Spatio-temporal modelling of hydro-meteorological derived risk using a Bayesian approach: a case study in Venezuela

Abstract

Extreme environmental events have considerable impacts on society. Preparation to mitigate or forecast accurately these events is a growing concern for governments. In this regard, policy and decision makers require accurate tools for risk estimation in order to take informed decisions. This work proposes a Bayesian framework for a unified treatment and statistical modeling of the main components of risk: hazard, vulnerability and exposure. Risk is defined as the expected economic loss or population affected as a consequence of a hazard event. The vulnerability is interpreted as the loss experienced by an exposed population due to hazard events. The framework combines data of different spatial and temporal supports. It produces a sequence of temporal risk maps for the domain of interest including a measure of uncertainty for the hazard and vulnerability. In particular, the considered hazard (rainfall) is interpolated from point-based measured rainfall data using a hierarchical spatio-temporal Kriging model, whose parameters are estimated using the Bayesian paradigm. Vulnerability is modeled using zero-inflated distributions with parameters dependent on climatic variables at local and large scales. Exposure is defined as the total population settled in the spatial domain and is interpolated using census data. The proposed methodology was applied to the Vargas state of Venezuela to map the spatio-temporal risk for the period 1970–2006. The framework highlights both high and low risk areas given extreme rainfall events.

A Appendix

1.1 A. 1 Spatial-temporal hierarchical Bayesian Kriging

The components of the rainfall posterior predictive distribution given by Eq. (4) are described as follows.

$$\begin{aligned} {{\,\mathrm{\mathrm {Y}}\,}}_f^{[g]}\,\bigg |\,\texttt {D}&\sim t_g\left( \mu ^{[g]},\;\frac{cc}{l}\;\widehat{\!\psi }_{gg},\;l\right) ,\\ {{\,\mathrm{\mathrm {Y}}\,}}_f^{[u]}\,\bigg |\,{{\,\mathrm{\mathrm {Y}}\,}}_f^{[g]},\texttt {D}&\sim t_u\left( \mu ^{[u]},\;\frac{d}{q}\;\widehat{\!\psi }_{u\,|\,g},\;q\right). \end{aligned}$$

(13)

where l defines the degrees of freedom from the marginal distribution for the observable values and q is the number of degrees of freedom for the conditional distribution of the non-observable values conditioned on the observable values. These should satisfy the conditions: $l=\delta +n-u-g+1$ and $q=\delta -u+1$. The values l and q must be positive to avoid degenerate probability distributions. Since l and q decrease when u increases, for a fixed number of observations, the number of new estimation u should be bounded. The components of Eq. (13) are defined as

$$\begin{aligned} \widehat{B}^{[g]}&= A^{-1} CC,\qquad A=\sum _{i=1}^n z_t^\top z_t, \qquad CC=\sum _{i=1}^n z_t^\top {{\,\mathrm{\mathrm {y}}\,}}_t^{[g]},\\ S&=\sum _{i=1}^n \left( {{\,\mathrm{\mathrm {y}}\,}}_t^{[g]}-z_t\widehat{B}^{[g]}\right) ^\top \left( {{\,\mathrm{\mathrm {y}}\,}}_t^{[g]}-z_t\widehat{B}^{[g]}\right) . \end{aligned}$$

$\widehat{B}^{[g]}$ and S are the usual least-squares estimators. Using the Bartlett’s decomposition (Bartlett 1934), the variance-covariance matrix $\varSigma$ can be re-parameterized as follows

$$ \varSigma = \begin{bmatrix}\varSigma _{gg}&\varSigma _{u\,|\,g}&\tau \end{bmatrix}, $$

where

$\varSigma _{gg}$:

is the $(g\times g)$ variance-covariance matrix of ${{\,\mathrm{\mathrm {Y}}\,}}_t^{[g]}$,

$\varSigma _{u\,|\,g}$:

is defined as $\varSigma _{u\,|\,g}=\varSigma _{uu}-\varSigma _{ug}\; \varSigma _{gg}^{-1}\; \varSigma _{gu}$ with dimensions $u\times u$, and

$\tau$:

is a $(u\times g)$ matrix defined by $\varSigma _{ug}\;\varSigma _{gg}^{-1}$.

