Greedy Gaussian Segmentation of Multivariate Time Series

D. Hallac, P. Nystrup, and S. Boyd

Advances in Data Analysis and Classification, 13(3), 727–751, November 2019. Originally posted October 2016.

We consider the problem of breaking a multivariate (vector) time series into segments over which the data is well explained as independent samples from a Gaussian distribution. We formulate this as a covariance-regularized maximum likelihood problem, which can be reduced to a combinatorial optimization problem of searching over the possible breakpoints, or segment boundaries. This problem can be solved using dynamic programming, with complexity that grows with the square of the time series length. We propose a heuristic method with linear complexity in the time series length, that approximately solves the problem, and always yields a locally optimal choice, in the sense that no change of any one breakpoint improves the objective. Our method, which we call greedy Gaussian segmentation (GGS), is quite efficient and easily scales to problems with vectors of dimension over 1000 and time series of arbitrary length. We discuss methods that can be used to validate such a model using data, and also to automatically choose good values of the two hyperparameters in the method. Finally, we illustrate the approach on various financial time series.