Emma Simpson. 6 September 2013

Size: px
Start display at page:

Download "Emma Simpson. 6 September 2013"

Transcription

1 6 September 2013 Test

2 What is? Beijing during periods of low and high air pollution Air pollution is composed of sulphur oxides, nitrogen oxides, carbon monoxide and particulates. Particulates are small particles of solid or liquid material in the air. PM 2.5 and PM 10 are particulates that are smaller than 2.5 and 10 micrometres respectively. Test

3 Measuring The US Embassy and Chinese Government release hourly PM 2.5 readings for Beijing. Some people believe that there is a discrepancy between the two sources of data. Test

4 Measuring The US Embassy and Chinese Government release hourly PM 2.5 readings for Beijing. Some people believe that there is a discrepancy between the two sources of data. Both use the same formula to calculate the PM 2.5 index I from the concentration C: I = I high I low C high C low (C C low ) + I low, (1) but the breakpoints are different for the US AQI and Chinese API. Test

5 Air Quality Index & Index US breakpoints breakpoints C low C high I low I high C low C high I low I high Table: PM 2.5 breakpoints for the US AQI and Chinese API. I = I high I low C high C low (C C low ) + I low Test

6 Air Quality Index & Index Plot of AQI/API vs Concentration Concentration US Test AQI/API

7 Our data consists of: six months of hourly PM 2.5 readings for Beijing from the US and Chinese sources; twelve years of daily PM 10 readings for Beijing, Tianjin, Shanghai and Suzhou from the Chinese Government. Test

8 There are two methods for deciding which data points are extreme: API Block Maxima Index API Threshold Exceedances Index Test

9 There are two methods for deciding which data points are extreme: API Block Maxima Index Threshold Exceedances Separate the data into blocks and take the maximum value in each block; API Index Test

10 There are two methods for deciding which data points are extreme: API Block Maxima Index Threshold Exceedances Separate the data into blocks and take the maximum value in each block; 2. Choose a suitable threshold above which points are considered extreme. API Index Test

11 Generalised Pareto Distribution (GPD) For the hourly PM 2.5 data, we first took the daily maxima and then applied a threshold to determine the extremes. A GPD distribution could then be fitted to the data. Test

12 Generalised Pareto Distribution (GPD) For the hourly PM 2.5 data, we first took the daily maxima and then applied a threshold to determine the extremes. A GPD distribution could then be fitted to the data. The GPD has distribution functions of the form: { 1 ( 1 + H(y) = ξy ) 1/ξ σ if ξ 0 1 exp( y σ ) if ξ = 0, for y > 0, and subject to the constraint (1 + ξy σ ) > 0. Test

13 Testing the Reliability of the US/Chinese Data We want to test whether there is a difference between the US AQI and Chinese API data. Since the US and Chinese data are measured on different scales, it cannot be compared directly. Instead, we fit GPD models to the US and Chinese data sets separately and compared the threshold exceedance probabilities. Then we used a bootstrapping technique to test for differences. Test

14 Suppose we have data x 1,...,x n, and a model fitted to this data with parameters θ. works as follows: 1. Resample (with replacement) from these n observations, obtaining another sample also of length n. 2. Fit the model to the resampled data to get a new set of parameters θ Repeat the process of resampling and fitting the model N times, obtaining new parameters θ i each time, for i = 1,..., N. 4. These θ 1,...,θ N, then allow us to make inferences about the parameter θ. 5. Block bootstrapping involves taking blocks of the original data when resampling rather than individual data points. Test

15 Result of the Test The block bootstrapping procedure was applied to the probabilities that the PM 2.5 concentrations exceed the 500 threshold, with: blocks of seven days; 1000 iterations. 95% confidence intervals were found for US and Chinese bootstrapped probabilities. If the confidence intervals overlap, there is no significant difference between the sets of data. Test

16 Result of the Test The block bootstrapping procedure was applied to the probabilities that the PM 2.5 concentrations exceed the 500 threshold, with: blocks of seven days; 1000 iterations. 95% confidence intervals were found for US and Chinese bootstrapped probabilities. If the confidence intervals overlap, there is no significant difference between the sets of data. The confidence intervals were: US: ( , ) : ( , ). Test

17 Result of the Test The block bootstrapping procedure was applied to the probabilities that the PM 2.5 concentrations exceed the 500 threshold, with: blocks of seven days; 1000 iterations. 95% confidence intervals were found for US and Chinese bootstrapped probabilities. If the confidence intervals overlap, there is no significant difference between the sets of data. The confidence intervals were: US: ( , ) : ( , ). The confidence intervals overlap, suggesting there is no significant difference in the two data sets. Test

18 Result of the Test The boxplots of the bootstrapped probabilities are also very similar. Test

19 Result of the Test The boxplots of the bootstrapped probabilities are also very similar US Figure: Boxplot of the bootstrapped probabilities Test

20 Result of the Test The boxplots of the bootstrapped probabilities are also very similar US Figure: Boxplot of the bootstrapped probabilities This reiterates that there is no significant difference between the data from the US and. Test

21 Asymptotic Dependence It is interesting to investigate whether high API/AQI levels in one city correlate with high readings elsewhere. Test

22 Asymptotic Dependence It is interesting to investigate whether high API/AQI levels in one city correlate with high readings elsewhere. Two sets of data, X 1 and X 2, are: asymptotically dependent if lim Pr(X 1 > u X 2 > u) = α > 0; u asymptotically independent if lim Pr(X 1 > u X 2 > u) = 0. u Test

