## 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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### References:

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