Non-parametric Determination of Real-Time Lag Structure between Two Time Series: the "Optimal Thermal Causal Path" Method

TL;DR

This paper presents a non-parametric, noise-robust method for detecting and analyzing the evolving lag structure between two time series using an optimal path search in a constructed distance matrix landscape.

Contribution

It introduces the 'Optimal Thermal Causal Path' method, combining optimal path search with fuzzy sampling to improve robustness against noise in lag detection.

Findings

Demonstrates effectiveness on synthetic data.

Outperforms standard cross-correlation methods.

Provides a dynamic view of causality over time.

Abstract

We introduce a novel non-parametric methodology to test for the dynamical time evolution of the lag-lead structure between two arbitrary time series. The method consists in constructing a distance matrix based on the matching of all sample data pairs between the two time series. Then, the lag-lead structure is searched as the optimal path in the distance matrix landscape that minimizes the total mismatch between the two time series, and that obeys a one-to-one causal matching condition. To make the solution robust to the presence of large noise that may lead to spurious structures in the distance matrix landscape, we then generalize this optimal search by introducing a fuzzy search by sampling over all possible paths, each path being weighted according to a multinomial logit or equivalently Boltzmann factor proportional to the exponential of the global mismatch of this path. We present the efficient transfer matrix method that solves the problem and test it on simple synthetic examples to demonstrate its properties and usefulness compared with the standard running-time cross-correlation method. We then apply our `Optimal Thermal Causal Path'' method to the question of the causality between ......