Equal Covariance Research Articles

Distances between normal populations when covariance matrices are unequal

The definition of distance between two populations of equal covariance matrices is extended to two and more than two populations with unequal covariance matrices and Rao’s U test for testing the conditional contribution of a subset of variables to the distance is extended to this situation, even when sample sizes are not necessarily the same.

Statistical process control of acoustic emission for cutting tool monitoring : Mechanical Systems and Signal Processing, Vol. 3, No. 4, pp. 405–425 (Oct. 1989)

Abstract The problem of cutting process monitoring has been investigated in recent years, with encouraging results, using pattern recognition analysis of acoustic emission (AE) signals. The analyses are based on linear discriminant functions, which assume that the observed data (from each class) are independent random samples from multivariate normal distributions with equal covariance matrices. However, in a number of practical situations some (or all) of these assumptions may not necessarily hold, resulting in errors in the analysis. In this paper, the distributions of AE spectra generated in earlier work are first analysed, and the results indicate departure from the assumptions, although the lack of normality was not too severe. Relaxing the assumption of equality of the covariance matrices, quadratic discriminant function analysis produced improved results for tool wear and chip noise monitoring while degrading tool fracture detection. The latter is due to inadequacy of the amount of data used in training the system. It is expected that increasing the data base would improve the results for all classes. The analysis until now has focused on reducing the dimensionality of the feature space by eliminating the features with the least discriminatory power. Even though this inevitably reduces the performance of the system, it is a necessary compromise for increased computational speed. To make use of the entire feature set with a reduced matrix rank, a principal component analysis is investigated. The result is a substantial improvement in correct classification of AE signals, even under different cuting conditions.

Saddlepoint Approximations for Tests of Block Independence, Sphericity and Equal Variances and Covariances

SUMMARY Approximations for the null distribution of various test statistics arising in multivariate analysis using Skovgaard's saddlepoint formula are discussed. In each case an approximation is found with such accuracy as to obviate the need for exact computation.

Feature extraction based on decision boundaries

A novel approach to feature extraction for classification based directly on the decision boundaries is proposed. It is shown how discriminantly redundant features and discriminantly informative features are related to decision boundaries. A procedure to extract discriminantly informative features based on a decision boundary is proposed. The proposed feature extraction algorithm has several desirable properties: (1) it predicts the minimum number of features necessary to achieve the same classification accuracy as in the original space for a given pattern recognition problem; and (2) it finds the necessary feature vectors. The proposed algorithm does not deteriorate under the circumstances of equal class means or equal class covariances as some previous algorithms do. Experiments show that the performance of the proposed algorithm compares favorably with those of previous algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

CLASSIFICATION AND PROPORTION ESTIMATION

SummaryAssume that a number of individuals are to be classified into one of two populations and that, at the same time, the proportion of members of each population needs to be estimated. The allocated proportions given by the Bayes classification rule are not consistent estimates of the true proportions, so a different classification rule is proposed; this rule yields consistent estimates with only a small increase in the probability of misclassification. As an illustration, the case of two normal distributions with equal covariance matrices is dealt with in detail.

COMPARISON OF QUANTITATIVE GENETIC PARAMETERS.

COMPARISON OF QUANTITATIVE GENETIC PARAMETERS.

Rao distance between multivariate linear normal models and their application to the classification of response curves

In this paper we have defined and obtained algebraic expressions for a distance between multivariate linear normal models, with identical covariance matrix, through the Fisher information matrix. Explicit algebraic expressions have been obtained of the estimator of the distance which is used to design a test of hypothesis concerning the comparison of various multivariate linear normal models of equal covariance matrices, establishing their relation with the classic Multivariate Variance Analysis hypotheses test. This distance has also been used to obtain a classification by means of Discriminant Analysis. The advantages have also been emphasized of the use of distances in the carrying out of hypotheses testing as this enables us to make graphic representations of the results, either by representing the linear models compared on a graph, dendogram or additive tree, or in a Euclidean space of low dimension (usually a plane). A method of classifying response curves is proposed for a classification of children subjected to an oral glucose tolerance test, showing the practical use of the distance obtained. Such classification enables us to detect groups of individuals which, without having appreciable clinical differences, do present significantly different responses to oral glucose tolerance test.

