Geometric consistency of principal component scores for high-dimensional mixture models and its application

In this article, we consider clustering based on principal component analysis (PCA) for high-dimensional mixture models. We present theoretical reasons why PCA is effective for clustering high-dimensional data. First, we derive a geometric representation of high-dimension, low-sample-size (HDLSS) data taken from a two-class mixture model. With the help of the geometric representation, we give geometric consistency properties of sample principal component scores in the HDLSS context. We develop ideas of the geometric representation and provide geometric consistency properties for multiclass mixture models. We show that PCA can cluster HDLSS data under certain conditions in a surprisingly explicit way. Finally, we demonstrate the performance of the clustering using gene expression datasets.     

Missing data in principal component analysis of questionnaire data: a comparison of methods

Principal component analysis (PCA) is a widely used statistical technique for determining subscales in questionnaire data. As in any other statistical technique, missing data may both complicate its execution and interpretation. In this study, six methods for dealing with missing data in the context of PCA are reviewed and compared: listwise deletion (LD), pairwise deletion, the missing data passive approach, regularized PCA, the expectation-maximization algorithm, and multiple imputation. Simulations show that except for LD, all methods give about equally good results for realistic percentages of missing data. Therefore, the choice of a procedure can be based on the ease of application or purely the convenience of availability of a technique.     

Cross-validation methods in principal component analysis: A comparison

In principal component analysis (PCA), it is crucial to know how many principal components (PCs) should be retained in order to account for most of the data variability. A class of “objective” rules for finding this quantity is the class of cross-validation (CV) methods. In this work we compare three CV techniques showing how the performance of these methods depends on the covariance matrix structure. Finally we propose a rule for the choice of the “best” CV method and give an application to real data.     

Functional principal component analysis via regularized Gaussian basis expansions and its application to unbalanced data

This paper introduces regularized functional principal component analysis for multidimensional functional data sets, utilizing Gaussian basis functions. An essential point in a functional approach via basis expansions is the evaluation of the matrix for the integral of the product of any two bases (cross-product matrix). Advantages of the use of the Gaussian type of basis functions in the functional approach are that its cross-product matrix can be easily calculated, and it creates a much more flexible instrument for transforming each individual's observation into a functional form. The proposed method is applied to the analysis of three-dimensional (3D) protein structural data that can be referred to as unbalanced data. It is shown that our method extracts useful information from unbalanced data through the application. Numerical experiments are conducted to investigate the effectiveness of our method via Gaussian basis functions, compared to the method based on B-splines. On performing regularized functional principal component analysis with B-splines, we also derive the exact form of its cross-product matrix. The numerical results show that our methodology is superior to the method based on B-splines for unbalanced data.     

Antedependence models in the analysis of multi-group high-dimensional data

Antedependence modelling has previously been shown to be useful for twogroup discriminant analysis of high-dimensional data. In this paper, the theory of such models is extended to multi-group discriminant analysis and to canonical variate analysis for data display. The application of antedependence models of orders 1, 2 and 3 to spectroscopic analyses of rice samples is described, and the results are compared with those from standard methods based on principal component scores calculated from the data.     

Exact principal component influence measures applied to the analysis of spectroscopic data on rice

Exact influence measures are applied in the evaluation of a principal component decomposition for high dimensional data. Some data used for classifying samples of rice from their near infra-red transmission profiles, following a preliminary principal component analysis, are examined in detail. A normalization of eigenvalue influence statistics is proposed which ensures that measures reflect the relative orientations of observations, rather than their overall Euclidean distance from the sample mean. Thus, the analyst obtains more information from an analysis of eigenvalues than from approximate approaches to eigenvalue influence. This is particularly important for high dimensional data where a complete investigation of eigenvector perturbations may be cumbersome. The results are used to suggest a new class of influence measures based on ratios of Euclidean distances in orthogonal spaces.     

Sensitivity analysis in principal component analysis:influence on the subspace spanned by principal components.

The problem of detecting influential observations in principalcomponent analysis was discussed by several authors. Radhakrishnan and kshirsagar ( 1981 ), Critchley ( 1985 ), jolliffe ( 1986 )among others discussed this topicby using the influence functions I(X;θ_s)and I(X;V_s)of eigenvalues and eigenvectors, which wwere derived under the assumption that the eigenvalues of interest were simple. In this paper we propose the influence functionsI(X;∑^q _s=1θ_sV_sV_s ^T)and I(x;∑^q _s=1V_sV_s ^t)(q<p;p:number of variables) to investigate the influence onthe subspace spanned by principal components. These influence functions are applicable not only to the case where the edigenvalues of interst are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

Predictive performance of missing data methods for logistic regression,classification trees and neural networks

