A similarity measure for second order properties of non-stationary functional time series with applications to clustering and testing

TL;DR

This paper introduces a novel similarity measure for non-stationary functional time series based on spectral density operators, enabling clustering and hypothesis testing to identify patterns in complex data.

Contribution

It develops the first spectral clustering algorithm for non-stationary functional time series using a new similarity measure and establishes its theoretical properties.

Findings

The similarity measure effectively captures second order properties.

The clustering algorithm successfully groups similar non-stationary series.

The hypothesis test accurately compares second order properties.

Abstract

Due to the surge of data storage techniques, the need for the development of appropriate techniques to identify patterns and to extract knowledge from the resulting enormous data sets, which can be viewed as collections of dependent functional data, is of increasing interest in many scientific areas. We develop a similarity measure for spectral density operators of a collection of functional time series, which is based on the aggregation of Hilbert-Schmidt differences of the individual time-varying spectral density operators. Under fairly general conditions, the asymptotic properties of the corresponding estimator are derived and asymptotic normality is established. The introduced statistic lends itself naturally to quantify (dis)-similarity between functional time series, which we subsequently exploit in order to build a spectral clustering algorithm. Our algorithm is the first of its kind in the analysis of non-stationary (functional) time series and enables to discover particular patterns by grouping together `similar' series into clusters, thereby reducing the complexity of the analysis considerably. The algorithm is simple to implement and computationally feasible. As a further application we provide a simple test for the hypothesis that the second order properties of two non-stationary functional time series coincide.