Infinite-dimensional Bayesian filtering for detection of quasiperiodic phenomena in spatiotemporal data

This paper introduces a spatiotemporal resonator model and an inference method for detection and estimation of nearly periodic temporal phenomena in spatiotemporal data. The model is derived as a spatial extension of a stochastic harmonic resonator model, which can be formulated in terms of a stochastic differential equation. The spatial structure is included by introducing linear operators, which affect both the oscillations and damping, and by choosing the appropriate spatial covariance structure of the driving time-white noise process. With the choice of the linear operators as partial differential operators, the resonator model becomes a stochastic partial differential equation, which is compatible with infinite-dimensional Kalman filtering. The resulting infinite-dimensional Kalman filtering problem allows for a computationally efficient solution as the computational cost scales linearly with measurements in the temporal dimension. This framework is applied to weather prediction and to physiological noise elimination in functional magnetic resonance imaging brain data.

Figure 1

This figure illustrates an example of a stochastic resonator realization in one spatial dimension. The upper left-hand figure shows the observation locations and the corresponding estimate of the oscillating field. The state at $t=0.50$s is shown in the right-hand figure, where the estimate is presented together with the dashed true values of the process.

Figure 2

Slow bias and resonator maps for temperatures as a snapshot on July 8, 2011 at 2 PM (GMT). The weather stations are marked with crosses and areas of uncertainty are hatched.

Figure 3

Mean amplitude maps for the physiological noise components. The results are shown both in isolation and overlaid on top of the corresponding anatomical image.