Discriminant analysis of survey data

We consider the problem of the effect of sample designs on discriminant analysis. The selection of the learning sample is assumed to depend on the population values of auxiliary variables. Under a superpopulation model with a multivariate normal distribution, unbiasedness and consistency are examined for the conventional estimators (derived under the assumptions of simple random sampling), maximum likelihood estimators, probability-weighted estimators and conditionally unbiased estimators of parameters. Four corresponding sampled linear discriminant functions are examined. The rates of misclassification of these four discriminant functions and the effect of sample design on these four rates of misclassification are discussed. The performances of these four discriminant functions are assessed in a simulation study.     

Discriminant coordinates analysis for multivariate functional data

Abstract

One of the basic statistical methods of dimensionality reduction is analysis of discriminant coordinates given by Fisher (1936 Fisher, R. A. 1936. The use of multiple measurements in taxonomic problem. Annals of Eugenics 7 (2):179–88. doi:10.1111/j.1469-1809.1936.tb02137.x.[Crossref] , [Google Scholar]) and Rao (1948). The space of discriminant coordinates is a space convenient for presenting multidimensional data originating from multiple groups and for the use of various classification methods (methods of discriminant analysis). In the present paper, we adapt the classical discriminant coordinates analysis to multivariate functional data. The theory has been applied to analysis of textural properties of apples of six varieties, measured over a period of 180?days, stored in two types of refrigeration chamber.     

Discriminant analysis of multivariate repeated measures data with a Kronecker product structured covariance matrices

This paper proposes new classifiers under the assumption of multivariate normality for multivariate repeated measures data with Kronecker product covariance structures. These classifiers are especially effective when the number of observations is not large enough to estimate the covariance matrices, and thus the traditional classifiers fail. Computational scheme for maximum likelihood estimates of required class parameters are also given. The quality of these new classifiers are examined on some real data.     

Discriminant and cluster analysis of possibly high-dimensional time series data by a class of disparities

Discriminant and cluster analysis of high-dimensional time series data have been an urgent need in more and more academic fields. To settle the always-existing problem of bias in distance-based classifiers for high-dimensional models, we consider a new classifier with jackknife-type bias adjustment for stationary time series data. The consistency of the classifier is theoretically shown under suitable conditions, including the situations of possibly high-dimensional data. We also conduct the cluster analysis for real financial data.     

Influence Analysis on Discriminant Coordinates

Discriminant analysis (DA), particularly Discriminant Coordinates (DC), is broadly applied in the scientific literature and included in many statistical software packages. DC is used to analyze biomedical data, especially for differential diagnosis on the basis of laboratory profiles. Articles handling influence analysis in DA can be found in the literature; however, this topic has been scarcely touched upon in DC. In this article, the case-deletion approach is followed to introduce a perturbation in the data and influence measures are proposed to assess the effect on three statistics of interest: the transformation matrix, canonical directions, and configuration, of the sample centroids.     

Discriminant analysis with stratified prior probabilities

This study investigates the use of stratification to improve discrimination when prior probabilities vary across strata of a population of interest. Sources of heterogeneity in prior probabilities include differences in geographic locale, age differences in the population studied, or differences in the time component of the data collected. The article suggests using logistic regression both to identify the underlying stratification and to estimate prior probabilities. A simulation study compares misclassification rates under two alternative stratification schemes with the traditional discriminant approach that ignores stratification in favor of pooled prior estimates. The simulations show that large asymptotic gains can be realized by stratification, and that these gains can be realized in finite samples, given moderate differences in prior probabilities.     

Logistic回归模型在判别分析中的应用

介绍Logistic回归模型用于判别的方法,利用给出的某期间华北地区和长江中下游降水年变化为判别对象,以这种判别方法确定界于两个地区中间地带的一些观测站属于何种年变化型,并且与传统用的最大概率法做了比较,发现Logistic的效果要比最大概率法好。     

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.     

