A zero-inflated model for spatiotemporal count data with extra zeros: application to 1950–2015 tornado data in Kansas

Abstract

In many tornado climate studies, the number of tornado touchdowns is often the primary outcome of interest. These outcome measures are usually generated under a spatiotemporal correlation structure and contains many zeros due to the rarity of tornado occurrence at a specific location and time interval. To model the spatiotemporal count data with excess zeros, we propose a spatiotemporal zero-inflated Poisson (ZIP) model, which lends itself to ease of interpretation and computational simplicity. Technically, we embed a modified conditional autoregressive model in the ZIP model to describe the spatial and temporal correlations, where the probability of a pure zero in the ZIP is purposely designed to depend on locations but independent of time. Illustrated with the longitudinal tornado touchdown data in the state of Kansas from 1950 to 2015, our model suggests that the spatial correlation among the counties and the corresponding temperature are significant factors attributed to the tornado touchdowns. Through the model, we can also estimate the probabilities of no tornado touchdowns for each county over time. These estimated probabilities substantially help us understand the pattern of touchdowns and further identify the risk areas across Kansas. Moreover, these estimates can be iteratively updated when more current touchdown data are available. The final model for Kansas tornado touchdown data is evaluated using more recent data.

Appendix: derivations of the full conditional distributions for the model parameters

According to the assumptions for the variance components $\sigma ^2_t$; $t=0,1,\ldots ,T$, and spatiotemporal random process $\delta _{0i}$; $i=1,\ldots ,N$, the corresponding full conditional densities can be exactly determined and then the Gibbs sampling schedule can be used to generate the posterior samples $\sigma ^2_t$ and $\delta _{0i}$, respectively. This supplement focuses on illustrating the full conditional distributions of $\sigma ^2_t$ and $\delta _{0i}$. For each $\sigma ^2_t$, $t=0,1,\ldots ,T$, the full conditional density can be derived by the following manner:

$$\begin{aligned}{} & {} P\big (\sigma ^2_t\mid \varvec{\theta }_{\big (-\sigma ^2_t\big )},\varvec{\delta }_1,\ldots ,\varvec{\delta }_T,\varvec{\delta }_0,\textbf{Y}\big ) \nonumber \\{} & {} \quad \propto P\big (\varvec{\delta }_{t}\mid \sigma ^2_t,\phi _t,\varvec{\delta }^{*}_t\big ) \times \pi \big (\sigma ^2_t\big ) \nonumber \\{} & {} \quad = \frac{1}{(2\pi )^{(N/2)}\left( \text{ det }\left( \sigma ^2_t\varvec{V}(\phi _t)\right) \right) ^{1/2}} \exp \left\{ -\frac{1}{2}(\varvec{\delta }_t-\varvec{\delta }^{*}_t)'\big (\sigma ^2_t\varvec{V}(\phi _t)\big )^{-1}(\varvec{\delta }_t-\varvec{\delta }^{*}_t)\right\} \nonumber \\{} & {} \qquad \times \frac{1}{\Gamma (a_t)b_t^{a_t}}\left( \frac{1}{\sigma ^2_t}\right) ^{a_t+1}\exp \left\{ -\frac{1}{\sigma ^2_tb_t}\right\} \nonumber \\{} & {} \quad \propto \left( \frac{1}{\sigma ^2_t}\right) ^{\frac{N}{2}+a_t+1} \exp \left\{ -\frac{1}{\sigma ^2_t}\left[ \frac{1}{2}(\varvec{\delta }_t-\varvec{\delta }^{*}_t)'(\varvec{V}(\phi _t))^{-1}(\varvec{\delta }_t-\varvec{\delta }^{*}_t)+\frac{1}{b_t}\right] \right\} , \end{aligned}$$

(A1)

where $\varvec{\delta }^{*}_t=\varvec{W\alpha }$ for $t=0$ and $\varvec{\delta }^{*}_t=\varvec{\delta }_0$ for $t=1,\ldots ,T$. According to (A1), the full conditional distribution of $\sigma ^2_t$, for $t=0,1,\ldots ,T$ is

$$\begin{aligned} \sigma ^2_t\mid \phi _t,\varvec{\delta }_t,\varvec{\delta }^{*}_t \sim IG \left( \frac{N}{2}+a_t,\left[ \frac{1}{2}(\varvec{\delta }_t-\varvec{\delta }^{*}_t)'(\varvec{V}(\phi _t))^{-1}(\varvec{\delta }_t-\varvec{\delta }^{*}_t)+\frac{1}{b_t}\right] ^{-1}\right) . \end{aligned}$$

