Discrete Stochastic Processes and Optimal Filtering

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:

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

In the same way, if we call:

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

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

with : coefficients of the predictor of order K − 1

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