Conditional Cauchy-Schwarz Divergence for Time Series Analysis: Kernelized Estimation and Applications in Clustering and Fraud Detection

TL;DR

This work tackles the challenge of comparing conditional distributions in time-series by introducing the conditional Cauchy--Schwarz divergence (C-CSD), a symmetric, density-free measure. A practical kernelized estimator is derived using Parzen-window concepts with $K_\tau$ on the condition space and $L_\sigma$ on the output space, augmented by numerical safeguards such as an $epsilon$ ridge, a symmetric logarithmic form, and effective-rank filtering. The method is demonstrated in two domains: time-series clustering (conditioning on time indices) and transactional fraud detection (conditioning on sliding windows with global vs local references), under strictly leak-proof evaluation. Results on UCR benchmarks and BankSim show stable estimation and competitive performance, highlighting C-CSD as a versatile, task-agnostic primitive for model-free comparisons of conditional laws. The work also discusses practical considerations and future directions, including scalability, bandwidth selection, multivariate extensions, and theoretical finite-sample analyses.

Abstract

We study the conditional Cauchy-Schwarz divergence (C-CSD) as a symmetric and density-free measure for time series analysis. We derive a practical kernel based estimator using radial basis function kernels on both the condition and output spaces, together with numerical stabilizations including a symmetric logarithmic form with an epsilon ridge and a robust bandwidth selection rule based on the interquartile range. Median heuristic bandwidths are applied to window vectors, and effective rank filtering is used to avoid degenerate kernels. We demonstrate the framework in two applications. In time series clustering, conditioning on the time index and comparing scalar series values yields a pairwise C-CSD dissimilarity. Bandwidths are selected on the training split, after which precomputed distance k-medoids clustering is performed on the test split and evaluated using normalized mutual information. In fraud detection, conditioning on sliding transaction windows and comparing the magnitude of value changes with categorical and merchant change indicators, each query window is scored by contrasting a global normal reference mixture against a same account local history mixture with recency decay and change flag weighting. Account level decisions are obtained by aggregating window scores using the maximum value. Experiments on benchmark time series datasets and a transactional fraud detection dataset demonstrate stable estimation and effective performance under a strictly leak free evaluation protocol.

Figures (1)

• Figure 1: Pipeline of fraud detection with conditional Cauchy--Schwarz divergence (C-CSD).