I am trying to estimate the parameters of the following linear dynamical system \begin{align} X_t &= \phi X_{t-1}+\varepsilon_t, \quad \varepsilon_t\sim N(0, \Sigma_\varepsilon)\\ Y_t & = h^TX_t + \delta_t, \quad \delta_t\sim N(0, \sigma_\delta^2) \end{align} where the hidden states $X_t\in\mathbb{R}^2$ are two-dimensional, $\phi$ is just a constant (no matrix) and the covariance matrix $\Sigma_\varepsilon$ of $\varepsilon_t$ is given by \begin{align} \Sigma_\varepsilon = \sigma_\varepsilon^2\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} \end{align} The observations $Y_t$ are one-dimensional, i.e. $h_t\in\mathbb{R}^2$. Further we can assume the distribution of $X_1$ is given. This is pretty much the standard definition of an LDS with the exception that we put some contraints on the transition matrix of the hidden states (it's just $\phi$ times the identity matrix) and the noise covariance matrix (variances are homogeneous). Usually such an LDS is estimated using the EM algorithm (described for example in Bishop's book PRML, http://users.isr.ist.utl.pt/~wurmd/Livros/school/Bishop%20-%20Pattern%20Recognition%20And%20Machine%20Learning%20-%20Springer%20%202006.pdf pages 642-643). In the formulation given in the book there is no constraint as we have above and hence one can analytically compute the maxima in the M-step of the algorithm. However, when I write out these steps for my constrained example it seems that I have to numerically solve the M-step which would dramatically increase the computational cost of my algorithm. Therefore, the question whether there are any standard methods for such constrained LDS formulations.


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