## 2 Specifications of Geographically Weighted Multivariate Poisson Inverse Gaussian Regression
According to [31], the selection of the optimal bandwidth value can be achieved using the cross validation (CV) criteria, which can be formulated mathematically as follows:  
\[CV(h)=\min\!\left\{\sum\limits_{i=1}^{n}\left[\mathbf{y}_{i}-\mathbf{\overset{ \boldsymbol{\lambda}}{Y}}_{(\
eq i)}(h)\right]^{T}\!\left[\mathbf{y}_{i}- \mathbf{\overset{\boldsymbol{\lambda}}{Y}}_{(\
eq i)}(h)\right]\right\} \tag{10}\]  
The ML estimator that maximizes the log-likelihood function of the GWMPIGR model in Equation (7) is obtained by solving the system of equations for all of the first partial derivatives of the log-likelihood function for each parameter and then equating it with zero as follows:  
\[\frac{\partial\ell^{*}(\theta_{i^{*}})}{\partial\theta_{i^{*}}}=0 \tag{11}\]  
The first-order partial derivatives of the log-likelihood function are as follows:  
\[\frac{\partial\ell(\theta_{i^{*}})}{\partial\beta_{1}(u_{i^{*}},v_{i^{*}})} =\sum\limits_{i=1}^{n}\Big{[}y_{i1}-M(y_{i1},y_{i2},\ldots,y_{im}) \exp\Big{(}\x_{i}^{\T}\beta_{1}(u_{i^{*}},v_{i^{*}})\Big{)}\Big{]}\x_{i^{*}}^{ \T}w_{ii^{*}} \tag{12}\]  
\[\frac{\partial\ell(\theta_{i^{*}})}{\partial\beta_{2}(u_{i^{*}},v_{i^{*}})} =\sum\limits_{i=1}^{n}\Big{[}y_{i2}-M(y_{i1},y_{i2},\ldots,y_{im}) \exp\Big{(}\x_{i}^{\T}\beta_{2}(u_{i^{*}},v_{i^{*}})\Big{)}\Big{]}\x_{i^{*}}^{ \T}w_{ii^{*}} \tag{13}\]  
\[\frac{\partial\ell(\theta_{i^{*}})}{\partial\beta_{m}(u_{i^{*}},v_{i^{*}})} =\sum\limits_{i=1}^{n}\Big{[}y_{im}-M(y_{i1},y_{i2},\ldots,y_{im} )\exp\Big{(}\x_{i}^{\T}\beta_{m}(u_{i^{*}},v_{i^{*}})\Big{)}\Big{]}\x_{i^{*}}^{ \T}w_{ii^{*}} \tag{14}\]  
\[ \begin{array}{ll}\frac{\partial\ell(\theta_{i^{*}})}{\partial\tau(u_{i^{*}}, \varphi_{i^{*}})}&=\frac{1}{\tau^{2}(u_{i^{*}},v_{i^{*}})}\sum\limits_{i=1}^{ n}(M(y_{i1},y_{i2},\ldots,y_{im})\times\\ &\left(1+\tau(u_{i^{*}},v_{i^{*}})\sum\limits_{j=1}^{m}\exp\Big{(}\x_{i}^{ \T}\beta_{j}(u_{i^{*}},v_{i^{*}})\Big{)}\right)-1-\frac{1}{\tau(u_{i^{*}},v_{ i^{*}})}\sum\limits_{j=1}^{m}y_{ij}\end{array}  \tag{15}\]  
where  
\[M(y_{i1},y_{i2},\ldots,y_{im})=\frac{1}{\sqrt{1+2\tau(u_{i^{*}},v_{i^{*}}) \sum\limits_{j=1}^{m}q_{i}\exp\Big{(}\x_{i}^{\T}\beta_{j}(u_{i^{*}},v_{i^{*}} )\Big{)}}}\frac{K\sum\limits_{j=1}^{m}y_{ij}+\frac{1}{2}(z(u_{i^{*}},v_{i^{*} }))}{K\sum\limits_{j=1}^{m}y_{ij}-\frac{1}{2}(z(u_{i^{*}},v_{i^{*}}))} \tag{16}\]  
The first derivative of the log-likelihood function of the GWMPIGR model parameters (Equations (12)-(15)) is non-closed-form equation. Thus, the estimator value is obtained through the Berndt-Hall-Hall-Hausman (BHHH) iteration method using the following algorithm:  
* Determine the initial value of each parameter using the results of the MPIGR model parameter estimator. * Define the gradient vector \(\mathbf{g}(\theta_{i})\) with the first derivative (Equation (11)) as the element. * Define the Hessian matrix \(\mathbf{H}(\theta_{i})\), \[\mathbf{H}(\theta_{i})=-\left[\sum\limits_{i=1}^{n}\mathbf{g}_{i}(\theta)( \mathbf{g}_{i}(\theta))^{T}\right]\] where \(\mathbf{g}_{i}(\theta)\) is the individual gradient vector. * Define the tolerance limits (\(\varepsilon=10^{-3}\)) and maximum iteration (\(t^{*}=1000\)). * Start the BHHH iteration using the following equation: \[\overset{\alpha}{\theta}_{i}^{(t+1)}=\overset{\alpha}{\theta}_{i}^{(t)}- \mathbf{H}^{-1}\left(\overset{\alpha}{\theta}_{i}^{(t)}\right)\mathbf{g} \left(\overset{\alpha}{\theta}_{i}^{(t)}\right),\] (17)
* The iteration will stop at the \(t^{*}\) iteration or at the value of \(\overset{\alpha}{\theta}_{i}^{(t+1)}-\overset{\alpha}{\theta}_{i}^{(t)}\|\leq\varepsilon\). Repeat this algorithm for each location \(i\) (\(i=1,2,\ldots,n\)). The simultaneous hypothesis testing of the GWMPIGR model parameters is carried out by testing \(\mathrm{H}_{0}\): \(\beta_{j1}(u_{i},v_{i})=\beta_{j2}(u_{i},v_{i})=...=\beta_{jk}(u_{i},v_{i})= \ldots=\beta_{jp}(u_{i},v_{i})=0\) versus \(\mathrm{H}_{1}\): at least one \(\beta_{jk}(u_{i},v_{i})\
eq 0\), where \(j=1,2,\ldots,m;k=1,2,\ldots,p\); and \(i=1,2,\ldots,n\).