Using the left Gram matrix to cluster high dimensional data

For high dimensional data, where P features for N objects (P >> N) are represented in an NxP matrix X, we describe a clustering algorithm based on the normalized left Gram matrix, G = XX'/P. Under certain regularity conditions, the rows in G that correspond to objects in the same cluster converge to the same mean vector. By clustering on the row means, the algorithm does not require preprocessing by dimension reduction or feature selection techniques and does not require specification of tuning or hyperparameter values. Because it is based on the NxN matrix G, it has a lower computational cost than many methods based on clustering the feature matrix X. When compared to 14 other clustering algorithms applied to 32 benchmarked microarray datasets, the proposed algorithm provided the most accurate estimate of the underlying cluster configuration more than twice as often as its closest competitors.

A prediction perspective on the Wiener-Hopf equations for discrete time series

The Wiener-Hopf equations are a Toeplitz system of linear equations that have several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The Wiener-Hopf technique is the classical tool for solving the equations, and is based on a comparison of coefficients in a Fourier series expansion. The purpose of this note is to revisit the (discrete) Wiener-Hopf equations and obtain an alternative expression for the solution that is more in the spirit of time series analysis. Specifically, we propose a solution to the Wiener-Hopf equations that combines linear prediction with deconvolution. The solution of the Wiener-Hopf equations requires one to obtain the spectral factorization of the underlying spectral density function. For general spectral density functions this is infeasible. Therefore, it is usually assumed that the spectral density is rational, which allows one to obtain a computationally tractable solution. This leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution together with Baxter's inequality to derive an error bound for the rational spectral density approximation.