Pearson chi^2-divergence Approach to Gaussian Mixture Reduction and its Application to Gaussian-sum Filter and Smoother

The Gaussian mixture distribution is important in various statistical problems. In particular it is used in the Gaussian-sum filter and smoother for linear state-space model with non-Gaussian noise inputs. However, for this method to be practical, an efficient method of reducing the number of Gaussian components is necessary. In this paper, we show that a closed form expression of Pearson chi^2-divergence can be obtained and it can apply to the determination of the pair of two Gaussian components in sequential reduction of Gaussian components. By numerical examples for one dimensional and two dimensional distribution models, it will be shown that in most cases the proposed criterion performed almost equally as the Kullback-Libler divergence, for which computationally costly numerical integration is necessary. Application to Gaussian-sum filtering and smoothing is also shown.