Generative Models for Periodicity Detection in Noisy Signals

Abstract

We present the Gaussian Mixture Periodicity Detection Algorithm (GMPDA), a novel method for detecting periodicity in the binary time series of event onsets. The GMPDA addresses the periodicity detection problem by inferring parameters of a generative model. We introduce two models, the Clock Model and the Random Walk Model, which describe distinct periodic phenomena and provide a comprehensive generative framework. The GMPDA demonstrates robust performance in test cases involving single and multiple periodicities, as well as varying noise levels. Additionally, we evaluate the GMPDA on real-world data from recorded leg movements during sleep, where it successfully identifies expected periodicities despite high noise levels. The primary contributions of this paper include the development of two new models for generating periodic event behavior and the GMPDA, which exhibits high accuracy in detecting multiple periodicities even in noisy environments.

Keywords: algorithm; generative models; periodic leg movements during sleep; periodicity; periodicity detection.

Figure A1

Random Walk Model performance with respect to $\beta$, for $\sigma =log\left(\mu \right)$ and $\left|\mu \right|=1$.

Figure A2

Clock Model performance with respect to $\beta$, for $\sigma =log\left(\mu \right)$ and $\left|\mu \right|=1$.

Figure A3

Random Walk Model performance with respect to $\beta$, averaged over $\sigma$, $\left|\mu \right|=1$.

Figure A3

Random Walk Model performance with respect to $\beta$, averaged over $\sigma$, $\left|\mu \right|=1$.

Figure A4

Clock Model performance with respect to $\beta$, averaged over $\sigma$, $\left|\mu \right|=1$.

Figure A5

Random Walk Model performance with respect to $\sigma$, averaged over noise, $\left|\mu \right|=1$.

Figure A6

Clock Model performance with respect to $\sigma$, averaged over noise, $\left|\mu \right|=1$.

Figure A6

Clock Model performance with respect to $\sigma$, averaged over noise, $\left|\mu \right|=1$.

Figure A7

Comparison of GMPA loss with and without curve fitting for individual nights (left panel) and single sleep bouts (right panel).

Figure 1

Example of a well-posed test case with two underlying periodicities and no noise. Histogram of the intervals $D\left(\mu \right)-\widehat{z}$ and generative curves $G(\mu ,\sigma )$ for the Random Walk Model with $n=30$ and $\beta =0$.

Figure 2

Example of an ill-posed test case with a signal-to-noise ratio of 1:2. Histogram of the intervals $D\left(\mu \right)-\widehat{z}$ and generative curves $G(\mu ,\sigma )$ for the Random Walk Model with $n=100$ and $\beta =2$.

Figure 3

Performance of the GMPDA without curve fitting for the Random Walk Model (a) and for the Clock Model (b), with $\beta =0$ and $\left|\mu \right|=1$ and varying number of events (n).

Figure 4

Performance of the GMPDA with curve fitting for the Random Walk Model (a) and for the Clock Model (b), with $\beta =0$ and $\left|\mu \right|=1$ and varying number of events (n).

Figure 5

Performance of the GMPDA with curve fitting for the Random Walk Model (a), and for the Clock Model (b), with $\left|\mu \right|=1$ and $\sigma =log\left(\mu \right)$ across varying levels of uniform noise beta and number of events.

Figure 6

Comparison of the GMPDA to alternative algorithms: for the Random Walk Model (a), and for the Clock Model (b). Accuracy is plotted for different levels of variance $\sigma$ for cases with one period ($\left|\mu \right|=1$), no noise ($\beta =0$) and number of events, $n=100$.

Figure 7

Comparison of the GMPDA to alternative methods: for the Random Walk Model (a), and for the Clock Model (b). Accuracy is plotted against increasing levels of noise $\beta$ for cases with one period ($\left|\mu \right|=1$), known variance, i.e., $\sigma =log\left(\mu \right)$) and number of events, $n=100$.

Figure 8

Comparison of the GMPDA to alternative methods for the Random Walk Model (a), and for the Clock Model (b). Accuracy is plotted against the number of events averaged over $\sigma =\{1,log\left(\mu \right),{\displaystyle \frac{\mu }{16}},{\displaystyle \frac{\mu }{8}}\}$ and $\beta \le 2$.

Figure 9

Detection of multiple periodicities ($\left|\mu \right|=2$) quantified as the number of correctly extracted periodicities by the GMPDA and alternative methods for the Random Walk Model.

Figure 10

Detection of multiple periodicities ($\left|\mu \right|=3$) quantified as the number of correctly extracted periodicities by the GMPDA and alternative methods for the Random Walk Model.

Figure 11

Detection of multiple periodicities ($\left|\mu \right|=2$) quantified as the number of correctly extracted periodicities by the GMPDA and alternative methods for the Clock Model.

Figure 12

Detection of multiple periodicities ($\left|\mu \right|=3$) quantified as the number of correctly extracted periodicities by the GMPDA and alternative methods for the Clock Model.

Figure 13

Effect of the GMPDA parameter $max\_periods$, the maximum number of searched for periodicities, on computational performance averaged over 1200 executions for the GMPDA, with (dark blue symbols) and without curve fitting (light blue symbols).

Figure 14

The GMPDA loss for 100 whole night time series plotted against the number of events (left panel) and length of time series (ts, in seconds, right panel).

Figure 15

The GMPDA loss for 579 sleep bouts of at least 5 min plotted against the number of events (left panel) and the length of the sleep bout (in seconds, right panel).

Figure 16

Histogram of significant periodicities identified in 100 whole night time series by the GMPDA.

Figure 17

Histogram of significant periodicities identified in 579 sleep bouts, 5 min or longer, by the GMPDA.