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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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7.2. Approach to estimation

7.2.1. Scalar case

It is clear that we can give an estimate of the magnitude of a process based on past observation of this process.

In the expression of the innovation:

images

YK represents the magnitude to be estimated (see predictor) and images represents the estimation.

images

In the same way, if we call:

images

the estimate of a process at instant K, starting from measurement y1, …, yK, … of process Y1, …, YK, …, we can write:

images

Let us write the innovation at instants 1, 2,…, K :

images

with images: coefficients of the predictor of order K − 1

images

This expression can be put in the form: I = M Y

with M, invertible triangular matrix because |det M| = 1.

Thus Y = M−1 I.

As a consequence, each vector I

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