本論文研究了全球暖化與台灣極端降雨之間的關係，探討各地區日極端降雨發生的頻率
及規模隨溫度上升的變化趨勢。本論文所分析的資料集包含國科會 (MOST) 主導的 TCCIP計畫所提供的台灣降雨數據及由 NOAA 主導的 NCEI 資料平台所提供的北半球平均相對溫度。本論文採用台灣全區 5×5 km2 網格在 1960-2020 年間之每年日最大降雨量資料進行資料分析，分析方法則採用 PGEV 模型 (Olafsdottir, et al., 2021) 來擬合極端值，並假設模型中的頻率參數與強度參數皆為地點與相對溫度的函數。分析結果顯示，隨著全球溫度上升，台灣近乎全區域的極端降雨強度都增強，而在極端降雨的頻率上，南臺灣的極端降雨頻率明顯增強。但不論是極端降雨強度或頻率受溫度上升的影響皆隨地點而有差異，在北部和南部地區，極端降雨的頻率逐年顯著改變，而在中部和東部地區，極端降雨的強度則有所改變。在設想未來溫度可能上升 0.5 度-3 度的四種情境下，本論文利用重現水準 (return level) 分析未來極端降雨的發生情況，結果顯示: 即使溫度只上升 1 度，極端降雨量事件的頻繁度將會倍增。

Global warming is a critical issue for the Earth's ecosystem and environment. Many studies have found evidence that global warming has an impact on more frequent and severe climate conditions. This study investigates the effects of global warming on extreme precipitation in Taiwan. In this study, two data sources are used for extreme value analysis. The Taiwan rainfall data are obtained from the TCCIP project sponsored by MOST. The global temperature anomaly data (North Hemisphere Reference Temperature) is obtained from NCEI sponsored by NOAA. This thesis analyses the annual maximum of daily precipitation using a PGEV model (Olafsdottir et al., 2021). In particular, both the intensity and frequency parameters are assumed to have a regression form of the temperature anomaly, and the regression coefficients are spatially varying. The model parameters are estimated by maximum likelihood for each spatial location, and the estimation variances are evaluated by bootstrap methods. Based on the PGEV fits, this thesis found evidence that: (1) the intensity of extreme rainfall becomes more severe with increasing temperature in the Taiwan area, especially in the southern and central regions. (2) The frequency of extreme precipitation also becomes higher as the temperature increases in the Taiwan area, especially in the southern and northern regions. To quantify the impacts more specifically, the return level analysis is further performed under 4 hypothetical scenarios of temperature rise ranging from $0.5^\circ$ C to $3^\circ$ C. It was found that even if the global temperature rise is only $1^\circ$ C, the chances of seeing the events of the current 20-year return level in Taiwan would be doubled.

Contents
摘要 i
Abstract ii
List of Figures v
List of Tables vi
1 Introduction 1
2 Data 4
2.1 Rainfall Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Temperature Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Models for Extreme Value Analysis 10
3.1 Generalized Extreme Value Models (GEV) . . . . . . . . . . . . . . . . . . . . . 10
3.2 Peak Over Threshold Models (PoT) . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 PGEV Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Temperature Effects on Intensity of Rainfall . . . . . . . . . . . . . . . . . . . . 13
3.4.1 Relative Frequency and Scale Changes . . . . . . . . . . . . . . . . . . . 13
3.4.2 Return Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Estimation 16
4.1 Determining Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Maximum Likelihood Estimation for PGEV . . . . . . . . . . . . . . . . . . . . 16
4.3 Hypothesis Testing and Model Selection . . . . . . . . . . . . . . . . . . . . . . 17
5 Application 20
5.1 GEV Parameters and Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.2 PGEV Fitting and Model Comparison . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 Inference on Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3.1 Inference Under P GEVσ Model . . . . . . . . . . . . . . . . . . . . . . . 24
5.3.2 Inference Under P GEVλ Model . . . . . . . . . . . . . . . . . . . . . . . 29
5.4 Hypothesis Testing on Models with Covariate . . . . . . . . . . . . . . . . . . . 32
6 Discussion 36
References 38

楊承道，林士堯 (2021). 網格化觀測雨量日資料生產履歷 (4.0 版). 取自臺灣氣候變遷推估資訊與調適知識平台.
Akaike, H. (1974). A new look at the statistical model identification. IEEE transactions on automatic control, 19(6):716–723.
Blanchet, J. and Davison, A. C. (2011). Spatial modeling of extreme snow depth. The Annals of Applied Statistics, pages 1699–1725.
Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American statistical association, 74(368):829–836.
Coles, S., Bawa, J., Trenner, L., and Dorazio, P. (2001). An introduction to statistical modeling of extreme values, volume 208. Springer.
Cotter, J. and Dowd, K. (2006). Extreme spectral risk measures: an application to futures clearinghouse margin requirements. Journal of Banking & Finance, 30(12):3469–3485.
Davison, A. C. and Huser, R. (2015). Statistics of extremes. Annual Review of Statistics and its Application, 2:203–235.
Davison, A. C., Padoan, S. A., and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statistical science, 27(2):161–186.
Gkillas, K. and Katsiampa, P. (2018). An application of extreme value theory to cryptocurrencies. Economics Letters, 164:109–111.
Henningsen, A. and Toomet, O. (2011). maxlik: A package for maximum likelihood estimation in r. Computational Statistics, 26(3):443–458. 38
Henny, L., Thorncroft, C. D., Hsu, H.-H., and Bosart, L. F. (2021). Extreme rainfall in taiwan: Seasonal statistics and trends. Journal of Climate, 34(12):4711–4731.
Huser, R. and Wadsworth, J. L. (2022). Advances in statistical modeling of spatial extremes. Wiley Interdisciplinary Reviews: Computational Statistics, 14(1):e1537.
IPCC (2021). Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, volume In Press. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.
Nolde, N. and Zhou, C. (2021). Extreme value analysis for financial risk management. Annual Review of Statistics and Its Application, 8:217–240.
Nortey, E. N., Asare, K., and Mettle, F. O. (2015). Extreme value modelling of ghana stock exchange index. SpringerPlus, 4(1):1–17.
Olafsdottir, H. K., Rootzén, H., and Bolin, D. (2021). Extreme rainfall events in the northeastern united states become more frequent with rising temperatures, but their intensity distribution remains stable. Journal of Climate, 34(22):8863–8877.
Sheraz, M., Nasir, I., and Dedu, S. (2021). Extreme value analysis and risk assessment: A case of pakistan stock market. Economic Computation & Economic Cybernetics Studies & Research, 55(3).
Sun, Y. and Genton, M. G. (2011). Functional boxplots. Journal of Computational and Graphical Statistics, 20(2):316–334.
Tung, Y.-S., Cheng-Ta, C., Seung-Ki, M., and Lee-Yaw, L. (2016). Evaluating extreme rainfall changes over taiwan using a standardized index. TAO: Terrestrial, Atmospheric and Oceanic Sciences, 27(5):7.