This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan

Chu, L-F. (Lan-Fen)

McAleer, M.J. (Michael)

Wang, S-H. (Szu-Hua)

Working paper No. 19/2012This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan

Chu, L-F.

McAleer, M.

Wang, S-H.

UC Research Repository

 1  DEPARTMENT OF ECONOMICS AND FINANCE  COLLEGE OF BUSINESS AND ECONOMICS  UNIVERSITY OF CANTERBURY  CHRISTCHURCH, NEW ZEALAND    Statistical Modeling of Recent Changes in Extreme Rainfall in Taiwan   Lan-Fen Chu Michael McAleer Szu-Hua Wang      WORKING PAPER   No. 19/2012    Department of Economics and Finance College of Business and Economics University of Canterbury Private Bag 4800, Christchurch   New Zealand            2    WORKING PAPER No. 19/2012    Statistical Modeling of Recent Changes in Extreme Rainfall in Taiwan   Lan-Fen Chu1 Michael McAleer2 Szu-Hua Wang3  18 December 2012   Abstract:  This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan.    Keywords: Generalized extreme value, Extreme rainfall, Return level, Statistical modelling.  JEL Classifications: Q54, Q51, Q57 Acknowledgements: For financial support, the first and third authors are most grateful to the Taiwan Climate Change Projection and Information Platform Project (NSC 100-2621-M-492-001), and the second author wishes to acknowledge the Australian Research Council, National Science Council, Taiwan, and the Japan Society for the Promotion of Science.     1. National Science and Technology Center for Disaster Reduction (NCDR). Taiwan 2. Econometric Institute, Erasmus School of Economics, Erasmus University, Rotterdam; Tinbergen Institute, The Netherlands; Institute of Economic Research, Kyoto University, Japan; Department of Quantitative Economics, Complutense University of Madrid, Spain 3. Department of Urban Affairs and Environmental Planning, Chinese Cultural University,Taiwan  *Corresponding Author: michael.mcaleer@gmail.com  3  WORKING PAPER No. 19/2012    Statistical Modeling of Recent Changes in Extreme Rainfall in Taiwan  1. Introduction.  Extreme rainfall is one of the main causes of natural disasters, especially in flood hazards worldwide. Not surprisingly, considerable attention has been paid in recent years to the modelling of extreme rainfall to help prevent flooding hazards, and for analysing water-related structures, agriculture, and monitoring climate changes. Taiwan is a small island in East-Asia. An average of 3.5 typhoons strike Taiwan each year, often in summer and autumn, and cause significant damage, especially in highly concentrated population and property areas. In the past few years, the Kalmaegi and Sonlaku Typhoons in 2008 and Morakot Typhoon in 2009 have brought to Taiwan many casualties and serious injuries. Morakot Typhoon struck Taiwan from 7-9 August 2009 with abundant rainfall, reaching 2,777mm, and surpassing the historical record of Typhoon Herb, which had brought rainfall of 1,736mm ([1];[2]). The extremely heavy rainfall triggered severe flooding (the worst in the past 50 years) and enormous mudslides throughout southern Taiwan, leading to around 700 deaths and roughly NT$110 billion in property damage ([3]). More typhoons seem to have affected Taiwan after 1990 than between 1961-1989 (see [5]), and increased sharply around 2000 (see [6]). Such an indication of increasing frequency and intensity of rainfall means that Taiwan will face a higher probability of huge damages from extreme rainfall in the future. Thus, understanding the patterns of extreme rainfall and their future behaviour is of increasing importance to policy makers in Taiwan. Several published papers have analysed extreme rainfall using the generalized extreme value (GEV) distribution in different parts of the world, including Canada ( [7, 8]); Greece [9]; India [10, 11]; Italy [12, 13, 14]; Malaysia [15]; New Zealand [16]; China [17]; Korea [18, 19], and West Central Florida [20]. These findings highlight the urgency to model extreme rainfall using the GEV distribution. However, there would seem to have been little or no published research that has attempted to detect extreme rainfall by using GEV. Therefore, this paper would seem to be the first application of the GEV distribution for extreme rainfall in Taiwan. We believe that such an analysis can provide a useful reference for climate change in the research of extreme rainfall.    4   2. Data.  The data consist of daily rainfall records for the years 1911-2010, which were provided by the Central Weather Bureau, Taiwan. Table 1 shows a statistical description of the six rainfall stations, including the station number, the longitude, the latitude, and a statistical summary of the data.   3. Methodology. The generalized extreme value (GEV) distribution is based on the Gumbel, Fréchet and the Weibull distributions. It was developed by Jenkinson (1955), who combined the above three distributions (see Hosking et al., 1985, Galambos, 1987). The cumulative distribution function (cdf) of the GEV distribution is given as:   Model GEV0:  1/( ) exp (1 ( ) )F x x        ,      1 ( ) / 0x                                            (1)   where  , 0 and   are the location, scale and shape parameters, respectively. The case of 0  in equation (1) is defined as the Gumbel distribution:   Model Gum0:    )(expexp)(  xxF ,           x                                                           (2)   and the sub-families defined by 0  and 0  correspond to the Fréchet family and the Weibull family, respectively.  The maximum likelihood method was used to estimate (1) and (2) for these data, and the maximization was performed using a quasi-Newton iterative algorithm. Assuming independence of the data, the likelihood function is given as the product of the assumed densities for the observations nxx ,...,, 21 . For the GEV0 model, we have:    )1/1(1)(11),,( niinuxL  niix1/11exp                                      (3)    5  provided that   ,01  ix 1,...,i n   The likelihood ratio test may be used as a standard way of determining the best fitting models. In order to investigate the existence of a deterministic trend in extreme rainfall over time, we consider the following variations of Models GEV0 and Gum01:   Model GEV1: a b   (Year-t0+1),   = constant,     = constant                                                       (4)   a four-parameter model with   allowed to vary linearly with respect to time, and “constant” means that the parameter is not time dependent but is to be estimated:   Model Gum1: a b   (Year-t0+1),    = constant,     = 0,                                                               (5)   After the best models for the data have been determined, the next step is to derive the return levels for rainfall. The T-year return level, Tx  , is the level exceeded on average only once every T years. If Model GEV0  is assumed, then inverting TxF T 11)(   leads to the  expression:                           )11log(1 TxT                                                                                    (6)  If Model Gum0 is assumed, then the corresponding expression is given as:                          )11log(log TxT                                                                                            (7)                                                           1 We follow the same statistical methods as Nadarajah (2005), Feng et al. (2007), and Park et al. (2010). Consequently, some descriptions throughout this paper are the same as the above.    6  On substituting ˆ ,ˆ  and ˆ  into equations (6)-(7), we have the maximum likelihood estimates of the respective return levels.  