13 Estimation of Stochastic Process Variance

13.1 Optimal Variance Estimate of Gaussian Stochastic Process

Let the stationary Gaussian stochastic process ξ(t) with the correlation function

R(t1,t2)=σ2(t1,t2)(13.1)

be observed at N equidistant discrete time instants ti, i = 1, 2,…, N in such a way that

ti+1tj=Δ=const.(13.2)

Then, at the measurer input we have a set of samples xi = x(ti). Furthermore, we assume that the mathematical expectation of observed stochastic process is zero. Then, the conditional N-dimensional pdf of Gaussian stochastic process can be presented in the following form:

p(x1,x2,...,xN|σ2)=1(2πσ2)N/2det ij exp{ 12σ2i=1,j=1NxixjCij }(13.3)

where

  • det ‖ℛij‖ is the determinant of matrix consisting of elements ...

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