Estimation of the mixing kernel and the disturbance covariance in IDE-based spatiotemporal systems

doi:10.1016/j.sigpro.2015.10.031

Signal Processing

Volume 121, April 2016, Pages 46-53

https://doi.org/10.1016/j.sigpro.2015.10.031 Get rights and content

Highlights

•
An efficient approach for identification of spatiotemporal IDE model is presented.
•
A closed-form solution for the spatial mixing kernel of the IDE model is derived.
•
A closed-form solution for the disturbance covariance function is calculated.
•
An upper bound on the observation noise variance is computed.

Abstract

The integro-difference equation (IDE) is an increasingly popular mathematical model of spatiotemporal processes, such as brain dynamics, weather systems, and disease spread. We present an efficient approach for system identification based on correlation techniques for linear temporal systems that extended to spatiotemporal IDE-based models. The method is derived from the average (over time) spatial correlations of observations to calculate closed-form estimates of the spatial mixing kernel and the disturbance covariance function. Synthetic data are used to demonstrate the performance of the estimation algorithm.

Introduction

The ability to describe the spatiotemporal dynamics of systems has a profound effect on the manner in which we deal with the natural and man-made world. Complex spatiotemporal behavior is found in many different fields, such as population ecology [1], computer vision [2], video fusion [3] and brain dynamics [4], [5]. In order to describe such dynamical systems, the spatial and temporal behavior can be described simultaneously by diffusion or propagation through space and evolution through time.

Techniques for modeling spatiotemporal systems are generating growing interest, both in the applied and theoretical literature. This interest is perhaps driven by increasing computational power and an ever increasing fidelity of spatiotemporal measurements of various systems. Computational models to describe spatiotemporal processes of particular interest in the wider literature include the cellular automata (CA) [6], partial differential equations (PDEs) [7], [8], lattice dynamical systems (LDSs) [9], coupled map lattices (CMLs) [10], [11], spatially correlated time series [12], [13], [14], and the integro-difference equation (IDE). These models have all been used in a system identification context [15], [16]. This paper deals in particular with the IDE-based spatiotemporal models.

The key feature of IDE models is that they combine discrete temporal dynamics with a continuous spatial representation, enabling predictions at any location of the spatial domain. The dynamics of this model are governed by a spatial mixing kernel, which defines the mapping between the consecutive spatial fields.

Estimating the spatial mixing kernel of the IDE and the underlying spatial field is of particular interest and can be achieved by using conventional state-space modeling [17], [18] or in a hierarchical Bayesian framework [19], [20], [21]. Wikle et al. [22] addressed the estimation problem by describing the IDE in a state-space formulation by decomposing the kernel and the field using a set of spectral basis functions. An alternative approach for the decomposition of the IDE and estimation of the spatial mixing kernel, using the expectation maximization (EM) algorithm, was introduced by Dewar et al. [17]. The key development of this work was a framework where the state and parameter space dimensions were independent of the number of observation locations. The problem of an efficient decomposition of the spatial mixing kernel and the field was addressed in Scerri et al. [18], by incorporating the estimated support of the spatial mixing kernel and the spatial bandwidth of the system from observations. It should be noted that these methods can be combined with improved versions of the EM algorithm such as [23] used in several recent identification work (see [24] for an example) to overcome the limitations of the standard EM algorithm such as sensitivity to initialization.

Despite the popularity of the IDE-based modeling framework, the aforementioned methods for system identification have not been widely used or cited. Perhaps the limited application of the methods is due to the complicated nature of the state-of-the-art algorithms. The algorithms require identification of the spatial mixing kernel support and the system׳s spatial bandwidth for the model reduction, followed by iterative algorithms for state and parameter estimation. The contribution of this paper is a closed-form system identification method that takes care of all of the steps and is easy to implement. It is hoped that the elegant solution will facilitate the development and refinement of models of many systems with spatiotemporal dynamics governed by the IDE.