4. Estimated Results. All the empirical results, including the estimates and their corresponding standard errors (SE), are given in Table 2. It is evident that the Taipei, Taichung, Tainan and Taitung stations have consistent distributions for the two phases. The non-stationary Gumbel distribution is suitable for the Taitung station, while the stationary Gumel distribution is applicable for the Taipei, Taichung, Tainan stations for the two phases. The Hengchun station does not have a consistent distribution for the two periods. However, the Hualien station has a non-stationary GEV distribution for the period 1910-2010, but has a stationary GEV distribution for the period 1960-2010.  The Taipei and Tainan stations have a greater location parameter for the period 1960-2010 than for the period 1910-2010, showing a slight increase in extreme rainfall for the period 1960-2010. Similarly, the Taichung station has a slight decrease in extreme rainfall for the period 1960-2010. It is clear that the empirical results indicate extreme rainfall for the Taitung and Hualien stations have had a manifest increase during the period 1960-2010. Table 3 gives the estimates of the return level, Tx , corresponding to 10, 20, 50, 100 years for the locations, where the best fitting and stationary model for the period 1960-2010 was chosen. The 95% confidence intervals for these return levels are also provided. The empirical results indicate that, given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan2.  Figure 1-3 show the contour maps of the return levels for 10, 20 and 50-years, respectively. These contour maps can assist in determining changes in extreme rainfall in a simple manner.  5. Conclusion. This is the first investigation to use the generalized extreme value to model extreme rainfall in Taiwan. We estimated extreme rainfall using stationary and non-stationary models for both 1910-2010 and 1960-2010 to detect changes in extreme rainfall for these two periods. The empirical results indicated that the stationary Gumbel distribution was suitable for the Taipei, Taichung, and Tainan stations, while the non-stationary Gumbel distribution was suitable for the Taitung station.                                                           2 When the daily rainfall is greater than 50, 130, 200 and 350 mm, the rainfall will accord with the definitions of heavy rainfall, extremely heavy rainfall, torrential rainfall, and extremely torrential rainfall, respectively.  7  Among these stations, the Taipei and Tainan stations have a slight increase in extreme rainfall, while the Taitung station shows a manifest increase in extreme rainfall for the period 1961-2010.  Given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan.    8   References.  [1] X. Ge, T. Li, S. Zhang, and M. Peng. What causes the extremely heavy rainfall in Taiwan during Typhoon Morakot (2009)?. Atmos. Sci. Lett. 2010, 11: 46-50. doi: 10.1002/asl.255.  [2] C.-C. Hong, M.-Y. Lee, H.-H. Hsu, and J.-L. Kuo. Role of submonthly disturbance and 40–50 day ISO on the extreme rainfall event associated with Typhoon Morakot (2009) in southern Taiwan. Geophys. Res. Lett. 2010, 37, L08805, doi:10.1029/2010GL042761.  [3] H.-J. Chu, T.-Y. Pan, and J.-J. Liou. Extreme precipitation estimation with typhoon Morakot using frequency and spatial analysis. Terr. Atmos. Ocean. Sci. 2011, 22(6): 549-558, December 2011. doi: 10.3319/TAO.2011.05.10.02(TM).  [4] H.-H. Chia and C.T. Lee. The long-term variability of typhoon activity. Taiwan Climate Change Conference. 2008. Taipei. (in Chinese).  [5] J.-Y. Tu, C. Chou, and P.-S. Chu. The abrupt shift of typhoon activity in the vicinity of Taiwan and its association with western North Pacific-East Asian climate change. J. Climate.2009, 22: 3617-3628.  [6] V.T.V., Nguyen, T.D., Nguyen and H. Wang. Regional estimation of short duration rainfall extremes. Wat. Sci. Technol. 1998, 37:15-19.  [7] V.T.V., Nguyen, T.D., Nguyen and F. Ashkar. Regional frequency analysis of extreme rainfalls, Wat. Sci. Technol. 2002, 45:75-81.  [8] D. Koutsoyiannis and G. Baloutsos. Analysis of a long record of annual maximum rainfall in Athens, Greece, and design rainfall inferences. Nat. Hazards. 2000, 22:29-48.  [9] V. Ferro. Rainfall intensity–duration–frequency formula for India – discussion; J. Hydr. Eng-ASCE. 1993, 119: 960–962.   [10]  B.P. Parida. Modelling of Indian summer monsoon rainfall using a four-parameter Kappa distribution; Int. J. Climatol. 1999, 19:1389–1398.  [11]  M. Cannarozzo, F. Dasaro, and V. Ferro. Regional rainfall and flood frequency-analysis for Sicily using the 2-component extreme-value distribution. Hydro. Sci. J. 1995, 40:19-42.  [12]  G. Aronica, M. Cannarozzo, and L. Noto. Investigating the changes in extreme rainfall series recorded in an urbanised area. Wat. Sci. Technol. 2002, 45: 49-54.  [13]  A, Crisci, B. Gozzini, F. Meneguzzo, S. Pagliara and G. Maracchi. Extreme rainfall in a changing climate: regional analysis and hydrological implications in Tuscany. Hydrol. Process. 2002, 16:1261-1274.  [14]  M.D. Zalina, M.N.M. Desa, V.T.V. Nguyen, and A.H.M. Kassim. Selecting a probability distribution for extreme rainfall series in Malaysia. Wat. Sci. Technol. 2002, 45: 63–68.  [15]  C.S. Withers and S. Nadarajah. Evidence of trend in return levels for daily rainfall in New Zealand, J. Hydrol. (NZ). 2000, 39(2):155-166.   9  [16]  S. Feng, S. Nadarajah, and Q. Hu. Modeling annual extreme precipitation in China using the generalized extreme value distribution. J. Meteorol. Soc. Jpn. 2007, 85:599–613.  [17]  S. Nadarajah and D. Choi. Maximum daily rainfall in South Korea. J. Earth Syst. Sci. 2007, 116(4): 311-320.  [18]  S. Nadarajah. Extremes of daily rainfall in West Central Florida. Climatic Change, 2005. 69: 325-342.  [19]  J.S. Park, H.-S. Kang, Y.-S., Lee and M.-K. Kim. Changes in extreme daily rainfall in South Korea, Int. J. Climatol. 2010.  [20]  A.F. Jenkinson. The frequency distribution of the annual maximum (or minimum) values of meteorological elements.  Quart. J. Roy. Meteor. Soc. 1955, 81: 158−171.  [21]  J.R.M. Hosking and J. R. Wallis. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 1987, 29(3): 339−349.  [22]  J. Galambos. The asymptotic theory of extreme order statistics, 2nd edn. (Melbourne, Florida: Krieger), 1987.  10   Table 1.  Descriptive statistics of data set (unit: mm).  Location Station No. Longitude Latitude Min Max Media Mean Taipei 46692 121°51' 25°04' 20.6 425.2 145.9 163.5 Taichung 46749 120°68' 24°15' 25.8 660.2 166.0 187.2 Tainan 46741 120°18' 22°20' 12.9 523.5 195.6 205.5 Hengchun 46759 120°74' 22°01' 17.4 484.8 214.8 232.2 Hualien 46699 121°61 23°98" 12.8 465.8 210.3 217.0 Taidung 46766 121°15' 22°76' 7.8 484.0 201.4 245.2   11   Table 2. Best fitting models and parameter estimates for 1910-2010 and 1960-2010.  Location Model  Period  (SE)  (SE)  (SE)  (SE) Taipei Gum0 1910-2010 131.41 (5.55)  52.84  (4.12)  Gum0 1960-2010 133.04 (8.39)  57.06  (6.34)  Taichung Gum0 1910-2010 147.26 (7.09)  67.56 (5.32)  Gum0 1960-2010 146.55 (9.58)  65.16 (7.40)  Tainan Gum0 1910-2010 165.40 (7.77)  73.60 (5.48)  Gum0 1960-2010 167.40 (9.81)  66.41  (7.03)  Hengchun GEV0 1910-2010 196.33 (8.78)  79.6 (6.14) -0.14 (0.06) Gum1 1960-2010 151.71 (18.68) 64.85  (32.23) 69.13  (7.72)  Hualien GEV1 1910-2010 157.29 (15.71) 52.21 (26.99) 76.16 (5.77) -0.15 (0.05) GEV0 1960-2010 197.55 (12.79)  80.99  (9.17)  Taitung Gum1 1910-2010 144.61 (14.66) 67.01  (25.40) 69.15  (5.07)  Gum1 1960-2010 166.67  (16.48) 51.34  (28.18) 61.36 (6.81)    12   Table 3.  Return levels estimates for 10, 20, 50, 100-years (based on 1960-2010).  Location 10-year 20-year 50-year 100-year Taipei 261(143,379) 302(184,420) 355(238,473) 395(277,513) Taichung 293(175,410) 340(222,457) 400(283,518) 446(328,563) Tainan 317(128,505) 365(175,553) 426(238,615) 472(284,661) Hualien 343(312,391) 377(343,442) 415(374,506) 438(391,554)                         Figure 1 10-year return level      Figure 2 20-year return level                Figure 3 50-year return level  

