Discriminant Analysis for the von Mises-Fisher Distribution

The von Mises-Fisher distribution is widely used for modeling directional data. In this article, we derive the discriminant rules based on this distribution to assign objects into pre-existing classes. We determine a distance between two von Mises-Fisher populations and we calculate estimates of the misclassification probabilities. We also analyze the behavior of the distance between two von Mises-Fisher populations and of the estimates of the misclassification probabilities when we modify the parameters of the populations or the samples size or the dimension of the sphere. Finally, we present an example with real spherical data available in the literature.     

Errors of misclassification in discrimination with data from truncated <Emphasis Type="Italic">t</Emphasis> populations

The distribution of the probabilities of misclassification is derived in this paper, which are reproduced by the use of the linear discriminant function. The statistical background is two independent doubly truncated t populations with distinct location parameters and common scale parameter and degrees of freedom. The behavior of the linear discriminant function is studied by comparing the distribution function of the errors of misclassification under the truncated t and truncated normal models.     

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 a distributionfree method in discriminant analysis

Linear discriminant rules for two symmetrical distributions, which only need the first and second moments of these distributions, are presented. The rules are based on Zhezhel's idea using the most unfavourable probabilities of misclassification as an optimality criterion. Also a rule is considered which deals with distributions differing in a location and scale parameter.     

Qdf misclassification probabilities for known population parameters

Approximated QDF misclassification probabilities have been derived for bivariate normal populations with known parameter values. Tne effect of unequal covariances and population distance on the misclassification probabilities are examined     

A COMPARISON STUDY OF SOME COMBINED CLASSIFIERS

ABSTRACT

In this article we consider two methods for combining a number of individual classifiers in order to construct more effective classification rules. The effectiveness of these methods, as measured by a comparison of their misclassification error rates with those of the individual classifiers, is assessed via a number of examples that involve simulated data. We also compare the results to those of two existing combining procedures.     

Estimation of misclassification probabilities by bootstrap methods

Several methods have been proposed to estimate the misclassification probabilities when a linear discriminant function is used to classify an observation into one of several populations. We describe the application of bootstrap sampling to the above problem. The proposed method has the advantage of not only furnishing the estimates of misclassification probabilities but also provides an estimate of the standard error of estimate. The method is illustrated by a small simulation experiment. It is then applied to three published, well accessible data sets, which are typical of large, medium and small data sets encountered in practice.     

Combinatoric classification of multivariate normal variates

Consider classifying an n × I observation vector as coming from one of two multivariate normal distributions which differ both in mean vectors and covariance matrices. A class of dis-crimination rules based upon n independent univariate discrim-inate functions is developed yielding exact misclassification probabilities when the population parameters are known. An efficient search of this class to select the procedure with minimum expected misclassification is made by employing an algorithm of the implicit enumeration type used in integer programming. The procedure is applied to the classification of male twins as either monozygotic or dizygotic.     

Online change detection of Markov chains with unknown post-change transition probabilities

ABSTRACT

In this paper, we investigate the performance of cumulative sum (CUSUM) stopping rules for the online detection of unknown change point in a time homogeneous Markov chain. Under the condition that the post-change transition probabilities are unknown, we proposed two CUSUM type schemes for the detection. The first scheme is based on the maximum likelihood estimates of the post-change transition probabilities. This scheme is limited by its computation burden, which is mitigated by another scheme based on the reference transition probabilities selected from a prior known region. We give the bounds of the mean delay time and the mean time between false alarms to illustrate the effectiveness of the proposed schemes. The results of the simulation also demonstrate the feasibility of the proposed schemes.     

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.     

A note on optimum linear feature extraction for gaussian populations with equal covariances and equal a priori probabilities

We consider the linear feature selection problem of obtaining a nonzero 1 × n matrix B which minimizes the probability of misclassification based on the Bayes decision rule applied to the random variable Y = BX, where X is a random n-vector arising from one of m Gaussian populations with equal covariances and equal apriori probabilities. It is shown that the optimal B satisfies a fixed point equation B = F(B) which can be solved by successive substitution.     

