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Modeling non-Gaussian time-varying vector autoregressive processes by particle filtering


We present a novel and general methodology for modeling time-varying vector autoregressive processes which are widely used in many areas such as modeling of chemical processes, mobile communication channels and biomedical signals. In the literature, most work utilize multivariate Gaussian models for the mentioned applications, mainly due to the lack of efficient analytical tools for modeling with non-Gaussian distributions. In this paper, we propose a particle filtering approach which can model non-Gaussian autoregressive processes having cross-correlations among them. Moreover, time-varying parameters of the process can be modeled as the most general case by using this sequential Bayesian estimation method. Simulation results justify the performance of the proposed technique, which potentially can model also Gaussian processes as a sub-case.

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Dimension of the vector autoregressive process

E t :

Index of performance in modified recursive least squares

f t :

Process function

h t :

Measurement function


Order of the vector autoregressive process

m η :

Mean vector of normally distributed parameter η

v t :

Process noise vector

\({\tilde {w}_t^{(i)}}\) :

The normalized importance weight of the ith particle at time t

W t :

Matrix containing delayed augmented measurement matrices in modified recursive least squares

x t :

State variable vector at time t

\({{\bf x}_{0:t}^{(i)}}\) :

Vector of ith particle from initial time up to time t

X K,t :

Kth vector autoregressive coefficient matrix at time t

y t :

Measurement vector (vector autoregressive process)

Y t :

Augmented measurement matrix

Z t :

Error matrix in modified recursive least squares


Probability density function (p.d.f)


Importance function

η t :

Measurement noise vector

Θ :

Parameter matrix in modified recursive least squares


Forgetting factor

Σ ηt :

Time-varying covariance matrix of normally distributed parameter η




Modified Recursive Least Squares


Vector autoregressive


Time-varying autoregressive


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Correspondence to Deniz Gençağa.

Additional information

This work was supported by TÜBİTAK-CNR projects 104E101, 102E027 and by Boğaziçi University Scientific Research Fund project number: 04A201. The first author was supported by NATO-TÜBİTAK A2 fellowship, throughout his research at ISTI-CNR, Italy. This work was done when the first author was at the Electrical and Electronic Engineering Department, Boğaziçi University, İstanbul, Turkey

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Gençağa, D., Kuruoğlu, E.E. & Ertüzün, A. Modeling non-Gaussian time-varying vector autoregressive processes by particle filtering. Multidim Syst Sign Process 21, 73 (2010).

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  • Vector autoregressive processes
  • Sequential Monte Carlo
  • Particle filtering
  • Cross-correlated processes