Showing papers on "Linear discriminant analysis published in 1976"

A Unifying Tool for Linear Multivariate Statistical Methods: The RV-Coefficient

Abstract: Consider two data matrices on the same sample of n individuals, X(p x n), Y(q x n). From these matrices, geometrical representations of the sample are obtained as two configurations of n points, in Rp and Rq It is shown that the RV‐coefficient (Escoufier, 1970, 1973) can be used as a measure of similarity of the two configurations, taking into account the possibly distinct metrics to be used on them to measure the distances between points. The purpose of this paper is to show that most classical methods of linear multivariate statistical analysis can be interpreted as the search for optimal linear transformations or, equivalently, the search for optimal metrics to apply on two data matrices on the same sample; the optimality is defined in terms of the similarity of the corresponding configurations of points, which, in turn, calls for the maximization of the associated RV‐coefficient. The methods studied are principal components, principal components of instrumental variables, multivariate regression, canonical variables, discriminant analysis; they are differentiated by the possible relationships existing between the two data matrices involved and by additional constraints under which the maximum of RV is to be obtained. It is also shown that the RV‐coefficient can be used as a measure of goodness of a solution to the problem of discarding variables.

On the Effects of Dimension in Discriminant Analysis

Abstract: Given fixed numbers of labeled objects on which training data can be obtained, how many variables should be used for a particular discriminant algorithm? This, of course, cannot be answeredin general since it depends on the characteristics of the populations, the sample sizes, and the algorithm. Some insight is gained in this article by studying Gaussian populations and five algorithms: linear discrimination with urlknown means and known covariance, linear discrimination with unknown means and unknown covariances, quadratic discrimination with unknown covariances and two nonparametric Bayes-type algorithms having density estimates using different, kernels (Gaussian and Cauchy).

A prediction model of landslips

Abstract: It is the intention of this investigation to develop a prime relation that makes a prediction of landslips possible. Therefore 250 stable and unstable objects were obtained all over the BRD and 31 variables were determined. They consist of morphometric, soil-mechanic, material and stratification characteristics. The principle component-analysis explains the relations between the variables. An F-test shows that nine independent variables exist. A bivariate discriminant analysis chosen with 150 at random objects yields a prediction function which efficiency later is tested by a procedure of the Euklid distance. From seven tested variable combinations that one was selected which showed that the slope gradient HN (o), the watershed to the considered object FL (km2) and a parameter measuring the density of the soil EIND are sufficient to predict a landslip. The significant discriminant function is T = 0,92222 · 10−5HN2 + 0,7926 lg(FL + 10) − 0,6098 lg(EIND + 10) with the centroids 0,1934 for slips 0,1781 for stable objects. The quality of the model was demonstrated by 100 objects from which 94 were predicted in the right group.

Further applications of bias to discriminant analysis

Abstract: This article extends the biased minimum x2 rule to the unequal covariance matrix case and to the case of several populations, The biased procedure is shown to improve the performance of the commonly used classification procedures. Results of sampling experiments over a broad range of conditions are provided to demonstrate this improvement.

Alternative Approaches to Missing Values in Discriminant Analysis

Abstract: This paper compares by simulations several methods of handling missing observations in discrimination. In an earlier paper, several methods were compared in discriminating by the usual linear discriminant function between two multivariate normal populations in which all pairs of variables are equally correlated. In the present study, a variety of population matrices was used and two additional methods were introduced: the first, a simpler regression technique and the second, a modified technique based on the first principal component. The new regression technique was found to give relatively high probabilities of correct classification.

Practical significance and partitioning variance in discriminant analysis

Abstract: This note sets forth a method for obtaining viable estimates of total discriminatory power in a discriminant analysis. In addition, a technique based upon this methodisgiven for assessing the overall relative importance of each predictor variable regardless of the number of discriminant functions. Two examples are given to illustrate application of the method.

The bias of the apparent error rate in discriminant analysis

Abstract: SUMMARY The apparent error rate is a commonly used estimator of the actual error rate in discriminant analysis. In this study the asymptotic bias of the apparent error rate is derived in the context of two multivariate normal populations with unknown different means and unknown common covariance matrix. From the derived expansion a correction term is available for reducing the bias of the apparent error rate from the first to the second order with respect to the reciprocals of the initial sample sizes. Also, some previously unanswered questions on inequalities between the average apparent, the optimal, and the average actual error rates are solved.

