# Tornado Occurrences in the United States: A Spatio-Temporal Point Process Approach

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Spatio-Temporal Log-Gaussian Cox Process

**C**and

**G**as

**w**, is

**w**is a GMRF with precision matrix defined by (15).

#### 2.2. Data

## 3. Results

#### Zero-Inflated Poisson

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Fitted and Observed Tornadoes without Spatial Component

## Appendix B. Spatial Random Effects

## Appendix C. Fitted Log-Intensity and Observed Tornadoes

## References

- Agee, Ernest, and Samuel Childs. 2014. Adjustments in tornado counts, f-scale intensity, and path width for assessing significant tornado destruction. Journal of Applied Meteorology and Climatology 53: 1494–1505. [Google Scholar] [CrossRef]
- Allen, Myles R., Vicente R. Barros, John Broome, Wolfgang Cramer, Renate Christ, John A. Church, Leon Clarke, Qin Dahe, Purnamita Dasgupta, Navroz K. Dubash, and et al. 2014. Climate Change 2014 Synthesis Report. Technical Report. Geneva: Intergovernmental Panel on Climate Change (IPCC). [Google Scholar]
- Bakka, Haakon. 2013. How to solve the stochastic partial differential equation that gives a Matérn random field using the finite element method. arXiv arXiv:1803.03765. [Google Scholar]
- Brooks, Harold, and Charles A. Doswell III. 2001. Some aspects of the international climatology of tornadoes by damage classification. Atmospheric Research 56: 191–201. [Google Scholar] [CrossRef] [Green Version]
- Brooks, Harold E. 2004. On the relationship of tornado path length and width to intensity. Weather and Forecasting 19: 310–19. [Google Scholar] [CrossRef] [Green Version]
- Cook, Ashton Robinson, Lance M. Leslie, David B. Parsons, and Joseph T. Schaefer. 2017. The impact of El Niño–Southern oscillation (ENSO) on winter and early spring US tornado outbreaks. Journal of Applied Meteorology and Climatology 56: 2455–78. [Google Scholar] [CrossRef]
- Diffenbaugh, Noah S., Martin Scherer, and Robert J. Trapp. 2013. Robust increases in severe thunderstorm environments in response to greenhouse forcing. Proceedings of the National Academy of Sciences 110: 16361–66. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Gensini, Vittorio A., and Harold E. Brooks. 2018. Spatial trends in United States tornado frequency. npj Climate and Atmospheric Science 1: 38. [Google Scholar] [CrossRef] [Green Version]
- Gomez-Rubio, Virgilio. 2020. Bayesian Inference with INLA. Boca Raton: Chapman & Hall-CRC. [Google Scholar]
- Harvey, Andrew C. 1989. Forecasting, Structural Time Series and the Kalman Filter. Cambridge: Cambridge University Press. [Google Scholar]
- Illian, Janine B., Sigrunn H. Sørbye, Håvard Rue, and Ditte K. Hendrichsen. 2010. Fitting a Log Gaussian Cox Process with Temporally Varying Effects–A Case Study. Technical Report. Trondheim: Norges Teknisk-Naturvitenskapelige Universitet. [Google Scholar]
- Illian, Janine B., Sigrunn H. Sørbye, and Håvard Rue. 2012. A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA). Annals of Applied Statistics 6: 1499–530. [Google Scholar] [CrossRef] [Green Version]
- Ingebrigtsen, Rikke, Finn Lindgren, and Ingelin Steinsland. 2014. Spatial models with explanatory variables in the dependence structure. Spatial Statistics 8: 20–38. [Google Scholar] [CrossRef] [Green Version]
- Krainski, Elias T., Virgilio Gómez-Rubio, Haakon Bakka, Amanda Lenzi, Daniela Castro-Camilo, Daniel Simpson, Finn Lindgren, and Håvard Rue. 2018. Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA. Boca Raton: Chapman and Hall/CRC. [Google Scholar]
- Kunkel, Kenneth E., Thomas R. Karl, Harold Brooks, James Kossin, Jay H. Lawrimore, Derek Arndt, Lance Bosart, David Changnon, Susan L. Cutter, Nolan Doesken, and et al. 2013. Monitoring and understanding trends in extreme storms: State of knowledge. Bulletin of the American Meteorological Society 94: 499–514. [Google Scholar] [CrossRef]
