Applied Math Seminar
Winter Quarter 2009
3:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C

Friday, March 6, 2009

Lek-Heng Lim
Mathematics
University of California at Berkeley

Analysis of cumulants


Abstract:

Statistical modeling has been getting some bad press recently, faulted by the media for having precipitated the present global financial crisis. New York Times blamed the crisis on the use of Value-at-Risk to measure risk exposure while Wired Magazine blamed it on the use of Copulas to price CDOs. Without agreeing with these views, we will argue that the common problem being highlighted is the lack of rigorous techniques for analyzing non-Gaussian multivariate data.

We will see that a Gaussian assumption is equivalent to ignoring terms beyond quadratic in a multivariate power series, the coefficients of which are symmetric tensors known as cumulants. In the univariate case, these are scalar-valued and the first four are well-known: mean, variance, skewness, kurtosis. In the multivariate case, mean and covariance are vector- and matrix-valued and may be effectively analyzed using linear algebra. However, higher order multivariate cumulants are hypermatrix-valued with no well-known methods of analysis.

We will discuss two ways to study symmetric hypermatrices akin to the spectral theorem for symmetric matrices: (1) decomposing a homogeneous polynomial into a linear combination of powers of linear forms; (2) decomposing a symmetric tensor into a multilinear combination of points on a Stiefel manifold. Both decompositions have beautiful underlying geometries: (1) secant varieties of the Veronese; (2) symmetric subspace varieties. We then propose a PCA-like technique for analyzing cumulants: It identifies "principal components" that simultaneously accounts for variations in all cumulants via optimization over a single Grassmannian.

This is joint work with Jason Morton.