## 3 Methods
### Random Forest for Spatial Data  
Random Forests [38] is a method for classification and regression based on the idea of bootstrapping, that is, randomly selecting with the replacement of multiple samples from the data, applying a classifier/regression model, and then averaging over the samples--this is called _bagging_. In a generic context, if \(x_{1},x_{2},\ldots,x_{n}\) are input measurements (independent variable, which could be multi-dimensional) and \(y_{1},y_{2},\ldots,y_{n}\) are corresponding outcomes (dependent variable), then a subset \(S\) of these data of size \(B\) is chosen randomly and a classification or regression tree \(t_{S}\) is computed, based on this subset \(S\). The result of the bagging is obtained by averaging \(t_{S}\) over all such random selections. We emphasize that this random selection is conducted with replacement. This methodology is known to reduce variance while not increasing bias. Recently there has been interest in applying RFs to geospatial temporal statistical problems [27]. One important issue to overcome is that, in many ways, the standard random forests approach is indifferent to the spatial nature of the data, specifically in relation to what geographical location it represents and its attributes at each location. [PERSON] et al. [27] extend the basic random forests model to overcome this defect. This was achieved by introducing a measure of distance between observation points into a random forest model. That is, for each point/location \(\ell\), with an accompanying observed value \(Y_{\ell}\), a covariate made up of the distances to all other locations is given. In practice, in the model this amounts to the distance matrix between all pairs of locations with a \(Y_{\ell}\). These are referred to as buffer distances. Figure 9: Time series plot of Total Sum Aerosol for each region. Random Forests for spatial data (RFsp) as proposed by [PERSON] et al. [27] describes the generic equation for an RFsp system, as follows, where  
\[Z_{\ell}=f(X_{G_{\ell}},X_{R_{\ell}},X_{P_{\ell}}), \tag{1}\]  
where \(Z_{\ell}\) is the predicted value at location \(\ell\), \(X_{G_{\ell}}\) contains the distances from location \(\ell\) to \(\ell_{j}\) for all \(j\in\{1,\ldots,N\}\), \(X_{R_{\ell}}\) and \(X_{P_{\ell}}\) represent possible additional covariates; two types--surface reflectance and process based (e.g., wetness index). The function \(f\) is learned by the random forest, thereby there is no covariance function used. [PERSON] et al. [27] combine the methods of random forests with multiple linear regression and kriging [39] to handle the spatial variation issues. As delineated in [PERSON] et al. [27] the RF is a data-driven statistical method. The final predictions are the average of predictions of individual trees [38, 40, 41]. Statistical issues of RF estimation have been detailed recently in [41, 42]. We also refer the reader to a recent study by [PERSON] et al. [43] focused on an applied comparison of four 'fast' geostatistical modeling methods and the software available to implement them. Along with Spatial Random Forests (SpRF) three other methods were Integrated Nested Laplace Approximation (INLA), tree boosting with Gaussian processes and mixed effect models (GPBoost)) and Fixed Rank Kriging (FRK). This application involved estimating malaria prevalence on two different spatial scales, at the country and continent-scale, with the performance of the four methods on these data based on accuracy, computation time, and ease of implementation. As noted by [PERSON] et al. [27], amongst the suite of machine learning algorithms (MLA), the RF has been confirmed to be efficient for spatial predictions. Additionally, RF is available in R through several packages such as randomForest [44] or the computationally faster alternative ranger [45], which is chosen in this study (see the results; Section 4.1).