Projection Based Models for High Dimensional Data

Abstract

In recent years, many machine learning applications have arisen which deal with the problem of finding patterns in high dimensional data. Principal component analysis (PCA) has become ubiquitous in this setting. PCA performs dimensionality reduction by estimating latent factors which minimise the reconstruction error between the original data and its low-dimensional projection. We initially consider a situation where influential observations exist within the dataset which have a large, adverse affect on the estimated PCA model. We propose a measure of “predictive influence” to detect these points based on the contribution of each point to the leave-one-out reconstruction error of the model using an analytic PRedicted REsidual Sum of Squares (PRESS) statistic. We then develop a robust alternative to PCA to deal with the presence of influential observations and outliers which minimizes the predictive reconstruction error. In some applications there may be unobserved clusters in the data, for which fitting PCA models to subsets of the data would provide a better fit. This is known as the subspace clustering problem. We develop a novel algorithm for subspace clustering which iteratively fits PCA models to subsets of the data and assigns observations to clusters based on their predictive influence on the reconstruction error. We study the convergence of the algorithm and compare its performance to a number of subspace clustering methods on simulated data and in real applications from computer vision involving clustering object trajectories in video sequences and images of faces. We extend our predictive clustering framework to a setting where two highdimensional views of data have been obtained. Often, only either clustering or 5 predictive modelling is performed between the views. Instead, we aim to recover clusters which are maximally predictive between the views. In this setting two block partial least squares (TB-PLS) is a useful model. TB-PLS performs dimensionality reduction in both views by estimating latent factors that are highly predictive. We fit TB-PLS models to subsets of data and assign points to clusters based on their predictive influence under each model which is evaluated using a PRESS statistic. We compare our method to state of the art algorithms in real applications in webpage and document clustering and find that our approach to predictive clustering yields superior results. Finally, we propose a method for dynamically tracking multivariate data streams based on PLS. Our method learns a linear regression function from multivariate input and output streaming data in an incremental fashion while also performing dimensionality reduction and variable selection. Moreover, the recursive regression model is able to adapt to sudden changes in the data generating mechanism and also identifies the number of latent factors. We apply our method to the enhanced index tracking problem in computational finance.