After this re-parameterization, the $\varSigma$ prior distribution defined in the Eq. (3), can be represented as

$$ \begin{aligned} \varSigma _{gg} \,| \psi ,\,\delta&\sim W_g^{-1}(\psi _{gg},\,\delta -u),\\ \varSigma _{u\,|\,g} \,|\, \psi ,\delta&\sim W_u^{-1}(\psi _{u\,|\,g},\,\delta ),\\ \tau \,|\,\varSigma _{u\,|\,g},\psi&\sim N_{ug}\Big (\tau _0,\,\varSigma _{u\,|\,g}\otimes \psi _{gg}^{-1}\Big ). \end{aligned}$$

The constants describing Eq. (4) are given by the formulas:

$$\begin{aligned} cc&\,=\, 1 + z\,(A+F)^{-1}\, z^{\mathsf {T}}, \\ d&\,=\,1 + z\, F^{-1} z^{\mathsf {T}} + \left( {{\,\mathrm{\mathrm {y}}\,}}_f^{[g]} - z_f\, B_0^{[g]}\right) \; \psi _{gg}^{-1}\; \left( {{\,\mathrm{\mathrm {y}}\,}}_f^{[g]} - z_f\, B_0^{[g]}\right) ^{\!\!T},\\ q&\,=\,\delta -u + 1. \end{aligned}$$

The remaining terms are given by

$$\begin{aligned} \widehat{\!\psi }_{gg}&= \psi _{gg} + S \\&\quad + \left( \widehat{B}^{[g]}-B_0^{[g]}\right) ^{\mathsf {T}}\left( A^{-1} + F^{-1} \right) ^{-1} \left( \widehat{B}^{[g]}-B_0^{[g]} \right) ,\\ \mu ^{[g]}&= (I-W)\;\widehat{B}^{[g]} + W B_0^{[g]}, \qquad W=(A+F)^{-1} F^{-1}\\ \mu ^{[u]}&= z_f\; B_0^{[u]} + \tau _0 \left( {{\,\mathrm{\mathrm {y}}\,}}_f^{[g]} - z_f B_0^{[g]}\right) , \end{aligned}$$

and the final model for ${{\,\mathrm{\mathrm {Y}}\,}}$ is

$$ \begin{aligned} {{\,\mathrm{\mathrm {Y}}\,}}\,|\, \beta ,\,\varSigma&\sim N(Z\,\beta ,\,I_n \otimes \varSigma ),\\ \beta \,|\, \varSigma ,\beta _0&\sim N(\beta _0,\,F^{-1}\otimes \varSigma ),\\ \varSigma&\sim \text{ GIW }(\varTheta ,\,\delta ). \end{aligned}$$

where $N(\,,\,)$ denotes a Gaussian Multivariate distribution; $\beta _0$ is the hyper-parameter mean vector of the $\beta$ coefficients, with dimensions $l\times (g+u)$; $F^{-1}$ is a $l\times l$ matrix describing the variance among the l rows of $\beta$; Z is the covariates matrix and GIW denotes the generalized inverse Wishart distribution with parameters $(\varTheta ,\,\delta )$. The hyper-parameter vector $\mathcal {H}$ is the set ${\varTheta ,\,\delta ,\,F,\,\beta _0}$. The coefficient matrix $\beta : l\times (u+g)$ and the covariance matrix $\varSigma : (u+g)\times (u+g)$ can be partitioned according to the block data structure.

1.2 A. 2 Conditional probability distributions for the hierarchical zero-inflated negative binomial model

The conditional posterior probability distribution for the parameter vector $\varvec{\beta }$ is given by

$$\begin{aligned} {{\,\mathrm{\mathsf {P}}\,}}(\varvec{\beta } \,|\,\varvec{C}, \varvec{\zeta }_{-\varvec{\beta }})&\propto {{\,\mathrm{\mathsf {P}}\,}}(\varvec{C} \,|\,\varvec{\zeta }){{\,\mathrm{\mathsf {P}}\,}}(\varvec{\beta } \,|\,\sigma ^2_{\beta })\\&\propto \sum _{\begin{array}{c} t\;|\; c_t=0 \end{array}} \log \left[ (1-\pi _{t})+ \pi _{t} \left( \frac{\theta _{t}}{\exp (\varvec{x}_t^{\mathsf {T}} \varvec{\beta })+ \theta _{t}}\right) ^{\!\!\theta _{t}} \right] \\&\quad +\sum _{\begin{array}{c} t\;|\; c_t>0 \end{array}}\left[ c_{t}\; \varvec{x}_t^{\mathsf {T}} \varvec{\beta }- \log (\exp (\varvec{x}_t^{\mathsf {T}} \varvec{\beta } +\theta _{t} ))\right] \\&\quad \times \frac{\exp (\frac{-\varvec{\beta }^2}{2 \sigma _{\varvec{\beta }}^2})}{\sigma _{\beta }^2}. \end{aligned}$$

