Оptimal linear prognosis

Abstract

In the article, we consider the speed-up algorithm of linear prognosis of the (n+1)-th random value by the first n known values. The values are uncentred and unstationary. The result we obtain with the help of a new scalar production in the linear subspace of n vectors (by analogy with the geometry of Lobachevsky). With the help of the production, we obtain a new form of linear projection of the additional vector to the subspace. The rapid algorithm of a finding of optimal linear estimation of an additional vector on the known vectors ensues from the projection. The new linear prognosis we obtain from the scheme of the algorithm. We can write the projection in the new rapid form. The proof of such a presentation is the main part of the article. The result is of interest from the point of three-dimensional linear space for the analytical geometry. We consider the main part of this article in terms of the three-dimensional space too. A result is of interest from the point of projections of vectors in space of three measures. One of the theorems is proved by the methods of analytical geometry without application of the notion of the cosine of a corner and construction of additional vectors in a subspace of the initial two vectors. In the second part of the article, the basic theorems are used to some tasks of prognosis and filtration: for filtration of «useful signal» in the random «noise». We obtain the speed-up algorithm of the filtering of a «useful» signal. Unlike the traditional algorithms, we do not search for new additional vectors.