Estimation of Linear Space-Invariant Dynamics

We propose a computationally efficient estimator for multi-dimensional linear space-invariant system dynamics with periodic boundary conditions that attains low mean squared error from very few temporal steps. By exploiting the inherent redundancy found in many real-world spatiotemporal systems, the estimator performance improves with the dimensionality of the system. This paper provides a detailed analysis of maximum likelihood estimation of the state transition operator in linear space-invariant systems driven by Gaussian noise. The key result of this work is that, by incorporating the space-invariance prior, the mean squared error of a estimator normalized to the number of parameters is upper bounded by N-1M-1 + O(N-1 M-2), where N is the number of spatial points, and M is the number of observed timesteps after the initial value.