How slow is slow? SFA detects signals that are slower than the driving force

TL;DR

This paper explores how Slow Feature Analysis (SFA) can detect signals slower than the driving force, influenced by parameters like embedding dimension and predictability, revealing a phase transition in detection regimes.

Contribution

It quantifies the conditions under which SFA detects slower components than the driving force, highlighting a phase transition and the influence of various parameters.

Findings

SFA can detect signals slower than the driving force

Detection depends on embedding dimension, predictability, and frequency

A phase transition exists between detecting the driving force and slower subcomponents

Abstract

Slow feature analysis (SFA) is a method for extracting slowly varying driving forces from quickly varying nonstationary time series. We show here that it is possible for SFA to detect a component which is even slower than the driving force itself (e.g. the envelope of a modulated sine wave). It is shown that it depends on circumstances like the embedding dimension, the time series predictability, or the base frequency, whether the driving force itself or a slower subcomponent is detected. We observe a phase transition from one regime to the other and it is the purpose of this work to quantify the influence of various parameters on this phase transition. We conclude that what is percieved as slow by SFA varies and that a more or less fast switching from one regime to the other occurs, perhaps showing some similarity to human perception.