Modeling Interacting Time-Series Signals

Many real-life systems consist of multiple information signals which might be potentially interacting with each other over time. These interactions can be estimated/modeled using techniques like Pearson Correlation (PC), Time lagged Cross Correlation (TLCC) and windowed TLCC, Dynamic Time Warping (DTW), and coupled Hidden Markov Model (cHMM). These techniques, excluding cHMM, cannot capture non-linear interactions and does not work well with multi-variate data. Although cHMM can capture the interactions effectively, it is bound by Markov property and other assumptions like latent variables, prior distributions, etc. These influence the performance of the model significantly. Recurrent Neural Network (RNN) is a variant of Neural Networks which can be used to model time-series data. RNN based architectures are the new state-of-the-art for complex tasks like machine translation. In this research, we explore techniques to extend RNNs to model interacting time-series signals. We propose architectures with coupling and attention mechanism. We evaluate the performance of the models on synthetically generated and real-life data sets. We compare the performance of our proposed architectures to similar ones in the literature. The goal of this exercise is to determine the most effective architecture to capture interaction information in the given interrelated time-series signals

An algebraic approach to product-form stationary distributions for some reaction networks

Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative ergodic CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.Comment: Accepted for publication in SIAM Journal on Applied Dynamical System