Bayesian Predictive Analysis of Natural Disaster Losses
Abstract
:1. Introduction
2. Natural Losses in the US
2.1. The Data
2.2. The i.i.d. Assumption for Loss Severity
2.3. Non-Parametric Distribution of Loss Severity
3. Composite Models
3.1. Three Composite Distributions
3.2. Model Selection for Loss Severity
- 1.
- Sort the sample of the natural disaster damage losses in an increasing order, i.e., , where n is the sample size. Let be the size of the partial sample of the first losses . Start from .
- 2.
- Compute the maximum likelihood estimates and as in Table 3 for the given . If is in between , we found ; otherwise, increase by 1.
- 3.
- Repeat Step 2 for till . The ML estimates of the parameters are found based on the correct .
4. The Bayesian Estimate
4.1. Bayesian Estimator of LN-Pareto
- Sort the sample of size n in increasing order, i.e., and let be the size of the partial sample of the first losses . Start from .
- Compute the Bayes estimate via (8) for the given .
- Compute the conditional Bayes estimate of via (7), given from Step 2. If , then we found . otherwise, increase by 1.
- Repeat Step 2 and 3 for till , and we found the correct .
4.2. Validation by Simulation
4.3. Bayesian Estimates of Three Composite Models
5. Risk Measures
Value at Risk and Tailed Value at Risk
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Number and Loss Amounts of Natural Events from 1900 to 2016
1900 to 1980 | 1980 to 2016 | 1900 to 2016 | % of All Events | |
---|---|---|---|---|
Drought | 11 | 4.26% | ||
Earthquake | 32 | 12.40% | ||
Epidemic | 4 | 1.55% | ||
Extreme temperature | 31 | 12.02% | ||
Flood | 52 | 20.16% | ||
Landslide | 5 | 1.94% | ||
Storm | 92 | 35.66% | ||
Volcanic activity | 1 | 0.39% | ||
Wildfire | 30 | 11.63% | ||
Total | 258 | 100.00% |
1900 to 1979 | 1980 to 2016 | 1900 to 2016 | % of All Damages | |
---|---|---|---|---|
Drought | 4371.623471 | 3.62% | ||
Earthquake | 7975.435381 | 6.60% | ||
Epidemic | 0 | 0.00% | ||
Extreme temperature | 3776.856375 | 3.13% | ||
Flood | 13,772.64483 | 11.40% | ||
Landslide | 2.027634158 | 0.00% | ||
Storm | 87,642.85441 | 72.54% | ||
Volcanic activity | 250.4927427 | 0.21% | ||
Wildfire | 3035.724612 | 2.51% | ||
Total | 120,827.6595 | 100.00% |
Appendix B. Damage Losses from Natural Events from 1980 to 2016
Year | # of Losses | Loss | Loss | Loss | Loss | Loss | … | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
1980 | 6 | 1019.45 | 87.38 | 58.25 | 2504.93 | 5825.41 | 2504.93 | ||||
1981 | 2 | 1056.14 | 1217.20 | ||||||||
