Patrick L. Odell

Linear dimension reduction and Bayes classification

Abstract This paper develops an explicit expression for a compression matrix T of smallest possible left dimension k consistent with preserving the n -variate normal Bayes assignment of x to a given one of a finite number of populations and the k -variate Bayes assignment of Tx to that population. The Bayes population assignment of x and Tx are shown to be equivalent for a compression matrix T explicitly calculated as a function of the means and covariances (known) of the given populations.

Concerning several methods for estimating crop acreages using remote sensing data

The objective of this paper is to present a comprehensive survey of the proportion estimation methods so far found in the literature and a newly proposed method based on the concept of statistically equivalent blocks. All methods are restricted to the case of normal mixtures only. Graphical and semigraphical techniques are excluded. Several proportion estimation methods have been described and their properties discussed.

Behavior of aggregate state variables in ecosystem models

Abstract Our lack of complete knowledge concerning ecosystems necessitates the use of aggregate (lumped) state variables in ecosystems models. Solutions to the state equations describing these aggregates typically deviate from reference true values found by solving equivalent disaggregated systems. Since the true value is unknown in practice, it is necessary to understand how aggregate state variables behave, so as to (1) understand the error they will generate and (2) be able to minimize unacceptably high error. Analyses are restricted to the ecologically most meaningful case of aggregating within single trophic levels. Results include a general solution for aggregate dynamics under free and forced simulation, other general and specific solutions for nonequilibrium behavior, a recursive method for reducing aggregation error, and a suggested strategy for aiding in sample design.

On parallel summability of matrices

Abstract It is shown that the generalized inverses characterize the parallel sum. The almost positive definite ( a . p . d .) matrices introduced by Duffin and Morley [2] are of two types, whose intersection is the class of quasi-positive-definite matrices (Mitra and Puri [7]). The a . p . d . matrices of any one type form a “saturated” subclass of pairwise parallel summable a . p . d . matrices.

Quadratic discrimination: Some results on optimal low-dimensional representation

Abstract A random vector is assumed to belong to one several multivariate normal distributions possibility having unequal covariance matrices. The goal is to find a low-dimensional hyperplane which preserves or nearly preserves the separation of the individual population. We present a computationally simple method of deriving a linear transformation for low-dimensional representation and give conditions under which the Bayes classification rule is preserved in the low-dimensional space. Finally, we give several examples to demonstrate the method.

The effects of intraclass correlation on certain significance tests when sampling from multivariate normal population

This paper shows how the presence of simple equicorrelation in a sample from a multivariate normal population affects the confidence coefficients of the confidence set for the mean of multivariate normal population and difference of means of two multivariate normal populations with equal dispersion matrices.

On the Fixed Point Probability Vector of Regular or Ergodic Transition Matrices

Abstract A simple form for the fixed point probability row vector of regular or ergodic transition matrices is developed using some of the theory of generalized matrix inversion, together with the well known existence and uniqueness of a fixed point probability row vector. The forms for both the regular and ergodic case are identical and afford the easy calculation of the fixed point without calculating powers of the transition matrix or considering the usual convergence methods.

Estimating the probability of misclassification and variate selection

Abstract This paper considers the problem of estimating the probability of misclassifying normal variates using the usual discriminant function when the parameters are unknown. The probability of misclassification is estimated, by Monte Carlo simulation, as a function of n 1 and n 2 (sample sizes), p (number of variates) and α (measure of separation between the two populations). The probability of misclassification is used to determine, for a given situation, the best number and subset of variates for various sample sizes. An example using real data is given.

A model for dimension reduction in pattern recognition using continuous data

Abstract An approach to transform continuous data to finite dimensional data is briefly outlined. A model to reduce the dimension of the finite dimensional data is developed for the case when the covariance matrices are not necessarily equal. Necessary and sufficient conditions with respect to the spatial properties of the means and covariance matrices are given so that the linear transformation of data of higher dimensions to lower dimensions does not increase the probabilities of misclassification.

Best Linear Recursive Estimation

Abstract This article presents a matrix formulation of recursive forms for best linear unbiased estimators [Xcirc]N of the parameter vector x in the linear model yi=hix+ei, i = 1,2, ···, N when the observation vectors yi are correlated. If data are collected in sequence one can formulate recursive forms of the estimator [Xcirc]N so that it is not necessary to store all the previous data but only previous estimates and current data. This requires less storage space to obtain best linear unbiased estimators. This is especially advantageous in real-time estimation problems.