Sampling-reconstruction procedure of discrete Markov processes with continuous time
Y. Goritskiy, Vladimir Kazakov · ICCES: International Conference on Computational & Experimental Engineering and Sciences · 2010
At the first time the statistical description of the Sampling-Reconstruction Procedure (SRP) of Discrete Markov Processes (Markov chains) with continuous time and with an arbitrary number of states is given. The mathematical models of Markov chains with continuous time are intensively used in the description of some real stochastic processes with jumps (in control systems and radio engineering), for instance, impulse noise [1, 2]. This is the reason that it is necessary to know: how to sample, how to reconstruct and how to calculate the reconstruction errors of such processes. (Jumps can be occurred in continuous time moments.) So, the usual method of the SRP investigation of continuous stochastic processes (i.e. the method of the conditional mathematical expectation rule) can not be applied directly. Markov chain ξ (t) with continuous time and with the states 1, 2,. . . ,N, is completely described by the intensities λ1, . . . , λN and by the matrix of the transfer probabilities Pi j(Pii = 0) at the jumps moments. Time ηi of stay in state i has an exponential distribution with pdf pη i = λi · exp(−λit), t >0. Let us designate t0, t1, ..., tn, tn+1 as sampling moments. Let us ξ (tn) = i. It is necessary to find the time interval Ti determining the next sampling moment tn+1 = tn + Ti under the next conditions: 1) condition of accuracy: the variance Vτi j of the estimation τi j of the jump moment τi j from the state i into the state j ( j 6= i) is not more than a given value σ2 (the same for all i and j); 2) condition of miss: probability of state miss on interval (tn, tn +Ti) is not more than a given value γ . It is obtained conditional probability density for the jump moment under condition {ξ (tn) = i, ξ (tn +Ti) = j}: p(t|i, j) = Ce−(λi−λ j)t , 0 < t < T, (1)