- Lambert, Diane. 1992. Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34: 1–14. [Google Scholar] [CrossRef]
- Laurini, Márcio P. 2019. A spatio-temporal approach to estimate patterns of climate change. Environmetrics 30: e2542. [Google Scholar] [CrossRef] [Green Version]
- Lee, Cameron C. 2012. Utilizing synoptic climatological methods to assess the impacts of climate change on future tornado-favorable environments. Natural Hazards 62: 325–43. [Google Scholar] [CrossRef]
- Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73: 423–98. [Google Scholar] [CrossRef] [Green Version]
- Møller, Jesper, Anne Randi Syversveen, and Rasmus Plenge Waagepetersen. 1998. Log Gaussian Cox processes. Scandinavian Journal of Statistics 25: 451–82. [Google Scholar] [CrossRef]
- Moore, Todd W. 2017. On the temporal and spatial characteristics of tornado days in the United States. Atmospheric Research 184: 56–65. [Google Scholar] [CrossRef]
- Moore, Todd W., and Tiffany A. DeBoer. 2019. A review and analysis of possible changes to the climatology of tornadoes in the United States. Progress in Physical Geography: Earth and Environment 43: 365–90. [Google Scholar] [CrossRef]
- Moore, Todd W., and Michael P. McGuire. 2019. Using the standard deviational ellipse to document changes to the spatial dispersion of seasonal tornado activity in the United States. npj Climate and Atmospheric Science 2: 1–8. [Google Scholar] [CrossRef]
- Morana, Claudio, and Giacomo Sbrana. 2019. Climate change implications for the catastrophe bonds market: An empirical analysis. Economic Modelling 81: 274–94. [Google Scholar] [CrossRef]
- Rue, Håvard, and Leonhard Held. 2005. Gaussian Markov Random Fields: Theory and Applications. Boca Raton: Chapman & Hall-CRC. [Google Scholar]
- Rue, Håvard, Sara Martino, and Nicolas Chopin. 2009. Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71: 319–92. [Google Scholar] [CrossRef]
- Serra, Laura, Marc Saez, Jorge Mateu, Diego Varga, Pablo Juan, Carlos Díaz-Ávalos, and Håvard Rue. 2014. Spatio-temporal log-Gaussian Cox processes for modelling wildfire occurrence: The case of Catalonia, 1994–2008. Environmental and Ecological Statistics 21: 531–63. [Google Scholar] [CrossRef] [Green Version]
- Simpson, Daniel, Janine Baerbel Illian, Finn Lindgren, Sigrunn H. Sørbye, and Havard Rue. 2016. Going off grid: Computationally efficient inference for log-Gaussian Cox processes. Biometrika 103: 49–70. [Google Scholar] [CrossRef] [Green Version]
- Spiegelhalter, David J., Nicola G. Best, Bradley P. Carlin, and Angelika Van Der Linde. 2010. Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society, Series B 64: 568–639. [Google Scholar]
- Tippett, Michael K., John T. Allen, Vittorio A. Gensini, and Harold E. Brooks. 2015. Climate and hazardous convective weather. Current Climate Change Reports 1: 60–73. [Google Scholar] [CrossRef] [Green Version]
- Tippett, Michael K., Chiara Lepore, and Joel E. Cohen. 2016. More tornadoes in the most extreme US tornado outbreaks. Science 354: 1419–23. [Google Scholar] [CrossRef] [Green Version]
- Verbout, Stephanie M., Harold E. Brooks, Lance M. Leslie, and David M. Schultz. 2006. Evolution of the us tornado database: 1954–2003. Weather and Forecasting 21: 86–93. [Google Scholar] [CrossRef]
- Watanabe, Sumio. 2010. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research 11: 3571–94. [Google Scholar]
- Watanabe, Sumio. 2013. A widely applicable Bayesian information criterion. Journal of Machine Learning Research 14: 867–97. [Google Scholar]
- Whittle, Peter. 1954. On stationary processes in the plane. Biometrika 41: 434–49. [Google Scholar] [CrossRef]

