Gaussian Sampling Techniques

Assume an N-dimensional Gaussian random variable xN(ˆx,C), with mean ˆx and covariance matrix C. In nonlinear filtering it is often required to approximate the probability density function of x as a Dirac mixture, i.e., a set of L weighted samples according to Li=1wiδ(xxi), where δ() denotes the Dirac delta distribution, xi the sample positions, and wi the sample weights. A special case is an equally weighted Dirac mixture which simplifies to 1LLi=1δ(xxi).

For example, consider the random variable y=g(x) that originates from a transformation of the Gaussian random variable x. The mean ˆy of y can be approximated with a proper Dirac mixture according to ˆy=g(x)N(x;ˆx,C)dxLi=1wig(xi). Most of the sample-based Kalman Filters are based on this technique and only differ in the way how they generate such a Dirac mixture approximation.