Danny W. Turner

A comparison of asymptotic error rate expansions for the sample linear discriminant function

Abstract Several asymptotic expansions for approximating the expected or unconditional probability of misclassification for the sample linear discriminant function are compared for accuracy in terms of yielding the smallest mean absolute deviation from the exact value for 104 population configurations. The actual expected probabilities of misclassification are found via Monte Carlo simulation. A simple and relatively obscure asymptotic expansion derived by Raudys ( Tech. Cybern. 4 , 168–174, 1972) is found to yield better approximation than the well-known asymptotic expansions.

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.

On clustering with chernoff-type faces

Chernoff (1973) introduced a new procedure for representing multidimensional data by using cartoon-like faces drawn by a pen plotter, while Turner and Tidmore (1977) introduced asyrimetric Chernoff-type faces which can be generated on a line printer. The use of such faces for clustering multivariate data is a well known technique. However, there have been few attempts, to evaluate this graphical procedure in a systematic fashion. This paper reports results obtained in a comparison of the 1ine printer faces clustering method with several nongraphical hierarchical clustering algorithms, including single , compete , and average linkage and Wards minimum variance method..

Improved Kolmogorov inequalities for the binomial distribution

Abstract This paper presents improved Kolmogorov inequalities for the binomial distribution.

A note on the rubustness of the lilliefors test for univariate normality with respect to equicorrelated data

The Lilliefors test, which was developed by Lilliefors (1967), is a well-known test for univariate normality when population parameters are unknown. The main assumption for implementing the test is the independent-data assumption. This paper demonstrates the robustness of the Lilliefors test against equicorrelated observations. More specifically, we show that the null distribution of the Lilliefors test statistic is invariant under the alternate assumption that the observations are equicorrelated.

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).

A ratio-of-uniforms method for generating exponential power variates

A ratio-of-uniforms method of generating exponential power variates is presented. It is compared to an established generalized rejection method developed by Tadikamalla (1980) and shown to be faster and more easily implemented.

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.

Comparisons of several graphical methods for representing multivariate data

Abstract Graphical displays provide a powerful tool for presenting and studying many types of data. This article presents an evaluation of several special graphical methods for representing multivariate data, including three face-type methods and a function-plot method. The evaluation utilizes a split-plot-factorial experimental design. Under the conditions of this experiment, the modified Chernoff-face data-representation method is clearly superior.