Matching correlated VAR time series

TL;DR

This paper addresses the problem of matching correlated VAR time series by proposing a probabilistic model, deriving estimators, analyzing their theoretical recovery guarantees, and demonstrating empirical effectiveness of linear assignment methods.

Contribution

It introduces a probabilistic framework for matching VAR time series, derives the MLE, analyzes linear assignment recovery thresholds, and compares relaxation-based methods empirically.

Findings

Linear assignment often matches or outperforms MLE relaxation methods.

Recovery guarantees are established for the linear assignment approach.

The proposed methods enable efficient matching of correlated VAR time series.

Abstract

We study the problem of matching correlated VAR time series databases, where a multivariate time series is observed along with a perturbed and permuted version, and the goal is to recover the unknown matching between them. To model this, we introduce a probabilistic framework in which two time series $(x_t)_{t\in[T]},(x^\#_t)_{t\in[T]}$ are jointly generated, such that $x^\#_t=x_{\pi^*(t)}+\sigma \tilde{x}_{\pi^*(t)}$, where $(x_t)_{t\in[T]},(\tilde{x}_t)_{t\in[T]}$ are independent and identically distributed vector autoregressive (VAR) time series of order $1$ with Gaussian increments, for a hidden $\pi^*$. The objective is to recover $\pi^*$, from the observation of $(x_t)_{t\in[T]},(x^\#_t)_{t\in[T]}$. This generalizes the classical problem of matching independent point clouds to the time series setting. We derive the maximum likelihood estimator (MLE), leading to a quadratic optimization over permutations, and theoretically analyze an estimator based on linear assignment. For the latter approach, we establish recovery guarantees, identifying thresholds for $\sigma$ that allow for perfect or partial recovery. Additionally, we propose solving the MLE by considering convex relaxations of the set of permutation matrices (e.g., over the Birkhoff polytope). This allows for efficient estimation of $\pi^*$ and the VAR parameters via alternating minimization. Empirically, we find that linear assignment often matches or outperforms MLE relaxation based approaches.