1. | |

2. | When ${\varphi}_{k}={({\kappa}^{2}=\mathsf{\Delta})}^{1/2}{\phi}_{k}$ for $\alpha =1$ and ${\varphi}_{k}={\phi}_{k}$ for $\alpha =2$, these two approximations are denoted the least squares and the Galerkin solution, respectively. |

Total | Annual Average | Std. Dev. | Annual Min | Annual Max | |
---|---|---|---|---|---|

(E)F0 | 28,844 | 443.754 | 267.438 | 86 | 1186 |

(E)F1 | 20,760 | 319.385 | 86.037 | 170 | 614 |

(E)F2 | 8726 | 134.246 | 53.257 | 61 | 299 |

(E)F3 | 2320 | 35.692 | 17.064 | 10 | 94 |

(E)F4 | 520 | 8.000 | 5.804 | 0 | 30 |

(E)F5 | 54 | 0.831 | 1.387 | 0 | 7 |

Mean | SD | 0.025Quant | 0.5Quant | 0.975Quant | Mode | |
---|---|---|---|---|---|---|

(E)F0 tornadoes | ||||||

Intercept | −0.91 | 0.07 | −1.05 | −0.91 | −0.77 | −0.91 |

Precision for trend | 82.81 | 13.45 | 59.62 | 81.70 | 112.33 | 79.48 |

Precision for cycle | 35.56 | 6.40 | 23.60 | 35.47 | 48.45 | 35.58 |

PACF1 for cycle | −0.00 | 0.08 | −0.17 | −0.00 | 0.16 | 0.01 |

PACF2 for cycle | −0.05 | 0.11 | −0.25 | −0.05 | 0.19 | −0.07 |

Log $\tau $ | −0.77 | 0.04 | −0.85 | −0.77 | −0.70 | −0.77 |

Log $\kappa $ | −0.67 | 0.04 | −0.74 | −0.68 | −0.60 | −0.68 |

Group $\mathsf{\Phi}$ | 0.90 | 0.01 | 0.88 | 0.90 | 0.91 | 0.90 |

(E)F1 tornadoes | ||||||

Intercept | −1.07 | 0.12 | −1.30 | −1.07 | −0.85 | −1.07 |

Precision for trend | 597.09 | 101.69 | 390.67 | 603.27 | 775.28 | 630.87 |

Precision for cycle | 19.15 | 2.02 | 15.16 | 19.19 | 23.04 | 19.44 |

PACF1 for cycle | 0.11 | 0.06 | 0.01 | 0.11 | 0.23 | 0.09 |

PACF2 for cycle | −0.02 | 0.05 | −0.11 | −0.02 | 0.08 | −0.02 |

Log $\tau $ | −0.51 | 0.04 | −0.59 | −0.51 | −0.42 | −0.51 |

Log $\kappa $ | −0.98 | 0.04 | −1.07 | −0.98 | −0.90 | −0.97 |

Group $\mathsf{\Phi}$ | 0.93 | 0.01 | 0.92 | 0.93 | 0.95 | 0.93 |

(E)F2 tornadoes | ||||||

Intercept | −1.60 | 0.15 | −1.89 | −1.60 | −1.30 | −1.60 |

Precision for trend | 514.76 | 136.17 | 262.99 | 513.72 | 781.10 | 512.46 |

Precision for cycle | 76.14 | 15.58 | 50.31 | 74.52 | 111.19 | 71.35 |

PACF1 for cycle | 0.22 | 0.12 | 0.01 | 0.21 | 0.48 | 0.16 |

PACF2 for cycle | −0.06 | 0.18 | −0.44 | −0.05 | 0.27 | 0.01 |

Log $\tau $ | −0.29 | 0.06 | −0.40 | −0.29 | −0.18 | −0.29 |

Log $\kappa $ | −1.24 | 0.05 | −1.35 | −1.24 | −1.13 | −1.24 |

Group $\mathsf{\Phi}$ | 0.88 | 0.04 | 0.77 | 0.89 | 0.93 | 0.92 |

(E)F3 tornadoes | ||||||

Intercept | −2.47 | 0.22 | −2.92 | −2.47 | −2.03 | −2.46 |

Precision for trend | 1133.60 | 396.70 | 460.09 | 1108.60 | 1969.11 | 1027.53 |

Precision for cycle | 32.85 | 9.22 | 18.78 | 31.49 | 54.61 | 28.94 |

PACF1 for cycle | −0.14 | 0.14 | −0.39 | −0.15 | 0.17 | −0.20 |

PACF2 for cycle | −0.06 | 0.21 | −0.48 | −0.05 | 0.33 | 0.00 |

Log $\tau $ | −0.06 | 0.09 | −0.25 | −0.05 | 0.11 | −0.04 |

Log $\kappa $ | −1.51 | 0.09 | −1.68 | −1.51 | −1.32 | −1.53 |

Group $\mathsf{\Phi}$ | 0.93 | 0.01 | 0.90 | 0.93 | 0.95 | 0.93 |

(E)F4 tornadoes | ||||||

Intercept | −3.62 | 0.37 | −4.38 | −3.60 | −2.93 | −3.57 |

Precision for trend | 24,904.71 | 25,883.49 | 2088.27 | 17,256.64 | 93,210.40 | 5798.00 |

