Ordinary Principal Component Analysis Research Articles

Simple structure in component analysis techniques for mixtures of qualitative and quantitative variables

Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.

The logratio approach to the classification of modern sediments and sedimentary environments in northern South China Sea

Grain-size measurements are a type of compositional data and thus subject to closure effects and nonnormality. The logratio transform of Aitchison successfully resolves these problems in compositional data analysis. An application to modern sediment data from the northern part of the South China Sea demonstrates that logratio principal components analysis provides a clear separation of data which cannot be obtained by ordinary principal components analysis, and that cluster analysis using logratio principal components gives a much better classification of sediments than does cluster analysis using raw data. The delineation of sedimentary environments on the basis of a logratio classification of sediment samples provides a better understanding of hydrodynamic conditions on the shelf.

Robust Principal Components

This paper proposes a new algorithm to obtain an eigenvalue decomposition for the sample covariance matrix of a multivariate dataset. The algorithm is based on the rotation technique employed by Ammann and Van Ness (1988a,b) to obtain a robust solution to an errors-in-variables problem. When this rotation technique is combined with an iterative reweighting of the data, a robust eigenvalue decomposition is obtained. This robust eigenvalue decomposition has important applications to principal component analysis. Monte Carlo simulations are performed to compare ordinary principal component analysis using the standard eigenvalue decomposition with this algorithm, referred to as ROPRC. It is seen that ROPRC is reasonably efficient compared to an eigenvalue decomposition when Gaussian data is available, and that ROPRC is much better than the eigenvalue decomposition if outliers are present or if the data has a heavy-tailed distribution. The algorithm returns useful numerical diagnostic information in the form o...

A simple procedure for finding a composite of several multidimensional scaling solutions

One of the better known procedures for finding a representative composite of several different MDS solutions is based on Carroll’s generalized canonical correlation model (Carroll 1968). However, the program for implementing the fitting of this model has not received wide distribution. The present paper provides a simple computational method for implementing Carroll’s approach and shows its relationship to ordinary principal components analysis. In addition, we provide an algorithm for computing the parameters of Carroll’xs model in the presence of missing data. Each method is illustrated with data sets drawn from a marketing research study.

Multi-way principal components-and PLS-analysis

The Lohmöller–Wold decomposition of multi-way (three-way, four-way, etc.) data arrays is combined with the non-linear partial least squares (NIPALS) algorithms to provide multi-way solutions of principal components analysis (PCA) and partial least squares modelling in latent variables (PLS). The decomposition of a multi-way array is developed as the product of a score vector and a loading array, where the score vectors have the same properties as those of ordinary two-way PCA and PLS. In image analysis, the array would instead be decomposed as the product of a loading vector and an image score matrix. The resulting methods are equivalent to the method of unfolding a multi-way array to a two-way matrix followed by ordinary PCA or PLS analysis. This automatically proves the eigenvector and least squares properties of the multi-way PCA and PLS methods. The methodology is presented; the algorithms are outlined and illustrated with a small chemical example.