Mixtures of probabilistic principal component analysers

Principal component analysis (PCA) is one of the most popular techniques for processing, compressing and visualising data, although its effectiveness is limited by its global linearity. While nonlinear variants of PCA have been proposed, an alternative paradigm is to capture data complexity by a combination of local linear PCA projections. However, conventional PCA does not correspond to a probability density, and so there is no unique way to combine PCA models. Previous attempts to formulate mixture models for PCA have therefore to some extent been ad hoc. In this paper, PCA is formulated within a maximum-likelihood framework, based on a specific form of Gaussian latent variable model. This leads to a well-defined mixture model for probabilistic principal component analysers, whose parameters can be determined using an EM algorithm. We discuss the advantages of this model in the context of clustering, density modelling and local dimensionality reduction, and we demonstrate its application to image compression and handwritten digit recognition.

Probabilistic principal component analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss the advantages conveyed by the definition of a probability density function for PCA.

Probabilistic principal component analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss the advantages conveyed by the definition of a probability density function for PCA.

Evaluation of catalytic pyrolysis of cassava rhizome by principal component analysis

Rhizome of cassava plants (Manihot esculenta Crantz) was catalytically pyrolysed at 500 °C using analytical pyrolysis–gas chromatography/mass spectrometry (Py–GC/MS) method in order to investigate the relative effect of various catalysts on pyrolysis products. Selected catalysts expected to affect bio-oil properties were used in this study. These include zeolites and related materials (ZSM-5, Al-MCM-41 and Al-MSU-F type), metal oxides (zinc oxide, zirconium (IV) oxide, cerium (IV) oxide and copper chromite) catalysts, proprietary commercial catalysts (Criterion-534 and alumina-stabilised ceria-MI-575) and natural catalysts (slate, char and ashes derived from char and biomass). The pyrolysis product distributions were monitored using models in principal components analysis (PCA) technique. The results showed that the zeolites, proprietary commercial catalysts, copper chromite and biomass-derived ash were selective to the reduction of most oxygenated lignin derivatives. The use of ZSM-5, Criterion-534 and Al-MSU-F catalysts enhanced the formation of aromatic hydrocarbons and phenols. No single catalyst was found to selectively reduce all carbonyl products. Instead, most of the carbonyl compounds containing hydroxyl group were reduced by zeolite and related materials, proprietary catalysts and copper chromite. The PCA model for carboxylic acids showed that zeolite ZSM-5 and Al-MSU-F tend to produce significant amounts of acetic and formic acids.