23 Modelling Bivariate Extremes The data, X 1 and X 2, first needs to be transformed to unit Fréchet random variables, Y 1 and Y 2, using a Probability Integral Transform. Test

24 Modelling Bivariate Extremes The data, X 1 and X 2, first needs to be transformed to unit Fréchet random variables, Y 1 and Y 2, using a Probability Integral Transform. Then the model is as follows: Pr(Y 1 > y, Y 2 > y) c(y)y 1/η, for y u, (2) where u is the threshold of interest, c is a slowly varying function of y, and η (0, 1]. Test

25 Modelling Bivariate Extremes The data, X 1 and X 2, first needs to be transformed to unit Fréchet random variables, Y 1 and Y 2, using a Probability Integral Transform. Then the model is as follows: Pr(Y 1 > y, Y 2 > y) c(y)y 1/η, for y u, (2) where u is the threshold of interest, c is a slowly varying function of y, and η (0, 1]. The parameter η can be used as a measure of asymptotic dependence: If η = 1, there is asymptotic dependence; if 0 < η < 1, there is asymptotic independence. Test

26 Comparison Between Beijing and Shanghai Initially, the asymptotic dependence of the PM 10 levels in Beijing and Shanghai was tested. Test

27 Comparison Between Beijing and Shanghai Initially, the asymptotic dependence of the PM 10 levels in Beijing and Shanghai was tested. The η value was , which relates to asymptotic independence. Applying block bootstrapping gave a 95% confidence interval of ( , ) for the η values. This confidence interval does not contain 1, suggesting that the PM 10 levels in Beijing and Shanghai are asymptotically independent. It is possible that the distance between Beijing and Shanghai is causing the asymptotic independence. Test

28 Time Series for Shanghai and Suzhou Time Series Plot of Shanghai API API Time Time Series Plot of Suzhou API Test API Time PM 10 levels are known to vary between seasons, so we focus on just the summer data for Shanghai and Suzhou.

29 Asymptotic Dependence: Plot of Suzhou and Shanghai Summer APIs Shanghai Suzhou The correlation between all the data is approximately There is some positive linear correlation between the PM 10 levels in Shanghai and Suzhou. Test

30 Asymptotic Dependence: Plot of Suzhou and Shanghai Summer APIs Shanghai Test Suzhou

31 Asymptotic Dependence: Plot of Suzhou and Shanghai Summer APIs Shanghai Test Suzhou

32 Asymptotic Dependence: Plot of Suzhou and Shanghai Summer APIs Shanghai Test Suzhou

33 Asymptotic Dependence: The results for the bootstrapping of the η values were as follows: Eta Bootstrapped Eta Values Test

34 Asymptotic Dependence: The results for the bootstrapping of the η values were as follows: Eta Bootstrapped Eta Values The 95% confidence interval for the η values was ( , ). Test

35 Asymptotic Dependence: The results for the bootstrapping of the η values were as follows: Eta Bootstrapped Eta Values The 95% confidence interval for the η values was ( , ). This suggests there is asymptotic independence between the air pollution levels in Shanghai and Suzhou. Test

36 The correlation coefficient of 0.82 shows that overall, there is a positive linear relationship between the PM 10 data from Shanghai and Suzhou. Test

37 The correlation coefficient of 0.82 shows that overall, there is a positive linear relationship between the PM 10 data from Shanghai and Suzhou. The bootstrapping test revealed that there is no asymptotic dependence between the two sets of data. Test

38 The correlation coefficient of 0.82 shows that overall, there is a positive linear relationship between the PM 10 data from Shanghai and Suzhou. The bootstrapping test revealed that there is no asymptotic dependence between the two sets of data. We can conclude that there are underlying factors that affect the pollution levels of cities in the same region, but that different factors contribute to the extreme air pollution levels in individual cities. Test

39 Coles, S. (2001) An to Statistical Modelling of Extreme Values, Springer, Ledford, A.W. and Tawn, J.A. (1996) Modelling Dependence within Joint Tail Regions, Journal of the Royal Statistical Society, Hill, B.M. (1975) A Simple General Approach to Inference About the Tail of a Distribution The Annals of Statistics, Test

40 Any Questions? Test

A Conditional Approach to Modeling Multivariate Extremes

A Conditional Approach to Modeling Multivariate Extremes A Approach to ing Multivariate Extremes By Heffernan & Tawn Department of Statistics Purdue University s April 30, 2014 Outline s s Multivariate Extremes s A central aim of multivariate extremes is trying

More information

Overview of Extreme Value Theory. Dr. Sawsan Hilal space

Overview of Extreme Value Theory. Dr. Sawsan Hilal space Overview of Extreme Value Theory Dr. Sawsan Hilal space Maths Department - University of Bahrain space November 2010 Outline Part-1: Univariate Extremes Motivation Threshold Exceedances Part-2: Bivariate

More information

Assessing Dependence in Extreme Values

Assessing Dependence in Extreme Values 02/09/2016 1 Motivation Motivation 2 Comparison 3 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations 4 Idea Results Motivation Given historical flood levels, how high should

More information

Financial Econometrics and Volatility Models Extreme Value Theory

Financial Econometrics and Volatility Models Extreme Value Theory Financial Econometrics and Volatility Models Extreme Value Theory Eric Zivot May 3, 2010 1 Lecture Outline Modeling Maxima and Worst Cases The Generalized Extreme Value Distribution Modeling Extremes Over

More information

Bayesian Point Process Modeling for Extreme Value Analysis, with an Application to Systemic Risk Assessment in Correlated Financial Markets