基于Fisher变换的Bayes判别方法探索

判别分析是三大多元统计分析方法之一,在许多领域都有广泛的应用。通常认为距离判别、Fisher判别和Bayes判别是三种不同的判别分析方法,本文的研究表明,距离判别与Bayes判别是两种实质的判别方法,前者实际依据的是百分位点或置信区间,后者实际依据的是概率。而著名的Fisher判别,只是依据方差分析的思想,对判别变量进行线性变换,然后用于距离判别,其实不能算是一种实质的判别方法。本文将Fisher变换与Bayes判别结合起来,即先做Fisher变换,再利用概率最大原则做Bayes判别,得到一种新的判别途径,可进一步提高判别效率。理论与实证分析表明,基于Fisher变换的Bayes判别,适用场合广泛,判别效率最高。     

Discriminant analysis under the common principal components model

For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015 Pepler, P. T., Uys, D. W. and Nel, D. G. (2015). Regularised covariance matrix estimation under the common principal components model. Communications in Statistics: Simulation and Computation. (In press). [Google Scholar]) proposed a regularized CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations, where the CPC model is applicable. This article extends their work to the context of discriminant analysis for two groups, by plugging the regularized CPC estimator into the ordinary quadratic discriminant function. Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations, where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures.     

Examination of Influential Observations to Improve Discriminant Analysis

Three simple transformations are proposed in the context of ratio and product methods of estimation, based on any probability sampling design, and the usual unbiased estimation under varying probability sampling. These transformations may be effected

after the data are collected in a survey. The objective is to obtain improved estimators of the population total     

Discriminant analysis under parameter restrictions statistical and computational aspects

In the paper, it is shown that the error rate of the discriminant analysis can be diminished when restrictions for the parameters are valid. To find suitable restrictions, at first the properties of hierarchical and other multivariate systems are investigated. Then, in a practical section, a modification of the discriminant analysis is offered that consists in decreasing the estimated partial correlations. Finally in the theoretical section, it is veri¬fied that an improvement of the discriminant analysis is attained by a suitable correctionof the positive and negative signs of the discriminant function.     

On the Spectral Decomposition in Normal Discriminant Analysis

This article enlarges the covariance configurations, on which the classical linear discriminant analysis is based, by considering the four models arising from the spectral decomposition when eigenvalues and/or eigenvectors matrices are allowed to vary or not between groups. As in the classical approach, the assessment of these configurations is accomplished via a test on the training set. The discrimination rule is then built upon the configuration provided by the test, considering or not the unlabeled data. Numerical experiments, on simulated and real data, have been performed to evaluate the gain of our proposal with respect to the linear discriminant analysis.     

The Dantzig Discriminant Analysis with High Dimensional Data

It is well known that linear discriminant analysis (LDA) works well and is asymptotically optimal under fixed-p-large-n situations. But Bickel and Levina (2004 Bickel, P.J., Levina, E. (2004). Some theory for Fishers linear discriminant function, naive Bayes, and some alternatives when there are many more variables than observations. Bernoulli 10:989–1010.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the LDA is as bad as random guessing when p > n. This article studies the sparse discriminant analysis via Dantzig penalized least squares. Our method avoids estimating the high-dimensional covariance matrix and does not need the sparsity assumption on the inverse of the covariance matrix. We show that the new discriminant analysis is asymptotically optimal theoretically. Simulation and real data studies show that the classifier performs better than the existing sparse methods.     

Identification of Influential Cases in Kernel Fisher Discriminant Analysis

We study the influence of a single data case on the results of a statistical analysis. This problem has been addressed in several articles for linear discriminant analysis (LDA). Kernel Fisher discriminant analysis (KFDA) is a kernel based extension of LDA. In this article, we study the effect of atypical data points on KFDA and develop criteria for identification of cases having a detrimental effect on the classification performance of the KFDA classifier. We find that the criteria are successful in identifying cases whose omission from the training data prior to obtaining the KFDA classifier results in reduced error rates.