Conditions for invariance of the multivariate versions of grubbs’ test and bartlett’s test under a general dependency structure

This paper generalizes a result by Stadje (1984) by deriving conditions for which a general dependency structure for multivariate observations, given in Pavur (1987), yields a positive definite covariance structure. This general dependency structure allows the sample covariance matrix to be distributed as a constant times a Wishart random matrix. It is then demonstrated that the maximum squared-radii test and a test for equal population covariance matrices have null distributions which remain unchanged when the new general dependency structure, rather than the usual independence structure, for the vector observations, is assumed. Moreover, under a general dependency structure for which the population covariance matrices are unequal, it is shown that the distribution of the test statistic for testing equal covariance matrices is identical to the distribution of the same test statistic when the population covariance matrices are equal and the observations are independent.

Classification of postoperative cardiac patients: Comparative evaluation of four algorithms

Four classification algorithms based on Bayes' rule for minimum error are compared by evaluating their ability to recognize high- and normal-risk cardio-surgical patients. These algorithms differ in the modelling of the probability density function (pdf) for each class and include: (a) two parametric algorithms based on the assumption of normal pdf; (b) two non-parametric algorithms using Parzen multidimensional approximation of pdf with normal kernels. In each case, classes with both equal and different covariance matrices were considered. A set of 200 patients in the 6 h immediately following cardiac surgery has been used to test the performance of the algorithms. For each patient the three measured variables most effective in representing the difference between the two classes were considered. We found that the two algorithms which explicitly incorporate the information on the different sample covariance between the physiological variables existing in the two classes generally provide better recognition of high- and normal-risk patients. Of these two algorithms the parametric one appears extremely attractive for practical applications, since it exhibits slightly better performance in spite of its great simplicity.

Statistical tests for structural relationship

It is an open problem to construct a test for structural relationship among the mean vectors of several multivariate normal populations with unequal covariance matrices. In this paper some solutions to this problem are provided when the unequal covariance matrices are either completely known or possess a special structure and are partially known. A class of admissible Bayes tests is characterized when the covariance matrices are completely known. When the covariance matrices Σ 1,…,Σ k have the structure Σ i = σ 2 i V, where σ 2 i are positive constants and V is a positive definite matrix, we have derived some meaningful tests using the union-intersection principle in conjunction with the Hotelling's T 2-test under the assumption that either σ 2 i 's are known or V is known. We have also provided an alternative derivation of the likelihood ratio test for the case of equal covariance matrices using the union-intersection principle, and in the sequel proposed a few new tests.

Evolution equations of the expected phase space ion densities and correlation functions in a D+T plasma

In this paper we formualte a master equation approach describing a D+T thermonuclear plasma in a lumped phase space. From the first moments of this master equation and performing the pass to the continuous limit the evolution equations for the expected phase space ion densities emerge. Also we have obtained the evolution equations of the equal time correlation and covariance functions. Finally we have deduced the hydrodynamic equations that arise from a master equation approach.

Set-values filtering and smoothing

A theory of discrete-time optimal filtering and smoothing based on convex sets of probability distributions is presented. Rather than propagating a single conditional distribution as does conventional Bayesian estimation, a convex set of conditional distributions is evolved. For linear Gaussian systems, the convex set can be generated by a set of Gaussian distributions with equal covariance with means in a convex region of state space. The conventional point-valued Kalman filter is generated to a set-valued Kalman filter consisting of equations of evolution of a convex set of conditional means and a conditional covariance. The resulting estimator is an exact solution to the problem of running an infinity of Kalman filters and fixed-interval smoothers, each with different initial conditions. An application is presented to illustrate and interpret the estimator results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust logistic discrimination