Precision for cycle | 13.30 | 8.86 | 4.14 | 10.85 | 36.69 | 7.75 |

PACF1 for cycle | −0.35 | 0.27 | −0.80 | −0.38 | 0.24 | −0.45 |

PACF2 for cycle | −0.22 | 0.31 | −0.76 | −0.23 | 0.40 | −0.24 |

Log $\tau $ | −0.32 | 0.13 | −0.57 | −0.32 | −0.06 | −0.33 |

Log $\kappa $ | −1.49 | 0.13 | −1.76 | −1.49 | −1.23 | −1.48 |

Group $\mathsf{\Phi}$ | 0.95 | 0.01 | 0.92 | 0.95 | 0.97 | 0.96 |

**Table 3.**Information Criteria. Note: M0-Model with trend, cycle, and time varying spatial random effects. M1-Model with cycle and time varying spatial random effects. M2-Model with trend and time varying spatial random effects. M3-Model with trend, cycle and fixed in time spatial random effects. M4-Model with fixed in time spatial random effects.

Crit. | M0 | M1 | M2 | M3 | M4 | ||
---|---|---|---|---|---|---|---|

(E)F0 | DIC | 40,835.42 | 40,829.03 | 40,830.83 | 48,622.88 | 58,878.41 | |

WAIC | 41,214.03 | 41,223.32 | 41,206.82 | 49,078.74 | 59,306.55 | ||

(E)F1 | DIC | 37,076.27 | 37,075.51 | 37,064.35 | 42,739.80 | 44,224.90 | |

WAIC | 37,413.70 | 37,410.93 | 37,407.28 | 43,043.76 | 44,418.91 | ||

(E)F2 | DIC | 21,976.90 | 21,974.27 | 21,975.14 | 23,968.95 | 25,239.81 | |

WAIC | 22,147.80 | 22,151.29 | 22,146.38 | 24,121.63 | 25,335.83 | ||

(E)F3 | DIC | 9340.40 | 9339.86 | 9341.27 | 9682.43 | 10,047.74 | |

WAIC | 9949.46 | 9951.30 | 10,716.05 | 9744.69 | 10,078.78 | ||

(E)F4 | DIC | 4964.12 | 5057.13 | 14,701.86 | 3302.51 | 3405.22 | |

WAIC | 7284.037 | 7111.25 | 7788.11 | 3224.74 | 3279.26 |

Mean | SD | 0.025Quant | 0.5Quant | 0.975Quant | Mode | |
---|---|---|---|---|---|---|

Intercept | −3.202 | 0.636 | −4.542 | −3.179 | −1.985 | −3.137 |

ZIP parameter | 0.480 | 5.00 × 10${}^{-2}$ | 0.381 | 0.480 | 0.576 | 0.482 |

Precision for trend | 22,484.136 | 2.01 × 10${}^{4}$ | 2716.213 | 16,904.253 | 76,040.634 | 7676.415 |

Precision for cycle | 10.103 | 5.95 × 10 | 3.092 | 8.670 | 25.554 | 6.482 |

PACF1 for cycle | −0.143 | 3.16 × 10${}^{-1}$ | −0.711 | −0.152 | 0.478 | −0.167 |

PACF2 for cycle | −0.047 | 1.85 × 10${}^{-1}$ | −0.417 | −0.042 | 0.301 | −0.017 |

Log $\tau $ | 0.021 | 1.69 × 10${}^{-1}$ | −0.311 | 0.021 | 0.353 | 0.022 |

Log $\kappa $ | −1.787 | 2.05 × 10${}^{-1}$ | −2.198 | −1.785 | −1.390 | −1.775 |

Group $\mathsf{\Phi}$ | 0.983 | 8.00 × 10${}^{-3}$ | 0.964 | 0.985 | 0.994 | 0.987 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Valente, F.; Laurini, M.
Tornado Occurrences in the United States: A Spatio-Temporal Point Process Approach. *Econometrics* **2020**, *8*, 25.
https://doi.org/10.3390/econometrics8020025

**AMA Style**

Valente F, Laurini M.
Tornado Occurrences in the United States: A Spatio-Temporal Point Process Approach. *Econometrics*. 2020; 8(2):25.
https://doi.org/10.3390/econometrics8020025

**Chicago/Turabian Style**

Valente, Fernanda, and Márcio Laurini.
2020. "Tornado Occurrences in the United States: A Spatio-Temporal Point Process Approach" *Econometrics* 8, no. 2: 25.
https://doi.org/10.3390/econometrics8020025