Distribution-Free Partial Discriminant Analysis

Abstract: A distribution-free rank procedure is proposed for use in partial discrimination problems involving two populations. It is shown that this procedure can be applied with virtually any discriminant function. Moreover, the discriminant function may be selected after observing the samples on which it is to be based. Using Monte Carlo methods the rank procedure is compared with a normal theory and a tolerance region procedure. The rank procedure was the only one that adequately controlled the probabilities of misclassification while maintaining relatively small probabilities of not classifying an observation.

Discrimination, Allocatory and Separatory, Linear Aspects

Abstract: Publisher Summary This chapter discusses the twin goals of linear discrimination, that is, allocation and separation. It reviews linearity in the mutivariate normal case and discusses the extent to which linearity is optimal. The exact linear theory is strictly appropriate for restricted sets of distributional assumptions, though the linear theory is somewhat wider than the logistic family. However, the linear theory gives reasonably robust, if less than optimal, solutions to many cases. However, there are situations where the linear theory should not be applied, for example, where two normal populations have the same mean but differ in their covariance matrices. In this situation, the linear discriminants will be quite inappropriate. However, the logistic model encompasses a variety of possible distributional assumptions. While presumably robust for its class when its parameters are estimated, it is not expected to yield as efficient a procedure when compared to one based on the true member of the class.

Criterion for Selecting Variables for Linear Discriminant Function

Abstract: This paper considers the problem of selection of variables for the linear discriminant function (A nderson's classification statistic, W) in the case of two multivariate normal populations with different means and a common covariance matrix. The two conditional errors of allocation associated with the application of W are averaged over the prior distribution of the populations to form the conditional risk. A method for selecting variables is then developed on the basis of the conditional risk which is of central interest in evaluating the performance of the discriminant function formed from the selected variables. A tolerance interval on the increased conditional risk consequent to deleting a subset of variables from the linear discriminant function is presented. The confidence coefficient corresponding to no increase in risk is considered as an indicator of the additional discrimination value of the subset deleted. The proposed method can be used either as a step-down or a step-up type selection procedure.

Investigating Group Differences An Explication of the Sociological Potential of Discriminant Analysis

Abstract: Frequently, sociologists encounter research problems which call for assessing the extent to which two or more categories of people or events can be maximally distinguished from one another with respect to a number of common variable attributes. Unfortunately, predictive models which evidently accomplish just this, in addition to providing rules for the classification of new entities, have been conspicuously absent from the literature. Discriminant analysis, a multivariate technique extensively employed in physical anthropology, clinical psychology, and the biological sciences, has made scant headway in sociology, even though many of our most interesting substantive investigations are clearly amenable to it. The purpose of this paper, then, is first to suggest numerous research situations in which two-group and multiple discriminant analyses might be productively applied. By way of illustration, a brief example of the use of discriminant analysis for data on student politics will be presented. Second, the ...

Geologic mapping using LANDSAT data

Abstract: The feasibility of automated classification for lithologic mapping with LANDSAT digital data was evaluated using three classification algorithms. The two supervised algorithms analyzed, a linear discriminant analysis algorithm and a hybrid algorithm which incorporated the Parallelepiped algorithm and the Bayesian maximum likelihood function, were comparable in terms of accuracy; however, classification was only 50 per cent accurate. The linear discriminant analysis algorithm was three times as efficient as the hybrid approach. The unsupervised classification technique, which incorporated the CLUS algorithm, delineated the major lithologic boundaries and, in general, correctly classified the most prominent geologic units. The unsupervised algorithm was not as efficient nor as accurate as the supervised algorithms. Analysis of spectral data for the lithologic units in the 0.4 to 2.5 microns region indicated that a greater separability of the spectral signatures could be obtained using wavelength bands outside the region sensed by LANDSAT.

Application of step-wise discriminant analysis and Bayesian classification procedure in determining prognosis of acute myocardial infarction.

