Nonlinear Filters

For a nonlinear filtering problem, the most heuristic andeasiest approximation is to use the Taylor series expansionand apply the conventional linear recursive Kalman filteralgorithm directly to the linearized nonlinear measurementand transition equations. First, it is discussed that theTaylor series expansion approach gives us the biasedestimators. Next, a Monte-Carlo simulation filter isproposed, where each expectation of the nonlinear functionsis evaluated generating random draws. It is shown fromMonte-Carlo experiments that the Monte-Carlo simulationfilter yields the unbiased but inefficient estimator. Anotherapproach to the nonlinear filtering problem is toapproximate the underlyingdensity functions of the statevector. In this monograph, a nonlinear and nonnormal filteris proposed by utilizing Monte-Carlo integration, in which arecursive algorithm of the weighting functions is derived. The densityapproximation approach gives us anasymptotically unbiased estimator. Moreover, in terms ofprogramming and computational time, the nonlinear filterusing Monte-Carlo integration can be easily extended tohigher dimensional cases, compared with Kitagawa's nonlinearfilter using numericalintegration.