Bayesian Point Process Modeling for Extreme Value Analysis, with an Application to Systemic Risk Assessment in Correlated Financial Markets Bayesian Point Process Modeling for Extreme Value Analysis, with an Application to Systemic Risk Assessment in Correlated Financial Markets Athanasios Kottas Department of Applied Mathematics and Statistics,

More information

New Classes of Multivariate Survival Functions

New Classes of Multivariate Survival Functions Xiao Qin 2 Richard L. Smith 2 Ruoen Ren School of Economics and Management Beihang University Beijing, China 2 Department of Statistics and Operations Research University of North Carolina Chapel Hill,

More information

APPLICATION OF EXTREMAL THEORY TO THE PRECIPITATION SERIES IN NORTHERN MORAVIA

APPLICATION OF EXTREMAL THEORY TO THE PRECIPITATION SERIES IN NORTHERN MORAVIA APPLICATION OF EXTREMAL THEORY TO THE PRECIPITATION SERIES IN NORTHERN MORAVIA DANIELA JARUŠKOVÁ Department of Mathematics, Czech Technical University, Prague; jarus@mat.fsv.cvut.cz 1. Introduction The

More information

Bayesian Modelling of Extreme Rainfall Data

Bayesian Modelling of Extreme Rainfall Data Bayesian Modelling of Extreme Rainfall Data Elizabeth Smith A thesis submitted for the degree of Doctor of Philosophy at the University of Newcastle upon Tyne September 2005 UNIVERSITY OF NEWCASTLE Bayesian

More information

Investigation of an Automated Approach to Threshold Selection for Generalized Pareto

Investigation of an Automated Approach to Threshold Selection for Generalized Pareto Investigation of an Automated Approach to Threshold Selection for Generalized Pareto Kate R. Saunders Supervisors: Peter Taylor & David Karoly University of Melbourne April 8, 2015 Outline 1 Extreme Value

More information

A conditional approach for multivariate extreme values

A conditional approach for multivariate extreme values J. R. Statist. Soc. B (2004) 66, Part 3, pp. 497 546 A conditional approach for multivariate extreme values Janet E. Heffernan and Jonathan A. Tawn Lancaster University, UK [Read before The Royal Statistical

More information

HIERARCHICAL MODELS IN EXTREME VALUE THEORY

HIERARCHICAL MODELS IN EXTREME VALUE THEORY HIERARCHICAL MODELS IN EXTREME VALUE THEORY Richard L. Smith Department of Statistics and Operations Research, University of North Carolina, Chapel Hill and Statistical and Applied Mathematical Sciences

More information

Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC

Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC EXTREME VALUE THEORY Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu AMS Committee on Probability and Statistics

More information

Statistics for extreme & sparse data

Statistics for extreme & sparse data Statistics for extreme & sparse data University of Bath December 6, 2018 Plan 1 2 3 4 5 6 The Problem Climate Change = Bad! 4 key problems Volcanic eruptions/catastrophic event prediction. Windstorms

More information

STATISTICAL MODELS FOR QUANTIFYING THE SPATIAL DISTRIBUTION OF SEASONALLY DERIVED OZONE STANDARDS

STATISTICAL MODELS FOR QUANTIFYING THE SPATIAL DISTRIBUTION OF SEASONALLY DERIVED OZONE STANDARDS STATISTICAL MODELS FOR QUANTIFYING THE SPATIAL DISTRIBUTION OF SEASONALLY DERIVED OZONE STANDARDS Eric Gilleland Douglas Nychka Geophysical Statistics Project National Center for Atmospheric Research Supported

More information

MFM Practitioner Module: Quantitiative Risk Management. John Dodson. October 14, 2015

MFM Practitioner Module: Quantitiative Risk Management. John Dodson. October 14, 2015 MFM Practitioner Module: Quantitiative Risk Management October 14, 2015 The n-block maxima 1 is a random variable defined as M n max (X 1,..., X n ) for i.i.d. random variables X i with distribution function

More information

ESTIMATING BIVARIATE TAIL

ESTIMATING BIVARIATE TAIL Elena DI BERNARDINO b joint work with Clémentine PRIEUR a and Véronique MAUME-DESCHAMPS b a LJK, Université Joseph Fourier, Grenoble 1 b Laboratoire SAF, ISFA, Université Lyon 1 Framework Goal: estimating

More information

MULTIVARIATE EXTREMES AND RISK

MULTIVARIATE EXTREMES AND RISK MULTIVARIATE EXTREMES AND RISK Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu Interface 2008 RISK: Reality Durham,

More information

Extreme Precipitation: An Application Modeling N-Year Return Levels at the Station Level

Extreme Precipitation: An Application Modeling N-Year Return Levels at the Station Level Extreme Precipitation: An Application Modeling N-Year Return Levels at the Station Level Presented by: Elizabeth Shamseldin Joint work with: Richard Smith, Doug Nychka, Steve Sain, Dan Cooley Statistics

More information

Peaks-Over-Threshold Modelling of Environmental Data

Peaks-Over-Threshold Modelling of Environmental Data U.U.D.M. Project Report 2014:33 Peaks-Over-Threshold Modelling of Environmental Data Esther Bommier Examensarbete i matematik, 30 hp Handledare och examinator: Jesper Rydén September 2014 Department of

More information

Bayesian Inference for Clustered Extremes

Bayesian Inference for Clustered Extremes Newcastle University, Newcastle-upon-Tyne, U.K. lee.fawcett@ncl.ac.uk 20th TIES Conference: Bologna, Italy, July 2009 Structure of this talk 1. Motivation and background 2. Review of existing methods Limitations/difficulties