Abstract: A retrospective study was carried out to assess the feasibility of computer-assisted prognostication by discriminant analysis and the Bayesian classification procedure based on clinical information collected on patients with acute myocardial infarction. The overall accuracy was 94.2% in predicting hospital death but the prediction of late death after discharge was less accurate. It was found that not all of the 44 variables used for analysis were necessary to reach the same level of predictive accuracy--16 to 20 variables would result in almost the identical prediction. The Bayesian classification procedure was applied to estimate probabilities of individual patients belonging to the different prognostic categories.

Simultaneous Procedures in Discriminant Analysis Involving Two Groups

Abstract: A simrdtaneous test, procedure is proposed for redrIcing the number of variables required to discriminate between two multivariate normal poprdatiorls. This procedure forms part of a wider simrdtaneolls procedure, for detailed descriptive analysis of group differences, which isolates subsets of variables providing significant discrimination as well as sltbsets of variables providing essentially as milch grollp separation as the original set of variables.

A discriminant analysis approach to the identification of human chromosomes.

Abstract: The identification of the various types of the 46 chromosomes in a normal human cell is formulated as a discriminant analysis problem. Derived are expressions for posterior probabilities, and rules are given for identification where the number of chromosomes of each type is exactly known. The resulting model is applied to experimental data from chromosomes of the Denver B-group. The results using the developed model compare favorably with those of the standard discriminant analysis approach.

Ridge Discriminant Analysis.

Abstract: : Discriminant analysis is one of the more commonly applied multivariate techniques. The performance of the sample linear discriminant function usually employed depends on the quality of the estimates of the parameters involved, but especially on the estimate of the covariance matrix of the population. It is demonstrated that an improved classification process is possible with techniques similar to those of ridge regression. A Monte Carlo study is undertaken to examine the extent of the improvement possible and to determine a reasonable criterion for selecting the ridge constant. Application of the method is discussed.

Identification of complex geometrical shapes by means of low-frequency radar returns

Abstract: A study of the effectiveness of low-frequency electromagnetic responses for identifying objects of complex shape is presented. The linear separability of a large variety of objects such as cubes, cylinders and aircraft were examined. Two classification algorithms, a linear discriminant and a nearest neighbour rule, were used to classify a set of four aircraft models. The classification performance is presented in terms of the probability of misclassification versus noise level. The effectiveness of the various combinations of electromagnetic features was evaluated. The results indicate that amplitude, phase and polarization all contribute substantial amounts of target information. Making the assumption of a priori knowledge of the target's approximate aspect angle, a reliable classification can be attained utilizing a rather small number of frequencies.

An application of a discriminant analysis, based on the weighted H-index—a reanalysis

Abstract: .— A discriminant-analysis method for dichotomized data, based on the weighted H-index as the similarity measure between two persons, is introduced. The weight assigned each item is a strictly increasing function of the absolute value of its D-estimate. Here, only power functions are used. The method which is called the WHIDD-analysis, is applied to some clinical data (Jonsson, 1975). The power of 3 produces a correct classification of all 32 persons in the validation group.

The Utility of a Standard Composite for Forecasting Academic Performance in Several Liberal Arts Colleges.

Abstract: In this study, one linear discriminant function of four admissions scores was found to exhaust more than 90% of their predictive power for separating 16 college/level-of-achievement groups — i.e., high- and low-performing freshman women at each of eight liberal arts colleges. Moreover, the corresponding discriminant score yielded validity coefficients vs freshman- and senior-level continuous grade-average criteria within each college comparable to coefficients yielded by group-specific, regression-weighted composites of the four scores in cross-validation samples. Results of the study suggest that multiple-discriminant analysis provides a rigorous and practical basis for adding an important and largely unexplored between-group dimension to studies in the prediction of within-group performance.

Computation of Concave Piecewise Linear Discriminant Functions Using Chebyshev Polynomials

Abstract: In two-class pattern classification problems, the use of piecewise linear discriminant functions (PLDF's) is often encountered. Following a consideration of the relative advantages of a PLDF above an optimal?high degree?discriminant function, a new procedure to compute a concave PLDF is presented. The characteristic property of this procedure is that the number of linear functions necessary for adequate results of the PLDF is determined adaptively during the procedure. The linear functions are computed with Chebyshev polynominals.