# Clustering piecewise stationary processes

## TL;DR

This paper introduces a novel approach to clustering time-series data generated by piecewise stationary processes, relaxing the stationarity assumption and providing consistent, efficient algorithms for the first time.

## Contribution

It formulates the clustering problem for piecewise stationary processes, introduces a notion of consistency, and proposes computationally efficient algorithms with proven consistency.

## Key findings

- Algorithms are consistent without extra assumptions.
- First application of clustering to piecewise stationary processes.
- Provides a natural formulation for the problem.

## Abstract

The problem of time-series clustering is considered in the case where each data-point is a sample generated by a piecewise stationary ergodic process. Stationary processes are perhaps the most general class of processes considered in non-parametric statistics and allow for arbitrary long-range dependence between variables. Piecewise stationary processes studied here for the first time in the context of clustering, relax the last remaining assumption in this model: stationarity. A natural formulation is proposed for this problem and a notion of consistency is introduced which requires the samples to be placed in the same cluster if and only if the piecewise stationary distributions that generate them have the same set of stationary distributions. Simple, computationally efficient algorithms are proposed and are shown to be consistent without any additional assumptions beyond piecewise stationarity.