Virgil R. Marco

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 Euclidean distance classifier: an alternative to the linear discriminant function

The sample linear discriminant function (LDF) is known to perform poorly when the number of features p is large relative to the size of the training samples, A simple and rarely applied alternative to the sample LDF is the sample Euclidean distance classifier (EDC). Raudys and Pikelis (1980) have compared the sample LDF with three other discriminant functions, including thesample EDC, when classifying individuals from two spherical normal populations. They have concluded that the sample EDC outperforms the sample LDF when p is large relative to the training sample size. This paper derives conditions for which the two classifiers are equivalent when all parameters are known and employs a Monte Carlo simulation to compare the sample EDC with the sample LDF no only for the spherical normal case but also for several nonspherical parameter configurations. Fo many practical situations, the sample EDC performs as well as or superior to the sample LDF, even for nonspherical covariance configurations.

Probability inequalities for continuous unimodal random variables with finite support

A paramecer-free Bernstein-type upper bound is derived for the probability that the sum S of n i.i.d, unimodal random variables with finite support, X1 ,X2,…,Xn, exceeds its mean E(S) by the positive value nt. The bound for P{S - nμ ≥ nt} depends on the range of the summands, the sample size n, the positive number t, and the type of unimodality assumed for Xi. A two-sided Gauss-type probability inequality for sums of strongly unimodal random variables is also given. The new bounds are contrasted to Hoeffdings inequality for bounded random variables and to the Bienayme-Chebyshev inequality. Finally, the new inequalities are applied to a classic probability inequality example first published by Savage (1961).

Predictive discrimination for autoregressive processes

Abstract A predictive method for discriminating between distinct univariate autoregressive classes is derived. The predictive classification rule is derived for the case of known class order, and a rule is given for the case where the orders of the competing autoregressive processes are unknown.

On the robustness of the equal-mean discrimination rule with uniform covariance structure against serially correlated training data

Abstract The effect of correlated training data on the error rates for the sample linear discriminant function has been studied by Basu and Odell,(1) McLachlan,(2) Tubbs,(3) and Lawoko and McLachlan.(4–6) This paper investigates the effect of serially correlated data on the expected error rate of the equal-mean classifier with uniform covariance structure.

A note on improved variance bounds for certain bounded unimodal distributions

Gray and Odell have proved that no symmetric continuous unimodal density on the interval [a,b], with modes interior to (a,b), can have variance exceeding (b - a)2/12. Jacobson has derived more general sufficient conditions for the application of this bound and also has shown that no unimodal distribution on [a,b] can have variance larger than (b - a)2/9. Seaman, Odell and Young have presented even more general sufficient conditions for the smaller bound. In this note, we make use of a dispersion ordering to show that the previous conditions for the smaller bound are far too restrictive. Indeed, no continuous unimodal density [latin small letter f with hook] on [a, b], with [latin small letter f with hook](a) [less-than-or-equals, slant]1/(b - a) and [latin small letter f with hook](b) [less-than-or-equals, slant] 1/(b - a), can have variance larger than (b - a)2/12.

Dimension reduction for predictive discrimination

Abstract Aitchison, Habbema, and Kay [2] have shown that the predictive discrimination method, formulated by Geisser [11], often outperforms the classical Bayes ‘estimative’ procedure when the parameters of the competing populations are unknown, especially when the sample sizes are small relative to the dimension of the data. However, the larger the data dimension, the more computationally cumbersome the classification algorithm, primarily due to the inversion of high-dimensional matrices. Furthermore, high-dimensional data requires large sample sizes to adequately estimate population parameters. To simultaneously reduce the computational effort and estimator variability, we derive a dimension-reduction technique applicable to predictive discrimination. A Monte Carlo simulation is performed to demonstrate the information-preserving capacity, computational simplicity, and performance properties of the proposed dimension-reduction method for predictive classification.

Bayesian outlier testing using the predictive distribution for a linear model op constant intraclass form

The problem of testing suspected outliers from a linear model with constant intraclass correlation is considered from a Bayesian viewpoint. The main objective of this paper is to develop an outlier test procedure based on the predictive distribution of suspected outlier observations given a set of existing inlier observations. The test procedure is easily performed with the usual F and t distributions.