# Statistics of spatio-temporal processes

September 11, 2020 — September 11, 2020

The dynamics of spatial processes evolving in time. This notebook covers the abstract case; many more specific examples can be found in oceanography, atmospheric science, computational fluid dynamics…

Clearly, there are many different problems one might wonder about here. I am thinking in particular of the kind of problem whose discretisation might look like this, as a graphical model.

This is highly stylized — I’ve imagined there is one spatial dimension, but usually there would be two or three. The observed notes are where we have sensors that can measure the state of some parameter of interest \(w\) which evolves in time \(t\). I am wondering what we need to control for to simultaneously learn the parameters of the spatial field \(r_i\), the (possibly emulated) process \(p\) and the state of the unobserved \(w\) nodes.

## 1 Intros

Cosma Shalizi’s Data Over Space and Time course.

## 2 Ensemble Kalman Filters

A classic; happens to work pretty well with spatial fields. See Ensemble Kalman Filters.

## 3 Laplace approximation in spatial fields

AFAICT usually the justification we use for applying Gaussian process formalism to inference. See Laplace approximation for a background.

## 4 Low rank spatial fields

Fixed-Rank Kriging etc.

## 5 Tools

See python spatial and R spatial software.

## 6 References

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