Analysis of Modulated Multivariate Oscillations

TL;DR

This paper formalizes the concept of a common modulated oscillation across multiple time series, proposes a wavelet ridge analysis extension for its recovery, and analyzes the bias properties of the method, with an oceanographic application.

Contribution

It introduces a multivariate wavelet ridge analysis method for detecting common oscillations and derives its bias properties, advancing signal processing techniques for multivariate time series.

Findings

The method effectively identifies common oscillations in noisy data.

Bias depends on the signal's instantaneous curvature and filter choice.

Application demonstrates detection of oceanic vortex motions.

Abstract

The concept of a common modulated oscillation spanning multiple time series is formalized, a method for the recovery of such a signal from potentially noisy observations is proposed, and the time-varying bias properties of the recovery method are derived. The method, an extension of wavelet ridge analysis to the multivariate case, identifies the common oscillation by seeking, at each point in time, a frequency for which a bandpassed version of the signal obtains a local maximum in power. The lowest-order bias is shown to involve a quantity, termed the instantaneous curvature, which measures the strength of local quadratic modulation of the signal after demodulation by the common oscillation frequency. The bias can be made to be small if the analysis filter, or wavelet, can be chosen such that the signal's instantaneous curvature changes little over the filter time scale. An application is presented to the detection of vortex motions in a set of freely-drifting oceanographic instruments tracking the ocean currents.