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Global and local diagnostic analytics for a geostatistical model based on a new approach to quantile regression

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Abstract

Data with spatial dependence are often modeled by geoestatistical tools. In spatial regression, the mean response is described using explanatory variables with georeferenced data. This modeling frequently considers Gaussianity assuming the response follows a symmetric distribution. However, when this assumption is not satisfied, it is useful to suppose distributions with the same asymmetric behavior of the data. This is the case of the Birnbaum–Saunders (BS) distribution, which has been considered in different areas and particularly in environmental sciences due to its theoretical arguments. We propose a geostatistical model based on a new approach to quantile regression considering the BS distribution. Global and local diagnostic analytics are derived for this model. The estimation of model parameters and its local influence are conducted by the maximum likelihood method. Global influence is based on the Cook distance and it is compared to local influence, in both cases to detect influential observations, whose detection and removal can modify the conclusions of a study. We illustrate the proposed methodology applying it to environmental data, which shows this situation changing the conclusions after removing potentially influential observations. A comparison with Gaussian spatial regression is conducted.

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Acknowledgements

The authors would like to thank the Editors and Reviewers for their constructive comments which led to improve the presentation of the manuscript. The research was partially supported by the project grants “FONDECYT 1200525” from the National Commission for Scientific and Technological Research of the Chilean government (V. Leiva) and “Puente 001/2019” from the Research Directorate of the Vice President for Research of the Pontificia Universidad Católica de Chile, Chile (M. Galea).

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Authors and Affiliations

  1. School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile

    Víctor Leiva

  2. Department of Mathematics and Statistics, Universidad de La Frontera, Temuco, Chile

    Luis Sánchez

  3. Department of Statistics, Pontificia Universidad Católica de Chile, Santiago, Chile

    Manuel Galea

  4. Department of Statistics, Universidade de Brasília, Brasília, Brazil

    Helton Saulo

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  1. Víctor Leiva
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  2. Luis Sánchez
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  3. Manuel Galea
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  4. Helton Saulo
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Correspondence to Víctor Leiva.

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Appendix: Perturbation matrices for the BS spatial model

Appendix: Perturbation matrices for the BS spatial model

For the model defined by (5) and its corresponding log-likelihood function given in (10), we have

$$\begin{aligned} \dfrac{\partial \ell ({\varvec{\theta }}; {\varvec{\omega }})}{\partial \omega _i} = -\tilde{{\varvec{A}}}_{{\varvec{\omega }}}^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i}+\dfrac{\partial }{\partial \omega _i} \left( \log (\tilde{a}_{{\varvec{\omega }}})\right) . \end{aligned}$$

The corresponding \((p+2) \times n\) perturbation matrix is given by \(\displaystyle {\varvec{\Delta }}= ({\partial \ell ^2 ({\varvec{\theta }}; {\varvec{\omega }})}/{\partial \theta _j \omega _i})\), where \(j={1,..., p+2}\) and \(i={1,..., n}\), with \(\theta _1=\beta _0, \ldots \theta _p=\beta _{p-1}\), \(\theta _{p+1}=\varphi \) and \(\theta _{p+2}=\alpha \). The elements of this matrix are given by

