On l_1 Mean and Variance Filtering

TL;DR

This paper proposes an l_1 regularized maximum likelihood approach for segmenting time-series data based on changes in mean or variance, leveraging convex formulations to unify variance and mean estimation methods.

Contribution

It introduces a convex formulation for variance segmentation by relating it to mean estimation, enabling the application of existing mean estimation techniques to variance change detection.

Findings

Convex formulation for variance change detection using inverse variance.

Unification of mean and variance segmentation methods via l_1 regularization.

Application of total variation denoising principles to variance estimation.

Abstract

This paper addresses the problem of segmenting a time-series with respect to changes in the mean value or in the variance. The first case is when the time data is modeled as a sequence of independent and normal distributed random variables with unknown, possibly changing, mean value but fixed variance. The main assumption is that the mean value is piecewise constant in time, and the task is to estimate the change times and the mean values within the segments. The second case is when the mean value is constant, but the variance can change. The assumption is that the variance is piecewise constant in time, and we want to estimate change times and the variance values within the segments. To find solutions to these problems, we will study an l_1 regularized maximum likelihood method, related to the fused lasso method and l_1 trend filtering, where the parameters to be estimated are free to vary at each sample. To penalize variations in the estimated parameters, the l_1-norm of the time difference of the parameters is used as a regularization term. This idea is closely related to total variation denoising. The main contribution is that a convex formulation of this variance estimation problem, where the parametrization is based on the inverse of the variance, can be formulated as a certain l_1 mean estimation problem. This implies that results and methods for mean estimation can be applied to the challenging problem of variance segmentation/estimation.