Optimal detection of a~given number of unknown quasiperiodic fragments in a~numerical sequence

A. V. Kel'manov ; S. A. Khamidullin

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 2, pp. 159-175

Résumé

The a posteriori approach to the problem of the noise-proof detection of unknown quasiperiodic fragments in a numerical sequence is studied. It is assumed that the number of elements in the fragments is given. The case is analyzed, where (1) the number of fragments is known; (2) the index of a sequence term corresponding to the beginning of a fragment is a deterministic (not random) value; (3) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is established that the problem under consideration is reduced to testing a set of hypotheses about the mean of a random Gaussian vector. It is shown that the search for a maximum-likelihood hypothesis is equivalent to finding the arguments which yield a maximum for auxiliary object function. It is proven that maximizing this auxiliary object function is a polynomial-solvable discrete optimization problem. An exact algorithm for solving this auxiliary problem is substantiated. We derive and prove an algorithm for the optimal (maximum-likelihood) detection of fragments. The kernel of this algorithm is the algorithm for solution to an auxiliary problem. The results of numerical simulation are presented.