Showing papers on "Linear discriminant analysis published in 1968"

Financial ratios, discriminant analysis and the prediction of corporate bankruptcy

Abstract: ACADEMICIANS SEEM to be moving toward the elimination of ratio analysis as an analytical technique in assessing the performance of the business enterprise. Theorists downgrade arbitrary rules of thumb, such as company ratio comparisons, widely used by practitioners. Since attacks on the relevance of ratio analysis emanate from many esteemed members of the scholarly world, does this mean that ratio analysis is limited to the world of \"nuts and bolts\"? Or, has the significance of such an approach been unattractively garbed and therefore unfairly handicapped? Can we bridge the gap, rather than sever the link, between traditional ratio \"analysis\" and the more rigorous statistical techniques which have become popular among academicians in recent years? The purpose of this paper is to attempt an assessment of this issue-the quality of ratio analysis as an analytical technique. The prediction of corporate bankruptcy is used as an illustrative case.' Specifically, a set of financial and economic ratios will be investigated in a bankruptcy prediction context wherein a multiple discriminant statistical methodology is employed. The data used in the study are limited to manufacturing corporations. A brief review of the development of traditional ratio analysis as a technique for investigating corporate performance is presented in section I. In section II the shortcomings of this approach are discussed and multiple discriminant analysis is introduced with the emphasis centering on its compatibility with ratio analysis in a bankruptcy prediction context. The discriminant model is developed in section III, where an initial sample of sixty-six firms is utilized to establish a function which best discriminates between companies in two mutually exclusive groups: bankrupt and non-bankrupt firms. Section IV reviews empirical results obtained from the initial sample and several secondary samples, the latter being selected to examine the reliability of the discriminant

Estimation of Error Rates in Discriminant Analysis

Abstract: Several methods of estimating error rates in Discriminant Analysis are evaluated by sampling methods. Multivariate normal samples are generated on a computer which have various true probabilities of misclassification for different combinations of sample sizes and different numbers of parameters. The two methods in most common use are found to be significantly poorer than some new methods that are proposed.

State of the art in pattern recognition

Abstract: This paper reviews statistical, adaptive, and heuristic techniques used in laboratory investigations of pattern recognition problems. The discussion includes correlation methods, discriminant analysis, maximum likelihood decisions minimax techniques, perceptron-like algorithms, feature extraction, preprocessing, clustering and nonsupervised learning. Two-dimensional distributions are used to illustrate the properties of the various procedures. Several experimental projects, representative of prospective applications, are also described.

On Discrimination Using Qualitative Variables

Abstract: The use of a linear function for discriminating with dichotomous variables is discussed and evaluated. Four such functions are considered: Fisher's linear discriminant function, two functions based upon a logistic model, and a function based upon the assumption of mutual independence of the variables. The evaluation of these functions as well as of a completely general multinomial procedure is carried out within the context of a 1st order interaction model by means of computer experiments. The product moment correlation of the optimal function with the linear function under evaluation plays a central role as a criterion for judging the relative merits of the procedures considered.

Pattern Classifier Design by Linear Programming

Abstract: —A common nonparametric method for designing linear discriminant functions for pattern classification is the iterative, or "adaptive," weight adjustment procedure, which designs the discriminant function to do well on a set of typical patterns. This paper presents a linear programming formulation of discriminant function design which minimizes the same objective function as the "fixed-increment" adaptive method. With this formulation, as with the adaptive methods, weights which tend to minimize the number of classification errors are computed for both separable and nonseparable pattern sets, and not just for separable pattern sets as has been the emphasis in previous linear programming formulations.

On Expected Probabilities of Misclassification in Discriminant Analysis, Necessary Sample Size, and a Relation with the Multiple Correlation Coefficient

Abstract: When a sample discriminant function D, is computed, it is desired to estimate the chance of misclassification using D8. This is often done by classifying the sample with the help of D8 or by computing 4((ID), where b is the cumulative normal distribution, and D2 is Mahalanobis' distance. When D. is applied to a new sample, the observed probabilities of misclassification are usually found to be greater than those computed from the initial sample. The purposes of this paper are to show that this increase in the probabilities of misclassification is directly related to the 'shrinkage' of the multiple correlation coefficient R2 in new samples and that these are related to the unbiased estimation of Mahalanobis' 62 using D2.