1982 | 8 | 572.04 | 2487.12 | 84.31 | 99.48 | 248.71 | 104.46 | 497.42 | 1243.56 | ||
1983 | 9 | 7229.13 | 2409.71 | 74.70 | 15.06 | 1265.10 | 722.91 | 240.97 | 313.26 | 36.15 | |
1984 | 10 | 1683.05 | 80.85 | 2309.98 | 69.30 | 46.20 | 39.27 | 80.85 | 265.65 | 69.30 | 1385.99 |
1985 | 10 | 2453.60 | 2007.49 | 3345.82 | 26.99 | 758.39 | 1784.44 | 446.11 | 22.31 | 516.59 | 0.89 |
1986 | 5 | 1.58 | 3832.23 | 87.59 | 65.70 | 54.75 | |||||
1987 | 7 | 450.01 | 8.45 | 242.96 | 33.80 | 122.54 | 21.13 | 10.56 | |||
1988 | 0 | ||||||||||
1989 | 4 | 13,548.78 | 10,839.03 | 735.51 | 967.77 | ||||||
1990 | 6 | 23.32 | 73.45 | 183.63 | 918.16 | 64.27 | 82.63 | ||||
1991 | 9 | 59.03 | 2643.25 | 52.86 | 4405.41 | 52.86 | 1762.17 | 1497.84 | 1762.17 | 590.33 | |
1992 | 8 | 128.30 | 171.07 | 145.41 | 8553.35 | 5132.01 | 153.96 | 171.07 | 45,332.75 | ||
1993 | 7 | 315.58 | 207.62 | 166.09 | 8304.74 | 19,931.38 | 1660.95 | 12.46 | |||
1994 | 7 | 161.95 | 3.24 | 404.87 | 48,584.41 | 1133.64 | 809.74 | 3.40 | |||
1995 | 12 | 196.86 | 3307.18 | 4724.55 | 1330.75 | 1102.39 | 3149.70 | 4724.55 | 157.48 | 15.75 | 15.75 |
3149.70 | 4724.55 | ||||||||||
1996 | 6 | 1070.78 | 5200.92 | 13.00 | 2294.52 | 764.84 | 30.59 | ||||
1997 | 17 | 269.17 | 3.74 | 2.99 | 366.37 | 74.77 | 373.84 | 224.31 | 299.07 | 747.69 | 149.54 |
224.31 | 299.07 | 89.72 | 747.69 | 747.69 | 2243.06 | 7476.85 | |||||
1998 | 25 | 2061.41 | 2945.61 | 1472.44 | 406.39 | 690.57 | 6294.66 | 220.87 | 147.24 | 88.35 | 6.63 |
73.62 | 0.88 | 92.03 | 1472.44 | 2.94 | 1774.21 | 544.80 | 663.33 | 92.03 | 295.22 | ||
2208.65 | 736.22 | 397.56 | 92.03 | 295.22 | |||||||
1999 | 18 | 648.28 | 144.06 | 3976.11 | 288.84 | 1440.62 | 216.09 | 132.54 | 100.84 | 10,084.33 | 288.84 |
144.06 | 1440.62 | 10.08 | 90.04 | 288.12 | 0.43 | 1584.68 | 432.19 | ||||
2000 | 16 | 627.20 | 292.69 | 2090.65 | 139.38 | 39.72 | 11.29 | 1393.77 | 231.37 | 69.69 | 125.44 |
305.24 | 13.94 | 27.88 | 487.82 | 1533.15 | 696.88 | ||||||
2001 | 12 | 338.80 | 31.17 | 2.44 | 8131.24 | 17.62 | 27.10 | 13.55 | 40.66 | 5.42 | 9.49 |
4.07 | 2710.41 | ||||||||||
2002 | 16 | 533.65 | 267.49 | 6.67 | 17.34 | 5.34 | 2935.05 | 26.68 | 267.49 | 1334.11 | 26.68 |
400.23 | 933.88 | 2668.23 | 601.02 | 8.81 | 4402.57 | ||||||
2003 | 13 | 6521.93 | 32.61 | 5217.54 | 138.26 | 4395.78 | 260.88 | 521.75 | 22.17 | 65.22 | 4565.35 |
4.43 | 2739.21 | 260.88 | |||||||||
2004 | 14 | 381.17 | 5.72 | 1397.61 | 889.39 | 76.23 | 0.22 | 20,328.81 | 79.41 | 13,976.06 | 22,869.91 |
10,164.40 | 2.67 | 1.27 | 635.28 | ||||||||
2005 | 11 | 307.23 | 245.78 | 36.87 | 430.12 | 6.55 | 430.12 | 2740.48 | 19,662.63 | 17,573.48 | 301.08 |
