Events Calendar

Topological Inference, Large-Scale Multiple Testing, and Random Positive Definite Matrices

Speaker: Dr. Armin Schwartzman, Assistant Professor, Department of Biostatistics, Harvard University

Abstract: The first concerns the problem of detecting local significant regions in signals and images, where the need is to make inferences about spatial features such as smooth peaks or compact regions of unknown location rather than individual pixels or voxels. Example include detection of spikes in neuronal recordings, finding protein binding sites in CHIP-Seq genomic data and finding regions of neural activation in brain imaging. Focusing on the 1D case, I propose a topological multiple testing approach involving kernel smoothing and testing of local maxima. Theory and simulation show that global error rates are controlled asymptotically and that the optimal bandwidth corresponds to the “matched filter” principle, where the kernel size should be close to that of the peaks to be detected.

Addressing more generally the problem of large-scale testing under arbitrary correlation, such as in gene expression data, I will describe some of the difficulties in making inferences in this setting such as the added bias and variance in the usual false discovery rate (FDR) estimator and the variability of the observed distribution of the test statistics, calling for the so-called empirical null correction.

Finally, relating to geometry and manifold-valued data, I will present statistical tools I have developed for making inferences about eigenvalues and eigenvectors of random symmetric positive definite (PD) matrices. This problem is relevant in the analysis of Diffusion Tensor Imaging data, where the observations themselves ate 3 x 3 PD matrices. The parameter sets involved in the inference problems are subsets of Euclidean space that are either affine subspaces, polyhedral convex cones, or embedded submanifolds that are invariant under orthogonal transformation. A key tool for working with random PD matrices is the matrix log transformation, leading by the central limit theorem to what may be called a matrix-variate log-normal distribution.

Biography: Armin Schwartzman is an Assistant Professor in the Department of Biostatistics at Harvard. He received his Ph.D. in Statistics from Stanford, working with Bradley Efron and Jonathan Taylor. He holds MS and BS degrees in Electrical Engineering and Science Education from Caltech and the Technion (Israel Institute of Technology).

Host: Information and Operations Management Department, Marshall School of Business

Audiences: Everyone Is Invited

Contact: Mayumi Thrasher