposed and proved. Moreover, a classification of Boolean functions based on the nature of deviation from linearity has been suggested with a view to grouping them together to classes/subclasses such that the members of a class/subclass satisfy certain similar properties. Next, an error vector, which cannot be captured by the Jacobian matrix, is identified and systematically classified. This leads us to the concept of modified Jacobian matrix whereby a quasi-affine representation of a non-linear cellular automaton is introduced.

On selecting the most likely event

In a multinomial population, when the procedure for selecting the most likely event is to choose the cell with maximum observations (ties broken by randomisation), the least favourable configuration is derived under certain restrictions on the probability vector. The results in particular disprove a conjecture of Marshall and Olkin (1979) and provide partial answers to all the four conjectures of Chen and Hwang (1984).

Goodness-of-fit testing in growth curve models: A general approach based on finite differences

Growth curve models are routinely used in various fields such as biology, ecology, demography, population dynamics, finance, econometrics, etc. to study the growth pattern of different populations and the variables linked with them. Many different kinds of growth patterns have been used in the literature to model the different types of realistic growth mechanisms. It is generally a matter of substantial benefit to the data analyst to have a reasonable idea of the nature of the growth pattern under study. As a result, goodness-of-fit tests for standard growth models are often of considerable practical value. In this paper we develop some natural goodness-of-fit tests based on finite differences of the size variables under consideration. The method is general in that it is not limited to specific parametric forms underlying the hypothesized model so long as an appropriate finite difference of some function of the size variables can be made to vanish. In addition it allows the testing process to be carried out under a set up which manages to relax most of the assumptions made by Bhattacharya et al. (2009); these assumptions are generally reasonable but not guaranteed to hold universally. Thus our proposed method has a very wide scope of application. The performance of the theory developed is illustrated numerically through several sets of real data and through simulations.

Selecting the t-best cells in a multinomial distribution

The problem of selecting the t-best cells in a multinomial distribution with t + k cells, k > 1, 2 <= t is considered under the fixed sample-size indifference zone approach. The least favourable configuration is derived for the usual procedure of selection, for large values of N (the sample size). The result settles Conjecture I (for large N) and Conjecture IV of Chen and Hwang (Commun. Statist. - Theory Meth. 13 (10), 1289-1298, 1984) in the affirmative.

MULTINOMIAL SUBSET SELECTION USING INVERSE SAMPLING AND ITS EFFICIENCY WITH RESPECT TO FIXED SAMPLING

The multinomial selection problem is considered in its general form where the objective is to select a subset of s cells which contain the t ‘best’ cells, s ≥ t. The inverse-sampling procedure is studied for this problem and the LFC is derived under the difference zone. An expression for the relative efficiency of this procedure with respect to the widely used fixed-sample-size selection procedure is obtained and theoretical bounds are derived for this efficiency. It is found that the inverse-sampling procedure performs uniformly better than the usual fixed-sampling procedure in the case s = t and is often more efficient for s > t. When the selection goal is to select any c of the t best cells, using a subset of s cells, expressions for efficiency may be similarly obtained.

An asymptotically minimax procedure for selecting the t-best multinomial cells

Abstract An asymptotically minimax simple sequential procedure for selection of the cells corresponding to the t highest multinomial cell probabilities has been given. It is found that the procedure performs uniformly better than the other two conventional procedures, namely the inverse sampling procedure and the fixed-sample-size procedure. For t=1, this proves some conjectures of Alam (Technometrics 13 (1971) 843–850). For t⩾1, derivations of least favorable configurations are immediate for the above-mentioned procedures with different indifference zones and with the number of observations tending to ∞. These also provide a simple proof of various main results in Bhandari and Bose (J. Statist. Plann. Inference 17 (1987) 227–240, Comm. Statist. Theory Methods 18 (1989) 3313–3326), Chen and Sobel (IMS Lecture Notes—Monograph Series 5 (1984) 206–210), Cacoullos and Sobel (Proc. 1 st Internat. Symp. on Multivariate Analysis (1966)) and Kesten and Morse (Ann. Math. Statist. 30 (1959) 120–127).

Multivariate Majorization and Directional Majorization : Negative Results

General relations between directional majorization and multi variate majorization are studied, and sufficient conditions for directional majorization not to imply multivariat majorization are obtained.

An efficient set estimator in high dimensions: consistency and applications to fast data visualization

Data visualization from a point set by estimating the underlying region is a problem of considerable practical interest and is an associated problem of set estimation. The most important issue in set estimation is consistency. Only a few existing point pattern shape descriptors that estimate the underlying region are consistent set estimators (a set estimator is consistent if it converges--in an appropriate sense--to the original set as the sample size increases). On the other hand, to be used as a shape descriptor, a set estimator should also satisfy several important criteria such as correct identification of number of components, robustness in the presence of noise and computational efficiency. Here we propose such a class of set estimators called s-shapes, which remain consistent in finite dimensions when the data are generated from any continuous distribution. These set estimators can be easily computed and effectively used for fast data visualization. Detailed studies on their performance such as error rates, robustness in presence of noise, run-time analysis, etc., are also performed.

On Gaussian Correlation Inequalities for Rectangles and Symmetric Sets

Pitts conjecture (1977) that P(A ∩ B) ≥ P(A)P(B) under the Nn (0, In) distribution of X, where A, B are symmetric convex sets in IRn still lacks a complete proof. This note establishes that the above result is true when A is a symmetric rectangle while B is any symmetric convex set, where A, B ∈ IRn. We give two different proofs of the result, the key component in the first one being a recent result by Hargé (1999). The second proof, on the other hand, is based on a rather old result of Šidák (1968), dating back a period before Pitts conjecture.

An Optimal Sequential Procedure for Ranking Pairwise Compared Treatments

Scores of k players have been observed. Tho problem of selecting the player with highest expected wins is considered. An optimal sequential rule has been derived in this context, Its efficiency with the bost fixed sample size rule is considered at the common least favourable configuration. AMS 1980 Subject Classification: 62F07.