A New Model for Multivariate Markov Chains

Abstract

We propose a new model for multivariate Markov chains of order one or higher on the basis of the mixture transition distribution (MTD) model. We call it the MTD‐Probit. The proposed model presents two attractive features: it is completely free of constraints, thereby facilitating the estimation procedure, and it is more precise at estimating the transition probabilities of a multivariate or higher‐order Markov chain than the standard MTD model.

Granger Causality: A Review and Recent Advances

“…Examples include voting records of politicians, discrete health states for a patient over time, or action labels for players on a team. Furthermore, even when the raw recording mechanism produces continuous-valued time series, to facilitate downstream analyses, the series may be quantized into a small set of discrete values; examples include weather data from multiple stations (Doshi-Velez et al, 2011), wind data (Raftery, 1985), stock returns (Nicolau, 2014), or sales volume for a collection of products (Ching et al, 2002). In these cases, the traditional VAR framework for Granger causal analysis, (6), is inappropriate.…”

“…The MTD model-originally proposed for parsimonious modeling of higher order Markov chainshas been plagued by a non-convex objective and unknown identifiability conditions that have limited its utility (Nicolau, 2014;Zhu and Ching, 2010;Berchtold, 2001). Tank et al (2021) instead propose a change-of-variables reparameterization of the MTD that straightforwardly addresses both issues, thus enabling practical application of the MTD model to Granger causality selection.…”

“…Although the MTD and mLTD are conceptually similar, the parameters of the mLTD are unfortunately harder to interpret. Another alternative formulation one might consider is based on the MTD-probit model of Nicolau (2014); however, this framework is not a natural fit for inferring Granger causality, both due to the non-convexity of the probit model and the non-convex constraints on Z j matrices.…”

Bernoulli vector autoregressive model

“…For stock indexes and economic data, a binary time series is defined by two states, one when the stock price is over a threshold, and one when it is below. Predicting the time when it is below a threshold influences the investment recommendations [15,16]. Lastly, in environmental sciences, binary time series are used to monitor drought or rain occurrences, which has a significant impact on agriculture and the preservation of ecosystems [13,19].…”

Exploiting context information in recommendation systems

“…some query nodes, e.g, user for which we recommend, in the heterogeneous graph for recommendations. However, existing MMC based methods either (i) need to manually set the influence weights between different types of entities [29,35,36], which is tedious and makes these methods less attractive when multiple types of entities exist, as in our case; or (ii) learn the model parameters from transition sequences sampled from data [37][38][39], which are not available in recommendation problems. To overcome these problems, we propose an optimization framework to automatically learn the influence weights by using a ranking method.…”