Sensor placement for optimal estimation of vector-valued diffusion processes

Diffusion processes are commonplace in many scientific disciplines, as they describe a broad range of physical phenomenon. Consider a diffusion process observed through linear sensors with additive white noise. We derive the optimal placement of these sensors for estimating this process, where optimality is defined in terms of the mean squared estimation error (MSE) of the state given past observations. We consider two cases. First, we assume the sensors to be orthogonal. We show in this case that the minimum MSE is related to the nuclear norm of the system matrix of the process. Second, we remove the orthogonality constraint and show that the MSE is related to the Schatten p-norm of the system matrix of the process and the optimal sensors are proportional its matrix cube root. We present simulation results illustrating the fact that the gain afforded by optimizing the choice of sensors depends on the ratio p∕n, where n is the dimension of the system and p the dimension of the Wiener processes driving it, and this gain is in general large, especially when p∕n is small.