The posterior conditional distribution for $\varvec{\gamma }$ is

$$\begin{aligned} {{\,\mathrm{\mathsf {P}}\,}}(\varvec{\gamma } \,|\,\varvec{C}, \varvec{\zeta }_{-\varvec{\gamma }})&\propto {{\,\mathrm{\mathsf {P}}\,}}(\varvec{C} \,|\,\varvec{\zeta }){{\,\mathrm{\mathsf {P}}\,}}(\varvec{\gamma } \,|\,\sigma ^2_{\gamma })\\&\propto \sum _{t\;|\; c_t>0} \log \left[ \frac{\exp (\varvec{g}_t^{\mathsf {T}}\varvec{\gamma })}{1+ \exp (\varvec{g}_t^{\mathsf {T}}\varvec{\gamma })}\right] \frac{\exp (\frac{-\varvec{\gamma }^2}{2 \sigma _{\gamma }^2})}{\sigma _{\gamma }^2}. \end{aligned}$$

The conditional posterior distributions for $\sigma ^2_{\beta }$ and $\sigma ^2_{\gamma }$ are

$$\begin{aligned} {{\,\mathrm{\mathsf {P}}\,}}(\sigma ^2_{\beta } \,|\,\varvec{C}, \zeta _{-\sigma ^2_{\beta }})&\propto {{\,\mathrm{\mathsf {P}}\,}}(\varvec{\beta } \,|\,\sigma ^2_{\beta }){{\,\mathrm{\mathsf {P}}\,}}(\sigma ^2_{\beta })\\&\propto \frac{1}{\sigma ^2_{\beta }} \exp \left( \frac{-\varvec{\beta }^2}{2 \sigma _{\beta }^2}\right) (\sigma ^2_{\beta })^{(a_{\beta }-1)} \exp (-b_{\beta }\sigma ^2_{\beta })\\&\propto (\sigma ^2_{\beta })^{(a_{\beta }-2)} \exp \left( \frac{-(\varvec{\beta }^{\mathsf {T}}\varvec{\beta }+2b_{\beta }\sigma ^4_{\beta })}{2b_{\beta }\sigma ^2_{\beta }} \right) .\\ {{\,\mathrm{\mathsf {P}}\,}}(\sigma ^2_{\gamma } \,|\,\varvec{C}, \varvec{\zeta }_{-\sigma ^2_{\gamma }})&\propto {{\,\mathrm{\mathsf {P}}\,}}(\varvec{\gamma }\,|\,\sigma ^2_{\gamma }){{\,\mathrm{\mathsf {P}}\,}}(\sigma ^2_{\gamma })\\&\propto (\sigma ^2_{\gamma })^{(a_{\gamma }-2)} \exp \left( \frac{-(\varvec{\gamma }^{\mathsf {T}}\varvec{\gamma }+2b_{\gamma }\sigma ^4_{\gamma })}{2b_{\gamma }\sigma ^2_{\gamma }} \right) . \end{aligned}$$

Finally, the posterior conditional distribution for $\varvec{\theta }$ is given by

$$\begin{aligned} {{\,\mathrm{\mathsf {P}}\,}}(\varvec{\theta } \,|\,\varvec{C},\varvec{\zeta }_{-\varvec{\theta }})&\propto {{\,\mathrm{\mathsf {P}}\,}}(\varvec{C} \,|\,\varvec{\zeta }){{\,\mathrm{\mathsf {P}}\,}}(\varvec{\theta })\\&\propto \sum _{\begin{array}{c} t= 1;{}\\ c_t=0 \end{array}}^{T} \log \left[ (1- \pi _t)+ \pi _t \left( \frac{\theta _t}{\exp (\varvec{x}_t^{\mathsf {T}}\varvec{\beta })+\theta _t}\right) ^{\!\!\theta _t}\right] \\&\quad +\sum _{t\;|\; c_t>0} \log \left[ \varGamma (c_t+\theta _t)\right] - \log (\varGamma (\theta _t))+ \theta _t\log (\theta _t)\\&\quad -\log (\exp (\varvec{x}_t^{\mathsf {T}}\varvec{\beta })+ \theta _t)(\theta _t+ c_{t}) \theta _{t}^{(a_{\theta }-1)}\exp ^{(-b_{\theta } \theta _{t})}. \end{aligned}$$