[Paper Review] Periodically stationary multivariate autoregressive models

Johannes Bracher, Leonhard Held|arXiv (Cornell University)|Jul 14, 2017

Hydrology and Drought AnalysisEnvironmental Science38 references3 citations

TL;DR

This paper proposes a class of multivariate periodic autoregressive models with linear mean functions to capture seasonality and cross-correlations in count and continuous time series, particularly for infectious disease surveillance. It develops analytical and iterative methods to compute unconditional moments and auto/cross-correlations, enabling improved model interpretation and assessment, with application to norovirus incidence in Germany showing significant fit improvement using distributed lags.

ABSTRACT

A class of multivariate periodic autoregressive models is proposed where coupling between time series is achieved through linear mean functions. Various response distributions with quadratic mean-variance relationships fit into the framework, including the negative binomial, gamma and Gaussian distributions. We develop an iterative algorithm to obtain unconditional means, variances and auto-/cross-covariances for models with higher order lags. Analytical solutions are given for the univariate model with lag one and multivariate models with linear mean-variance relationship. A special case of the model class is an established framework for modelling multivariate time series of counts from routine surveillance of infectious diseases. We extend this model class to allow for distributed lags and apply it to a dataset on norovirus gastroenteritis in two German states. The availability of unconditional moments and auto/cross-correlations enhances model assessment and interpretation.

Motivation & Objective

• To develop a general framework for multivariate periodic autoregressive models that incorporate seasonality and cross-series dependence through linear conditional mean functions.
• To extend existing endemic-epidemic models for infectious disease surveillance to include higher-order lags and distributed lags.
• To provide analytical and numerical methods for computing unconditional means, variances, and auto/cross-covariances in periodic models.
• To improve model assessment and interpretation by deriving theoretical properties of unconditional moments and correlations.
• To demonstrate the utility of the framework through an application to norovirus incidence data in two German states with distributed lag structures.

Proposed method

• Proposes a multivariate periodic autoregressive model (MPAR) with periodic coefficients and conditional independence given past information.
• Uses a quadratic variance function (QVF) framework to accommodate distributions like negative binomial, gamma, and Gaussian.
• Derives recursive algorithms to compute unconditional first and second moments for models with higher-order lags and QVF.
• Establishes analytical solutions for univariate models with lag one and multivariate models with linear mean-variance relationships.
• Applies a recursive algorithm based on vectorized matrix equations to compute moments under periodic stationarity.
• Uses a modified vectorization approach to solve the system of equations for the unconditional second-order moments, handling the rank deficiency via a normalization constraint.

Experimental results

Research questions

• RQ1 How can multivariate periodic autoregressive models be extended to include higher-order lags and distributed lags while preserving interpretability and computational feasibility?
• RQ2 What are the analytical and numerical conditions under which unconditional moments (means, variances, auto- and cross-covariances) can be derived in periodic multivariate models with quadratic variance functions?
• RQ3 To what extent do the unconditional correlations in the model reflect true dynamic dependencies between time series, even when conditional independence is assumed?
• RQ4 How does the inclusion of distributed lags improve model fit for multivariate count time series of infectious disease incidence?
• RQ5 Can the proposed framework be applied to real-world surveillance data to enhance long-term forecasting and model interpretation?

Key findings

• The model framework successfully generalizes the endemic-epidemic model to allow for distributed lags, significantly improving fit to norovirus incidence data in Bremen and Lower Saxony.
• Analytical solutions for unconditional moments are derived for univariate models with lag one and multivariate models with linear mean-variance relationships.
• For models with higher-order lags and quadratic variance functions, a recursive algorithm is developed to compute unconditional moments numerically.
• The unconditional auto- and cross-correlations arise from the coupling in the mean structure, even under conditional independence, demonstrating the model's ability to capture complex dependence.
• The model's unconditional moments align well with observed seasonal dynamics, supporting the assumption of stable seasonality in the data.
• The inclusion of distributed lags leads to a more accurate representation of the temporal dynamics of norovirus transmission between the two German states.