CLASSIFICATION, DISTRIBUTIONS AND ADMISSIBILITY IN AUTOREGRESSIVE PROCESS

The problem considered here is to classify a unit into one of two populations based on a vector of measurements on the unit. The observation vector is assumed to follow an auto-regressive process. Samples from the process are used to construct classification rules. The distributions of some classification statistics are obtained. The admissibility of some classification rules is established.     

Classification into two normal populations with a common mean and unequal variances

The problem of classification into two univariate normal populations with a common mean is considered. Several classification rules are proposed based on efficient estimators of the common mean. Detailed numerical comparisons of probabilities of misclassifications using these rules have been carried out. It is shown that the classification rule based on the Graybill-Deal estimator of the common mean performs the best. Classification rules are also proposed for the case when variances are assumed to be ordered. Comparison of these rules with the rule based on the Graybill-Deal estimator has been done with respect to individual probabilities of misclassification.     

On discrimination and classification with multivariate repeated measures data

We study the problem of classification with multiple q-variate observations with and without time effect on each individual. We develop new classification rules for populations with certain structured and unstructured mean vectors and under certain covariance structures. The new classification rules are effective when the number of observations is not large enough to estimate the variance–covariance matrix. Computational schemes for maximum likelihood estimates of required population parameters are given. We apply our findings to two real data sets as well as to a simulated data set.     

A classification rule for ordered exponential populations

In this paper, we consider classification procedures for exponential populations when an order on the populations parameters is known. We define and study the behavior of a classification rule which takes into account the additional information and outperforms the likelihood-ratio-based rule when two populations are considered. Moreover, we study the behavior of this rule in each of the two populations and compare the misclassification probabilities with the classical ones. Type II censorship, which is usual in practice, is considered and results obtained. The performance for more than two populations is evaluated by simulation.     

Proportion estimation in mixtures with covariates

Linear maps of a single unclassified observation are used to estimate the mixing proportion in a mixture of two populations with homogeneous variances in the presence of covariates. with complete knowledge of the parameters of the individual populations, the linear map for which the estimator is unbiased and has minimum variance amongst all similar estimators can be determined. Plug-in estimator based on independent training samples from the component populations can be constructed and is asymptotically equivalent to Cochran's classification statistic V* for covariate classification; see Memon and Okamoto (1970). Under normality assumptions, asymptotic expansion of the distribution of the plug-in estimator is available. In the absence of covariates, our estimator reduces to that suggested by Walker (1980) who has investigated the problem based on information on large unclassified samples from a mixture of two populations with heterogeneous variances. In contrast, distribution of Walker's estimator seems intractable in moderate sample sizes even with normality assumption.     

Sampling strategy for optimal classification into one of two correlated normal populations

A unit ω is to be classified into one of two correlated homoskedastic normal populations by linear discriminant function known as W classification statistic [T.W. Anderson, An asymptotic expansion of the distribution of studentized classification statistic, Ann. Statist. 1 (1973), pp. 964–972; T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd edn, Wiley, New York, 1984; G.J. Mclachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley and Sons, New York, 1992]. The two populations studied here are two different states of the same population, like two different states of a disease where the population is the population of diseased patient. When a sample unit is observed in both the states (populations), the observations made on it (which form a pair) become correlated. A training sample is unbalanced when not all sample units are observed in both the states. Paired and also unbalanced samples are natural in studies related to correlated populations. S. Bandyopadhyay and S. Bandyopadhyay [Choosing better training sample for classifying an individual into one of two correlated normal populations, Calcutta Statist. Assoc. Bull. 54(215–216) (2003), pp. 167–180] studied the effect of unbalanced training sample structure on the performance of W statistics in the univariate correlated normal set-up for finding optimal sampling strategy for a better classification rate. In this study, the results are extended to the multivariate case with discussion on application in real scenario.     

Prior elicitation in the classification problem

Results are developed concerning the asymptotic behaviour of the Bayes classification rule as the number of unclassified observations grows without bound. It is shown that unclassified observations serve only to estimate the individual population parameters in an unlabeled sense and do not provide information about the labels that are attached to the populations. Prior construction is approached through investigation of prior odds over regions of the joint parameter space (across all populations) deemed likely to contain the true joint parameter vector. It is shown that consideration of these prior odds can lead to more robust a posteriori classification of individual observations.     

Classification rules for two parameter exponential populations under order restrictions on parameters

In this paper, we suggest classification procedures of an observation into one of two exponential populations assuming a known ordering between population parameters. We propose classification rules when either location or scale parameters are ordered. Some of these classification rules under ordering are better than usual classification rules with respect to the expected probability of correct classification. We also derive likelihood ratio-based classification rules. Comparison of these classification rules has been done using Monte Carlo simulations.     

Sequential discrimination

An analysis of the 1-stage classification decision with two candidate populations is provided in this paper. When the successive posterior probabilities follow a first order markov process it it shown that the optimal classification rules are greatly simplified. A detailed analysis and example are provided for the important case of multivariate normality with equal covariance matrices.     

How non-normality affects the quadratic discriminant function

The quadratic discriminant function is commonly used for the two group classification problem when the covariance matrices in the two populations are substantially unequal. This procedure is optimal when both populations are multivariate normal with known means and covariance matrices. This study examined the robustness of the QDF to non-normality. Sampling experiments were conducted to estimate expected actual error rates for the QDF when sampling from a variety of non-normal distributions. Results indicated that the QDF was robust to non-normality except when the distributions were highly skewed, in which case relatively large deviations from optimal were observed. In all cases studied the average probabilities of misclassification were relatively stable while the individual population error rates exhibited considerable variability.     

