# Element analysis: a wavelet-based method for analyzing time-localized events in noisy time series

## TL;DR

Element analysis is a wavelet-based technique for detecting and analyzing time-localized events in noisy signals, enabling signal reconstruction and significance testing, demonstrated on ocean current data.

## Contribution

The paper introduces element analysis, a novel wavelet-based method for identifying and reconstructing localized events in noisy time series, with applications to oceanography.

## Key findings

- Effective detection of localized events in noisy signals.
- Ability to reconstruct signals from minimal wavelet transform points.
- Quantitative significance testing for event detection.

## Abstract

A method is derived for the quantitative analysis of signals that are composed of superpositions of isolated, time-localized "events". Here these events are taken to be well represented as rescaled and phase-rotated versions of generalized Morse wavelets, a broad family of continuous analytic functions. Analyzing a signal composed of replicates of such a function using another Morse wavelet allows one to directly estimate the properties of events from the values of the wavelet transform at its own maxima. The distribution of events in general power-law noise is determined in order to establish significance based on an expected false detection rate. Finally, an expression for an event's "region of influence" within the wavelet transform permits the formation of a criterion for rejecting spurious maxima due to numerical artifacts or other unsuitable events. Signals can then be reconstructed based on a small number of isolated points on the time/scale plane. This method, termed element analysis, is applied to the identification of long-lived eddy structures in ocean currents as observed by along-track measurements of sea surface elevation from satellite altimetry