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Linear optimal prediction and innovations representations of hidden Markov models. (English) Zbl 1075.62626

Let \(\{x_t\}\) be an unobservable Markov chain with states \(e_1, \dots , e_n\) and \(\{y_t\}\) an observable process which takes values \(d_1, \dots , d_{\ell }\). Assume that given \(\{x_t\}\), \(\{y_t\}\) is a sequence of conditionally independent random variables and that the conditional distribution of \(y_t\) depends on \(x_t\) only. Then we have hidden a Markov model. Define \(a_{ij}=P(x_{t+1}=e_i\mid x_t=e_j)\), \(c_{ij}=P(y_{t+1}=d_i\mid x_t=e_j)\), \(A=(a_{ij})\), \(C=(c_{ij})\). The authors transform the state space system \(x_{t+1}=A x_t + \xi _{t+1}\), \(y_{t+1}=C x_t + \eta _{t+1}\) to innovations representation, which is a recursive representation of the optimal linear predictor. The derivation requires the solution of algebraic Riccati equations under non-minimality assumptions. Two numerical examples with \(n=\ell =2\) are presented. It is shown that the optimal predictor and optimal linear predictor perform similarly. The optimal linear predictor has a smaller computational advantage.

MSC:

62M20 Inference from stochastic processes and prediction
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