Statistical Modeling of Recent Changes in Extreme Rainfall in Taiwan

Erasmus University Digital Repository

1   Statistical Modelling of Recent Changes in Extreme Rainfall  in Taiwan*   Lan-Fen Chu  National Science and Technology Center for Disaster Reduction (NCDR)  Taiwan   Michael McAleer  Econometric Institute  Erasmus School of Economics  Erasmus University Rotterdam and Tinbergen Institute  The Netherlands and Institute of Economic Research  Kyoto University  Japan and Department of Quantitative Economics  Complutense University of Madrid  Spain   Szu-Hua Wang  Department of Urban Affairs and Environmental Planning Chinese Cultural University Taiwan   EI 2012-36  December 2012    * For financial support, the first and third authors are most grateful to the Taiwan Climate Change Projection and Information Platform Project (NSC 100-2621-M-492-001), and the second author wishes to acknowledge the Australian Research Council, National Science Council, Taiwan, and the Japan Society for the Promotion of Science. 2    Abstract  This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan.    Keywords: Generalized extreme value, Extreme rainfall, Return level, Statistical modelling.  JEL: Q54, Q51, Q57. 3   1. Introduction   Extreme rainfall is one of the main causes of natural disasters, especially in flood hazards worldwide. Not surprisingly, considerable attention has been paid in recent years to the modelling of extreme rainfall to help prevent flooding hazards, and for analysing water-related structures, agriculture, and monitoring climate changes.  Taiwan is a small island in East-Asia. An average of 3.5 typhoons strike Taiwan each year, often in summer and autumn, and cause significant damage, especially in highly concentrated population and property areas. In the past few years, the Kalmaegi and Sonlaku Typhoons in 2008 and Morakot Typhoon in 2009 have brought to Taiwan many casualties and serious injuries. Morakot Typhoon struck Taiwan from 7-9 August 2009 with abundant rainfall, reaching 2,777mm, and surpassing the historical record of Typhoon Herb, which had brought rainfall of 1,736mm ([1];[2]). The extremely heavy rainfall triggered severe flooding (the worst in the past 50 years) and enormous mudslides throughout southern Taiwan, leading to around 700 deaths and roughly NT$110 billion in property damage ([3]).  More typhoons seem to have affected Taiwan after 1990 than between 1961-1989 (see [5]), and increased sharply around 2000 (see [6]). Such an indication of increasing frequency and intensity of rainfall means that Taiwan will face a higher probability of huge damages from extreme rainfall in the future. Thus, understanding the patterns of extreme rainfall and their future behaviour is of increasing importance to policy makers in Taiwan.  Several published papers have analysed extreme rainfall using the generalized extreme value (GEV) distribution in different parts of the world, including Canada ( [7, 8]); Greece [9]; India [10, 11]; Italy [12, 13, 14]; Malaysia [15]; New Zealand [16]; China [17]; Korea [18, 19], and West Central Florida [20]. These findings highlight the urgency to model extreme rainfall using the GEV distribution. However, there would seem to have been little or no published research that has attempted to detect extreme rainfall by using GEV. Therefore, this paper would seem to be the first application of the GEV distribution for extreme rainfall in Taiwan. We believe that such an analysis can provide a useful reference for climate change in the research of extreme rainfall.   4  2. Data   The data consist of daily rainfall records for the years 1911-2010, which were provided by the Central Weather Bureau, Taiwan. Table 1 shows a statistical description of the six rainfall stations, including the station number, the longitude, the latitude, and a statistical summary of the data.    3. Methodology  The generalized extreme value (GEV) distribution is based on the Gumbel, Fréchet and the Weibull distributions. It was developed by Jenkinson (1955), who combined the above three distributions (see Hosking et al., 1985, Galambos, 1987). The cumulative distribution function (cdf) of the GEV distribution is given as:   Model GEV0:  1/( ) exp (1 ( ) )F x x        ,      1 ( ) / 0x                                            (1)   where  , 0 and   are the location, scale and shape parameters, respectively. The case of 0  in equation (1) is defined as the Gumbel distribution:   Model Gum0:    )(expexp)(  xxF ,           x                                                           (2)   and the sub-families defined by 0  and 0  correspond to the Fréchet family and the Weibull family, respectively.   The maximum likelihood method was used to estimate (1) and (2) for these data, and the maximization was performed using a quasi-Newton iterative algorithm. Assuming independence of the data, the likelihood function is given as the product of the assumed densities for the observations nxx ,...,, 21 . For the GEV0 model, we have:   5    )1/1(1)(11),,(  niin uxL    niix1/11exp                                      (3)   provided that   ,01   ix 1,...,i n   The likelihood ratio test may be used as a standard way of determining the best fitting models. In order to investigate the existence of a deterministic trend in extreme rainfall over time, we consider the following variations of Models GEV0 and Gum01:   Model GEV1: a b   (Year-t0+1),   = constant,     = constant                                                       (4)   a four-parameter model with   allowed to vary linearly with respect to time, and “constant” means that the parameter is not time dependent but is to be estimated:   Model Gum1: a b   (Year-t0+1),    = constant,     = 0,                                                               (5)   After the best models for the data have been determined, the next step is to derive the return levels for rainfall. The T-year return level, Tx  , is the level exceeded on average only once every T years. If Model GEV0  is assumed, then inverting TxF T 11)(   leads to the  expression:                             )11log(1 TxT                                                                                     (6)                                                           1 We follow the same statistical methods as Nadarajah (2005), Feng et al. (2007), and Park et al. (2010). Consequently, some descriptions throughout this paper are the same as the above.   6    If Model Gum0 is assumed, then the corresponding expression is given as:                           )11log(log TxT                                                                                            (7)   On substituting ˆ ,ˆ  and ˆ  into equations (6)-(7), we have the maximum likelihood estimates of the respective return levels.   4. Estimated Results  All the empirical results, including the estimates and their corresponding standard errors (SE), are given in Table 2. It is evident that the Taipei, Taichung, Tainan and Taitung stations have consistent distributions for the two phases. The non-stationary Gumbel distribution is suitable for the Taitung station, while the stationary Gumel distribution is applicable for the Taipei, Taichung, Tainan stations for the two phases. The Hengchun station does not have a consistent distribution for the two periods. However, the Hualien station has a non-stationary GEV distribution for the period 1910-2010, but has a stationary GEV distribution for the period 1960-2010.   