Preferential subspace identification (PSID) with forward-backward smoothing

System identification methods for multivariate time-series, such as neural and behavioral recordings, have been used to build models for predicting one from the other. For example, Preferential Subspace Identification (PSID) builds a state-space model of a primary time-series (e.g., neural activity) to optimally predict a secondary time-series (e.g., behavior). However, PSID focuses on optimal prediction using past primary data, even though in offline applications, better estimation can be achieved by incorporating concurrent data (filtering) or all available data (smoothing). Here, we extend PSID to enable optimal filtering and smoothing. First, we show that the presence of a secondary signal makes it possible to uniquely identify a model with an optimal Kalman update step (to enable filtering) from a family of otherwise equivalent state-space models. Our filtering solution augments PSID with a reduced-rank regression step that directly learns the optimal gain required for the update step from data. We refer to this extension of PSID as PSID with filtering. Second, inspired by two-filter Kalman smoother formulations, we develop a novel forward-backward PSID smoothing algorithm where we first apply PSID with filtering and then apply it again in the reverse time direction on the residuals of the filtered secondary signal. We validate our methods on simulated data, showing that our approach recovers the ground-truth model parameters for filtering, and achieves optimal filtering and smoothing decoding performance of the secondary signal that matches the ideal performance of the true underlying model. This work provides a principled framework for optimal linear filtering and smoothing in the two-signal setting, significantly expanding the toolkit for analyzing dynamic interactions in multivariate time-series.