$$\begin{aligned} \dfrac{\partial \ell ^2({\varvec{\theta }}; {\varvec{\omega }})}{\partial \beta _j \, \partial \omega _i}= & {} - \left( \left( \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \beta _j} \right) ^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i}+\tilde{{\varvec{A}}}_{{\varvec{\omega }}}^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial ^2 \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i \partial \beta _j}\right) \nonumber \\&+\dfrac{\partial }{\partial \beta _j} \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right) , \nonumber \\ \dfrac{\partial \ell ^2({\varvec{\theta }}; {\varvec{\omega }})}{\partial \varphi \, \partial \omega _i}= & {} - \left( \left( \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \varphi } \right) ^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i}+ \tilde{{\varvec{A}}}_{{\varvec{\omega }}}^{\top } \dfrac{\partial \, {\varvec{\Gamma }}^{-1}}{\partial \varphi } \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i}\right. \nonumber \\&\left. +\tilde{{\varvec{A}}}_{{\varvec{\omega }}}^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial ^2 \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i \partial \varphi }\right) +\dfrac{\partial }{\partial \beta _j} \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right) , \nonumber \\ \dfrac{\partial \ell ^2({\varvec{\theta }}; {\varvec{\omega }})}{\partial \alpha \, \partial \omega _i}= & {} - \left( \left( \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \alpha } \right) ^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i}\right. \nonumber \\&\left. +\tilde{{\varvec{A}}}_{{\varvec{\omega }}}^{\top } {\varvec{\Gamma }}^{-1} \dfrac{\partial ^2 \tilde{{\varvec{A}}}_{{\varvec{\omega }}}}{\partial \omega _i \partial \alpha }\right) +\dfrac{\partial }{\partial \alpha } \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right) . \end{aligned}$$
(23)

Perturbation in the response: In the case of perturbation in the response and based on its corresponding log-likelihood function given in (20), we have

