The EM-based hyper-paramters identification should be modified to perform parametric identification. This requires the definition of the base parameters, i.e. the invariance of the log-likelihood function L with respect to the dynamic parameters theta. In particular, there exists a matrix N such that L(theta) = L(theta + N . n) for all choices of theta and n. Numerically, it is convenient to restrict choices of theta in the subspace orthogonal to the space generated by the columns of N. Let's denote with B the matrix whose columns generate the space orthogonal to N. We then restrict the choices of theta as follows: theta = B . gamma for arbitrary choices of gamma. Optimisation is then performed on gamma. Instead of optimising L(theta) we optimise L(B . gamma) = L_bar(gamma). Derivatives are defined as follows:
The EM-based hyper-paramters identification should be modified to perform parametric identification. This requires the definition of the base parameters, i.e. the invariance of the log-likelihood function
Lwith respect to the dynamic parameterstheta. In particular, there exists a matrixNsuch thatL(theta) = L(theta + N . n)for all choices ofthetaandn. Numerically, it is convenient to restrict choices ofthetain the subspace orthogonal to the space generated by the columns ofN. Let's denote withBthe matrix whose columns generate the space orthogonal toN. We then restrict the choices ofthetaas follows:theta = B . gammafor arbitrary choices ofgamma. Optimisation is then performed ongamma. Instead of optimisingL(theta)we optimiseL(B . gamma) = L_bar(gamma). Derivatives are defined as follows:I will work on this issue with @traversaro.