More information

CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS

CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS EVA IV, CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS Jose Olmo Department of Economics City University, London (joint work with Jesús Gonzalo, Universidad Carlos III de Madrid) 4th Conference

More information

Generalized additive modelling of hydrological sample extremes

Generalized additive modelling of hydrological sample extremes Generalized additive modelling of hydrological sample extremes Valérie Chavez-Demoulin 1 Joint work with A.C. Davison (EPFL) and Marius Hofert (ETHZ) 1 Faculty of Business and Economics, University of

More information

Extreme Value Analysis and Spatial Extremes

Extreme Value Analysis and Spatial Extremes Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models

More information

Overview of Extreme Value Analysis (EVA)

Overview of Extreme Value Analysis (EVA) Overview of Extreme Value Analysis (EVA) Brian Reich North Carolina State University July 26, 2016 Rossbypalooza Chicago, IL Brian Reich Overview of Extreme Value Analysis (EVA) 1 / 24 Importance of extremes

More information

EVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH. Rick Katz

EVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH. Rick Katz 1 EVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu Home page: www.isse.ucar.edu/staff/katz/

More information

Sharp statistical tools Statistics for extremes

Sharp statistical tools Statistics for extremes Sharp statistical tools Statistics for extremes Georg Lindgren Lund University October 18, 2012 SARMA Background Motivation We want to predict outside the range of observations Sums, averages and proportions

More information

Zwiers FW and Kharin VV Changes in the extremes of the climate simulated by CCC GCM2 under CO 2 doubling. J. Climate 11:

Zwiers FW and Kharin VV Changes in the extremes of the climate simulated by CCC GCM2 under CO 2 doubling. J. Climate 11: Statistical Analysis of EXTREMES in GEOPHYSICS Zwiers FW and Kharin VV. 1998. Changes in the extremes of the climate simulated by CCC GCM2 under CO 2 doubling. J. Climate 11:2200 2222. http://www.ral.ucar.edu/staff/ericg/readinggroup.html

More information

Modelação de valores extremos e sua importância na

Modelação de valores extremos e sua importância na Modelação de valores extremos e sua importância na segurança e saúde Margarida Brito Departamento de Matemática FCUP (FCUP) Valores Extremos - DemSSO 1 / 12 Motivation Consider the following events Occurance

More information

RISK AND EXTREMES: ASSESSING THE PROBABILITIES OF VERY RARE EVENTS

RISK AND EXTREMES: ASSESSING THE PROBABILITIES OF VERY RARE EVENTS RISK AND EXTREMES: ASSESSING THE PROBABILITIES OF VERY RARE EVENTS Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu

More information

Models and estimation.

Models and estimation. Bivariate generalized Pareto distribution practice: Models and estimation. Eötvös Loránd University, Budapest, Hungary 7 June 2011, ASMDA Conference, Rome, Italy Problem How can we properly estimate the

More information

Estimating Bivariate Tail: a copula based approach

Estimating Bivariate Tail: a copula based approach Estimating Bivariate Tail: a copula based approach Elena Di Bernardino, Université Lyon 1 - ISFA, Institut de Science Financiere et d'assurances - AST&Risk (ANR Project) Joint work with Véronique Maume-Deschamps

More information

Bivariate generalized Pareto distribution

Bivariate generalized Pareto distribution Bivariate generalized Pareto distribution in practice Eötvös Loránd University, Budapest, Hungary Minisymposium on Uncertainty Modelling 27 September 2011, CSASC 2011, Krems, Austria Outline Short summary

More information

EXTREMAL MODELS AND ENVIRONMENTAL APPLICATIONS. Rick Katz

EXTREMAL MODELS AND ENVIRONMENTAL APPLICATIONS. Rick Katz 1 EXTREMAL MODELS AND ENVIRONMENTAL APPLICATIONS Rick Katz Institute for Study of Society and Environment National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu Home page: www.isse.ucar.edu/hp_rick/

More information

RISK ANALYSIS AND EXTREMES

RISK ANALYSIS AND EXTREMES RISK ANALYSIS AND EXTREMES Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu Opening Workshop SAMSI program on

More information

Frequency Estimation of Rare Events by Adaptive Thresholding

Frequency Estimation of Rare Events by Adaptive Thresholding Frequency Estimation of Rare Events by Adaptive Thresholding J. R. M. Hosking IBM Research Division 2009 IBM Corporation Motivation IBM Research When managing IT systems, there is a need to identify transactions

More information

STATISTICAL METHODS FOR RELATING TEMPERATURE EXTREMES TO LARGE-SCALE METEOROLOGICAL PATTERNS. Rick Katz

STATISTICAL METHODS FOR RELATING TEMPERATURE EXTREMES TO LARGE-SCALE METEOROLOGICAL PATTERNS. Rick Katz 1 STATISTICAL METHODS FOR RELATING TEMPERATURE EXTREMES TO LARGE-SCALE METEOROLOGICAL PATTERNS Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder,

More information

Data. Climate model data from CMIP3

Data. Climate model data from CMIP3 Data Observational data from CRU (Climate Research Unit, University of East Anglia, UK) monthly averages on 5 o x5 o grid boxes, aggregated to JJA average anomalies over Europe: spatial averages over 10

More information

Multivariate generalized Pareto distributions

Multivariate generalized Pareto distributions Multivariate generalized Pareto distributions Holger Rootzén and Nader Tajvidi Abstract Statistical inference for extremes has been a subject of intensive research during the past couple of decades. One

More information

Accommodating measurement scale uncertainty in extreme value analysis of. northern North Sea storm severity

Accommodating measurement scale uncertainty in extreme value analysis of. northern North Sea storm severity Introduction Model Analysis Conclusions Accommodating measurement scale uncertainty in extreme value analysis of northern North Sea storm severity Yanyun Wu, David Randell, Daniel Reeve Philip Jonathan,

More information

Using statistical methods to analyse environmental extremes.

