Spatio-Temporal Modelling and Adaptive Sampling

The previous chapter showed the similarity of models for time series and for spatial variation. So we should not expect that spatio-temporal modelling adds any completely new algorithms. Spatio-temporal modelling aims to estimate the changes over time of a spatially distributed dynamic system. We could actually use pure time series models z ( t ) $$z(t)$$ or spatial models z ( s ) $$z(s)$$ to achieve that, by defining z as location or time, respectively. So z ( t ) $$z(t)$$ could represent an individual’s changing location over time t, and z ( s ) $$z(s)$$ could be the time at which a wildfire reaches location s. But the term ‘spatio-temporal modelling’ usually refers to models where the state variable z changes as a function of both spatial and temporal coordinates and is written as z ( s , t ) $$z(s,t)$$ . Still, that does not require us to move to a new kind of model. We already saw that kernel-based approaches such as Gaussian Process modelling (GP) can be used for both spatial and temporal modelling, with the difference between the two mainly being one of dimensionality. A spatio-temporal system could certainly be modelled by means of a GP in three-dimensional space-time if we can define a sensible covariance function C [ ( s , t ) , ( s′ , t′ ) ] $$C[(s,t),(s',t')]$$ . However, that is generally a difficult thing to do. It is very difficult to imagine what the proper covariance should be between our state variable z at location s and time t with the value of z at another place and time. Moroever, even if we could do so, this would lead to very large matrices that are difficult to invert and thus would cause computational problems for GP-estimation and -prediction. Therefore most spatio-temporal models employ separable covariance functions that can be decomposed as C [ ( s , t ) , ( s′ , t′ ) ] = C s [ s , s′ ] C t [ t , t′ ] $$C[(s,t),(s',t')]= C_s[s,s'] C_t[t,t']$$ . The question of how similar we expect the system to be at nearby locations is thereby separated from the question of how we expect the system to evolve over time.