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The Markov approximation of the periodic multivariate Poisson autoregression

Mahmoud Khabou, Edward A. K. Cohen, Almut E. D. Veraart

TL;DR

This work advances count time-series modeling by introducing a periodic multivariate Poisson autoregression with potentially infinite memory. It develops a contraction-based stability theory and a universal Markov approximation using exponential-polynomial kernels to achieve linear-time simulation and likelihood computation, while proving strong consistency of the Markov MLE. The authors provide both theoretical guarantees and empirical validation, including extensive simulations under well-specified and misspecified settings and a real-data application to Berlin Rotavirus counts that outperforms the existing PNAR model in several districts and horizons. The results demonstrate the method's utility for periodic, networked count data with long memory and hint at broader extensions to risk modeling and exogenous inputs.

Abstract

This paper introduces a periodic multivariate Poisson autoregression with potentially infinite memory, with a special focus on the network setting. Using contraction techniques, we study the stability of such a process and provide upper bounds on how fast it reaches the periodically stationary regime. We then propose a computationally efficient Markov approximation using the properties of the exponential function and a density result. Furthermore, we prove the strong consistency of the maximum likelihood estimator for the Markov approximation and empirically test its robustness in the case of misspecification. Our model is applied to the prediction of weekly Rotavirus cases in Berlin, demonstrating superior performance compared to the existing PNAR model.

The Markov approximation of the periodic multivariate Poisson autoregression

TL;DR

This work advances count time-series modeling by introducing a periodic multivariate Poisson autoregression with potentially infinite memory. It develops a contraction-based stability theory and a universal Markov approximation using exponential-polynomial kernels to achieve linear-time simulation and likelihood computation, while proving strong consistency of the Markov MLE. The authors provide both theoretical guarantees and empirical validation, including extensive simulations under well-specified and misspecified settings and a real-data application to Berlin Rotavirus counts that outperforms the existing PNAR model in several districts and horizons. The results demonstrate the method's utility for periodic, networked count data with long memory and hint at broader extensions to risk modeling and exogenous inputs.

Abstract

This paper introduces a periodic multivariate Poisson autoregression with potentially infinite memory, with a special focus on the network setting. Using contraction techniques, we study the stability of such a process and provide upper bounds on how fast it reaches the periodically stationary regime. We then propose a computationally efficient Markov approximation using the properties of the exponential function and a density result. Furthermore, we prove the strong consistency of the maximum likelihood estimator for the Markov approximation and empirically test its robustness in the case of misspecification. Our model is applied to the prediction of weekly Rotavirus cases in Berlin, demonstrating superior performance compared to the existing PNAR model.

Paper Structure

This paper contains 33 sections, 13 theorems, 154 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The model
  3. Notations
  4. Type I periodic Poisson autoregression
  5. Type II periodic Poisson autoregression
  6. Periodic stability of the model
  7. General results for periodic infinite memory autoregressions
  8. Application to multivariate periodic Poisson autoregressions
  9. The universal Markov approximation of Poisson autoregressions
  10. Continuity of the Poisson autoregression with respect to the kernels
  11. The Markov properties of the Poisson autoregression with an exponential polynomial kernel
  12. Statistical inference
  13. Properties of the Markov maximum likelihood estimator
  14. Numerical illustrations
  15. Estimation for the well specified model
  16. ...and 18 more sections

Key Result

Proposition 3.3

Let $m\in{\mathbb N}^*$ and $(\zeta)_{t\in {\mathbb Z}}$ be an i.p.d family of random variables. Under Assumption ass:contraction_periodic, the regression eq:finite_reg has a unique periodically stationary and periodically weakly dependent solution $(\tilde{X}^{(m)}_t)_{t\in {\mathbb Z}}$. Moreover, then there exists $C>0$ and $r\in (0,1)$ such that

Figures (17)

  • Figure 1: The counts drop significantly when $t$ is not divisible by 4.
  • Figure 2: The peak of the average number of counts no longer coincides with the instants $t=0\text{ mod}[4]$, but occurs right after. We can also see that it decreases more slowly from its peak than in the Type I periodicity.
  • Figure 3: The counts $Y$ differ occasionally from the stationary solution $\tilde{Y}$ for $t\in \{0,\cdots,7\}$ but then the two become identical. Similarly, after a transitory period the curve ${\mathbb E} [Y^{(3)}_t]$ joins the strictly periodic curve ${\mathbb E}[\tilde{Y}^{(3)}_t]$.
  • Figure 4: The linear decay of $\log {\mathbb E} |\tilde{Y}_t - Y_t|$. The empty history count process $Y$ joins its periodically stationary version $\tilde{Y}$ exponentially fast. The expected value is approximated by averaging $N_{MC}=2500$ trajectories.
  • Figure 5: The $\ell_1({\mathbb N})$ approximation of $\phi^2$ using $q=3$ exponential functions with $\tau=36$, in orange.
  • ...and 12 more figures

Theorems & Definitions (30)

  • Remark 2.1
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • proof
  • Corollary 3.6
  • proof
  • Proposition 3.8
  • ...and 20 more