Estimation of the mixing kernel and the disturbance covariance in IDE-based spatiotemporal systems
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 bywhere , is the spatial mixing kernel and Ts is the sampling time. The index denotes discrete time and is the position in an n-dimensional physical space, where . The continuous spatial field at time t and at location is denoted . The model dynamics are governed by the homogeneous, time invariant spatial mixing kernel, , 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 and the spatial convolution and the spatial cross-correlation are, respectively, denoted asandwhere denotes the spatial expectation and is the spatial shift. We also denote the spatial cross-spectrum as
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 ofwhere θi is the weight andIn each example 20 s of data was generated using (1), (3) over the spatial region . 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.
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Cited by (1)
Estimation and Identification of Spatio-Temporal Models with Applications in Engineering, Healthcare and Social Science
2016, Annual Reviews in ControlCitation 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.
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Authors contributed equally to this work.