Classification of Higher-order Data with Separable Covariance and Structured Multiplicative or Additive Mean Models

Although devised in 1936 by Fisher, discriminant analysis is still rapidly evolving, as the complexity of contemporary data sets grows exponentially. Our classification rules explore these complexities by modeling various correlations in higher-order data. Moreover, our classification rules are suitable to data sets where the number of response variables is comparable or larger than the number of observations. We assume that the higher-order observations have a separable variance-covariance matrix and two different Kronecker product structures on the mean vector. In this article, we develop quadratic classification rules among g different populations where each individual has κth order (κ ≥2) measurements. We also provide the computational algorithms to compute the maximum likelihood estimates for the model parameters and eventually the sample classification rules.     

Penalized optimal scoring for the classification of multi-dimensional functional data

Many fields of research need to classify individual systems based on one or more data series, which are obtained by sampling an unknown continuous curve with noise. In other words, the underlying process is an unknown function which the observed variables represent only imperfectly. Although functional logistic regression has many attractive features for this classification problem, this method is applicable only when the number of individuals to be classified (or available to estimate the model) is large compared to the number of curves sampled per individual.To overcome this limitation, we use penalized optimal scoring to construct a new method for the classification of multi-dimensional functional data. The proposed method consists of two stages. First, the series of observed discrete values available for each individual are expressed as a set of continuous curves. Next, the penalized optimal scoring model is estimated on the basis of these curves. A similar penalized optimal scoring method was described in my previous work, but this model is not suitable for the analysis of continuous functions. In this paper we adopt a Gaussian kernel approach to extend the previous model. The high accuracy of the new method is demonstrated on Monte Carlo simulations, and used to predict defaulting firms on the Japanese Stock Exchange.     

Classification with discrete and continuous variables via general mixed-data models

We study the problem of classifying an individual into one of several populations based on mixed nominal, continuous, and ordinal data. Specifically, we obtain a classification procedure as an extension to the so-called location linear discriminant function, by specifying a general mixed-data model for the joint distribution of the mixed discrete and continuous variables. We outline methods for estimating misclassification error rates. Results of simulations of the performance of proposed classification rules in various settings vis-à-vis a robust mixed-data discrimination method are reported as well. We give an example utilizing data on croup in children.     

A NEW APPROACH TO DISCRIMINATION AND CLASSIFICATION USING A HAUSDORFF TYPE DISTANCE

A new method of discrimination and classification based on a Hausdorff type distance is proposed. In two groups, the Hausdorff distance is defined as the sum of the furthest distance of the nearest elements of one set to another. This distance has some useful properties and is exploited in developing a discriminant criterion between individual objects belonging to two groups based on a finite number of classification variables. The discrimination criterion is generalized to more than two groups in a couple of ways. Several data sets are analysed and their classification accuracy is compared to that obtained from linear discriminant function and the results are encouraging. The method in simple, lends itself to parallel computation and imposes less stringent conditions on the data.     

The Equivalence of Two Rules of Classification for Two Populations

In many applied classification problems, the populations of interest are defined in terms of ranges for the dependent variable. In these situations, it is intuitively appealing to classify individuals into the respective populations based on their estimated conditional expectation. On the other hand, based on theoretical considerations, one may wish to use the classification rule based on the posterior probabilities. This article shows that under certain conditions these two classification rules are equivalent.     

A Classification Method for Directional Data with Application to the Human Skull

There are many well-known methods applied in classification problem for linear data with both known and unknown distribution. Here, we deal with classification involving data on torus and cylinder. A new method involving a generalized likelihood ratio test is developed for classifying in two populations using directional data. The approach assumes that one of the probabilities of misclassification is known. The procedure is constructed by applying Gibbs sampler on the conditionally specified distribution. A parametric bootstrap approach is also presented. An application to data involving linear and circular measurements on human skull from two tribal populations is given.     

Anderson's classification statistic based on a post-stratified training sample

The performance of Anderson's classification statistic based on a post-stratified random sample is examined. It is assumed that the training sample is a random sample from a stratified population consisting of two strata with unknown stratum weights. The sample is first segregated into the two strata by post-stratification. The unknown parameters for each of the two populations are then estimated and used in the construction of the plug-in discriminant. Under this procedure, it is shown that additional estimation of the stratum weight will not seriously affect the performance of Anderson's classification statistic. Furthermore, our discriminant enjoys a much higher efficiency than the procedure based on an unclassified sample from a mixture of normals investigated by Ganesalingam and McLachlan (1978).     

Point Estimates of Test Sensitivity and Specificity from Sample Means and Variances

In a wide variety of biomedical and clinical research studies, sample statistics from diagnostic marker measurements are presented as a means of distinguishing between two populations, such as with and without disease. Intuitively, a larger difference between the mean values of a marker for the two populations, and a smaller spread of values within each population, should lead to more reliable classification rules based on this marker. We formalize this intuitive notion by deriving practical, new, closed-form expressions for the sensitivity and specificity of three different discriminant tests defined in terms of the sample means and standard deviations of diagnostic marker measurements. The three discriminant tests evaluated are based, respectively, on the Euclidean distance and the Mahalanobis distance between means, and a likelihood ratio analysis. Expressions for the effects of measurement error are also presented. Our final expressions assume that the diagnostic markers follow independent normal distributions for the two populations, although it will be clear that other known distributions may be similarly analyzed. We then discuss applications drawn from the medical literature, although the formalism is clearly not restricted to that application.     

CLASSIFICATION AND PROPORTION ESTIMATION

Assume 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.