Next, we show that the full conditional density of $\delta _{0i}$; $i=1,\ldots ,N$ is a Gaussian density. That is,

$$\begin{aligned}{} & {} P(\delta _{0i}\mid \varvec{\theta },\varvec{\delta }_1,\ldots ,\varvec{\delta }_T,\varvec{\delta }_{0(-i)},\textbf{Y}) \nonumber \\{} & {} \quad \propto P\big (\delta _{0i}\mid \sigma ^2_0,\phi _0,\varvec{\alpha },\varvec{\delta }_{0(-i)}\big )\times \prod ^{T}_{t=1} P\big (\delta _{ti}\mid \sigma ^2_t,\phi _t,\varvec{\delta }_0,\varvec{\delta }_{t(-i)}\big ) \nonumber \\{} & {} \quad = \frac{1}{\sqrt{2\pi \sigma ^2_0}}\exp \left\{ -\frac{1}{2\sigma ^2_0}\left[ \delta _{0i}-\Bigg (\varvec{W}'_{j}\varvec{\alpha }+\phi _0\sum _{j\in N_i}c_{ij}\Big (\delta _{0j}-\varvec{W}'_{j}\varvec{\alpha }\Big )\Bigg )\right] ^2\right\} \nonumber \\{} & {} \qquad \times \prod ^{T}_{t=1} \frac{1}{\sqrt{2\pi \sigma ^2_t}} \exp \left\{ -\frac{1}{2\sigma ^2_t}\left[ \delta _{ti}-\Bigg (\delta _{0i}+\phi _t\sum _{j\in N_i}c_{ij}\Big (\delta _{tj}-\delta _{0j}\Big )\Bigg )\right] ^2\right\} \nonumber \\{} & {} \quad \propto \exp \left\{ -\frac{1}{2} \sum ^T_{t=0}\frac{1}{\sigma ^2_{t}} \left[ \delta _{0i}-\Bigg (\delta ^{*}_{ti}+\phi _t\sum _{j\in N_i}c_{ij}\Big (\delta _{0j}-\delta ^{*}_{ti}\Big )\Bigg )\right] ^2\right\} \nonumber \\{} & {} \quad = \exp \left\{ -\frac{1}{2}\left[ \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t}\times \frac{\displaystyle \prod ^{T}_{t=0}\sigma ^2_t}{\displaystyle \sum ^{T}_{t=0}\prod _{k\ne t}\sigma ^2_k}\right] \times \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t}(\delta _{0i}-d^*_{ti})^2 \right\} \nonumber \\{} & {} \quad = \exp \left\{ -\frac{1}{2} \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t} \times \sum ^{T}_{t=0}\frac{\displaystyle \prod _{k\ne t}\sigma ^2_k}{\displaystyle \sum ^{T}_{t=0}\prod _{k\ne t}\sigma ^2_k} \big (\delta ^2_{0i}-2\delta _{0i}d^*_{ti}+d^{*2}_{ti}\big ) \right\} \nonumber \\{} & {} \quad \propto \exp \left\{ -\frac{1}{2} \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t} \times \left[ \delta ^2_{0i}-2\delta _{0i}\sum ^{T}_{t=0}\frac{\displaystyle \prod _{k\ne t}\sigma ^2_k}{\displaystyle \sum ^{T}_{t=0}\prod _{k\ne t}\sigma ^2_k}d^*_{ti}\right] \right\} \nonumber \\{} & {} \quad \propto \exp \left\{ -\frac{1}{2} \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t} \times \left[ \delta _{0i}-\sum ^{T}_{t=0}\frac{d^*_{ti}/\sigma ^2_t}{\displaystyle \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t}}\right] ^2\right\} , \end{aligned}$$

(A2)

where $\delta ^{*}_{ti} = \varvec{W}'_{i}\varvec{\alpha }$ for $t=0$ and $\delta ^{*}_{ti} = \delta _{ti}$ for $t=1,\ldots ,T$, and $d^*_{ti}\equiv \delta ^{*}_{ti}+\phi _t\displaystyle \sum _{j\in N_i}c_{ij}(\delta _{0j}-\delta ^{*}_{tj})$. The last result of (A2) follows from

$$\begin{aligned} \sum ^{T}_{t=0}\frac{\displaystyle \prod _{k\ne t}\sigma ^2_k}{\displaystyle \sum ^{T}_{t=0}\prod _{k\ne t}\sigma ^2_k}\times d^*_{ti} = \sum ^{T}_{t=0}\frac{\displaystyle \prod ^T_{t=0}\sigma ^2_k}{\displaystyle \sum ^{T}_{t=0}\prod _{k\ne t}\sigma ^2_k}\times \frac{d^*_{ti}}{\sigma ^2_t} = \sum ^{T}_{t=0}\left( \displaystyle \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t}\right) ^{-1}\times \frac{d^*_{ti}}{\sigma ^2_t}. \end{aligned}$$

Thus, the full conditional distribution of $\delta _{0i}$ is a Gaussian distribution with mean $A_{2i}A^{-1}_1$ and variance $A^{-1}_1$, where $A_1=\displaystyle \sum ^{T}_{t=0}\frac{1}{\sigma ^2_t}$ and $A_{2i}=\displaystyle \sum ^T_{t=0}\frac{d^*_{ti}}{\sigma ^2_t}=\displaystyle \sum ^T_{t=0}\frac{1}{\sigma ^2_t}\left( \delta ^{*}_{ti}+\phi _t\displaystyle \sum _{j\in N_i}c_{ij}(\delta _{0j}-\delta ^{*}_{tj})\right)$ for $t=1,\ldots ,T$.