Parameter inference from a non-stationary unknown process

TL;DR

The paper defines Parameter Inference from a Non-stationary Unknown Process ($\text{PINUP}$) as recovering time-varying parameters $\boldsymbol{\theta}(t)$ from observations generated by an unknown system $\mathcal{M}$, without modeling the underlying dynamics. It unifies six methodological families—dimension reduction, time-series features, recurrence quantification analysis, prediction error, phase-space partitioning, and Bayesian inference—and demonstrates that traditional benchmarks like the Lorenz system and logistic map can be solved with simple windowed statistics, challenging their value as benchmarks. It introduces more difficult test problems (Langford, sine map) and shows that selective features from the catch22 set can track TVPs where simple statistics fail, thereby guiding future PINUP developments. Overall, the work provides a cohesive framing, critical benchmarking insights, and a roadmap to advance PINUP and the broader study of non-stationary phenomena.

Abstract

Non-stationary systems are found throughout the world, from climate patterns under the influence of variation in carbon dioxide concentration, to brain dynamics driven by ascending neuromodulation. Accordingly, there is a need for methods to analyze non-stationary processes, and yet most time-series analysis methods that are used in practice, on important problems across science and industry, make the simplifying assumption of stationarity. One important problem in the analysis of non-stationary systems is the problem class that we refer to as Parameter Inference from a Non-stationary Unknown Process (PINUP). Given an observed time series, this involves inferring the parameters that drive non-stationarity of the time series, without requiring knowledge or inference of a mathematical model of the underlying system. Here we review and unify a diverse literature of algorithms for PINUP. We formulate the problem, and categorize the various algorithmic contributions. This synthesis will allow researchers to identify gaps in the literature and will enable systematic comparisons of different methods. We also demonstrate that the most common systems that existing methods are tested on - notably the non-stationary Lorenz process and logistic map - are surprisingly easy to perform well on using simple statistical features like windowed mean and variance, undermining the practice of using good performance on these systems as evidence of algorithmic performance. We then identify more challenging problems that many existing methods perform poorly on and which can be used to drive methodological advances in the field. Our results unify disjoint scientific contributions to analyzing non-stationary systems and suggest new directions for progress on the PINUP problem and the broader study of non-stationary phenomena.

Figures (9)

• Figure 1: Parameter Inference from a Non-stationary Unknown Process (PINUP). a. A non-stationary process is generated by an unknown model $\mathcal{M}$ under the influence of some time-varying parameter(s) (TVPs) $\boldsymbol{\theta}(t)$. A selection of example processes are depicted with TVPs denoted in red: (i) brain activity under the time-varying influence of ascending neuromodulation by dopamine; (ii) climate (e.g., mean temperature) under the influence of changing $\text{CO}_2$; and (iii) the Lorenz process with a time-varying parameter, $\rho(t)$ [see Eq. \ref{['eq:lorenz']}]. b. A time-series realization $X$ of the non-stationary process is observed. c. A PINUP algorithm is used to infer the time-varying parameter(s) $\hat{\boldsymbol{\theta}}(t)$ directly from the time series $X$.
• Figure 3: Individual time-series features match or exceed the TVP reconstruction accuracy of several PINUP methods across four chaotic systems. The processes (and varied parameters) are: a-c. the Lorenz process (parameter $\rho$); d-f. the logistic map (parameter $r$); g-i. the Langford process (parameter $\omega$); and j-l. the sine map (parameter $r$); each of which displays results from a parameter inference experiment. A noise-free example of each process is visualized (in panels a, d, g, and j) and colored according to the corresponding TVP value at each point, noting that the flows are visualized as 2-dimensional spatial projections, whereas the 1-dimensional maps are visualized using space and time. The time course of the associated sinusoidal TVP is visualized for each system (b, e, h, and k) with color corresponding to the TVP value for comparison to the corresponding process plots. The numerical experiment result panels (c, f, i, and l) show mean and standard deviation of TVP reconstruction accuracy (Pearson $R^2$) for each method across multiple trials using two levels of signal-to-noise-ratio (SNR) ($0$ and $20$ dB) and a noise-free condition. The PINUP methods were quadratic slow feature analysis (SFA2), a prediction-error-based method using echo state networks (ESN error), and characteristic distance (CD). The single time-series features that were used were mean, standard deviation (std), the first time lag at which autocorrelation function falls below $1/e$ (feature name: acf_timescalelubbaCatch22CAnonicalTimeseries2019a), and the centroid of the power spectrum (feature name: centroid_freqlubbaCatch22CAnonicalTimeseries2019a).
• Figure : a. Dimension reduction
• Figure : a. Dimension reduction
• Figure : b. Statistical time-series features