Improved algorithms for initial condition parameter estimation

Consider an ensemble of discrete-time stochastic processes. They arise from state-space models that may have entirely different parameters, but the initial conditions of their State vectors are Gaussian, i.i.d, across the ensemble. Existing algorithms to find the maximum likelihood estimates (m.l.e.´s) of the mean vector and variance matrix of the initial state are improved in this paper by (1) allowing for the possibility that in some of the stochastic processes there may be insufficient information to estimate the entire initial condition vector, (2) permitting the m.l.e. of the variance matrix to be singular (or nearly so), and (3) improving numerical stability and computational speed by the use of Cholesky decompositions. Included also are modifications to the EM algorithm for finding the m.l.e.´s and an efficient form for calculating the value of the likelihood function that is independent of the choice of origin and coordinate frame.