Although the effect of missing data on regression estimates has received considerable attention, their effect on predictive performance has been neglected. We studied the performance of three missing data strategies—omission of records with missing values, replacement with a mean and imputation based on regression—on the predictive performance of logistic regression (LR), classification tree (CT) and neural network (NN) models in the presence of data missing completely at random (MCAR). Models were constructed using datasets of size 500 simulated from a joint distribution of binary and continuous predictors including nonlinearities, collinearity and interactions between variables. Though omission produced models that fit better on the data from which the models were developed, imputation was superior on average to omission for all models when evaluating the receiver operating characteristic (ROC) curve area, mean squared error (MSE), pooled variance across outcome categories and calibration X ² on an independently generated test set. However, in about one-third of simulations, omission performed better. Performance was also more variable with omission including quite a few instances of extremely poor performance. Replacement and imputation generally produced similar results except with neural networks for which replacement, the strategy typically used in neural network algorithms, was inferior to imputation. Missing data affected simpler models much less than they did more complex models such as generalized additive models that focus on local structure For moderate sized datasets, logistic regressions that use simple nonlinear structures such as quadratic terms and piecewise linear splines appear to be at least as robust to randomly missing values as neural networks and classification trees.     

Statistical methods for the analysis of high-throughput data based on functional profiles derived from the Gene Ontology

The increasing availability of high-throughput data, that is, massive quantities of molecular biology data arising from different types of experiments such as gene expression or protein microarrays, leads to the necessity of methods for summarizing the available information. As annotation quality improves it is becoming common to rely on biological annotation databases, such as the Gene Ontology (GO), to build functional profiles which characterize a set of genes or proteins using the distribution of their annotations in the database. In this work we describe a statistical model for such profiles, provide methods to compare profiles and develop inferential procedures to assess this comparison. An R-package implementing the methods will be available at publication time.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.     

Multivariate process capability using principal component analysis in the presence of measurement errors

Various different definitions of multivariate process capability indices have been proposed in the literature. Most of the research works related to multivariate process capability indices assume no gauge measurement errors. However, in industrial applications, despite the use of highly advanced measuring instruments, account needs to be taken of gauge imprecision. In this paper we are going to examine the effects of measurement errors on multivariate process capability indices computed using the principal components analysis. We show that measurement errors alter the results of a multivariate process capability analysis, resulting in either a decrease or an increase in the capability of the process. In order to achieve accurate process capability assessments, we propose a method useful for overcoming the effects of gauge measurement errors.     

A comparison of regularization methods applied to the linear discriminant function with high-dimensional microarray data

Classification of gene expression microarray data is important in the diagnosis of diseases such as cancer, but often the analysis of microarray data presents difficult challenges because the gene expression dimension is typically much larger than the sample size. Consequently, classification methods for microarray data often rely on regularization techniques to stabilize the classifier for improved classification performance. In particular, numerous regularization techniques, such as covariance-matrix regularization, are available, which, in practice, lead to a difficult choice of regularization methods. In this paper, we compare the classification performance of five covariance-matrix regularization methods applied to the linear discriminant function using two simulated high-dimensional data sets and five well-known, high-dimensional microarray data sets. In our simulation study, we found the minimum distance empirical Bayes method reported in Srivastava and Kubokawa [Comparison of discrimination methods for high dimensional data, J. Japan Statist. Soc. 37(1) (2007), pp. 123–134], and the new linear discriminant analysis reported in Thomaz, Kitani, and Gillies [A Maximum Uncertainty LDA-based approach for Limited Sample Size problems – with application to Face Recognition, J. Braz. Comput. Soc. 12(1) (2006), pp. 1–12], to perform consistently well and often outperform three other prominent regularization methods. Finally, we conclude with some recommendations for practitioners.     

Mixture model on the variance for the differential analysis of gene expression data

Summary.  In microarray experiments, accurate estimation of the gene variance is a key step in the identification of differentially expressed genes. Variance models go from the too stringent homoscedastic assumption to the overparameterized model assuming a specific variance for each gene. Between these two extremes there is some room for intermediate models. We propose a method that identifies clusters of genes with equal variance. We use a mixture model on the gene variance distribution. A test statistic for ranking and detecting differentially expressed genes is proposed. The method is illustrated with publicly available complementary deoxyribonucleic acid microarray experiments, an unpublished data set and further simulation studies.     

Identification of genomic markers correlated with sensitivity in solid tumors to Dasatinib using sparse principal components

Differential analysis techniques are commonly used to offer scientists a dimension reduction procedure and an interpretable gateway to variable selection, especially when confronting high-dimensional genomic data. Huang et al. used a gene expression profile of breast cancer cell lines to identify genomic markers which are highly correlated with in vitro sensitivity of a drug Dasatinib. They considered three statistical methods to identify differentially expressed genes and finally used the results from the intersection. But the statistical methods that are used in the paper are not sufficient to select the genomic markers. In this paper we used three alternative statistical methods to select a combined list of genomic markers and compared the genes that were proposed by Huang et al. We then proposed to use sparse principal component analysis (Sparse PCA) to identify a final list of genomic markers. The Sparse PCA incorporates correlation into account among the genes and helps to draw a successful genomic markers discovery. We present a new and a small set of genomic markers to separate out the groups of patients effectively who are sensitive to the drug Dasatinib. The analysis procedure will also encourage scientists in identifying genomic markers that can help to separate out two groups.     

