Local Component Analysis

Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectation-minimization (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semi-parametric Gaussian-Parzen density model, where some of the variables are modelled through a jointly Gaussian density, while others are modelled through Parzen windows. For these two models, EM leads to simple closed-form updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher test-likelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on such metrics, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a large-scale setting

Parametric and nonparametric inference in equilibrium job search models

Equilibrium job search models allow for labor markets with homogeneous workers and firms to yield nondegenerate wage densities. However, the resulting wage densities do not accord well with empirical regularities. Accordingly, many extensions to the basic equilibrium search model have been considered (e.g., heterogeneity in productivity, heterogeneity in the value of leisure, etc.). It is increasingly common to use nonparametric forms for these extensions and, hence, researchers can obtain a perfect fit (in a kernel smoothed sense) between theoretical and empirical wage densities. This makes it difficult to carry out model comparison of different model extensions. In this paper, we first develop Bayesian parametric and nonparametric methods which are comparable to the existing non-Bayesian literature. We then show how Bayesian methods can be used to compare various nonparametric equilibrium search models in a statistically rigorous sense

The generalized shrinkage estimator for the analysis of functional connectivity of brain signals

We develop a new statistical method for estimating functional connectivity between neurophysiological signals represented by a multivariate time series. We use partial coherence as the measure of functional connectivity. Partial coherence identifies the frequency bands that drive the direct linear association between any pair of channels. To estimate partial coherence, one would first need an estimate of the spectral density matrix of the multivariate time series. Parametric estimators of the spectral density matrix provide good frequency resolution but could be sensitive when the parametric model is misspecified. Smoothing-based nonparametric estimators are robust to model misspecification and are consistent but may have poor frequency resolution. In this work, we develop the generalized shrinkage estimator, which is a weighted average of a parametric estimator and a nonparametric estimator. The optimal weights are frequency-specific and derived under the quadratic risk criterion so that the estimator, either the parametric estimator or the nonparametric estimator, that performs better at a particular frequency receives heavier weight. We validate the proposed estimator in a simulation study and apply it on electroencephalogram recordings from a visual-motor experiment.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS396 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org