When large multivariate datasets are analyzed, it is often desirable to reduce their dimensionality. Principal component analysis is one technique for doing this. It replaces the p original variables by a smaller number, q, of derived variables, the principal components, which are linear combinations of the original variables. Often, it is possible to retain most of the variability in the original variables with q very much smaller than p. Despite its apparent simplicity, principal component analysis has a number of subtleties, and it has many uses and extensions. A number of choices associated with the technique are briefly discussed, namely, covariance or correlation, how many components, and different normalization constraints, as well as confusion with factor analysis. Various uses and extensions are outlined.


Keywords:

dimension reduction;
factor analysis;
multivariate analysis;
variance maximization

Principal Component Analysis

Variable and feature selection have become the focus of much research in areas of application for which datasets with tens or hundreds of thousands of variables are available. These areas include text processing of internet documents, gene expression array analysis, and combinatorial chemistry. The objective of variable selection is three-fold: improving the prediction performance of the predictors, providing faster and more cost-effective predictors, and providing a better understanding of the underlying process that generated the data. The contributions of this special issue cover a wide range of aspects of such problems: providing a better definition of the objective function, feature construction, feature ranking, multivariate feature selection, efficient search methods, and feature validity assessment methods.

/pdf/an-introduction-to-variable-and-feature-selection-22eu49r0zd.pdf

An introduction to variable and feature selection

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

/pdf/gaussian-processes-for-machine-learning-40fp9c190d.pdf

Gaussian Processes for Machine Learning

The success of machine learning algorithms generally depends on data representation, and we hypothesize that this is because different representations can entangle and hide more or less the different explanatory factors of variation behind the data. Although specific domain knowledge can be used to help design representations, learning with generic priors can also be used, and the quest for AI is motivating the design of more powerful representation-learning algorithms implementing such priors. This paper reviews recent work in the area of unsupervised feature learning and deep learning, covering advances in probabilistic models, autoencoders, manifold learning, and deep networks. This motivates longer term unanswered questions about the appropriate objectives for learning good representations, for computing representations (i.e., inference), and the geometrical connections between representation learning, density estimation, and manifold learning.

/pdf/representation-learning-a-review-and-new-perspectives-4sidphwbhg.pdf

Representation Learning: A Review and New Perspectives

This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the 'relevance vector machine' (RVM), a model of identical functional form to the popular and state-of-the-art 'support vector machine' (SVM) We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while offering a number of additional advantages These include the benefits of probabilistic predictions, automatic estimation of 'nuisance' parameters, and the facility to utilise arbitrary basis functions (eg non-'Mercer' kernels) We detail the Bayesian framework and associated learning algorithm for the RVM, and give some illustrative examples of its application along with some comparative benchmarks We offer some explanation for the exceptional degree of sparsity obtained, and discuss and demonstrate some of the advantageous features, and potential extensions, of Bayesian relevance learning

/pdf/sparse-bayesian-learning-and-the-relevance-vector-machine-4z6kv3jpiu.pdf

Sparse bayesian learning and the relevance vector machine

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.

/pdf/probabilistic-principal-component-analysis-2xwcqnp8gi.pdf

Probabilistic Principal Component Analysis

Principal component analysis (PCA) is one of the most popular techniques for processing, compressing, and visualizing 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. Therefore, previous attempts to formulate mixture models for PCA have been ad hoc to some extent. In this article, 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 analyzers, whose parameters can be determined using an expectationmaximization algorithm. We discuss the advantages of this model in the context of clustering, density modeling, and local dimensionality reduction, and we demonstrate its application to image compression and handwritten digit recognition.

/pdf/mixtures-of-probabilistic-principal-component-analyzers-jzirwm3q84.pdf

Mixtures of probabilistic principal component analyzers

The support vector machine (SVM) is a state-of-the-art technique for regression and classification, combining excellent generalisation properties with a sparse kernel representation. However, it does suffer from a number of disadvantages, notably the absence of probabilistic outputs, the requirement to estimate a trade-off parameter and the need to utilise 'Mercer' kernel functions. In this paper we introduce the Relevance Vector Machine (RVM), a Bayesian treatment of a generalised linear model of identical functional form to the SVM. The RVM suffers from none of the above disadvantages, and examples demonstrate that for comparable generalisation performance, the RVM requires dramatically fewer kernel functions.

/pdf/the-relevance-vector-machine-3561o1hs8d.pdf

The Relevance Vector Machine

The ‘sparse Bayesian’ modelling approach, as exemplified by the ‘relevance vector machine’, enables sparse classification and regression functions to be obtained by linearly-weighting a small number of fixed basis functions from a large dictionary of potential candidates Such a model conveys a number of advantages over the related and very popular ‘support vector machine’, but the necessary ‘training’ procedure — optimisation of the marginal likelihood function — is typically much slower We describe a new and highly accelerated algorithm which exploits recently-elucidated properties of the marginal likelihood function to enable maximisation via a principled and efficient sequential addition and deletion of candidate basis functions

/pdf/fast-marginal-likelihood-maximisation-for-sparse-bayesian-5cajprx41j.pdf

Michael E. Tipping

Papers

Sparse bayesian learning and the relevance vector machine

Probabilistic Principal Component Analysis

Mixtures of probabilistic principal component analyzers

The Relevance Vector Machine

Fast Marginal Likelihood Maximisation for Sparse Bayesian Models