$$\begin{aligned} \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i}= & {} a(t_k({\varvec{\omega }}); \alpha , Q_k) A_{ki}, \\ \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \beta _j}= & {} -\dfrac{1}{\alpha \gamma _{\alpha } \sqrt{t_k({\varvec{\omega }}) Q_k}}\nonumber \\&\left( \dfrac{t_k({\varvec{\omega }}) \gamma _{\alpha }^2}{4Q_k}+1\right) \dfrac{1}{h^{\prime }(Q_k)} x_{kj}, \\ \dfrac{\partial ^2 \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i \partial \beta _j}= & {} \left( -\dfrac{\gamma _{\alpha }^2}{4Q_k}+\dfrac{1}{t_k({\varvec{\omega }})}\right) \nonumber \\&\dfrac{1}{2 \alpha \gamma _{\alpha } \sqrt{Q_k t_k({\varvec{\omega }})}} \dfrac{1}{h^{\prime }(Q_k)} x_{kj} A_{kj},\\ \dfrac{\partial }{\partial \beta _j} \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right)= & {} -4 \sum _{k=1}^n\,\left( \dfrac{t_k({\varvec{\omega }}) \gamma _{\alpha }}{t_k^2({\varvec{\omega }}) \gamma _{\alpha }^2 + 4Q_k t_k({\varvec{\omega }})}\right) ^2\nonumber \\&\dfrac{1}{h^{\prime }(Q_k)} x_{kj} A_{kj},\\ \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \varphi }= & {} a(t_k({\varvec{\omega }}); \alpha , Q_k) \dfrac{\partial t_k({\varvec{\omega }})}{\partial \varphi },\\ \dfrac{\partial t_k({\varvec{\omega }})}{\partial \varphi }= & {} \left( k\text {-th row of } \dfrac{\partial {{\varvec{A}}}}{\partial \varphi }\right) {\varvec{\omega }}, \\ \dfrac{\partial {{\varvec{A}}}}{\partial \varphi }= & {} {{\varvec{A}}}\left( \dfrac{1}{\alpha } {\varvec{\Gamma }}^{-1/2} \dfrac{\partial \, {\varvec{\Gamma }}^{1/2}}{\partial \varphi } {\varvec{\Gamma }}^{-1/2} + \dfrac{\alpha }{4} \dfrac{\partial \, {\varvec{\Gamma }}^{1/2}}{\partial \varphi } \right) {{\varvec{A}}}, \\ \dfrac{\partial ^2 \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i \partial \varphi }= & {} \dfrac{1}{\alpha \gamma _{\alpha } \sqrt{4Q_k}} \\&\left( -\dfrac{1}{2t_k^{3/2}({\varvec{\omega }})} \left( \dfrac{\gamma _{\alpha }^2}{2}+\dfrac{2Q_k}{t_k({\varvec{\omega }})}\right) -\dfrac{2Q_k}{t_k^{3/2}({\varvec{\omega }})}\right) \\&\times \dfrac{\partial t_k({\varvec{\omega }})}{\partial \varphi } A_{ki} + a(t_k({\varvec{\omega }}); \alpha , Q_k) \dfrac{\partial A_{ki}}{\partial \varphi },\\ \dfrac{\partial }{\partial \varphi } \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right)= & {} \sum _{k=1}^n \, \left( \dfrac{1}{2t_k^2({\varvec{\omega }})} \dfrac{\partial t_k({\varvec{\omega }})}{\partial \varphi }\right. \\&\left. + \dfrac{4Q_k}{(t_k^2({\varvec{\omega }}) \gamma _{\alpha }^2+4Q_k t_k({\varvec{\omega }}))^2} \right. \\&\left. \left( 2t_k({\varvec{\omega }}) \dfrac{\partial t_k({\varvec{\omega }})}{\partial \varphi } \gamma _{\alpha }^2 + 2t_k^2({\varvec{\omega }}) \gamma _{\alpha } \gamma _{\alpha }^{\prime }\right. \right. \\&\left. \left. + 4Q_k \dfrac{\partial t_k({\varvec{\omega }})}{\partial \varphi }\right) \right) A_{ki} \\&+ \sum _{k=1}^n\, \left( -\dfrac{1}{2t_k({\varvec{\omega }})}-\dfrac{4Q_k}{t_k^2({\varvec{\omega }}) \gamma _{\alpha }^2 + 4Q_k t_k({\varvec{\omega }})} \right) \\&\dfrac{\partial A_{ki}}{\partial \varphi }, \\ \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \alpha }= & {} \dfrac{1}{\sqrt{4Q_k}} \left( -\dfrac{\gamma _{\alpha }+\alpha \gamma _{\alpha }^{\prime }}{\alpha ^2 \gamma _{\alpha }^2}\right. \\&\left. \left( \sqrt{t_k({\varvec{\omega }})} \gamma _{\alpha }^2-4Q_k t_k^{-1/2}({\varvec{\omega }})\right) \right. \\&\left. + \dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{1}{2 \sqrt{t_k({\varvec{\omega }})}} \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha } \gamma _{\alpha }^2\right. \right. \\&\left. \left. + 2 \sqrt{t_k({\varvec{\omega }})} \gamma _{\alpha } \gamma _{\alpha }^{\prime } \right. \right. \\&\left. \left. + 2 Q_k t_k^{-3/2}({\varvec{\omega }}) \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha } \right) \right) ,\\ \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha }= & {} \left( k\text {-th row of } {{\varvec{A}}}\left( \dfrac{1}{\alpha ^2} {\varvec{\Gamma }}^{-1/2} + \dfrac{1}{4} {\varvec{\Gamma }}^{1/2} \right) {{\varvec{A}}}\right) {\varvec{\omega }}, \\ \dfrac{\partial ^2 \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i \partial \alpha }= & {} \dfrac{1}{\sqrt{4Q_k}} \left( \left( -\dfrac{\gamma _{\alpha } + \alpha \gamma _{\alpha }^{\prime }}{\alpha \gamma _{\alpha }} \dfrac{1}{\sqrt{t_k({\varvec{\omega }})}}\right. \right. \\&\left. \left. -\dfrac{1}{2\alpha \gamma _{\alpha } t_k^{3/2}({\varvec{\omega }})} \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha }\right) \right. \\&\left. \times \left( \dfrac{\gamma _{\alpha }^2}{2}+\dfrac{2Q_k}{t_k({\varvec{\omega }})}\right) +\dfrac{1}{\alpha \gamma _{\alpha }}\right. \\&\left. \left( \gamma _{\alpha } \gamma _{\alpha }^{\prime } - \dfrac{2Q_k}{t_k^2({\varvec{\omega }})} \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha }\right) \right) A_{ki}\\&+ a(t_k({{\varvec{\omega }}}); \alpha , Q_k) \left( ki\text {-th\ element\ of } \dfrac{\partial {{\varvec{A}}}}{\partial \alpha }\right) ,\\ \dfrac{\partial }{\partial \alpha } \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right)= & {} \sum _{k=1}^n\, \left( \dfrac{1}{2t_k^2({\varvec{\omega }})} \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha }\right. \\&\left. +\dfrac{4Q_k}{(t_k^2({\varvec{\omega }}) \gamma _{\alpha }^2+4Q_k t_k({\varvec{\omega }}))^2} \right. \\&\left. \times \left( 2 t_k({\varvec{\omega }}) \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha } \gamma _{\alpha }^2 + 2 \gamma _{\alpha } \gamma _{\alpha }^{\prime } t_k^2({\varvec{\omega }}) \right. \right. \\&\left. \left. + 4Q_k \dfrac{\partial t_k({\varvec{\omega }})}{\partial \alpha }\right) \right) A_{ki} \\&+ \sum _{k=1}^n\, \left( -\dfrac{1}{2t_k({\varvec{\omega }})}-\dfrac{4Q_k}{t_k^2({\varvec{\omega }}) \gamma _{\alpha }^2+4Q_k t_k({\varvec{\omega }})} \right) \\&\dfrac{\partial A_{ki}}{\partial \alpha }, \end{aligned}$$