SUMMARY Logistic discrimination is a well established method for allocating observations to one of two or more populations. In this paper we show how the logistic model can be adapted to make it robust against outlying observations from the populations. The new model is successfully tested using simulated data against ordinary logistic discrimination and also against logistic discrimination where parameters are fitted using resistant methods. The model is then used on data collected from the teeth of rats. Logistic discrimination is a partially parametric method for discrimination applicable in many situations and under several sampling schemes. Notable references to the method are Day & Kerridge (1967), Anderson (1972, 1979, 1982) and Anderson & Blair (1982). The appeal of the logistic method for discrimination is that only the ratio of likelihoods has to be modelled and not the full distributional form of the underlying variables. Press & Wilson (1978) and Byth & McLachlan (1980) show that in many circumstances logistic discrimination is preferable to the usual linear discrimination based on either normal populations with equal covariance matrices or Fisher's linear discriminant. For the two group case, suppose x is a vector of observations, H1 and H2 are the two groups and a a parameter vector. Then the log likelihood ratio is modelled as

The test anxiety inventory: A confirmatory factor analysis

Abstract The study was a confirmatory investigation of the factor structure of the Test Anxiety Inventory (TAI) proposed by Spielberger, Gonzalez, Taylor, Algaze, and Anton (1978). The subjects were 752 college students (200 men and 542 women; 10 subjects did not report gender). The viability of a two-factor (emotionality and worry) oblique model was examined as compared to a two-factor orthogonal model, a single-factor model, and a null model. Results clearly supported the two-factor oblique solution. In addition, the invariance of the TAI factor structure across gender was supported as results indicated that a two-factor oblique solution with equal loadings and equal factor covariances provided a good model fit across both men and women. It was noted that two particular items, though empirically confirmed, appeared to be placed on the wrong factors. In addition, the necessity for a twenty-item, as opposed to a sixteen-item, TAI was questioned as the internal consistencies (coefficient alpha) of the two ...

Uniformly minimum variance unbiased estimation for symmetric normal distributions

We give uniformly minimum variance unbiased estimators of a mean, a variance, and a covariance for a symmetric normal distribution which is a multivariate normal distribution with equal means, equal variances, and equal covariances. The properties of the estimators are discussed.

Latent features with maximal correlation between two equal sets of variables

AbstractIf in two data tables X and Y objects are characterized by the same variables (measured at different occasions), then looking for common latent features should be more appropriate than choosing latent features in X and Y separately (e.g. as in canonical correlation or PLS). The procedure to be proposed here is a slight modification of the method of linear characteristics (according to De Groot and Li) by disclaiming the assumption of equal inner‐block covariance matrices. In order to find weights defining a latent feature with maximal correlation between X and Y, a system of non‐linear equations has to be solved. The procedure is applied to the investigation of heavy metal concentrations in different human tissues.

An empirical investigation of methods for interval estimation of the log odds ratio in discriminant analysis

SUMMARY Methods for interval estimation of the log odds ratio in discrimination are reviewed and assessed in a simulation study. The methods based on the approach of Rigby (1982) are shown to have very good properties in both the equal and unequal covariance matrix cases.

Transition to motherhood and the self: Measurement, stability, and change.

Different ways of conceptualizing and measuring change in attitudes during transition to motherhood are examined. A series of analyses was performed on data from a cross-sectional sample (N = 667) and a smaller longitudinal sample (n = 48) to demonstrate sound psychometric properties for 2 new scales and to show construct comparability across different phases of childbearing. For Childbearing Attitudes Questionnaire, results demonstrated equality of covariance for 16 scales and comparability of structure and meaning of 4 higher order factors--identification with motherhood, social orientation, self-confidence, and negative aspects of giving birth. For Mothering Self-Definition Questionnaire, results demonstrated equality of covariance of 5 scales and comparability of structure and meaning of a single higher order factor, interpreted as reflecting positive feelings about one's mothering characteristics. Analyses of correlations and mean differences identified areas of change and stability.