Canonical Correlation and Its Relationship to Discriminant Analysis and Multiple Regression

Abstract: Canonical correlation analysis is concerned with the determination of a linear combination of each of two sets of variables such that the correlation between the two functions is a maximum. Under certain conditions this analysis is equivalent to discriminant analysis and under other conditions it is equivalent to multiple regression. In this paper the relationships among these techniques are discussed, equations relating to prediction by canonical variates are derived, a generalized correlation coefficient is proposed, and an example of canonical correlation analysis is presented.

Missing Values in Linear Multiple Discriminant Analysis

Abstract: This paper presents the results of an empirical study of handling the problem of missing values in a discriminant function analysis where both number of variables and number of individuals were very large. Elimination of individuals with missing values is considered and rejected. Estimation of missing values using means and estimation by an iterative regression technique are essayed, and the results compared. In this particular study, the far simpler method of using means for missing values gives comparable results with the regression estimation technique. Additional studies of the estimation techniques are recommended.

Classification algorithms in pattern recognition

Abstract: Linear and nonlinear methods of pattern classification which have been found useful in laboratory investigations of various recognition tasks are reviewed. The discussion includes correlation methods, maximum likelihood formulations with independence or normality assumptions, the minimax Anderson-Bahadur formula, trainable systems, discriminant analysis, optimal quadratic boundaries, tree and chain expansions of binary probability density functions, and sequential decision schemes. The area of applicability, basic assumptions, manner of derivation, and relative computational complexity of each algorithm are described. Each method is illustrated by means of the same two-class two-dimensional numerical example. The "training set" in this example comprises four samples from either class; the "test set" is the set of all points in the normal distributions characterized by the sample means and sample covariance matrices of the training set. Procedural difficulties stemming from an insufficient number of samples, various violations of the underlying statistical models, linear nonseparability, noninvertible covariance matrices, multimodal distributions, and other experimental facts of life are touched on.

A survey of pattern recognition

Abstract: This paper surveys the developments in the field of pattern classification to describe the current state-of-the-art The author divides the pattern classification problem into deterministic and statistical approaches although, in many instances, both converge to the same result The approaches are further subdivided into: Adalines and Madalines, linear discriminant functions, mathematical programming, mode-seeking, nearest neighbor, Bayes and minimax, statistical criteria, and fuzzy sets subclasses A major effort is devoted to show relationships among procedures The conditions under which procedures are equivalent are discussed Such relationships are summarized through a graph The problems of supervision, adaptive property, sample size, and sequential analysis are discussed

The distributions of certain factors occurring in discriminant analysis

Abstract: Significance tests for several hypothetical discriminant functions have been developed by Williams (7,8) and considered further by the author (6). The test criteria consist of the factors in certain factorizations of the residual likelihood criterion, when the effect of the hypothetical discriminant functions has been eliminated. The independence and distributions of the factors can be seen by geometrical considerations, to be a consequence of the manner in which the factors are constructed in the sample space. In the case of a single hypothetical discriminant function Kshirsagar (5) has produced analytic proofs, by means of matrix transformations, of the independence and distributions of the factors. In this paper we shall give analytic proofs of the independence and distributions of the factors, given in sections 4 and 5 of the authors' paper (6), by extending Kshirsagar's proof to the case of several hypothetical discriminant functions.

On the upper bound of the probability of error of a linear pattern classifier for probabilistic pattern classes

Abstract: Given a set of sample patterns for two pattern classes, some simple expressions for the upper bound of the probability of error for a linear pattern classifier and the optimal linear discriminant function minimizing the upper bound are obtained. Using these results, if the tolerable probability of error of classifying patterns in the two pattern classes is not smaller than this upper bound, not only a linear pattern classifier is known to be feasible, but also a satisfactory linear discriminant function is given. The results presented here are independent of the probability distribution of the patterns in the pattern classes. For some special cases, a smaller upper bound is found.

Classification and discrimination in the analysis of credit risks: ii.

Abstract: : The report introduces a decision-theoretic approach to the problem of discriminating among several categories of credit risks. Bayesian decision rules are discussed and illustrated by a numerical example. Limitations of linear discriminant methods are pointed out. Practical problems concerning the unbiased testing of classification rules, the discriminations of rare patterns and the updating of the available statistical information are also discussed. The report concludes with a brief treatment of new approaches to sequential classification. (Author)