Linear system identification methods based on temporal correlation techniques and frequency analysis are well documented [25], [26], [27], [28]. However, such methods are often under utilized when studying complex spatiotemporal systems. Here we extend such techniques to IDE based spatiotemporal models, where the spatial mixing kernel and the covariance of the field disturbance are estimated. The estimates are given by closed-form equations, based on the average (over time) spatial auto-correlation and cross-correlation of the observed field. An upper bound on the observation noise variance is also computed. This way we eliminate the computational load of the methods in [17], [18], [21]. Furthermore, we relax assumptions in the previous work where it was assumed that disturbance characteristics were known to the estimator.

The paper is structured as follows. In Section 2 the stochastic IDE model is briefly reviewed. New methods for closed-form estimations of the spatial mixing kernel and the covariance function of the disturbance signal are derived in Section 3. In Section 4 synthetic examples are given to demonstrate the performance of the proposed method using both isotropic and anisotropic spatial mixing kernels. The paper is summarized in Section 5.

Section snippets

Stochastic IDE model

The linear spatially homogeneous IDE is given by $z_{t + 1} (r) = \int_{Ω} k (r - r^{'}) z_{t} (r^{'}) d r^{'} + e_{t} (r),$ where $k (\cdot) = T_{s} κ (\cdot)$ , $κ (\cdot)$ is the spatial mixing kernel and T_s is the sampling time. The index $t \in Z_{0}$ denotes discrete time and $r \in Ω \subset R^{n}$ is the position in an n-dimensional physical space, where $n \in {1, 2, 3}$ . The continuous spatial field at time t and at location $r$ is denoted $z_{t} (r)$ . The model dynamics are governed by the homogeneous, time invariant spatial mixing kernel, $k (r - r^{'})$ , that maps the spatial field through time via

Estimation method

To derive the estimator for the spatial mixing kernel and the disturbance covariance function of the IDE model, we adopt a more compact notation to define convolution and correlation operators. For stationary functions $f (\cdot)$ and $g (\cdot)$ the spatial convolution and the spatial cross-correlation are, respectively, denoted as $\int_{Ω} f (r - r^{'}) g (r^{'}) d r^{'} = (f ⁎ g) (r),$ and $E [f (r) g (r + τ)] = (f ⋆ g) (τ) = R_{f, g} (τ),$ where $E [\cdot]$ denotes the spatial expectation and $τ$ is the spatial shift. We also denote the spatial cross-spectrum as $S_{f,}$

Simulation and results

This section demonstrates the performance of the proposed estimation scheme. Examples are shown where different spatial mixing kernels (isotropic and anisotropic) were adopted. In our forward simulations, these kernels are defined as a sum of Gaussian basis functions in the form of $k (r) = \sum_{i = 0}^{n_{θ}} θ_{i} ψ_{i} (r),$ where θ_i is the weight and $ψ_{i} = \exp (- \frac{{(r - μ_{i})}^{⊤} (r - μ_{i})}{σ_{i}^{2}}) .$ In each example 20 s of data was generated using (1), (3) over the spatial region $Ω = {[- 10, 10]}^{2}$ . Periodic boundary conditions (PBC) were used. Data

Conclusion

A novel and efficient approach for creating data-driven models of spatiotemporal systems has been presented. We have presented a derivation of an estimator that can identify the spatial mixing kernel, disturbance and noise characteristics from measured data. The estimation problem is solved with closed-form solutions, which extends linear systems theory to a broader class of spatiotemporal systems. The closed-form solutions enable straight forward application of the theory, which will

Acknowledgements

The research reported herein was partly supported by the Australian Research Council (LP100200571). Dr. Freestone acknowledges the support of the Australian American Fulbright Commission. The authors also acknowledge valuable support and feedback from Prof. Liam Paninski, Prof. David Grayden, and Prof. Visakan Kadirkamanathan.