Three estimators of the Mahalanobis distance in high-dimensional data

This paper treats the problem of estimating the Mahalanobis distance for the purpose of detecting outliers in high-dimensional data. Three ridge-type estimators are proposed and risk functions for deciding an appropriate value of the ridge coefficient are developed. It is argued that one of the ridge estimator has particularly tractable properties, which is demonstrated through outlier analysis of real and simulated data.     

A comparison of different procedures for principal component analysis in the presence of outliers

Principal component analysis (PCA) is a popular technique that is useful for dimensionality reduction but it is affected by the presence of outliers. The outlier sensitivity of classical PCA (CPCA) has caused the development of new approaches. Effects of using estimates obtained by expectation–maximization – EM and multiple imputation – MI instead of outliers were examined on the artificial and a real data set. Furthermore, robust PCA based on minimum covariance determinant (MCD), PCA based on estimates obtained by EM instead of outliers and PCA based on estimates obtained by MI instead of outliers were compared with the results of CPCA. In this study, we tried to show the effects of using estimates obtained by MI and EM instead of outliers, depending on the ratio of outliers in data set. Finally, when the ratio of outliers exceeds 20%, we suggest the use of estimates obtained by MI and EM instead of outliers as an alternative approach.     

Treatments of non-metric variables in partial least squares and principal component analysis

This paper reviews various treatments of non-metric variables in partial least squares (PLS) and principal component analysis (PCA) algorithms. The performance of different treatments is compared in an extensive simulation study under several typical data generating processes and associated recommendations are made. Moreover, we find that PLS-based methods are to prefer in practice, since, independent of the data generating process, PLS performs either as good as PCA or significantly outperforms it. As an application of PLS and PCA algorithms with non-metric variables we consider construction of a wealth index to predict household expenditures. Consistent with our simulation study, we find that a PLS-based wealth index with dummy coding outperforms PCA-based ones.     

Bayesian Weibull tree models for survival analysis of clinico-genomic data

An important goal of research involving gene expression data for outcome prediction is to establish the ability of genomic data to define clinically relevant risk factors. Recent studies have demonstrated that microarray data can successfully cluster patients into low- and high-risk categories. However, the need exists for models which examine how genomic predictors interact with existing clinical factors and provide personalized outcome predictions. We have developed clinico-genomic tree models for survival outcomes which use recursive partitioning to subdivide the current data set into homogeneous subgroups of patients, each with a specific Weibull survival distribution. These trees can provide personalized predictive distributions of the probability of survival for individuals of interest. Our strategy is to fit multiple models; within each model we adopt a prior on the Weibull scale parameter and update this prior via Empirical Bayes whenever the sample is split at a given node. The decision to split is based on a Bayes factor criterion. The resulting trees are weighted according to their relative likelihood values and predictions are made by averaging over models. In a pilot study of survival in advanced stage ovarian cancer we demonstrate that clinical and genomic data are complementary sources of information relevant to survival, and we use the exploratory nature of the trees to identify potential genomic biomarkers worthy of further study.     

A Monte Carlo examination of the broken-stick distribution to identify components to retain in principal component analysis

ABSTRACT

The broken-stick (BS) is a popular stopping rule in ecology to determine the number of meaningful components of principal component analysis. However, its properties have not been systematically investigated. The purpose of the current study is to evaluate its ability to detect the correct dimensionality in a data set and whether it tends to over- or underestimate it. A Monte Carlo protocol was carried out. Two main correlation matrices deemed usual in practice were used with three levels of correlation (0, 0.10 and 0.30) between components (generating oblique structure) and with different sample sizes. Analyses of the population correlation matrices indicated that, for extremely large sample sizes, the BS method could be correct for only one of the six simulated structure. It actually failed to identify the correct dimensionality half the time with orthogonal structures and did even worse with some oblique ones. In harder conditions, results show that the power of the BS decreases as sample size increases: weakening its usefulness in practice. Since the BS method seems unlikely to identify the underlying dimensionality of the data, and given that better stopping rules exist it appears as a poor choice when carrying principal component analysis.     

Robust methods for the analysis of spatially autocorrelated data

In this paper we propose a new robust technique for the analysis of spatial data through simultaneous autoregressive (SAR) models, which extends the Forward Search approach of Cerioli and Riani (1999) and Atkinson and Riani (2000). Our algorithm starts from a subset of outlier-free observations and then selects additional observations according to their degree of agreement with the postulated model. A number of useful diagnostics which are monitored along the search help to identify masked spatial outliers and high leverage sites. In contrast to other robust techniques, our method is particularly suited for the analysis of complex multidimensional systems since each step is performed through statistically and computationally efficient procedures, such as maximum likelihood. The main contribution of this paper is the development of joint robust estimation of both trend and autocorrelation parameters in spatial linear models. For this purpose we suggest a novel definition of the elemental sets of the Forward Search, which relies on blocks of contiguous spatial locations.