Gaussian Sampling Techniques
Assume an N-dimensional Gaussian random variable x∼N(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 L∑i=1wi⋅δ(x−xi), 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 1LL∑i=1δ(x−xi).
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)dx≈L∑i=1wi⋅g(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.