Rank procedures in many population forced discrimination problems

In this paper the rank method for forced discrimination in two population problems, introduced by Randies, Broffitt, Ramberg and Hogg (1978), is extended to cover settings involving more than two populations. Several methods of ranking are compared to the normal theory procedure in a Monte Carlo study. Asymptotic theory is included which confirms that the rank method does balance the limiting probabilities of misclassification in a two population setting.     

Systematically misclassified binary dependent variables

ABSTRACT

When a binary dependent variable is misclassified, that is, recorded in the category other than where it really belongs, probit and logit estimates are biased and inconsistent. In some cases, the probability of misclassification may vary systematically with covariates, and thus be endogenous. In this paper, we develop an estimation approach that corrects for endogenous misclassification, validate our approach using a simulation study, and apply it to the analysis of a treatment program designed to improve family dynamics. Our results show that endogenous misclassification could lead to potentially incorrect conclusions unless corrected using an appropriate technique.     

The application of bias to discriminant analysis

When classification rules are constructed using sample estimatest it is known that the probability of misclassification is not minimized. This article introduces a biased minimum X² rule to classify items from a multivariate normal population. Using the principle of variance reduction, the probability of misclassification is reduced when the biased procedure is employed. Results of sampling experiments over a broad range of conditions are provided to demonstrate this improvement.     

A prior-valued estimator applied to multinomial classification

A multinomial classification rule is proposed based on a prior-valued smoothing for the state probabilities. Asymptotically, the proposed rule has an error rate that converges uniformly and strongly to that of the Bayes rule. For a fixed sample size the prior-valued smoothing is effective in obtaining reason¬able classifications to the situations such as missing data. Empirically, the proposed rule is compared favorably with other commonly used multinomial classification rules via Monte Carlo sampling experiments     

Errors of misclassification associated with the inverse gaussion distribution

Errors of misclassification and their probabilities are studied for classification problems associated with univariate inverse Gaussian distributions. The effects of applying the linear discriminant function (LDF), based on normality, to inverse Gaussian populations are assessed by comparing probabilities (optimum and conditional) based on the LDF with those based on the likelihood ratio rule (LR) for the inverse Gaussian, Both theoretical and empirical results are presented     

Comparison of combinatoric and likelihood ratio procedures for classifying samples

A random sample is to be classified as coming from one of two normally distributed populations with known parameters. Combinatoric procedures which classify the sample based upon the sample mean(s) and variance(s) are described for the univariate and multivariate problems. Comparisons of misclassification probabilities are made between the combinatoric and the likelihood ratio procedure in the univariate case and between two alternative combinatoric procedures in the bivariate case.     

Bayesian sample size determination for estimating binomial parameters from data subject to misclassification

We investigate the sample size problem when a binomial parameter is to be estimated, but some degree of misclassification is possible. The problem is especially challenging when the degree to which misclassification occurs is not exactly known. Motivated by a Canadian survey of the prevalence of toxoplasmosis infection in pregnant women, we examine the situation where it is desired that a marginal posterior credible interval for the prevalence of width w has coverage 1−α, using a Bayesian sample size criterion. The degree to which the misclassification probabilities are known a priori can have a very large effect on sample size requirements, and in some cases achieving a coverage of 1−α is impossible, even with an infinite sample size. Therefore, investigators must carefully evaluate the degree to which misclassification can occur when estimating sample size requirements.     

Inference for misclassified multinomial data with covariates

This article considers multinomial data subject to misclassification in the presence of covariates which affect both the misclassification probabilities and the true classification probabilities. A subset of the data may be subject to a secondary measurement according to an infallible classifier. Computations are carried out in a Bayesian setting where it is seen that the prior has an important role in driving the inference. In addition, a new and less problematic definition of nonidentifiability is introduced and is referred to as hierarchical nonidentifiability.     

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.