The typical particle filtering approximation error is exponentially dependenton the dimension of the model. Therefore, to control this error, an enormousnumber of particles are required, which means a heavy computational burden thatis often so great it is simply prohibitive. Rebeschini and van Handel (2013)consider particle filtering in a large-scale dynamic random field. Through asuitable localisation operation, they prove that a modified particle filteringalgorithm can achieve an approximation error that is mostly independent of theproblem dimension. To achieve this feat, they inadvertently introduce asystematic bias that is spatially dependent (in that the bias at one site isdependent on the location of that site). This bias consequently variesthroughout field. In this work, a simple extension to the algorithm ofRebeschini and van Handel is introduced which acts to average this bias termover each site in the field through a kind of spatial smoothing. It is shownthat for a certain class of random field it is possible to achieve a completelyspatially uniform bound on the bias and that in any general random field thespatial inhomogeneity is significantly reduced when compared to the case inwhich spatial smoothing is not considered. While the focus is on spatialaveraging in this work, the proposed algorithm seemingly exhibits otheradvantageous properties such as improved robustness and accuracy in those casesin which the underlying dynamic field is time varying.
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