where \( {\partial \, {\varvec{\Gamma }}^{-1}}/{\partial \varphi } = -{\varvec{\Gamma }}^{-1}({\partial \, {\varvec{\Gamma }}}/{\partial \varphi }) {\varvec{\Gamma }}^{-1}\), \({\partial {{\varvec{A}}}}/{\partial \alpha } = {{\varvec{A}}}( ({1}/{\alpha ^2}) {\varvec{\Gamma }}^{-1/2}+({1}/{4}) {\varvec{\Gamma }}^{1/2}) {{\varvec{A}}}\) and \(A_{ki}\) corresponds to the ki-th element of the matrix \({{\varvec{A}}}\). To calculate \({\partial \, {\varvec{\Gamma }}^{1/2}}/{\partial \varphi }\), see De Bastiani et al. (2015).

Perturbation in the explanatory variable: Based on the corresponding log-likelihood function given in (22), the elements defined in (23) have as components the expressions stated as

$$\begin{aligned} \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i}= & {} -\dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \\&\left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \beta _l A_{ki}, \\ \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \beta _j}= & {} -\dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \\&\left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} Z_{kj},\\ \dfrac{\partial ^2 \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i \partial \beta _j}= & {} \dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \\&\left( \left( \dfrac{3 \gamma _{\alpha }^2}{8 Q_k^2({\varvec{\omega }})}+\dfrac{1}{2Q_k({\varvec{\omega }})}\right) \right. \\&\left. + \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \dfrac{h^{\prime \prime } (Q_k({\varvec{\omega }}))}{(h^{\prime }(Q_k({\varvec{\omega }})))^2} \right) \\&\dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \beta _l A_{ki} Z_{kj},\\ \dfrac{\partial }{\partial \beta _j} \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i}\right)= & {} \sum _{k=1}^n\, \left( \left( \dfrac{1}{2 Q_k^2({\varvec{\omega }})}-\dfrac{16}{(t_k \gamma _{\alpha }^2 + 4Q_k({\varvec{\omega }}))^2} \right) \right. \\&\left.\times \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} + \left( \dfrac{1}{2Q_k({\varvec{\omega }})}-\dfrac{4}{t_k\gamma _{\alpha }^2+4Q_k({\varvec{\omega }})}\right) \right. \\&\left. \times \dfrac{h^{\prime \prime }(Q_k({\varvec{\omega }}))}{(h^{\prime }(Q_k({\varvec{\omega }})))^2} \right) \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \beta _l A_{ki} Z_{kj} \\&+ \sum _{k=1}^n\, \left( -\dfrac{1}{2 Q_k({\varvec{\omega }})}+\dfrac{4}{t_k \gamma _{\alpha }^2+4Q_k({\varvec{\omega }})}\right) \\&\dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} A_{ki} \rho _{jl},\\ \dfrac{\partial \tilde{A}_{k}({\varvec{\omega }})}{\partial \varphi }= & {} -\dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \\&\dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial {{\varvec{A}}}_k}{\partial \varphi } {\varvec{\omega }},\\ \dfrac{\partial ^2 \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i \partial \varphi }= & {} \beta _l \left( \dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2}\right. \\&\left. \left( \left( \dfrac{3 \gamma _{\alpha }^2}{8Q_k^2({\varvec{\omega }})}+\dfrac{1}{2Q_k({\varvec{\omega }})}\right) \right. \right. \\&\left. \left. \times \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} + \right. \right. \\&\left. \left. + \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1 \right) \dfrac{h^{\prime \prime }(Q_k({\varvec{\omega }}))}{(h^{\prime }(Q_k({\varvec{\omega }})))^2} \right) \right. \\&\left. \times \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \varphi } \dfrac{\partial {{\varvec{A}}}_k}{\partial \varphi } {\varvec{\omega }}A_{ki} \right. \\&\left. - \dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \right. \\&\left. \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial A_{ki}}{\partial \varphi } \right) ,\\ \dfrac{\partial }{\partial \varphi } \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i} \right)= & {} \beta _l \sum _{k=1}^n\, \left( \left( \left( \dfrac{1}{2 Q_k^2({\varvec{\omega }})}-\dfrac{16}{(t_k \gamma _{\alpha }^2+4Q_k({\varvec{\omega }}))^2} \right) \right. \right. \\&\left. \left. \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \right. \right. \\&\left. \left. +\left( \dfrac{1}{2Q_k({\varvec{\omega }})}-\dfrac{4}{t_k \gamma _{\alpha }^2+4Q_k({\varvec{\omega }})} \right) \right. \right. \\&\left. \left. \dfrac{h^{\prime \prime }(Q_k({\varvec{\omega }}))}{(h^{\prime }(Q_k({\varvec{\omega }})))^2} \right) \right. \\&\left. \times \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \varphi } A_{ki} \right. \\&+ \left. \left( -\dfrac{1}{2Q_k({\varvec{\omega }})}+\dfrac{4}{t_k \gamma _{\alpha }^2+4Q_k({\varvec{\omega }})}\right) \right. \\&\left. \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial A_{ki}}{\partial \varphi } \right) ,\\ \dfrac{\partial \tilde{A}_k({\varvec{\omega }})}{\partial \alpha }= & {} \dfrac{1}{\sqrt{4t_k}} \left( -\dfrac{\gamma _{\alpha }+\alpha \gamma _{\alpha }^{\prime }}{\alpha ^2 \gamma _{\alpha }^2} \left( \dfrac{t_k \gamma _{\alpha }^2 - 4Q_k({\varvec{\omega }})}{\sqrt{Q_k({\varvec{\omega }})}} \right) \right. \\&\left. +\dfrac{1}{\alpha \gamma _{\alpha } Q_k({\varvec{\omega }})} \left( \left( 2t_k \gamma _{\alpha } \gamma _{\alpha }^{\prime }-\frac{4\partial Q_k({\varvec{\omega }})}{\partial \alpha } \right) \sqrt{Q_k({\varvec{\omega }})}\right. \right. \\&\left. \left. -\frac{1}{2\sqrt{Q_k({\varvec{\omega }})}} \frac{\partial Q_k({\varvec{\omega }})}{\partial \alpha } (t_k \gamma _{\alpha }^2-4Q_k({\varvec{\omega }})) \right) \right) ,\\ \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \alpha }= & {} \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \beta _l \dfrac{\partial {{\varvec{A}}}_k}{\partial \alpha } {\varvec{\omega }},\\ \dfrac{\partial ^2 \tilde{A}_k({\varvec{\omega }})}{\partial \omega _i \partial \alpha }= & {} -\beta _l \left( -\dfrac{\gamma _{\alpha }+\alpha \gamma _{\alpha }^{\prime }}{\alpha ^2 \gamma _{\alpha }^2} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})} \right) ^{1/2}\right. \\&\left. \times \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} A_{ki} \right. \\&\left. + \dfrac{1}{\alpha \gamma _{\alpha }} \left( -\dfrac{1}{2} \left( \dfrac{Q_k({\varvec{\omega }})}{t_k}\right) ^{-3/2} \dfrac{1}{t_k} \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \alpha }\right) \right. \\&\left. \times \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} A_{ki} \right. \\&\left. + \dfrac{1}{\alpha \gamma _{\alpha }}\right. \\&\left. \times \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \left( \dfrac{8 \gamma _{\alpha } \gamma _{\alpha }^{\prime } Q_k({\varvec{\omega }}) - \frac{4\partial Q_k({\varvec{\omega }})}{\partial \alpha } \gamma _{\alpha }^2}{16 Q_k^2({\varvec{\omega }})}\right) \right. \\&\left. \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} A_{ki} \right. \\&\left. + \dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \right. \\&\left. \times \left( -\dfrac{1}{(h^{\prime }(Q_k({\varvec{\omega }})))^2}\right) h^{\prime \prime }(Q_k({\varvec{\omega }})) \right. \\&\left. \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \alpha } A_{ki} \right. \\&\left. + \dfrac{1}{\alpha \gamma _{\alpha }} \left( \dfrac{t_k}{Q_k({\varvec{\omega }})}\right) ^{1/2} \left( \dfrac{\gamma _{\alpha }^2}{4Q_k({\varvec{\omega }})}+1\right) \right. \\&\left. \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial A_{ki}}{\partial \alpha } \right) ,\\ \dfrac{\partial }{\partial \alpha } \left( \dfrac{\partial \log (\tilde{a}_{{\varvec{\omega }}})}{\partial \omega _i} \right)= & {} \beta _l \sum _{k=1}^n\, \left( \left( \dfrac{1}{2Q_k^2({\varvec{\omega }})} \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \alpha } \right. \right. \\&\left. \left. - \dfrac{8t_k \gamma _{\alpha } \gamma _{\alpha }^{\prime }+4\frac{\partial Q_k({\varvec{\omega }})}{\partial \alpha }}{(t_k \gamma _{\alpha }^2+4 Q_k({\varvec{\omega }}))^2} \right) \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} A_{ki} \right. \\&\left. +\left( -\dfrac{1}{2Q_k({\varvec{\omega }})} + \dfrac{4}{t_k \gamma _{\alpha }^2+4Q_k({\varvec{\omega }})}\right) \right. \\&\left. \times \left( -\dfrac{h^{\prime \prime }(Q_k({\varvec{\omega }}))}{(h^{\prime }(Q_k({\varvec{\omega }})))^2}\right) \dfrac{\partial Q_k({\varvec{\omega }})}{\partial \alpha } A_{ki} \right. \\&\left. + \left( -\dfrac{1}{2Q_k({\varvec{\omega }})}+\dfrac{4}{t_k \gamma _{\alpha }^2+4Q_k({\varvec{\omega }})}\right) \right. \\&\left. \dfrac{1}{h^{\prime }(Q_k({\varvec{\omega }}))} \dfrac{\partial A_{ki}}{\partial \alpha } \right) , \end{aligned}$$

where \(\rho _{jl}=1\), if \(j=l\); \(\rho _{jl}=0\), if \(j\ne l\); \(Z_{kj}=1\) for \(j=1\); \(Z_{kj}=X_{kl}({\varvec{\omega }})\), if \(j=l\); \(Z_{kj}=X_{kj}\), for \(j\ne 1,l\); and \({{\varvec{A}}}_k\) corresponds to the k-th row of \({{\varvec{A}}}\).

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Leiva, V., Sánchez, L., Galea, M. et al. Global and local diagnostic analytics for a geostatistical model based on a new approach to quantile regression. Stoch Environ Res Risk Assess 34, 1457–1471 (2020). https://doi.org/10.1007/s00477-020-01831-y

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