ABSTRACT
Since we have adopted the convention that all <x>=0, we can see that x =0 is as good an initial guess as any. Although this regularized solution depends to some (hopefully small) extent on x(0), I propose that we neglect that dependence here. If it were important, the whole method would be useless. By looking at equation (60), we can see that the choice x(0)=0 minimizes the magnitude of x(k) So I shall pursue the analysis of this regularization method only for this special case. Now there is a true value of x, call it u. The measured value is y'=Au+n. Putting this into equation (60), and letting B=AtA,
therefore
Remember that the covariance matrix of a vector w was defined to be <ww t>. A little matrix algebra shows that the covariance matrix of Aw is <AwwtAt>. In the special case that <ww t> is of the form qI for some scalar q, <AwwtAt> =qAAt. Now let us assume that the covariance matrix of u is
Neither this assumption nor the one that x(0)=0 is necessary, but a more general analysis would be more complicated than this subject warrants. As usual, we also assume that the noise covariance matrix is SN=sN
2I. Then the error covariance
matrix of x(k)- u is
