## 3 Materials and Methods
### Methodology  
#### 3.2.1 Machine Learning Regression (Elastic-Net)  
There are three well known penalized regression approaches, Ridge, LASSO and Elastic-net, that performs variable selection and regularization both simultaneously. All these types of ML approaches apply a shrinking method by using estimators with smaller variance modifying the cost function in Ordinary Least Squares (OLS) to penalize additional variables in the model, or complexity. The difference between each approach is how they perform their L\({}_{1}\)-L\({}_{2}\) regularization (See [PERSON], 1976 for a detailed discussion on Ridge regression and [PERSON] 1996 for LASSO regression). Within this study, we concentrate on the newer form of penalized regression proposed by [PERSON] and [PERSON] [62], the Elastic-net method, which is a hybrid regularization variable selection method that linearly combines the ridge and lasso regression techniques within a more flexible framework, with the regularization parameter allowing to fluctuate. The approach therefore switches the lambda penalty, when zero is selected it applies the LASSO approach, and when the regularization parameter is one, it becomes a Ridge regression model. Under the Elastic-net regression, the regression coefficient of (1), \(\beta\) is estimated by:  
\[\hat{\beta}_{ElasticNet}=argmin\beta\left[(y-X\beta)^{\prime}(y-X\beta)+ \lambda(\sum_{j=1}^{p}\left[(1-\alpha)\beta_{j}^{2}+\alpha\big{|}\beta_{j} \big{|}\right])\right] \tag{1}\]  
where the hyperparameter \(\alpha\) controls how much L1- and L2-norm are used. If \(\lambda=0\), there is no penalty term and \(\beta_{ElasticNet}=\beta OLS\). If \(\alpha=0\), \(\beta_{ElasticNet}=\beta Ridge\), and \(\alpha=1\), \(\beta_{ElasticNet}=\beta_{LASSO}\). We incorporate the ESF into the Elastic-net model via the set of selected eigenvectors to address any potential issues of spatial correlation. The ESF method introduces a set of spatial matrix eigenvectors (\(E_{j}\beta\)) into the regression framework to mitigate SA [63] by applying geographical coordinates that are subject to an eigen analysis of geographical distances to establish a set of spatial filters (eigenvectors) expressing the spatial structure of the region at different scales (for a full explanation see [64]). This interaction of eigenvalues and spatially systematic covariates culminates in eigenvector decomposition which extracts orthogonal and uncorrelated numerical components from the given contiguity matrix [65]. Eigenvectors can be extracted from a doubly centered spatial weights matrix C, expressed as:  
\[\text{MCM}=\left(\text{I}-\frac{11^{T}}{n}\right)\text{C}\left(\text{I}-\frac {11^{T}}{n}\right), \tag{2}\]  
where I is an \(n\)\(x\)\(n\) identity matrix, 1 is an n \(\times\) 1 vector of ones, \(n\) is the number of areal units, T the matrix transpose operator (For a full methodological overview see [PERSON] (2003)). The set of eigenvectors of MCM, E_full_ = {e1,..., eN}, provides all the possible distinct map pattern descriptions of latent spatial dependence, with each magnitude being indexed by its corresponding eigenvalue [65]. As discussed by [PERSON] et al. [65], this subset can be identified from a candidate eigenvector set with a stepwise regression procedure ([PERSON] (2008: 2761) further extended the basic linear model rather than using the final EVs to correct for spatial autocorrelation (SAC) on a global level. Interaction terms are introduced between the selected eigenvectors and the predictors to model spatially varying \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Variable** & **Mean** & **Std. Dev** & **Min** & **Max** \\ \hline Beds & 3.032 & 0.786 & 1 & 6 \\ Baths & 1.025 & 0.168 & 1 & 4 \\ Year Built & 1950 & 27.872 & 1810 & 2017 \\ Type & 3.08 & 0.875 & 1 & 4 \\ Build type & 0.84 & 0.371 & 0 & 1 \\ Story & 1.898 & 0.5490 & 1 & 3 \\ Garage & 7.709 & 11.180 & 0 & 119.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._coefficients. See [PERSON] 2008 for a full methodological discussion).