Using statistical methods to analyse environmental extremes. Using statistical methods to analyse environmental extremes. Emma Eastoe Department of Mathematics and Statistics Lancaster University December 16, 2008 Focus of talk Discuss statistical models used to

More information

Estimation of spatial max-stable models using threshold exceedances

Estimation of spatial max-stable models using threshold exceedances Estimation of spatial max-stable models using threshold exceedances arxiv:1205.1107v1 [stat.ap] 5 May 2012 Jean-Noel Bacro I3M, Université Montpellier II and Carlo Gaetan DAIS, Università Ca Foscari -

More information

Fin285a:Computer Simulations and Risk Assessment Section 6.2 Extreme Value Theory Daníelson, 9 (skim), skip 9.5

Fin285a:Computer Simulations and Risk Assessment Section 6.2 Extreme Value Theory Daníelson, 9 (skim), skip 9.5 Fin285a:Computer Simulations and Risk Assessment Section 6.2 Extreme Value Theory Daníelson, 9 (skim), skip 9.5 Overview Extreme value distributions Generalized Pareto distributions Tail shapes Using power

More information

MULTIDIMENSIONAL COVARIATE EFFECTS IN SPATIAL AND JOINT EXTREMES

MULTIDIMENSIONAL COVARIATE EFFECTS IN SPATIAL AND JOINT EXTREMES MULTIDIMENSIONAL COVARIATE EFFECTS IN SPATIAL AND JOINT EXTREMES Philip Jonathan, Kevin Ewans, David Randell, Yanyun Wu philip.jonathan@shell.com www.lancs.ac.uk/ jonathan Wave Hindcasting & Forecasting

More information

Change Point Analysis of Extreme Values

Change Point Analysis of Extreme Values Change Point Analysis of Extreme Values TIES 2008 p. 1/? Change Point Analysis of Extreme Values Goedele Dierckx Economische Hogeschool Sint Aloysius, Brussels, Belgium e-mail: goedele.dierckx@hubrussel.be

More information

On the estimation of the heavy tail exponent in time series using the max spectrum. Stilian A. Stoev

On the estimation of the heavy tail exponent in time series using the max spectrum. Stilian A. Stoev On the estimation of the heavy tail exponent in time series using the max spectrum Stilian A. Stoev (sstoev@umich.edu) University of Michigan, Ann Arbor, U.S.A. JSM, Salt Lake City, 007 joint work with:

More information

Classical Extreme Value Theory - An Introduction

Classical Extreme Value Theory - An Introduction Chapter 1 Classical Extreme Value Theory - An Introduction 1.1 Introduction Asymptotic theory of functions of random variables plays a very important role in modern statistics. The objective of the asymptotic

More information

Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution

Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution

More information

Regression, Curve Fitting and Optimisation

Regression, Curve Fitting and Optimisation Supervised by Elena Zanini STOR-i, University of Lancaster 4 September 2015 1 Introduction Root Finding 2 3 Simulated Annealing 4 5 The Rosenbrock Banana Function 6 7 Given a set of data, what is the optimum

More information

Wei-han Liu Department of Banking and Finance Tamkang University. R/Finance 2009 Conference 1

Wei-han Liu Department of Banking and Finance Tamkang University. R/Finance 2009 Conference 1 Detecting Structural Breaks in Tail Behavior -From the Perspective of Fitting the Generalized Pareto Distribution Wei-han Liu Department of Banking and Finance Tamkang University R/Finance 2009 Conference

More information

Estimation of the Angular Density in Multivariate Generalized Pareto Models

Estimation of the Angular Density in Multivariate Generalized Pareto Models in Multivariate Generalized Pareto Models René Michel michel@mathematik.uni-wuerzburg.de Institute of Applied Mathematics and Statistics University of Würzburg, Germany 18.08.2005 / EVA 2005 The Multivariate

More information

Tail dependence in bivariate skew-normal and skew-t distributions

Tail dependence in bivariate skew-normal and skew-t distributions Tail dependence in bivariate skew-normal and skew-t distributions Paola Bortot Department of Statistical Sciences - University of Bologna paola.bortot@unibo.it Abstract: Quantifying dependence between

More information

Spatial and temporal extremes of wildfire sizes in Portugal ( )

Spatial and temporal extremes of wildfire sizes in Portugal ( ) International Journal of Wildland Fire 2009, 18, 983 991. doi:10.1071/wf07044_ac Accessory publication Spatial and temporal extremes of wildfire sizes in Portugal (1984 2004) P. de Zea Bermudez A, J. Mendes

More information

UNIVERSITY OF CALGARY. Inference for Dependent Generalized Extreme Values. Jialin He A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

UNIVERSITY OF CALGARY. Inference for Dependent Generalized Extreme Values. Jialin He A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES UNIVERSITY OF CALGARY Inference for Dependent Generalized Extreme Values by Jialin He A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

More information

Kevin Ewans Shell International Exploration and Production

Kevin Ewans Shell International Exploration and Production Uncertainties In Extreme Wave Height Estimates For Hurricane Dominated Regions Philip Jonathan Shell Research Limited Kevin Ewans Shell International Exploration and Production Overview Background Motivating