The Taipei and Tainan stations have a greater location parameter for the period 1960-2010 than for the period 1910-2010, showing a slight increase in extreme rainfall for the period 1960-2010. Similarly, the Taichung station has a slight decrease in extreme rainfall for the period 1960-2010. It is clear that the empirical results indicate extreme rainfall for the Taitung and Hualien stations have had a manifest increase during the period 1960-2010.  Table 3 gives the estimates of the return level, Tx , corresponding to 10, 20, 50, 100 years for the locations, where the best fitting and stationary model for the period 1960-2010 was chosen. The 95% confidence intervals for these return levels are also provided. The empirical results indicate that, given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the 7  Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan2.   Figure 1-3 show the contour maps of the return levels for 10, 20 and 50-years, respectively. These contour maps can assist in determining changes in extreme rainfall in a simple manner.   5. Conclusion  This is the first investigation to use the generalized extreme value to model extreme rainfall in Taiwan. We estimated extreme rainfall using stationary and non-stationary models for both 1910-2010 and 1960-2010 to detect changes in extreme rainfall for these two periods. The empirical results indicated that the stationary Gumbel distribution was suitable for the Taipei, Taichung, and Tainan stations, while the non-stationary Gumbel distribution was suitable for the Taitung station. Among these stations, the Taipei and Tainan stations have a slight increase in extreme rainfall, while the Taitung station shows a manifest increase in extreme rainfall for the period 1961-2010.   Given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan.                                                             2  When the daily rainfall is greater than 50, 130, 200 and 350 mm, the rainfall will accord with the definitions of heavy rainfall, extremely heavy rainfall, torrential rainfall, and extremely torrential rainfall, respectively. 8   References   [1] X. Ge, T. Li, S. Zhang, and M. Peng. What causes the extremely heavy rainfall in Taiwan during Typhoon Morakot (2009)?. Atmos. Sci. Lett. 2010, 11: 46-50. doi: 10.1002/asl.255.  [2] C.-C. Hong, M.-Y. Lee, H.-H. Hsu, and J.-L. Kuo. Role of submonthly disturbance and 40–50 day ISO on the extreme rainfall event associated with Typhoon Morakot (2009) in southern Taiwan. Geophys. Res. Lett. 2010, 37, L08805, doi:10.1029/2010GL042761.  [3] H.-J. Chu, T.-Y. Pan, and J.-J. Liou. Extreme precipitation estimation with typhoon Morakot using frequency and spatial analysis. Terr. Atmos. Ocean. Sci. 2011, 22(6): 549-558, December 2011. doi: 10.3319/TAO.2011.05.10.02(TM).  [4] H.-H. Chia and C.T. Lee. The long-term variability of typhoon activity. Taiwan Climate Change Conference. 2008. Taipei. (in Chinese).  [5] J.-Y. Tu, C. Chou, and P.-S. Chu. The abrupt shift of typhoon activity in the vicinity of Taiwan and its association with western North Pacific-East Asian climate change. J. Climate.2009, 22: 3617-3628.  [6] V.T.V., Nguyen, T.D., Nguyen and H. Wang. Regional estimation of short duration rainfall extremes. Wat. Sci. Technol. 1998, 37:15-19.  [7] V.T.V., Nguyen, T.D., Nguyen and F. Ashkar. Regional frequency analysis of extreme rainfalls, Wat. Sci. Technol. 2002, 45:75-81.  [8] D. Koutsoyiannis and G. Baloutsos. Analysis of a long record of annual maximum rainfall in Athens, Greece, and design rainfall inferences. Nat. Hazards. 2000, 22:29-48.  [9] V. Ferro. Rainfall intensity–duration–frequency formula for India – discussion; J. Hydr. Eng-ASCE. 1993, 119: 960–962.   [10]  B.P. Parida. Modelling of Indian summer monsoon rainfall using a four-parameter Kappa distribution; Int. J. Climatol. 1999, 19:1389–1398.  [11]  M. Cannarozzo, F. Dasaro, and V. Ferro. Regional rainfall and flood frequency-analysis for Sicily using the 2-component extreme-value distribution. Hydro. Sci. J. 1995, 40:19-42.  [12]  G. Aronica, M. Cannarozzo, and L. Noto. Investigating the changes in extreme rainfall series recorded in an urbanised area. Wat. Sci. Technol. 2002, 45: 49-54.  [13]  A, Crisci, B. Gozzini, F. Meneguzzo, S. Pagliara and G. Maracchi. Extreme rainfall in a changing climate: regional analysis and hydrological implications in Tuscany. Hydrol. Process. 2002, 16:1261-1274.  [14]  M.D. Zalina, M.N.M. Desa, V.T.V. Nguyen, and A.H.M. Kassim. Selecting a probability distribution for extreme rainfall series in Malaysia. Wat. Sci. Technol. 2002, 45: 63–68.  9  [15]  C.S. Withers and S. Nadarajah. Evidence of trend in return levels for daily rainfall in New Zealand, J. Hydrol. (NZ). 2000, 39(2):155-166.  [16]  S. Feng, S. Nadarajah, and Q. Hu. Modeling annual extreme precipitation in China using the generalized extreme value distribution. J. Meteorol. Soc. Jpn. 2007, 85:599–613.  [17]  S. Nadarajah and D. Choi. Maximum daily rainfall in South Korea. J. Earth Syst. Sci. 2007, 116(4): 311-320.  [18]  S. Nadarajah. Extremes of daily rainfall in West Central Florida. Climatic Change, 2005. 69: 325-342.  [19]  J.S. Park, H.-S. Kang, Y.-S., Lee and M.-K. Kim. Changes in extreme daily rainfall in South Korea, Int. J. Climatol. 2010.  [20]  A.F. Jenkinson. The frequency distribution of the annual maximum (or minimum) values of meteorological elements.  Quart. J. Roy. Meteor. Soc. 1955, 81: 158−171.  [21]  J.R.M. Hosking and J. R. Wallis. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 1987, 29(3): 339−349.  [22]  J. Galambos. The asymptotic theory of extreme order statistics, 2nd edn. (Melbourne, Florida: Krieger), 1987. 10   Table 1  Descriptive statistics of data set (unit: mm)  Location Station No. Longitude Latitude Min Max Media Mean Taipei 46692 121°51' 25°04' 20.6 425.2 145.9 163.5 Taichung 46749 120°68' 24°15' 25.8 660.2 166.0 187.2 Tainan 46741 120°18' 22°20' 12.9 523.5 195.6 205.5 Hengchun 46759 120°74' 22°01' 17.4 484.8 214.8 232.2 Hualien 46699 121°61 23°98" 12.8 465.8 210.3 217.0 Taidung 46766 121°15' 22°76' 7.8 484.0 201.4 245.2  11   Table 2  Best fitting models and parameter estimates for 1910-2010 and 1960-2010  Location Model  Period  (SE)  (SE)  (SE)  (SE) Taipei Gum0 1910-2010 131.41 (5.55)52.84  (4.12) Gum0 1960-2010 133.04 (8.39)57.06  (6.34) Taichung Gum0 1910-2010 147.26 (7.09)67.56 (5.32) Gum0 1960-2010 146.55 (9.58)65.16 (7.40) Tainan Gum0 1910-2010 165.40 (7.77)73.60 (5.48) Gum0 1960-2010 167.40 (9.81)66.41  (7.03) Hengchun GEV0 1910-2010 196.33 (8.78)79.6 (6.14) -0.14 (0.06)Gum1 1960-2010 151.71 (18.68)64.85 (32.23)69.13  (7.72) Hualien GEV1 1910-2010 157.29 (15.71)52.21 (26.99)76.16 (5.77) -0.15 (0.05)GEV0 1960-2010 197.55 (12.79)80.99  (9.17) Taitung Gum1 1910-2010 144.61 (14.66)67.01 (25.40)69.15  (5.07) Gum1 1960-2010 166.67  (16.48)51.34  (28.18)61.36 (6.81)  12   Table 3  Return levels estimates for 10, 20, 50, 100-years (based on 1960-2010)  Location 10-year 20-year 50-year 100-year Taipei 261(143,379) 302(184,420) 355(238,473) 395(277,513) Taichung 293(175,410) 340(222,457) 400(283,518) 446(328,563) Tainan 317(128,505) 365(175,553) 426(238,615) 472(284,661) Hualien 343(312,391) 377(343,442) 415(374,506) 438(391,554)                         Figure 1 10-year return level      Figure 2 20-year return level                Figure 3 50-year return level  