122.89 | |||||||||||
2006 | 20 | 1428.61 | 714.31 | 1904.82 | 308.34 | 14.29 | 535.73 | 101.19 | 1190.51 | 8.33 | 19.05 |
119.05 | 18.45 | 39.12 | 29.76 | 113.10 | 357.15 | 29.76 | 178.58 | 107.15 | 428.58 | ||
2007 | 15 | 32.41 | 578.77 | 810.28 | 150.48 | 2893.85 | 364.63 | 1157.54 | 578.77 | 162.06 | 405.14 |
2315.08 | 347.26 | 347.26 | 694.52 | 347.26 | |||||||
2008 | 16 | 501.63 | 2.23 | 780.32 | 1783.58 | 113.70 | 1337.69 | 200.65 | 33,442.22 | 1449.16 | 2229.48 |
668.84 | 1114.74 | 1226.21 | 11,147.41 | 122.62 | 401.31 | ||||||
2009 | 12 | 185.71 | 1901.83 | 111.87 | 268.49 | 2796.80 | 559.36 | 1230.59 | 671.23 | 2237.44 | 1118.72 |
950.91 | 1678.08 | ||||||||||
2010 | 7 | 2586.57 | 2971.80 | 13.76 | 2201.33 | 110.07 | 1651.00 | 550.33 | |||
2011 | 14 | 2133.97 | 195.26 | 11,736.86 | 14,937.82 | 2027.28 | 7789.01 | 1066.99 | 3200.96 | 800.24 | 3734.45 |
4908.14 | 213.40 | 2133.97 | 8535.90 | ||||||||
2012 | 22 | 182.94 | 1620.30 | 1881.64 | 219.52 | 181.89 | 627.21 | 2090.71 | 52,267.70 | 2.09 | 52.27 |
4181.42 | 209.07 | 522.68 | 5226.77 | 4704.09 | 104.54 | 3554.20 | 1463.50 | 1986.17 | 731.75 | ||
219.52 | 20,907.08 | ||||||||||
2013 | 26 | 1648.42 | 1133.29 | 309.08 | 3193.82 | 309.08 | 2163.55 | 22.05 | 515.13 | 927.24 | 2.06 |
25.76 | 25.76 | 2.06 | 334.84 | 180.30 | 309.08 | 1957.50 | 10.30 | 1339.34 | 2.06 | ||
206.05 | 103.03 | 2266.58 | 103.03 | 103.03 | 1133.29 | ||||||
2014 | 19 | 2.03 | 2027.63 | 101.38 | 3953.89 | 273.73 | 66.91 | 1622.11 | 709.67 | 172.35 | 101.38 |
253.45 | 91.24 | 212.90 | 253.45 | 1622.11 | 760.36 | 2534.54 | 2230.40 | 20.28 | |||
2015 | 28 | 172.14 | 506.31 | 1417.66 | 961.98 | 1012.62 | 162.02 | 1417.66 | 2734.06 | 658.20 | 101.26 |
81.01 | 2.03 | 708.83 | 101.26 | 961.98 | 151.89 | 1417.66 | 2.03 | 1721.45 | 101.26 | ||
273.41 | 141.77 | 911.35 | 607.57 | 405.05 | 151.89 | 3037.85 | 1822.71 | ||||
2016 | 25 | 550.00 | 125.00 | 3900.00 | 2000.00 | 2400.00 | 1000.00 | 1100.00 | 300.00 | 1000.00 | 150.00 |
50.00 | 10,000.00 | 100.00 | 600.00 | 550.00 | 10,000.00 | 1200.00 | 275.00 | 20.00 | 1200.00 | ||
100.00 | 2300.00 | 1600.00 | 1200.00 | 2300.00 |
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1 | |
2 | |
3 | The CPI data was download from Bureau of Labor Statistics https://data.bls.gov/pdq/SurveyOutputServlet. |
Pairs | Kendall Tau Statistics (p-Value) | Spearman Statistics (p-Value) |
---|---|---|
The Composite pdf | |||
---|---|---|---|
Exp-Pareto | , | ||
IG-Pareto | , | where , , | |
LN-Pareto | where |
Composite Model | ML Estimates of Parameters |