More information

Lecture 2 APPLICATION OF EXREME VALUE THEORY TO CLIMATE CHANGE. Rick Katz

Lecture 2 APPLICATION OF EXREME VALUE THEORY TO CLIMATE CHANGE. Rick Katz 1 Lecture 2 APPLICATION OF EXREME VALUE THEORY TO CLIMATE CHANGE Rick Katz Institute for Study of Society and Environment National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu Home

More information

On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions

On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions J. L. Wadsworth Department of Mathematics and Statistics, Fylde College, Lancaster

More information

High-frequency data modelling using Hawkes processes

High-frequency data modelling using Hawkes processes Valérie Chavez-Demoulin joint work with High-frequency A.C. Davison data modelling and using A.J. Hawkes McNeil processes(2005), J.A EVT2013 McGill 1 /(201 High-frequency data modelling using Hawkes processes

More information

Statistics of Extremes

Statistics of Extremes Statistics of Extremes Anthony Davison c 211 http://stat.epfl.ch Multivariate Extremes 19 Componentwise maxima.................................................. 194 Standardization........................................................

More information

Math 576: Quantitative Risk Management

Math 576: Quantitative Risk Management Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 11 Haijun Li Math 576: Quantitative Risk Management Week 11 1 / 21 Outline 1

More information

On the Estimation and Application of Max-Stable Processes

On the Estimation and Application of Max-Stable Processes On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang

More information

Large sample distribution for fully functional periodicity tests

Large sample distribution for fully functional periodicity tests Large sample distribution for fully functional periodicity tests Siegfried Hörmann Institute for Statistics Graz University of Technology Based on joint work with Piotr Kokoszka (Colorado State) and Gilles

More information

Bivariate Analysis of Extreme Wave and Storm Surge Events. Determining the Failure Area of Structures

Bivariate Analysis of Extreme Wave and Storm Surge Events. Determining the Failure Area of Structures The Open Ocean Engineering Journal, 2011, 4, 3-14 3 Open Access Bivariate Analysis of Extreme Wave and Storm Surge Events. Determining the Failure Area of Structures Panagiota Galiatsatou * and Panagiotis

More information

Spatial Extremes in Atmospheric Problems

Spatial Extremes in Atmospheric Problems Spatial Extremes in Atmospheric Problems Eric Gilleland Research Applications Laboratory (RAL) National Center for Atmospheric Research (NCAR), Boulder, Colorado, U.S.A. http://www.ral.ucar.edu/staff/ericg

More information

On the Estimation and Application of Max-Stable Processes

On the Estimation and Application of Max-Stable Processes On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706 USA Richard L Smith Department of Statistics University of North

More information

PREPRINT 2005:38. Multivariate Generalized Pareto Distributions HOLGER ROOTZÉN NADER TAJVIDI

PREPRINT 2005:38. Multivariate Generalized Pareto Distributions HOLGER ROOTZÉN NADER TAJVIDI PREPRINT 2005:38 Multivariate Generalized Pareto Distributions HOLGER ROOTZÉN NADER TAJVIDI Department of Mathematical Sciences Division of Mathematical Statistics CHALMERS UNIVERSITY OF TECHNOLOGY GÖTEBORG

More information

Statistical modelling of extreme ocean environments for marine design : a review

Statistical modelling of extreme ocean environments for marine design : a review Statistical modelling of extreme ocean environments for marine design : a review Philip Jonathan Shell Projects and Technology Thornton, CH1 3SH, UK. Kevin Ewans Sarawak Shell Bhd., 50450 Kuala Lumpur,

More information

Some conditional extremes of a Markov chain

Some conditional extremes of a Markov chain Some conditional extremes of a Markov chain Seminar at Edinburgh University, November 2005 Adam Butler, Biomathematics & Statistics Scotland Jonathan Tawn, Lancaster University Acknowledgements: Janet

More information

Physically-Based Statistical Models of Extremes arising from Extratropical Cyclones

Physically-Based Statistical Models of Extremes arising from Extratropical Cyclones Lancaster University STOR603: PhD Proposal Physically-Based Statistical Models of Extremes arising from Extratropical Cyclones Author: Paul Sharkey Supervisors: Jonathan Tawn Jenny Wadsworth Simon Brown

More information

Shape of the return probability density function and extreme value statistics

Shape of the return probability density function and extreme value statistics Shape of the return probability density function and extreme value statistics 13/09/03 Int. Workshop on Risk and Regulation, Budapest Overview I aim to elucidate a relation between one field of research

More information

ON THE ESTIMATION OF EXTREME TAIL PROBABILITIES. By Peter Hall and Ishay Weissman Australian National University and Technion

ON THE ESTIMATION OF EXTREME TAIL PROBABILITIES. By Peter Hall and Ishay Weissman Australian National University and Technion The Annals of Statistics 1997, Vol. 25, No. 3, 1311 1326 ON THE ESTIMATION OF EXTREME TAIL PROBABILITIES By Peter Hall and Ishay Weissman Australian National University and Technion Applications of extreme

More information

of the 7 stations. In case the number of daily ozone maxima in a month is less than 15, the corresponding monthly mean was not computed, being treated

of the 7 stations. In case the number of daily ozone maxima in a month is less than 15, the corresponding monthly mean was not computed, being treated Spatial Trends and Spatial Extremes in South Korean Ozone Seokhoon Yun University of Suwon, Department of Applied Statistics Suwon, Kyonggi-do 445-74 South Korea syun@mail.suwon.ac.kr Richard L. Smith

