# Active Data Selection and Information Seeking
## 5 Dynamic Processes  
In this section, we consider processes that evolve in time. For example, while the approach in Section 3 might be appropriate for mapping out the heights of a mountain range, it would not be suitable for measuring (for example) weather conditions or tectonic activity in different locations across that terrain, as these will change with time (although see [36; 37] for closely related approaches to placing static sensors for environmental monitoring). To move beyond a temporal snapshot, we must extend our models to include dynamics. Figure 6: Function approximation with unresolvable uncertainty. This figure deals with the situation in which the variance associated with the likelihood distribution is heterogeneous along the \(x\)-axis. Specifically, it increases quadratically with distance from zero. The result of this is that less information gain is available at the extremes of the \(x\)-axis. This leads to termination of sampling despite the relatively larger uncertainty about the function in these extremes. This is optimal in this context as there is a limit to the amount of additional uncertainty that can be resolved. There are many ways to do this. One could set up a hidden Markov model, whose states probabilistically evolve according to discrete transition probabilities [38]. An alternative would be to formulate a model in terms of a differential equation that determines the time evolution of hidden states generating data [39]. A final option is to extend the model we used in Section 4 to include a temporal dimension. Specifically, in place of our one-dimensional set of basis functions, we now use two-dimensional basis functions with both temporal and spatial components. While we employ the third of these methods in this paper, each of these three approaches is perfectly valid. \[ \begin{array}{cc}P(y,\theta|\pi)=P(y|\theta,\pi)P(\theta)\\ P(\theta)=\mathcal{N}(\eta_{\theta},\Sigma_{\theta})\\ P(y|\theta,\pi)=\mathcal{N}\big{(}(\phi_{\tau}\otimes\phi_{\pi})\theta, \Sigma_{y}\big{)}\end{array} \quad\Phi=\left[ \begin{array}{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\\ \vdots\end{array} \right] \tag{12}\]  
Equation (12) can be interpreted similarly to Equation (8), in which the expectation of the data is treated as a function approximation, which now includes a time argument. \[ \begin{array}{c}\mathbb{E}[y|\theta,\pi,\tau]=\left(\Phi\Theta\Phi^{\text{T} }\right)_{\tau,\pi}=\sum_{i,j}\Phi_{\tau,i}\Theta_{i,j}\Phi_{\pi,j}\triangleq f _{\theta}(x=\pi,t=\tau)\\ vec(\Phi\Theta\Phi^{\text{T}})=(\Phi\otimes\Phi)vec(\Theta)\\ \theta=vec(\Theta)\end{array}  \tag{13}\]  
However, the form of Equation (12) means we can use the same inferential machinery, and the same information gain calculations, as we did for the static inference examples. Figure 7 shows a graphical representation of the matrices involved in generating our data and the inferences obtained after sampling. Every eight time steps, a sample is chosen to maximise the expected information gain, and this is used to update posterior beliefs exactly as previously. As can be seen from the lower-right panel in Figure 7, an accurate inference has been obtained of the evolution of our function over time. The choices made when sampling provide a relatively even coverage of the spatial dimension. Note the implicit inhibition of return, which prevents resampling near recently sampled locations. However, once sufficient time has elapsed, it is worth returning to previously sampled locations as they may have changed since previously being sampled. Figure 8 shows several snapshots of the inferences drawn at different time points during the simulation reported in Figure 7. This is designed to provide some intuition as to the filling in of predictions as more observations are made.