A Consistent Method for Learning OOMs from Asymptotically Stationary Time Series Data Containing Missing Values

TL;DR

This paper presents a refined spectral learning algorithm for Observable Operator Models (OOMs) that can handle complex missing data patterns in time series, relaxing previous strong assumptions and demonstrating consistency through numerical validation.

Contribution

The authors improve the spectral OOM learning algorithm to relax strong conditions, enabling it to handle more realistic missing data scenarios in time series.

Findings

The refined algorithm is consistent under weaker assumptions.

It can handle missingness patterns interacting with the visible process.

Numerical experiments confirm the theoretical results.

Abstract

In the traditional framework of spectral learning of stochastic time series models, model parameters are estimated based on trajectories of fully recorded observations. However, real-world time series data often contain missing values, and worse, the distributions of missingness events over time are often not independent of the visible process. Recently, a spectral OOM learning algorithm for time series with missing data was introduced and proved to be consistent, albeit under quite strong conditions. Here we refine the algorithm and prove that the original strong conditions can be very much relaxed. We validate our theoretical findings by numerical experiments, showing that the algorithm can consistently handle missingness patterns whose dynamic interacts with the visible process.