More information

Method of Conditional Moments Based on Incomplete Data

Method of Conditional Moments Based on Incomplete Data , ISSN 0974-570X (Online, ISSN 0974-5718 (Print, Vol. 20; Issue No. 3; Year 2013, Copyright 2013 by CESER Publications Method of Conditional Moments Based on Incomplete Data Yan Lu 1 and Naisheng Wang

More information

Bayesian nonparametrics for multivariate extremes including censored data. EVT 2013, Vimeiro. Anne Sabourin. September 10, 2013

Bayesian nonparametrics for multivariate extremes including censored data. EVT 2013, Vimeiro. Anne Sabourin. September 10, 2013 Bayesian nonparametrics for multivariate extremes including censored data Anne Sabourin PhD advisors: Anne-Laure Fougères (Lyon 1), Philippe Naveau (LSCE, Saclay). Joint work with Benjamin Renard, IRSTEA,

More information

Modelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions

Modelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions Modelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions Anna Kiriliouk 1 Holger Rootzén 2 Johan Segers 1 Jennifer L. Wadsworth 3 1 Université catholique de Louvain (BE) 2 Chalmers

More information

A Bayesian Spatial Model for Exceedances Over a Threshold

A Bayesian Spatial Model for Exceedances Over a Threshold A Bayesian Spatial odel for Exceedances Over a Threshold Fernando Ferraz do Nascimento and Bruno Sansó June 2, 2017 Abstract Extreme value theory focuses on the study of rare events and uses asymptotic

More information

U.K. Ozone and UV Trends and Extreme Events

U.K. Ozone and UV Trends and Extreme Events U.K. Ozone and UV Trends and Extreme Events Andrew C. Moss, Andrew R.D. Smedley and John S. Rimmer University of Manchester Acknowledgements: UKMO, DEFRA Workshop, Beijing 211 Introduction Overview of

More information

Change Point Analysis of Extreme Values

Change Point Analysis of Extreme Values Change Point Analysis of Extreme Values DGVFM Stuttgart 27 APRIL 2012 p. 1/3 Change Point Analysis of Extreme Values Goedele Dierckx Economische Hogeschool Sint Aloysius, Brussels, Belgium Jef L. Teugels

More information

Uncertainties in extreme surge level estimates from observational records

Uncertainties in extreme surge level estimates from observational records Uncertainties in extreme surge level estimates from observational records By H.W. van den Brink, G.P. Können & J.D. Opsteegh Royal Netherlands Meteorological Institute, P.O. Box 21, 373 AE De Bilt, The

More information

Semi-parametric estimation of non-stationary Pickands functions

Semi-parametric estimation of non-stationary Pickands functions Semi-parametric estimation of non-stationary Pickands functions Linda Mhalla 1 Joint work with: Valérie Chavez-Demoulin 2 and Philippe Naveau 3 1 Geneva School of Economics and Management, University of

More information

What Can We Infer From Beyond The Data? The Statistics Behind The Analysis Of Risk Events In The Context Of Environmental Studies

What Can We Infer From Beyond The Data? The Statistics Behind The Analysis Of Risk Events In The Context Of Environmental Studies What Can We Infer From Beyond The Data? The Statistics Behind The Analysis Of Risk Events In The Context Of Environmental Studies Sibusisiwe Khuluse, Sonali Das, Pravesh Debba, Chris Elphinstone Logistics

More information

Bivariate generalized Pareto distribution in practice: models and estimation

Bivariate generalized Pareto distribution in practice: models and estimation Bivariate generalized Pareto distribution in practice: models and estimation November 9, 2011 Pál Rakonczai Department of Probability Theory and Statistics, Eötvös University, Hungary e-mail: paulo@cs.elte.hu

More information

High-frequency data modelling using Hawkes processes

High-frequency data modelling using Hawkes processes High-frequency data modelling using Hawkes processes Valérie Chavez-Demoulin 1 joint work J.A McGill 1 Faculty of Business and Economics, University of Lausanne, Switzerland Boulder, April 2016 Boulder,

More information

The Spatial Variation of the Maximum Possible Pollutant Concentration from Steady Sources

The Spatial Variation of the Maximum Possible Pollutant Concentration from Steady Sources International Environmental Modelling and Software Society (iemss) 2010 International Congress on Environmental Modelling and Software Modelling for Environment s Sake, Fifth Biennial Meeting, Ottawa,

More information

Extreme Event Modelling

Extreme Event Modelling Extreme Event Modelling Liwei Wu, SID: 52208712 Department of Mathematics City University of Hong Kong Supervisor: Dr. Xiang Zhou March 31, 2014 Contents 1 Introduction 4 2 Theory and Methods 5 2.1 Asymptotic

More information

Change Point Analysis of Extreme Values

Change Point Analysis of Extreme Values Change Point Analysis of Extreme Values Lisboa 2013 p. 1/3 Change Point Analysis of Extreme Values Goedele Dierckx Economische Hogeschool Sint Aloysius, Brussels, Belgium Jef L. Teugels Katholieke Universiteit

More information

NEW METHOD FOR ESTIMATING DIRECTIONAL EXTREME WIND SPEED BY CONSIDERING THE CORRELATION AMONG EXTREME WIND SPEED IN DIFFERENT DIRECTIONS

NEW METHOD FOR ESTIMATING DIRECTIONAL EXTREME WIND SPEED BY CONSIDERING THE CORRELATION AMONG EXTREME WIND SPEED IN DIFFERENT DIRECTIONS The Eighth Asia-Pacific Conference on Wind Engineering, December 0 4, 203, Chennai, India NEW ETHOD FO ESTIATING DIECTIONAL EXTEE WIND SPEED BY CONSIDEING THE COELATION AONG EXTEE WIND SPEED IN DIFFEENT