---|---|
Exp-Pareto | , where . |
IG-Pareto | , where , , . |
LN-Pareto | If , , , where ; otherwise, , , where , . |
Model | ML Estimates of Parameters | |||
---|---|---|---|---|
Exp-Pareto | 2698.02 | 5398.05 | 5402.19 | |
IG-Pareto | 2719.35 | 5440.7 | 5444.83 | |
LN-Pareto | 2327.74 | 4659.49 | 4667.76 | |
Exponential Exp() | ||||
Inverse Gamma | 2813.29 | 5630.58 | 5638.85 | |
IG( ) | ||||
Lognormal | 3058.66 | 6121.31 | 6129.58 | |
LN( ) |
() | |||||||||
n | |||||||||
20 | 7.8604 | 4.0640 | 0.4747 | 0.1611 | 5.6933 | 2.0959 | 0.4641 | 0.0402 | |
50 | 7.8942 | 3.8006 | 0.4435 | 0.1121 | 5.7941 | 1.2265 | 0.4338 | 0.0688 | |
100 | 7.6421 | 3.6415 | 0.4333 | 0.1056 | 5.8745 | 1.1874 | 0.4058 | 0.0960 | |
() | |||||||||
n | |||||||||
20 | 21.8966 | 12.2540 | 0.5878 | 0.2626 | 20.5062 | 5.5390 | 0.5341 | 0.0345 | |
50 | 19.9977 | 4.9775 | 0.5738 | 0.2431 | 20.5262 | 3.3892 | 0.5025 | 0.0065 | |
100 | 19.5730 | 3.6691 | 0.5467 | 0.1830 | 21.2182 | 3.0585 | 0.4629 | 0.0379 | |
() | |||||||||
n | |||||||||
20 | 5.342 | 1.2443 | 1.510 | 0.3811 | 5.076 | 0.3877 | 1.480 | 0.02047 | |
50 | 5.041 | 0.5606 | 1.528 | 0.2181 | 5.038 | 0.2526 | 1.451 | 0.0492 | |
100 | 5.035 | 0.2344 | 1.516 | 0.1389 | 5.111 | 0.2196 | 1.406 | 0.0941 | |
( ) | |||||||||
n | |||||||||
20 | 20.698 | 3.3162 | 1.560 | 0.4157 | 20.556 | 1.8798 | 1.460 | 0.04030 | |
50 | 20.763 | 3.0164 | 1.468 | 0.2479 | 20.460 | 1.0797 | 1.404 | 0.09601 | |
100 | 20.218 | 2.6237 | 1.466 | 0.2839 | 20.637 | 0.9563 | 1.324 | 0.1762 |
Model | Prior Distributions | Bayesian Estimates | |||
---|---|---|---|---|---|
Exp-Pareto | Inverse-Gamma(10, 5) | 2697.57 | 5397.13 | 5401.27 | |
IG-Pareto | Gamma(50, 1) | 2699.17 | 5400.34 | 5404.48 | |
LN-Pareto | LN(1.61352, 2.857) Gamma(20,500, 1.1 × 10) | 2327.63 | 4659.27 | 4667.54 |
Models | |||
---|---|---|---|
Bayesian Estimation | Exp-Pareto | 1073 | 30,723 |
IG-Pareto | 58,212 | 98,041 | |
LN-Pareto | 11,070 | 75,860 | |
ML Estimation | Exp-Pareto | 1185 | 31,770 |
IG-Pareto | 38,029 | 94,619 | |
LN-Pareto | 11,963 | 76,946 |
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Deng, M.; Aminzadeh, M.; Ji, M. Bayesian Predictive Analysis of Natural Disaster Losses. Risks 2021, 9, 12. https://doi.org/10.3390/risks9010012
Deng M, Aminzadeh M, Ji M. Bayesian Predictive Analysis of Natural Disaster Losses. Risks. 2021; 9(1):12. https://doi.org/10.3390/risks9010012
Chicago/Turabian StyleDeng, Min, Mostafa Aminzadeh, and Min Ji. 2021. "Bayesian Predictive Analysis of Natural Disaster Losses" Risks 9, no. 1: 12. https://doi.org/10.3390/risks9010012