Statistical process control of acoustic emission for cutting tool monitoring

The problem of cutting process monitoring has been investigated in recent years, with encouraging results, using pattern recognition analysis of acoustic emission (AE) signals. The analyses are based on linear discriminant functions, which assume that the observed data (from each class) are independent random samples from multivariate normal distributions with equal covariance matrices. However, in a number of practical situations some (or all) of these assumptions may not necessarily hold, resulting in errors in the analysis. In this paper, the distributions of AE spectra generated in earlier work are first analysed, and the results indicate departure from the assumptions, although the lack of normality was not too severe. Relaxing the assumption of equality of the covariance matrices, quadratic discriminant function analysis produced improved results for tool wear and chip noise monitoring while degrading tool fracture detection. The latter is due to inadequacy of the amount of data used in training the system. It is expected that increasing the data base would improve the results for all classes. The analysis until now has focused on reducing the dimensionality of the feature space by eliminating the features with the least discriminatory power. Even though this inevitably reduces the performance of the system, it is a necessary compromise for increased computational speed. To make use of the entire feature set with a reduced matrix rank, a principal component analysis is investigated. The result is a substantial improvement in correct classification of AE signals, even under different cuting conditions.

Regional discrimination between NTS explosions and western U.S. earthquakes

AbstractA number of earthquake/explosion discriminants outlined in Pomeroy et al. (1982) are evaluated in the western United States. The data consist of 233 NTS explosions and 130 western U.S. earthquakes in the magnitude range of 2.5 to 6.5 recorded at four broadband seismic stations operated by Lawrence Livermore National Laboratory (LLNL). The stations surround NTS at distances of about 200 to 400 km. The propagation paths for the earthquakes range from approximately 175 to 1300 km and are confined mainly to the Basin and Range. The discriminants tested include mb − Ms, mb, − Msh (higher mode magnitude), long-period Love and Rayleigh wave energy density and their ratio, relative excitation of short-period SH waves, Lg/Pg, Lg/Rg, Lg/Sm short-period amplitude ratios, spectral ratios in PN, Pg and Lg, and third moment of frequency.In general, the long-period discriminants exhibit the best separation between the earthquake and explosion populations. However, the long-period measurements are often difficult to make, especially at small magnitudes. Thus, it is necessary to supplement the long-period discriminants with short-period time- and frequency-domain discriminants. The short-period time-domain discriminants generally were characterized by very poor discriminant performance. However, because of the good signal to noise ratio for many of the regional phases (such as Lg and Pg), a large number of measurements could be acquired down to small magnitudes. The usefulness of the short-period time-domain discriminants lies in the fact that when an event shows a ratio above a certain threshold, it can generally be correctly identified. Spectral discriminants appear to have good potential, even at relatively small magnitudes. The measurements are simple to make, and first-order distance corrections can easily be applied. However, a number of uncertainties regarding the lack of a physical understanding of how the spectral discriminants work makes their utilization questionable at this point.A multivariate discrimination technique is described that is designed to handle missing data while providing optimum classification of an unknown event based on a given set of discrimination variables. The discrimination variables are first transformed to be insensitive to magnitude, path, or features of the test program. Tests show that the commonly used assumption of equal covariance matrices for the earthquake and explosion populations (resulting in a linear discrimination function) is not valid. However, a quadratic discrimination function appears to be appropriate. An optimality criterion, misclassification cost minimization, is used to select the best set of discrimination variables at each station. Application of the multivariate technique to individual stations for the entire dataset shows misclassification rates ranging from 3 to 10 per cent and 4 to 26 per cent for explosions and earthquakes, respectively. Multistation discrimination shows misclassification probabilities of 0 to 2 per cent for explosions and 3 to 4 percent for earthquakes.Most misclassifications occur for mb &amp;lt; 4.0 where it is often difficult to obtain long-period measurements that perform well. Earthquakes show higher misclassification rates than explosions presumably because of their wider range of paths, depth, and focal mechanisms. Significant station variability in discrimination performance is observed, but we are unable to separate out effects of receiver structure from path effects.