More information

The extremal elliptical model: Theoretical properties and statistical inference

The extremal elliptical model: Theoretical properties and statistical inference 1/25 The extremal elliptical model: Theoretical properties and statistical inference Thomas OPITZ Supervisors: Jean-Noel Bacro, Pierre Ribereau Institute of Mathematics and Modeling in Montpellier (I3M)

More information

Reliable Inference in Conditions of Extreme Events. Adriana Cornea

Reliable Inference in Conditions of Extreme Events. Adriana Cornea Reliable Inference in Conditions of Extreme Events by Adriana Cornea University of Exeter Business School Department of Economics ExISta Early Career Event October 17, 2012 Outline of the talk Extreme

More information

EXTREMAL QUANTILES OF MAXIMUMS FOR STATIONARY SEQUENCES WITH PSEUDO-STATIONARY TREND WITH APPLICATIONS IN ELECTRICITY CONSUMPTION ALEXANDR V.

EXTREMAL QUANTILES OF MAXIMUMS FOR STATIONARY SEQUENCES WITH PSEUDO-STATIONARY TREND WITH APPLICATIONS IN ELECTRICITY CONSUMPTION ALEXANDR V. MONTENEGRIN STATIONARY JOURNAL TREND WITH OF ECONOMICS, APPLICATIONS Vol. IN 9, ELECTRICITY No. 4 (December CONSUMPTION 2013), 53-63 53 EXTREMAL QUANTILES OF MAXIMUMS FOR STATIONARY SEQUENCES WITH PSEUDO-STATIONARY

More information

Driving Restriction, Traffic Congestion, and Air Pollution: Evidence from Beijing

Driving Restriction, Traffic Congestion, and Air Pollution: Evidence from Beijing Driving Restriction, Traffic Congestion, and Air Pollution: Evidence from Beijing Chen Liu Junjie Zhang UC San Diego Camp Resources XXI August 10-12, 2014 Traffic Congestion and Air Pollution 1 Motivation

More information

AREP GAW. AQ Forecasting

AREP GAW. AQ Forecasting AQ Forecasting What Are We Forecasting Averaging Time (3 of 3) PM10 Daily Maximum Values, 2001 Santiago, Chile (MACAM stations) 300 Level 2 Pre-Emergency Level 1 Alert 200 Air Quality Standard 150 100

More information

Estimation of Quantiles

Estimation of Quantiles 9 Estimation of Quantiles The notion of quantiles was introduced in Section 3.2: recall that a quantile x α for an r.v. X is a constant such that P(X x α )=1 α. (9.1) In this chapter we examine quantiles

More information

Analysis methods of heavy-tailed data

Analysis methods of heavy-tailed data Institute of Control Sciences Russian Academy of Sciences, Moscow, Russia February, 13-18, 2006, Bamberg, Germany June, 19-23, 2006, Brest, France May, 14-19, 2007, Trondheim, Norway PhD course Chapter

More information

Inference for clusters of extreme values

Inference for clusters of extreme values J. R. Statist. Soc. B (2003) 65, Part 2, pp. 545 556 Inference for clusters of extreme values Christopher A. T. Ferro University of Reading, UK and Johan Segers EURANDOM, Eindhoven, the Netherlands [Received

More information

Discussion on Human life is unlimited but short by Holger Rootzén and Dmitrii Zholud

Discussion on Human life is unlimited but short by Holger Rootzén and Dmitrii Zholud Extremes (2018) 21:405 410 https://doi.org/10.1007/s10687-018-0322-z Discussion on Human life is unlimited but short by Holger Rootzén and Dmitrii Zholud Chen Zhou 1 Received: 17 April 2018 / Accepted:

More information

Two practical tools for rainfall weather generators

Two practical tools for rainfall weather generators Two practical tools for rainfall weather generators Philippe Naveau naveau@lsce.ipsl.fr Laboratoire des Sciences du Climat et l Environnement (LSCE) Gif-sur-Yvette, France FP7-ACQWA, GIS-PEPER, MIRACLE

More information

FRAPPÉ/DISCOVER-AQ (July/August 2014) in perspective of multi-year ozone analysis

FRAPPÉ/DISCOVER-AQ (July/August 2014) in perspective of multi-year ozone analysis FRAPPÉ/DISCOVER-AQ (July/August 2014) in perspective of multi-year ozone analysis Project Report #2: Monitoring network assessment for the City of Fort Collins Prepared by: Lisa Kaser kaser@ucar.edu ph:

More information

Spatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012

Spatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012 Spatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012 November 8, 2012 15:00 Clement Dombry Habilitation thesis defense (in french) 17:00 Snack buet November

More information

Central Ohio Air Quality End of Season Report. 111 Liberty Street, Suite 100 Columbus, OH Mid-Ohio Regional Planning Commission

Central Ohio Air Quality End of Season Report. 111 Liberty Street, Suite 100 Columbus, OH Mid-Ohio Regional Planning Commission 217 218 Central Ohio Air Quality End of Season Report 111 Liberty Street, Suite 1 9189-2834 1 Highest AQI Days 122 Nov. 217 Oct. 218 July 13 Columbus- Maple Canyon Dr. 11 July 14 London 11 May 25 New Albany

More information

Estimation of Stress-Strength Reliability for Kumaraswamy Exponential Distribution Based on Upper Record Values

Estimation of Stress-Strength Reliability for Kumaraswamy Exponential Distribution Based on Upper Record Values International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 2, 59-71 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7210 Estimation of Stress-Strength Reliability for

More information