Optimal state estimation of Generalized MAP under conditions of non-extendable dead time

The paper deals with Generalized MAP (GMAP) with intensity being piecewise constant stationary process X(t) with two states: X(t) = X₁ and X(t) = X₂ (X₁ > X₂ > 0). The duration of state i (i = 1,2) is an exponentially distributed random variable with distribution function F_i = 1 - e^-X,t, i = 1, 2. When the state i ends the process switches to state j with probability P₁ (Xj | X_i) at an event, and switches to state j with probability P₀ (Xj | X,), i, j = 1, 2, £?=1 (p (x j | x, )+ P₀ (x j | x, ))= 1 without an event. Each registered event at time instant tk generates dead time period of duration T when the other occurring events of GMAP are not observable. After ending of dead time the first occurring event again generates dead time T. The process X(t) is not observable, only time instants of events t₁, t₂ ... are observable. It is necessary to estimate states of X(t) (or GMAP) by only these time instants t₁, t₂ ... . It is assumed that X(t) is stationary. The observation of the process is performed over the period (t₀, t), where t₀ is the beginning of observation, and t is the end of observation. To estimate states of X(t) it is necessary to calculate a posteriori probabilities w(X, | t) that at time instant t the process' state X(t) = X,, i = 1, 2. If w(X, | t) > w(Xj | t), i, j = 1, 2, i Ф j, then x(t) = X,, otherwise X (t) = Xj. The explicit formula for a posteriori probability w(X₁ | t) (w(X₁ | t) = 1 - w(X₁ | t)) at time intervals is derived when GMAP is observable. The recalculation formula of a posteriori probability at time instants tk of occurring an event is derived as well as explicit formula for w(X₁ | t) at time intervals, where the GMAP is not observable, i.e. during dead time T. The described numerical results demonstrate high quality of estimation.

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