On the variance of identified SIMO systems with spatially correlated output noise

In this paper, we study the problem of evaluating the accuracy of identified linear single-input multi-output (SIMO) dynamical models, where the disturbances affecting the output measurements are spatially correlated. Assuming that the input is an observed white noise sequence, we provide an expression for the covariance matrix of the parameter estimates when weighted least-squares (WLS) are adopted to identify the parameters. Then, we show that, by describing one of the subsystems composing the SIMO structure using less parameters than the other subsystems, substantial improvement on the accuracy of the estimates of some parameters can be obtained. The amount of such an improvement depends critically on the covariance matrix of the output noise and we provide a condition on the noise correlation structure under which the mentioned model parametrization gives the lowest variance in the identified model. We illustrate the derived results through some numerical experiments.