Error Rates In Discriminant Analysis Research Articles

Optimal training sample allocation and asymptotic expansions for error rates in discriminant analysis

The sample-based rule obtained from Bayes classification rule by replacing the unknown parameters by ML estimates from a stratified training sample is used for the classification of a random observationX into one ofL populations. The asymptotic expansions in terms of the inverses of the training sample sizes for cross-validation, apparent and plug-in error rates are found. These are used to compare estimation methods of the error rate for a wide range of regular distributions as probability models for considered populations. The optimal training sample allocation minimizing the asymptotic expected error regret is found in the cases of widely applicable, positively skewed distributions (Rayleigh and Maxwell distributions). These probability models for populations are often met in ecology and biology. The results indicate that equal training sample sizes for each populations sometimes are not optimal, even when prior probabilities of populations are equal.

Estimationg error rates in discriminant analysis with correlated training observations: a simulation study

This article reports results of an extensive simulation study which investigated the performances of some commonly used methods of estimating error rates in discriminant analysis. Earlier research papers limited their comparisons of these methods to independent training data. This study allows for a simple auto-regressive dependence among the training data. The results suggest that the estimation methods based on the normal distribution perform adequately well under conditions of negative or mild positive correlation in the data, and small dimensions (p) of the observation vectors. For large p or strong positive correlation structures the conclusion is that one of the better non-parametric methods should be used. Special circumstances and conditions which notably affect the relative performances of the methods are identified.

Comparison of two asymptotic techniques for estimating error rates in discriminant analysis

The Qos and Qm are two leading estimators of the probability of misclassification which are based on the asymptotic expansion of the the expected value of the Error Rate, Pi. The estimators are, however, not suitable for estimating the Error rates for certain ranges of the parameters p , n1, n2 and s.We investigate the regions in which they produce unacceptable estimates , and show that the Qos is, in general, better than the Qm in producing acceptable estimates

An investigation of sample influence functions for the ds estimate of the optimum error rate in discriminant analysis

Approximate and exact influence functions are investigated for the Shrunken D or DS method of estimating the optimum error rate in a linear discriminant analysis with two populations. Two- and three-dimensional plots of these functions are constructed to illustrate regions where influential observations are most harmful. The extension of these results to higher dimensional cases are discussed. As expected, influence function values reveal that the impact of a perturbing point increases as sample size decreases. Influence function values are more extreme (higher and lower over the range of perturbed observations studied) when the Mahalanobis1 distance between the means are smaller. For sample sizes of 100 or less from each group, the exact expression for the inverted perturbed covariance matrix should be used when constructing the influence function.

Comparison of procedures for estimation of error rates in discriminant analysis under nonnormal population

The parametric and nonparametric methods for estimating the error rates in linear discriminant analysis are examined both in normal and in nonnormal situations. A Monte Carlo experiment was carried out under the assumption that two population distributions were characterized by a mixture of two multivariate normal distributions. The bootstrap bias-corrected apparent error rate compares favourably to other available estimators for nonnormal populations with small Mahalanobis distance. The methods for error estimation are also applied to a practical problem in medical diagnosis

Estimation of Error Rates in Discriminant Analysis with Selection of Variables

Accurate estimation of misclassification rates in discriminant analysis with selection of variables by, for example, a stepwise algorithm, is complicated by the large optimistic bias inherent in standard estimators such as those obtained by the resubstitution method. Application of a bootstrap adjustment can reduce the bias of the resubstitution method; however, the bootstrap technique requires the variable selection procedure to be repeated many times and is therefore difficult to compute. In this paper we propose a smoothed estimator that requires relatively little computation and which, on the basis of a Monte Carlo sampling study, is found to perform generally at least as well as the bootstrap method.

Error-Rate Estimation in Discriminant Analysis

A new derivation for a method to estimate error rates in discriminant analysis is presented. Known as the Shrunken D or DS method, the technique is evaluated in a sampling experiment, along with seven other parametric methods, as a means for estimating both the optimal and conditional error rates. The results show that the “best” estimators are not the same for the two types of error rates and that sample size should influence the choice of an estimator.

An efficient- estimator for the error rate in biscriromant analysis1

In this paper we discuss different approaches to the estimation of error rates in discriminant analysis. Especially we prove the asymptotic efficiency of an estimator of the bootstrap type and give some numerical results for the comparison of the resubstitution, the jackknife and the asymptotically efficient estimator

Classification Error Rate Estimators Evaluated by Unconditional Mean Squared Error

In this article the criterion of unconditional mean squared error is used to compare four commonly used estimators of error rates in discriminant analysis. The leave-one-out estimator, which has relatively small bias, is found to perform well relative to the other estimators when a large number of explanatory variables are used in the discriminant function. With a small number of explanatory variables, the large variance of this estimator results in poor performance. We also find the estimators that assume normally distributed explanatory variables to be nonrobust when the parent distributions are skewed or have large tails.

Classification Error Rate Estimators Evaluated by Unconditional Mean Squared Error

In this article the criterion of unconditional mean squared error is used to compare four commonly used estimators of error rates in discriminant analysis. The leave-one-out estimator, which has relatively small bias, is found to perform well relative to the other estimators when a large number of explanatory variables are used in the discriminant function. With a small number of explanatory variables, the large variance of this estimator results in poor performance. We also find the estimators that assume normally distributed explanatory variables to be nonrobust when the parent distributions are skewed or have large tails.

The efficiency of Efron's “Bootstrap” Approach Applied to Error Rate Estimation in Discriminant Analysis

The “bootstrap” approach of Efron is considered in its application to the estimation of error rates in discriminant analysis. Its efficiency relative to parametric estimation is investigated by simulation for Fisher's linear discriminant function in the context of two multivariate normal populations with a common covariance matrix.

The Bias of the Apparent Error Rate in Discriminant Analysis

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 ia 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.

The bias of the apparent error rate in discriminant analysis

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.

An Asymptotic Unbiased Technique for Estimating the Error Rates in Discriminant Analysis

A new technique for estimating the error rates in discriminant analysis is suggested. The estimates obtained using this technique are asymptotically less biased than existing estimates. The performance of this technique on the basis of criteria other than that of bias is studied using Monte Carlo methods to simulate practical situations and also the criterion of asymptotic mean square error. It is concluded that the all-round performance of the proposed technique is comparable to that of any available technique.

Estimation of Error Rates in Discriminant Analysis

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.

Commentary on “Estimation of Error Rates in Discriminant Analysis”

Commentary on “Estimation of Error Rates in Discriminant Analysis”

Commentary on "Estimation of Error Rates in Discriminant Analysis"

Commentary on "Estimation of Error Rates in Discriminant Analysis"

Estimation of Error Rates in Discriminant Analysis

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.