Estimation II

The problem we are seeking to solve is the continual estimation of a set of parameters whose values change over time. Updating is achieved by combining a set of observations or measurements z(t) which contain information about the signal of interest x(t). The role of the estimator is to provide an estimate x̂(t+ ) at some time t+ . If > 0 we have a prediction filter, if < 0 a smoothing filter and if = 0 the operation is simply called filtering. Recall that an estimator is said to be unbiased if the expectation of its output is the expectation of the quantity being estimated, E[x̂℄ = E[x℄. Also recall that a minimum variance unbiased estimator (MVUE) is an estimator which is unbiased and minimises the mean square error: x̂ = argmin x̂ E[jjx̂ xjj2jz℄ = E[xjz℄ The term E[jjx x̂jj2℄, the so-called variance of error, is closely related to the error covariance matrix, E[(x x̂)(x x̂)T ℄. Specifically, the variance of error of an estimator is equal to the trace of the error covariance matrix, E[jjx x̂jj2℄ = traceE[(x x̂)(x x̂)T ℄: