Extreme Event Modelling Liwei Wu Supervisor: Dr. Xiang Extreme Event Modelling Zhou Introduction Theory and Liwei Wu Methods Asymptotic Supervisor: Dr. Xiang Zhou Models Threshold Models Application Simulation of data from normal April 28, 2014 distribution Application in stock market Application in Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Table of Contents Extreme Event Modelling 1 Introduction Liwei Wu Supervisor: Dr. Xiang 2 Theory and Methods Zhou Asymptotic Models Introduction Threshold Models Theory and Methods Asymptotic 3 Application Models Threshold Simulation of data from normal distribution Models Application Application in stock market Simulation of data from Application in Hong Kong climate data normal distribution Application in stock market Application in 4 Summary and Outlook Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Introduction Extreme Event Modelling Extreme Event: event occurring very rarely, such as the 100-year Liwei Wu floor. Supervisor: Dr. Xiang We need the Extreme Event Modelling techniques to know what the Zhou extreme levels can be within a certain period of time. Introduction For example, the past 10 years of data → predict how large the Theory and 100-year flood can be Methods Asymptotic Two Methods available Models Threshold Models Block Maxima Method 1 Application Threshold Method 2 Simulation of data from normal Application to real data distribution Application in stock market Dow Jones Index data 1 Application in Hong Kong Hong Kong climate data climate data 2 Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Extreme Event Modelling The models introduced later focuses on the statistical behavior of Liwei Wu Supervisor: M n = max { X 1 , X 2 , ..., X n } , Dr. Xiang Zhou where X 1 , X 2 , ..., X n , is a sequence of independent random variables Introduction having a common distribution function F . Theory and Methods In Applications, the X i usually represents values of a process Asymptotic measured on a regular time-scale – perhaps hourly measurements of Models Threshold sea-level, or daily mean temperatures Models Application M n represents the maximum of the process over n time units of Simulation of observation. data from normal distribution If n is the number of observations in a year, then M n corresponds to Application in stock market the annual maximum. Application in Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Table of Contents Extreme Event Modelling 1 Introduction Liwei Wu Supervisor: Dr. Xiang 2 Theory and Methods Zhou Asymptotic Models Introduction Threshold Models Theory and Methods Asymptotic 3 Application Models Threshold Simulation of data from normal distribution Models Application Application in stock market Simulation of data from Application in Hong Kong climate data normal distribution Application in stock market Application in 4 Summary and Outlook Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Table of Contents Extreme Event Modelling 1 Introduction Liwei Wu Supervisor: Dr. Xiang 2 Theory and Methods Zhou Asymptotic Models Introduction Threshold Models Theory and Methods Asymptotic 3 Application Models Threshold Simulation of data from normal distribution Models Application Application in stock market Simulation of data from Application in Hong Kong climate data normal distribution Application in stock market Application in 4 Summary and Outlook Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Extremal Types Theorem Extreme Theorem Event Modelling If there exist sequences of constants { a n } and { b n } such that Liwei Wu � M n − b n � Supervisor: ≤ z → G ( z ) as n → ∞ P Dr. Xiang a n Zhou where G is a non-degenerate distribution function, then G belongs to one Introduction of the following families: Theory and Methods I Asymptotic � � z − b �� Models G ( z ) = exp {− exp − } Threshold a Models Application II � 0 Simulation of if z ≤ b ; data from G ( z ) = normal � z − b � − α } exp {− distribution if z > b . a Application in stock market III Application in � z − b Hong Kong � − α } � exp {− if z ≤ b ; climate data G ( z ) = a 0 if z > b . Summary and Outlook for parameters a > 0 , b and, in the case of families II and III, α > 0 . Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Extreme Event Modelling Liwei Wu Supervisor: Dr. Xiang Zhou Collectively, these three classes of distribution are termed the extreme Introduction value distributions, with types I, II and III widely known as the Theory and Gumbel, Fr´ echet and Weibull families respectively. Methods Asymptotic an extreme value analog of the central limit theorem. Models Threshold Models Application Simulation of data from normal distribution Application in stock market Application in Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
The Generalized Extreme Value Distribution Extreme Event Theorem Modelling If there exist sequences of constants { a n > 0 } and { b n } such Liwei Wu Supervisor: that Dr. Xiang Zhou � M n − b n � P ≤ z → G ( z ) as n → ∞ (1) a n Introduction Theory and for a non-degenerate distribution function G, then G is a Methods Asymptotic member of the GEV family Models Threshold Models � �� − 1 /ξ � � � z − µ Application G ( z ) = exp − 1 + ξ Simulation of data from σ normal distribution Application in stock market Application in defined on the set { z : 1 + ξ ( z − µ ) /σ > 0 } , where Hong Kong climate data −∞ < µ < ∞ , σ > 0 and −∞ < ξ < ∞ . Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Modelling extremes using Block Maxima Extreme Event Modelling A series of independent observations X 1 , X 2 ...are blocked into sequences of Liwei Wu observations of length n, for some large value of n , generating a series of Supervisor: Dr. Xiang block maxima, M n , 1 , ..., M n , m , say, to which the GEV distribution can be Zhou fitted. Introduction Return Period Theory and Return Level Methods Asymptotic z p is the return level associated with the return period 1 / p , which means Models Threshold the level z p is expected to be exceeded on average once every 1 / p years. Models Define y p = − log (1 − p ), then we have Application Simulation of � µ − σ data from � 1 − y − ξ � , for ξ � = 0; normal p distribution z p = ξ µ − σ log y p , for ξ = 0 . Application in stock market Application in Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Return Level Plot Extreme Event Modelling Liwei Wu Supervisor: Dr. Xiang The return level plot is the graph in which if Z p is plotted against y p on a Zhou logarithmic scale. Introduction If ξ = 0, the plot is linear. Theory and Methods If ξ < 0, the plot is convex with asymptotic limit as p → 0 at Asymptotic Models µ − σ/ξ . Threshold Models If ξ > 0, the plot is concave and has no finite bound. Application Simulation of data from normal distribution Application in stock market Application in Hong Kong climate data Summary and Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
Block Maxima Method Extreme Blocking the data into blocks of equal length. (Trade-off between bias and variance) 1 Event Fitting the GEV to the set of block maxima Z 1 , ..., Z m . 2 Modelling Obtaining the log-likelihood for the GEV parameters. When ξ � = 0, 3 Liwei Wu Supervisor: m m � z i − µ � z i − µ �� ( − 1 /ξ ) � �� � � � l ( µ, σ, ξ ) = − m log ( σ ) − (1+1 /ξ ) − Dr. Xiang log 1 + ξ 1 + ξ , σ σ Zhou i =1 i =1 (2) provided that Introduction � z i − µ � 1 + ξ > 0 , ∀ i = 1 , .., m (3) Theory and σ Methods Maximization of the equation with respect to the parameter vector ( µ, σ, ξ ) leads to the 4 Asymptotic Models maximum likelihood estimate of the parameters. Threshold 5 Obtaining the maximum likelihood estimate of z p for the 1/p return level. Models 6 Using the delta method to obtain the variance of the maximum likelihood estimate Application Simulation of z p ) ≈ ∇ z T Var ( ˆ p V ∇ z p data from normal distribution Application in � � ∇ z T ∂ z p ∂ z p ∂ z p stock market p = ∂µ ∂σ ∂ξ Application in Hong Kong � − ξ − 1 � 1 − y − ξ σξ − 2 � 1 − y − ξ − σξ − 1 y − ξ � = 1 � � log y p climate data p p p Summary and � � σ, ˆ evaluated at ˆ µ, ˆ ξ . Outlook Liwei WuSupervisor: Dr. Xiang Zhou Extreme Event Modelling
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