Statistical Modelling of Recent Changes in Extreme Rainfall in Taiwan

Chu, L. (LanFen)

TI 2013-004/III Tinbergen Institute Discussion Paper  Statistical Modelling of Recent Changes in Extreme Rainfall in Taiwan     Lan-Fen Chu1 Michael McAleer2 Szu-Hua Wang3     1 National Science and Technology Center for Disaster Reduction (NCDR), Taiwan; 2 Erasmus School of Economics, Erasmus University Rotterdam, Econometric and Tinbergen Institutes, Kyoto University, Japan, and Complutense University of Madrid ;  3 Chinese Cultural University, Taiwan.    Tinbergen Institute is the graduate school and research institute in economics of Erasmus University Rotterdam, the University of Amsterdam and VU University Amsterdam.  More TI discussion papers can be downloaded at http://www.tinbergen.nl  Tinbergen  Institute has two locations:  Tinbergen Institute Amsterdam Gustav Mahlerplein 117 1082 MS Amsterdam The Netherlands Tel.: +31(0)20 525 1600  Tinbergen Institute Rotterdam Burg. Oudlaan 50 3062 PA Rotterdam The Netherlands Tel.: +31(0)10 408 8900 Fax: +31(0)10 408 9031  Duisenberg school of finance is a collaboration of the Dutch financial sector and universities, with the ambition to support innovative research and offer top quality academic education in core areas of finance. DSF research papers can be downloaded at: http://www.dsf.nl/  Duisenberg school of finance Gustav Mahlerplein 117 1082 MS Amsterdam The Netherlands Tel.: +31(0)20 525 8579   1   Statistical Modelling of Recent Changes in Extreme Rainfall  in Taiwan*   Lan-Fen Chu  National Science and Technology Center for Disaster Reduction (NCDR)  Taiwan   Michael McAleer  Econometric Institute  Erasmus School of Economics  Erasmus University Rotterdam and Tinbergen Institute  The Netherlands and Institute of Economic Research  Kyoto University  Japan and Department of Quantitative Economics  Complutense University of Madrid  Spain   Szu-Hua Wang  Department of Urban Affairs and Environmental Planning Chinese Cultural University Taiwan     December 2012    * For financial support, the first and third authors are most grateful to the Taiwan Climate Change Projection and Information Platform Project (NSC 100-2621-M-492-001), and the second author wishes to acknowledge the Australian Research Council, National Science Council, Taiwan, and the Japan Society for the Promotion of Science. 2    Abstract  This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan.    Keywords: Generalized extreme value, Extreme rainfall, Return level, Statistical modelling.  JEL: Q54, Q51, Q57. 3   1. Introduction   Extreme rainfall is one of the main causes of natural disasters, especially in flood hazards worldwide. Not surprisingly, considerable attention has been paid in recent years to the modelling of extreme rainfall to help prevent flooding hazards, and for analysing water-related structures, agriculture, and monitoring climate changes.  Taiwan is a small island in East-Asia. An average of 3.5 typhoons strike Taiwan each year, often in summer and autumn, and cause significant damage, especially in highly concentrated population and property areas. In the past few years, the Kalmaegi and Sonlaku Typhoons in 2008 and Morakot Typhoon in 2009 have brought to Taiwan many casualties and serious injuries. Morakot Typhoon struck Taiwan from 7-9 August 2009 with abundant rainfall, reaching 2,777mm, and surpassing the historical record of Typhoon Herb, which had brought rainfall of 1,736mm ([1];[2]). The extremely heavy rainfall triggered severe flooding (the worst in the past 50 years) and enormous mudslides throughout southern Taiwan, leading to around 700 deaths and roughly NT$110 billion in property damage ([3]).  More typhoons seem to have affected Taiwan after 1990 than between 1961-1989 (see [5]), and increased sharply around 2000 (see [6]). Such an indication of increasing frequency and intensity of rainfall means that Taiwan will face a higher probability of huge damages from extreme rainfall in the future. Thus, understanding the patterns of extreme rainfall and their future behaviour is of increasing importance to policy makers in Taiwan.  Several published papers have analysed extreme rainfall using the generalized extreme value (GEV) distribution in different parts of the world, including Canada ( [7, 8]); Greece [9]; India [10, 11]; Italy [12, 13, 14]; Malaysia [15]; New Zealand [16]; China [17]; Korea [18, 19], and West Central Florida [20]. These findings highlight the urgency to model extreme rainfall using the GEV distribution. However, there would seem to have been little or no published research that has attempted to detect extreme rainfall by using GEV. Therefore, this paper would seem to be the first application of the GEV distribution for extreme rainfall in Taiwan. We believe that such an analysis can provide a useful reference for climate change in the research of extreme rainfall.   4  2. Data   The data consist of daily rainfall records for the years 1911-2010, which were provided by the Central Weather Bureau, Taiwan. Table 1 shows a statistical description of the six rainfall stations, including the station number, the longitude, the latitude, and a statistical summary of the data.    3. Methodology  The generalized extreme value (GEV) distribution is based on the Gumbel, Fréchet and the Weibull distributions. It was developed by Jenkinson (1955), who combined the above three distributions (see Hosking et al., 1985, Galambos, 1987). The cumulative distribution function (cdf) of the GEV distribution is given as:   Model GEV0:  1/( ) exp (1 ( ) )F x x        ,      1 ( ) / 0x                                            (1)   where  , 0 and   are the location, scale and shape parameters, respectively. The case of 0  in equation (1) is defined as the Gumbel distribution:   Model Gum0:    )(expexp)(  xxF ,           x                                                           (2)   and the sub-families defined by 0  and 0  correspond to the Fréchet family and the Weibull family, respectively.   The maximum likelihood method was used to estimate (1) and (2) for these data, and the maximization was performed using a quasi-Newton iterative algorithm. Assuming independence of the data, the likelihood function is given as the product of the assumed densities for the observations nxx ,...,, 21 . For the GEV0 model, we have:   5    )1/1(1)(11),,(  niin uxL    niix1/11exp                                      (3)   provided that   ,01   ix 1,...,i n   The likelihood ratio test may be used as a standard way of determining the best fitting models. In order to investigate the existence of a deterministic trend in extreme rainfall over time, we consider the following variations of Models GEV0 and Gum01:   Model GEV1: a b   (Year-t0+1),   = constant,     = constant                                                       (4)   a four-parameter model with   allowed to vary linearly with respect to time, and “constant” means that the parameter is not time dependent but is to be estimated:   Model Gum1: a b   (Year-t0+1),    = constant,     = 0,                                                               (5)   After the best models for the data have been determined, the next step is to derive the return levels for rainfall. The T-year return level, Tx  , is the level exceeded on average only once every T years. If Model GEV0  is assumed, then inverting TxF T 11)(   leads to the  expression:                             )11log(1 TxT                                                                                     (6)                                                           1 We follow the same statistical methods as Nadarajah (2005), Feng et al. (2007), and Park et al. (2010). Consequently, some descriptions throughout this paper are the same as the above.   6    If Model Gum0 is assumed, then the corresponding expression is given as:                           )11log(log TxT                                                                                            (7)   On substituting ˆ ,ˆ  and ˆ  into equations (6)-(7), we have the maximum likelihood estimates of the respective return levels.   4. Estimated Results  All the empirical results, including the estimates and their corresponding standard errors (SE), are given in Table 2. It is evident that the Taipei, Taichung, Tainan and Taitung stations have consistent distributions for the two phases. The non-stationary Gumbel distribution is suitable for the Taitung station, while the stationary Gumel distribution is applicable for the Taipei, Taichung, Tainan stations for the two phases. The Hengchun station does not have a consistent distribution for the two periods. However, the Hualien station has a non-stationary GEV distribution for the period 1910-2010, but has a stationary GEV distribution for the period 1960-2010.   The Taipei and Tainan stations have a greater location parameter for the period 1960-2010 than for the period 1910-2010, showing a slight increase in extreme rainfall for the period 1960-2010. Similarly, the Taichung station has a slight decrease in extreme rainfall for the period 1960-2010. It is clear that the empirical results indicate extreme rainfall for the Taitung and Hualien stations have had a manifest increase during the period 1960-2010.  Table 3 gives the estimates of the return level, Tx , corresponding to 10, 20, 50, 100 years for the locations, where the best fitting and stationary model for the period 1960-2010 was chosen. The 95% confidence intervals for these return levels are also provided. The empirical results indicate that, given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the 7  Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan2.   Figure 1-3 show the contour maps of the return levels for 10, 20 and 50-years, respectively. These contour maps can assist in determining changes in extreme rainfall in a simple manner.   5. Conclusion  This is the first investigation to use the generalized extreme value to model extreme rainfall in Taiwan. We estimated extreme rainfall using stationary and non-stationary models for both 1910-2010 and 1960-2010 to detect changes in extreme rainfall for these two periods. The empirical results indicated that the stationary Gumbel distribution was suitable for the Taipei, Taichung, and Tainan stations, while the non-stationary Gumbel distribution was suitable for the Taitung station. Among these stations, the Taipei and Tainan stations have a slight increase in extreme rainfall, while the Taitung station shows a manifest increase in extreme rainfall for the period 1961-2010.   Given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan.                                                             2  When the daily rainfall is greater than 50, 130, 200 and 350 mm, the rainfall will accord with the definitions of heavy rainfall, extremely heavy rainfall, torrential rainfall, and extremely torrential rainfall, respectively. 8   References   [1] X. Ge, T. Li, S. Zhang, and M. Peng. What causes the extremely heavy rainfall in Taiwan during Typhoon Morakot (2009)?. Atmos. Sci. Lett. 2010, 11: 46-50. doi: 10.1002/asl.255.  [2] C.-C. Hong, M.-Y. Lee, H.-H. Hsu, and J.-L. Kuo. Role of submonthly disturbance and 40–50 day ISO on the extreme rainfall event associated with Typhoon Morakot (2009) in southern Taiwan. Geophys. Res. Lett. 2010, 37, L08805, doi:10.1029/2010GL042761.  [3] H.-J. Chu, T.-Y. Pan, and J.-J. Liou. Extreme precipitation estimation with typhoon Morakot using frequency and spatial analysis. Terr. Atmos. Ocean. Sci. 2011, 22(6): 549-558, December 2011. doi: 10.3319/TAO.2011.05.10.02(TM).  [4] H.-H. Chia and C.T. Lee. The long-term variability of typhoon activity. Taiwan Climate Change Conference. 2008. Taipei. (in Chinese).  [5] J.-Y. Tu, C. Chou, and P.-S. Chu. The abrupt shift of typhoon activity in the vicinity of Taiwan and its association with western North Pacific-East Asian climate change. J. Climate.2009, 22: 3617-3628.  [6] V.T.V., Nguyen, T.D., Nguyen and H. Wang. Regional estimation of short duration rainfall extremes. Wat. Sci. Technol. 1998, 37:15-19.  [7] V.T.V., Nguyen, T.D., Nguyen and F. Ashkar. Regional frequency analysis of extreme rainfalls, Wat. Sci. Technol. 2002, 45:75-81.  [8] D. Koutsoyiannis and G. Baloutsos. Analysis of a long record of annual maximum rainfall in Athens, Greece, and design rainfall inferences. Nat. Hazards. 2000, 22:29-48.  [9] V. Ferro. Rainfall intensity–duration–frequency formula for India – discussion; J. Hydr. Eng-ASCE. 1993, 119: 960–962.   [10]  B.P. Parida. Modelling of Indian summer monsoon rainfall using a four-parameter Kappa distribution; Int. J. Climatol. 1999, 19:1389–1398.  [11]  M. Cannarozzo, F. Dasaro, and V. Ferro. Regional rainfall and flood frequency-analysis for Sicily using the 2-component extreme-value distribution. Hydro. Sci. J. 1995, 40:19-42.  [12]  G. Aronica, M. Cannarozzo, and L. Noto. Investigating the changes in extreme rainfall series recorded in an urbanised area. Wat. Sci. Technol. 2002, 45: 49-54.  [13]  A, Crisci, B. Gozzini, F. Meneguzzo, S. Pagliara and G. Maracchi. Extreme rainfall in a changing climate: regional analysis and hydrological implications in Tuscany. Hydrol. Process. 2002, 16:1261-1274.  [14]  M.D. Zalina, M.N.M. Desa, V.T.V. Nguyen, and A.H.M. Kassim. Selecting a probability distribution for extreme rainfall series in Malaysia. Wat. Sci. Technol. 2002, 45: 63–68.  9  [15]  C.S. Withers and S. Nadarajah. Evidence of trend in return levels for daily rainfall in New Zealand, J. Hydrol. (NZ). 2000, 39(2):155-166.  [16]  S. Feng, S. Nadarajah, and Q. Hu. Modeling annual extreme precipitation in China using the generalized extreme value distribution. J. Meteorol. Soc. Jpn. 2007, 85:599–613.  [17]  S. Nadarajah and D. Choi. Maximum daily rainfall in South Korea. J. Earth Syst. Sci. 2007, 116(4): 311-320.  [18]  S. Nadarajah. Extremes of daily rainfall in West Central Florida. Climatic Change, 2005. 69: 325-342.  [19]  J.S. Park, H.-S. Kang, Y.-S., Lee and M.-K. Kim. Changes in extreme daily rainfall in South Korea, Int. J. Climatol. 2010.  [20]  A.F. Jenkinson. The frequency distribution of the annual maximum (or minimum) values of meteorological elements.  Quart. J. Roy. Meteor. Soc. 1955, 81: 158−171.  [21]  J.R.M. Hosking and J. R. Wallis. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 1987, 29(3): 339−349.  [22]  J. Galambos. The asymptotic theory of extreme order statistics, 2nd edn. (Melbourne, Florida: Krieger), 1987. 10   Table 1  Descriptive statistics of data set (unit: mm)  Location Station No. Longitude Latitude Min Max Media Mean Taipei 46692 121°51' 25°04' 20.6 425.2 145.9 163.5 Taichung 46749 120°68' 24°15' 25.8 660.2 166.0 187.2 Tainan 46741 120°18' 22°20' 12.9 523.5 195.6 205.5 Hengchun 46759 120°74' 22°01' 17.4 484.8 214.8 232.2 Hualien 46699 121°61 23°98" 12.8 465.8 210.3 217.0 Taidung 46766 121°15' 22°76' 7.8 484.0 201.4 245.2  11   Table 2  Best fitting models and parameter estimates for 1910-2010 and 1960-2010  Location Model  Period  (SE)  (SE)  (SE)  (SE) Taipei Gum0 1910-2010 131.41 (5.55)52.84  (4.12) Gum0 1960-2010 133.04 (8.39)57.06  (6.34) Taichung Gum0 1910-2010 147.26 (7.09)67.56 (5.32) Gum0 1960-2010 146.55 (9.58)65.16 (7.40) Tainan Gum0 1910-2010 165.40 (7.77)73.60 (5.48) Gum0 1960-2010 167.40 (9.81)66.41  (7.03) Hengchun GEV0 1910-2010 196.33 (8.78)79.6 (6.14) -0.14 (0.06)Gum1 1960-2010 151.71 (18.68)64.85 (32.23)69.13  (7.72) Hualien GEV1 1910-2010 157.29 (15.71)52.21 (26.99)76.16 (5.77) -0.15 (0.05)GEV0 1960-2010 197.55 (12.79)80.99  (9.17) Taitung Gum1 1910-2010 144.61 (14.66)67.01 (25.40)69.15  (5.07) Gum1 1960-2010 166.67  (16.48)51.34  (28.18)61.36 (6.81)  12   Table 3  Return levels estimates for 10, 20, 50, 100-years (based on 1960-2010)  Location 10-year 20-year 50-year 100-year Taipei 261(143,379) 302(184,420) 355(238,473) 395(277,513) Taichung 293(175,410) 340(222,457) 400(283,518) 446(328,563) Tainan 317(128,505) 365(175,553) 426(238,615) 472(284,661) Hualien 343(312,391) 377(343,442) 415(374,506) 438(391,554)                         Figure 1 10-year return level      Figure 2 20-year return level                Figure 3 50-year return level  

Chu, LF

McAleer, Michael

Wang, S-H

EUR Research Repository

Statistical modelling of recent changes in extreme rainfall

JEL: Q54, Q51, Q57.This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan.Taiwan Climate Change Projection and Information Platform ProjectAustralian Research CouncilNational Science Council, TaiwanJapan Society for the Promotion of ScienceFac. de Ciencias Económicas y EmpresarialesInstituto Complutense de Análisis Económico (ICAE)FALSEunpu

Chu, Lan-Fen

Wang, Szu-Hua

Docta Complutense

 1   Statistical Modelling of Recent Changes in Extreme Rainfall  in Taiwan*   Lan-Fen Chu  National Science and Technology Center for Disaster Reduction (NCDR)  Taiwan   Michael McAleer  Econometric Institute  Erasmus School of Economics  Erasmus University Rotterdam and Tinbergen Institute  The Netherlands and Institute of Economic Research  Kyoto University  Japan and Department of Quantitative Economics  Complutense University of Madrid  Spain   Szu-Hua Wang  Department of Urban Affairs and Environmental Planning Chinese Cultural University Taiwan     December 2012    * For financial support, the first and third authors are most grateful to the Taiwan Climate Change Projection and Information Platform Project (NSC 100-2621-M-492-001), and the second author wishes to acknowledge the Australian Research Council, National Science Council, Taiwan, and the Japan Society for the Promotion of Science.  2    Abstract  This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95% confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan.    Keywords: Generalized extreme value, Extreme rainfall, Return level, Statistical modelling.  JEL: Q54, Q51, Q57.  3   1. Introduction   Extreme rainfall is one of the main causes of natural disasters, especially in flood hazards worldwide. Not surprisingly, considerable attention has been paid in recent years to the modelling of extreme rainfall to help prevent flooding hazards, and for analysing water-related structures, agriculture, and monitoring climate changes.  Taiwan is a small island in East-Asia. An average of 3.5 typhoons strike Taiwan each year, often in summer and autumn, and cause significant damage, especially in highly concentrated population and property areas. In the past few years, the Kalmaegi and Sonlaku Typhoons in 2008 and Morakot Typhoon in 2009 have brought to Taiwan many casualties and serious injuries. Morakot Typhoon struck Taiwan from 7-9 August 2009 with abundant rainfall, reaching 2,777mm, and surpassing the historical record of Typhoon Herb, which had brought rainfall of 1,736mm ([1];[2]). The extremely heavy rainfall triggered severe flooding (the worst in the past 50 years) and enormous mudslides throughout southern Taiwan, leading to around 700 deaths and roughly NT$110 billion in property damage ([3]).  More typhoons seem to have affected Taiwan after 1990 than between 1961-1989 (see [5]), and increased sharply around 2000 (see [6]). Such an indication of increasing frequency and intensity of rainfall means that Taiwan will face a higher probability of huge damages from extreme rainfall in the future. Thus, understanding the patterns of extreme rainfall and their future behaviour is of increasing importance to policy makers in Taiwan.  Several published papers have analysed extreme rainfall using the generalized extreme value (GEV) distribution in different parts of the world, including Canada ( [7, 8]); Greece [9]; India [10, 11]; Italy [12, 13, 14]; Malaysia [15]; New Zealand [16]; China [17]; Korea [18, 19], and West Central Florida [20]. These findings highlight the urgency to model extreme rainfall using the GEV distribution. However, there would seem to have been little or no published research that has attempted to detect extreme rainfall by using GEV. Therefore, this paper would seem to be the first application of the GEV distribution for extreme rainfall in Taiwan. We believe that such an analysis can provide a useful reference for climate change in the research of extreme rainfall.    4  2. Data   The data consist of daily rainfall records for the years 1911-2010, which were provided by the Central Weather Bureau, Taiwan. Table 1 shows a statistical description of the six rainfall stations, including the station number, the longitude, the latitude, and a statistical summary of the data.    3. Methodology  The generalized extreme value (GEV) distribution is based on the Gumbel, Fréchet and the Weibull distributions. It was developed by Jenkinson (1955), who combined the above three distributions (see Hosking et al., 1985, Galambos, 1987). The cumulative distribution function (cdf) of the GEV distribution is given as:   Model GEV0:  1/( ) exp (1 ( ) )F x x        ,      1 ( ) / 0x                                            (1)   where  , 0 and   are the location, scale and shape parameters, respectively. The case of 0  in equation (1) is defined as the Gumbel distribution:   Model Gum0:    )(expexp)(  xxF ,           x                                                           (2)   and the sub-families defined by 0  and 0  correspond to the Fréchet family and the Weibull family, respectively.   The maximum likelihood method was used to estimate (1) and (2) for these data, and the maximization was performed using a quasi-Newton iterative algorithm. Assuming independence of the data, the likelihood function is given as the product of the assumed densities for the observations nxx ,...,, 21 . For the GEV0 model, we have:    5   )1/1(1)(11),,( niinuxL  niix1/11exp                                      (3)   provided that   ,01  ix 1,...,i n   The likelihood ratio test may be used as a standard way of determining the best fitting models. In order to investigate the existence of a deterministic trend in extreme rainfall over time, we consider the following variations of Models GEV0 and Gum01:   Model GEV1: a b   (Year-t0+1),   = constant,     = constant                                                       (4)   a four-parameter model with   allowed to vary linearly with respect to time, and “constant” means that the parameter is not time dependent but is to be estimated:   Model Gum1: a b   (Year-t0+1),    = constant,     = 0,                                                               (5)   After the best models for the data have been determined, the next step is to derive the return levels for rainfall. The T-year return level, Tx  , is the level exceeded on average only once every T years. If Model GEV0  is assumed, then inverting TxF T 11)(   leads to the  expression:                            )11log(1 TxT                                                                                    (6)                                                           1 We follow the same statistical methods as Nadarajah (2005), Feng et al. (2007), and Park et al. (2010). Consequently, some descriptions throughout this paper are the same as the above.    6    If Model Gum0 is assumed, then the corresponding expression is given as:                           )11log(log TxT                                                                                            (7)   On substituting ̂ ,̂  and ̂  into equations (6)-(7), we have the maximum likelihood estimates of the respective return levels.   4. Estimated Results  All the empirical results, including the estimates and their corresponding standard errors (SE), are given in Table 2. It is evident that the Taipei, Taichung, Tainan and Taitung stations have consistent distributions for the two phases. The non-stationary Gumbel distribution is suitable for the Taitung station, while the stationary Gumel distribution is applicable for the Taipei, Taichung, Tainan stations for the two phases. The Hengchun station does not have a consistent distribution for the two periods. However, the Hualien station has a non-stationary GEV distribution for the period 1910-2010, but has a stationary GEV distribution for the period 1960-2010.   The Taipei and Tainan stations have a greater location parameter for the period 1960-2010 than for the period 1910-2010, showing a slight increase in extreme rainfall for the period 1960-2010. Similarly, the Taichung station has a slight decrease in extreme rainfall for the period 1960-2010. It is clear that the empirical results indicate extreme rainfall for the Taitung and Hualien stations have had a manifest increase during the period 1960-2010.  Table 3 gives the estimates of the return level, Tx , corresponding to 10, 20, 50, 100 years for the locations, where the best fitting and stationary model for the period 1960-2010 was chosen. The 95% confidence intervals for these return levels are also provided. The empirical results indicate that, given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the  7  Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan2.   Figure 1-3 show the contour maps of the return levels for 10, 20 and 50-years, respectively. These contour maps can assist in determining changes in extreme rainfall in a simple manner.   5. Conclusion  This is the first investigation to use the generalized extreme value to model extreme rainfall in Taiwan. We estimated extreme rainfall using stationary and non-stationary models for both 1910-2010 and 1960-2010 to detect changes in extreme rainfall for these two periods. The empirical results indicated that the stationary Gumbel distribution was suitable for the Taipei, Taichung, and Tainan stations, while the non-stationary Gumbel distribution was suitable for the Taitung station. Among these stations, the Taipei and Tainan stations have a slight increase in extreme rainfall, while the Taitung station shows a manifest increase in extreme rainfall for the period 1961-2010.   Given 50-year return level (for the year 2060), the return levels of daily extreme rainfall of the Taipei, Taichung, Tainan and Hualien stations were all greater than 350mm, reaching a warning line of extremely torrential rainfall, as defined by the CWB in Taiwan.                                                             2  When the daily rainfall is greater than 50, 130, 200 and 350 mm, the rainfall will accord with the definitions of heavy rainfall, extremely heavy rainfall, torrential rainfall, and extremely torrential rainfall, respectively.  8   References   [1] X. Ge, T. Li, S. Zhang, and M. Peng. What causes the extremely heavy rainfall in Taiwan during Typhoon Morakot (2009)?. Atmos. Sci. Lett. 2010, 11: 46-50. doi: 10.1002/asl.255.  [2] C.-C. Hong, M.-Y. Lee, H.-H. Hsu, and J.-L. Kuo. Role of submonthly disturbance and 40–50 day ISO on the extreme rainfall event associated with Typhoon Morakot (2009) in southern Taiwan. Geophys. Res. Lett. 2010, 37, L08805, doi:10.1029/2010GL042761.  [3] H.-J. Chu, T.-Y. Pan, and J.-J. Liou. Extreme precipitation estimation with typhoon Morakot using frequency and spatial analysis. Terr. Atmos. Ocean. Sci. 2011, 22(6): 549-558, December 2011. doi: 10.3319/TAO.2011.05.10.02(TM).  [4] H.-H. Chia and C.T. Lee. The long-term variability of typhoon activity. Taiwan Climate Change Conference. 2008. Taipei. (in Chinese).  [5] J.-Y. Tu, C. Chou, and P.-S. Chu. The abrupt shift of typhoon activity in the vicinity of Taiwan and its association with western North Pacific-East Asian climate change. J. Climate.2009, 22: 3617-3628.  [6] V.T.V., Nguyen, T.D., Nguyen and H. Wang. Regional estimation of short duration rainfall extremes. Wat. Sci. Technol. 1998, 37:15-19.  [7] V.T.V., Nguyen, T.D., Nguyen and F. Ashkar. Regional frequency analysis of extreme rainfalls, Wat. Sci. Technol. 2002, 45:75-81.  [8] D. Koutsoyiannis and G. Baloutsos. Analysis of a long record of annual maximum rainfall in Athens, Greece, and design rainfall inferences. Nat. Hazards. 2000, 22:29-48.  [9] V. Ferro. Rainfall intensity–duration–frequency formula for India – discussion; J. Hydr. Eng-ASCE. 1993, 119: 960–962.   [10]  B.P. Parida. Modelling of Indian summer monsoon rainfall using a four-parameter Kappa distribution; Int. J. Climatol. 1999, 19:1389–1398.  [11]  M. Cannarozzo, F. Dasaro, and V. Ferro. Regional rainfall and flood frequency-analysis for Sicily using the 2-component extreme-value distribution. Hydro. Sci. J. 1995, 40:19-42.  [12]  G. Aronica, M. Cannarozzo, and L. Noto. Investigating the changes in extreme rainfall series recorded in an urbanised area. Wat. Sci. Technol. 2002, 45: 49-54.  [13]  A, Crisci, B. Gozzini, F. Meneguzzo, S. Pagliara and G. Maracchi. Extreme rainfall in a changing climate: regional analysis and hydrological implications in Tuscany. Hydrol. Process. 2002, 16:1261-1274.  [14]  M.D. Zalina, M.N.M. Desa, V.T.V. Nguyen, and A.H.M. Kassim. Selecting a probability distribution for extreme rainfall series in Malaysia. Wat. Sci. Technol. 2002, 45: 63–68.   9  [15]  C.S. Withers and S. Nadarajah. Evidence of trend in return levels for daily rainfall in New Zealand, J. Hydrol. (NZ). 2000, 39(2):155-166.  [16]  S. Feng, S. Nadarajah, and Q. Hu. Modeling annual extreme precipitation in China using the generalized extreme value distribution. J. Meteorol. Soc. Jpn. 2007, 85:599–613.  [17]  S. Nadarajah and D. Choi. Maximum daily rainfall in South Korea. J. Earth Syst. Sci. 2007, 116(4): 311-320.  [18]  S. Nadarajah. Extremes of daily rainfall in West Central Florida. Climatic Change, 2005. 69: 325-342.  [19]  J.S. Park, H.-S. Kang, Y.-S., Lee and M.-K. Kim. Changes in extreme daily rainfall in South Korea, Int. J. Climatol. 2010.  [20]  A.F. Jenkinson. The frequency distribution of the annual maximum (or minimum) values of meteorological elements.  Quart. J. Roy. Meteor. Soc. 1955, 81: 158−171.  [21]  J.R.M. Hosking and J. R. Wallis. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 1987, 29(3): 339−349.  [22]  J. Galambos. The asymptotic theory of extreme order statistics, 2nd edn. (Melbourne, Florida: Krieger), 1987.  10   Table 1  Descriptive statistics of data set (unit: mm)  Location Station No. Longitude Latitude Min Max Media Mean Taipei 46692 121°51' 25°04' 20.6 425.2 145.9 163.5 Taichung 46749 120°68' 24°15' 25.8 660.2 166.0 187.2 Tainan 46741 120°18' 22°20' 12.9 523.5 195.6 205.5 Hengchun 46759 120°74' 22°01' 17.4 484.8 214.8 232.2 Hualien 46699 121°61 23°98" 12.8 465.8 210.3 217.0 Taidung 46766 121°15' 22°76' 7.8 484.0 201.4 245.2   11   Table 2  Best fitting models and parameter estimates for 1910-2010 and 1960-2010  Location Model  Period  (SE)  (SE)  (SE)  (SE) Taipei Gum0 1910-2010 131.41 (5.55)  52.84  (4.12)  Gum0 1960-2010 133.04 (8.39)  57.06  (6.34)  Taichung Gum0 1910-2010 147.26 (7.09)  67.56 (5.32)  Gum0 1960-2010 146.55 (9.58)  65.16 (7.40)  Tainan Gum0 1910-2010 165.40 (7.77)  73.60 (5.48)  Gum0 1960-2010 167.40 (9.81)  66.41  (7.03)  Hengchun GEV0 1910-2010 196.33 (8.78)  79.6 (6.14) -0.14 (0.06) Gum1 1960-2010 151.71 (18.68) 64.85  (32.23) 69.13  (7.72)  Hualien GEV1 1910-2010 157.29 (15.71) 52.21 (26.99) 76.16 (5.77) -0.15 (0.05) GEV0 1960-2010 197.55 (12.79)  80.99  (9.17)  Taitung Gum1 1910-2010 144.61 (14.66) 67.01  (25.40) 69.15  (5.07)  Gum1 1960-2010 166.67  (16.48) 51.34  (28.18) 61.36 (6.81)    12   Table 3  Return levels estimates for 10, 20, 50, 100-years (based on 1960-2010)  Location 10-year 20-year 50-year 100-year Taipei 261(143,379) 302(184,420) 355(238,473) 395(277,513) Taichung 293(175,410) 340(222,457) 400(283,518) 446(328,563) Tainan 317(128,505) 365(175,553) 426(238,615) 472(284,661) Hualien 343(312,391) 377(343,442) 415(374,506) 438(391,554)                         Figure 1 10-year return level      Figure 2 20-year return level                Figure 3 50-year return level  

English

Statistical Modelling of Recent Changes in Extreme Rainfall
in Taiwan

This paper has two primary purposes. First, we fit the annual maximum daily rainfall data for 6 rainfall stations, both with stationary and non-stationary generalized extreme value (GEV) distributions for the periods 1911-2010 and 1960-2010 in Taiwan, and detect the changes between the two phases for extreme rainfall. The non-stationary model means that the location parameter in the GEV distribution is a linear function of time to detect temporal trends in maximum rainfall. Second, we compute the future behavior of stationary models for the return levels of 10, 20, 50 and 100-years based on the period 1960-2010. In addition, the 95 % confidence intervals of the return levels are provided. This is the first investigation to use generalized extreme value distributions to model extreme rainfall in Taiwan

Statistical Modelling of Recent Changes in Extreme Rainfall in Taiwan

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