References (36)

X. Zhou et al.
Dynamic spatio-temporal modeling for example-based human silhouette recovery
Signal Process.
(2015)
Q. Zhang et al.
Multisensor video fusion based on spatial-temporal salience detection
Signal Process.
(2013)
F. Amor et al.
Imaging brain synchrony at high spatio-temporal resolutionapplication to MEG signals during absence seizures
Signal Process.
(2005)
A. Hegde et al.
Quantifying spatio-temporal dependencies in epileptic ECoG
Signal Process.
(2005)
B. Datsko et al.
Chaotic dynamics in Bonhoffer–Van der pol fractional reaction-diffusion systems
Signal Process.
(2011)
D.R. Freestone et al.
A data-driven framework for neural field modeling
NeuroImage
(2011)
P. Aram et al.
Spatiotemporal multi-resolution approximation of the Amari type neural field model
NeuroImage
(2013)
M. Kot et al.
Dispersal data and the spread of invading organisms
Ecology
(1996)
S. Wolfram
Cellular Automata and Complexity
(1994)
J. Wu
(1996)

S. Chow et al.

Pattern formation and spatial chaos in lattice dynamical systems. I

IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.

(1995)

S.A. Billings et al.

Identification of coupled map lattice models of deterministic distributed parameter systems

Int. J. Syst. Sci.

(2002)

L. Lin et al.

Convergence analysis for initial condition estimation in coupled map lattice systems

IEEE Trans. Signal Process.

(2012)

P.E. Pfeifer et al.

Identification and interpretation of first order space–time ARMA models

Technometrics

(1980)

M. Dewar et al.

A canonical space–time state space modelstate and parameter estimation

IEEE Trans. Signal Process.

(2007)

P. Aram et al.

Spatiotemporal system identification with continuous spatial maps and sparse estimation

IEEE Trans. Neural Netw. Learn. Syst.

(2015)

N. Cressie et al.

(2011)

S.A. Billings

Nonlinear System IdentificationNARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains

(2013)

Cited by (1)

Estimation and Identification of Spatio-Temporal Models with Applications in Engineering, Healthcare and Social Science
2016, Annual Reviews in Control
Citation Excerpt :
In particular it was shown that the loss of surround inhibition in the connectivity structure can contain epileptic events. The estimated algorithm based on spatial correlation technique was further developed to solve general IDE model of the form described in equation (8) (Aram & Freestone, 2016). This work does not provide an estimate of the spatial field, however, if one is interested in the field reconstruction, the kernel estimate can be used as an initialisation for state-space estimation frameworks developed by Dewar et al. (2009) and Scerri et al. (2009), improving the speed and the convergence of the estimation procedure.
Several natural phenomena are known to exhibit a spatio-temporal evolution process. The study of such processes, which is pivotal to our understanding of how best to predict and control spatio-temporal systems, has motivated researchers to develop appropriate tools that infer models and their parameters from observed data. This paper reviews this active area of research by providing an insight into the fundamental ideas spanning the development of spatio-temporal models, dimensionality reduction methods and techniques for state and parameter estimation. Recent advances are discussed in the context of novel spatio-temporal approaches proposed for applications in three specific domains – engineering, healthcare and social science. They illustrate the wide applicability of estimation and identification of spatio-temporal processes as novel advances in sensor systems and data collection are used to observe them.

¹: Authors contributed equally to this work.

View full text

Estimation of the mixing kernel and the disturbance covariance in IDE-based spatiotemporal systems

Highlights

Abstract

Introduction

Section snippets

Stochastic IDE model

Estimation method

Simulation and results

Conclusion

Acknowledgements

Signal Process.

Signal Process.

Signal Process.

Signal Process.

Signal Process.

NeuroImage

NeuroImage

Dispersal data and the spread of invading organisms

Ecology

Cellular Automata and Complexity

Pattern formation and spatial chaos in lattice dynamical systems. I

IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.

Identification of coupled map lattice models of deterministic distributed parameter systems

Int. J. Syst. Sci.

Convergence analysis for initial condition estimation in coupled map lattice systems

IEEE Trans. Signal Process.

Identification and interpretation of first order space–time ARMA models

Technometrics

A canonical space–time state space modelstate and parameter estimation

IEEE Trans. Signal Process.

Spatiotemporal system identification with continuous spatial maps and sparse estimation

IEEE Trans. Neural Netw. Learn. Syst.

Nonlinear System IdentificationNARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains