Practical Issues in Applications of Multivariate Extreme Values - PowerPoint PPT Presentation

Practical Issues in Applications of Multivariate Extreme Values Jonathan Tawn with Caroline Keef and Mark Latham Lancaster, UK

Two Applications • Sea-surge data Modelling of surge process over space for joint flood risk assessment for coastal sites and for offshore sites needed for insurance industry

Two Applications • Sea-surge data Modelling of surge process over space for joint flood risk assessment for coastal sites and for offshore sites needed for insurance industry • River flow data Modelling of river flow for network for joint flood risk assessment for planning purposes and insurance

Surge Data Hindcast output from the CSX model, a 2d numerical surge model for the European Continental Shelf forced by DNMI pressure data for the period 1955-2000 Data are: hourly maxima over 5-day blocks for 46 years at 259 sites

River Flow Data Daily river flows for a network of sites in River Thames catchment in UK Altitude < 100m Great 250 Altitude > 100m Britain River Flow gauge Rain gauge Northing (km) 200 150 400 450 500 550 Easting (km)

Marginal Standardisation and Notation X : univariate variable of most interest Y: d -dimensional variable Transform marginals to Gumbel distributions Pr( X > x ) = Pr( Y i > x ) ∼ exp( − x ) as x → ∞ for i = 1 , . . . , d Lack of Memory Property Pr( X > t + x ) ∼ exp( − t ) Pr( X > x ) for large x Allows focus on dependence structure

Standardisation for Surge Data A large surge event on the Danish coast in original and transformed margins 5.15 4.317 3.663 + North 2.175 0.688 −0.8 East 8.2 6.744 5.6 + North 3 0.4 −2.2 East

What is the Aim of Analysis? • Sea-surge data Simulation of surge events large at a given location Estimation of spatial risk measure E (# { Y > x } | X > x ) Dimension reduction for physical understanding

What is the Aim of Analysis? • Sea-surge data Simulation of surge events large at a given location Estimation of spatial risk measure E (# { Y > x } | X > x ) Dimension reduction for physical understanding • River flow data Estimation of Pr( Y > x | X > x )

Schematic of Threshold Approach Under assumption of asymptotic dependence x →∞ Pr( Y > x | X > x ) > 0 lim the following homogeneity property holds for all sets A extreme in at least one variable Pr(( X , Y ) ∈ t + A ) ≈ exp( − t ) Pr(( X , Y ) ∈ A ) Y t+ A A t x xx x xx x x x xx x x x xx xx xx x x xxx x x x x x x x x x u x x x x x x x x x x x x xx xx x x x xx x x x x xx xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X u

Is Surge Process Asymptotically Dependent? X : Danish Site 8.2 + North 6.744 5.6 East + North 3 East 0.4 + North −2.2 East

Is Surge Process Asymptotically Dependent? X : UK Site 8.2 North + 6.744 5.6 East North 3 + East 0.4 North + −2.2 East

Sites Significant on Testing for Asymptotic Dependence X : Danish Site O O O * * * * * * * O O O O * * * * * * * * O O O O O O * * * * * * * * * * O O O O O O O O O O O * * * * * * * * * * * O O O O O O O O O O O O O * * * * * * * * * * * * * O O O O O O O O O O O O * * * * * * * * * * * * * O O O O O O O O O O O O * * * * * * * * * * * * O O O O O O O O O O O O O O * * * * * * * * * * * * * * North O O O O O O O O O O O O O * * * * * * * * * * * * * O O O O O O O O * * * * * * * * * * * * * * * * * East

Sites Significant on Testing for Asymptotic Dependence X : UK Site * * O O O * * * * * * * * * O O O O O O * * * * * * * O O O O O O O O O * * * * * * * * * * O O O O O O O O * * * * * * * * North O O O O O O O O O * * * * * * * * * O O O O O O O O * * * * * * * * O O O O O O O O * * * * * * * * * O O O O * * * * * O * O * * O * O * East

Problems for River Flow Application Plot of data availability for Thames catchment sites dmf39130 dmf39081 dmf39046 dmf39025 dmf39020 dmf39019 dmf39016 dmf39008 dmf39001 1960 1969 1980 1990 2000 Year

Regression Interpretation of Threshold Method For X > u Y = X + Z where Z is independent of X � ∞ m 1 ˆ � Pr(( X , Y ) ∈ t + A ) = exp( − v ) 1 { ( x , x + z i ) ∈ t + A } exp( − x ) dx m v i =1 Y t+ A A t x xx x xx x x x xx x x x xx xx xx x x xxx x x x x x x x x x u x x x x x x x x x x x x xx xx x x x xx x x x x xx xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X u

Extension of Regression/Conditional Method Heffernan and Tawn (2004,JRSS B) For X > u Y = a X + X b Z where Z is independent of X d -dimensional parameters 0 ≤ a ≤ 1 and b Nonparametric model for Z Y x xx x xx x x x xx x x x xx xx xx x x xxx x x x x x x u x x x x x x x x x x x x x x x xx xx x x x xx x x x x xx xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Z x X u

Theoretical Examples Y = a X + X b Z Asymptotic Dependence a = 1 and b = 0 Asymptotic Independence with Y j a j < 1 Multivariate Normal Copula j and b j = 1 a j = ρ 2 2 for j = 1 , . . . , d

Estimates of a X : Danish Site 1 0.86 0.75 North 0.5 0.25 0 East

Estimates of a X : UK Site 1 0.86 0.75 North 0.5 0.25 0 East

Which Sites are Asymptotically Dependent? Test a j = 1 , b j = 0 X : Danish Site O O O * * * * * * * O O O O * * * * * * * * O O O O O O * * * * * * * * * * O O O O O O O O O O O * * * * * * * * * * * O O O O O O O O O O O O O * * * * * * * * * * * * * O O O O O O O O O O O O * * * * * * * * * * * * * O O O O O O O O O O O O * * * * * * * * * * * * O O O O O O O O O O O O O O * * * * * * * * * * * * * * North O O O O O O O O O O O O O * * * * * * * * * * * * * O O O O O O O O * * * * * * * * * * * * * * * * * East

Search for Parsimonious Model Dimension of model parameters currently 259 × 258 × 2 Dimension Reduction helpful/insightful

Search for Parsimonious Model Dimension of model parameters currently 259 × 258 × 2 Dimension Reduction helpful/insightful How many sites do we need to condition on to get all sites asymptotically dependent on a conditioning site?

Search for Parsimonious Model Dimension of model parameters currently 259 × 258 × 2 Dimension Reduction helpful/insightful How many sites do we need to condition on to get all sites asymptotically dependent on a conditioning site? * * * North * * * East

Parsimonious Spatial Model Partition ( X , Y ) = ( X C , Y C ) where X C the six conditioning sites Y C the remaining sites Then [ X C , Y C ] = [ X C ][ Y C | X C ] where [ X C ] is low dimensional, and [ Y C | X C ] is simpler due to asymptotic dependence property Extremes for [ Y C ] only arise when [ X C ] is extreme in at least only component

Spatial Risk Measure E (# { Y > x } | X > x ) where x is the 97% quantile Comparison of empirical, global model, parsimonious model 190 North 167.46 149.75 East North 109.5 East 69.25 North 29 East

Extrapolation of Spatial Risk Measure E (# { Y > x } | X > x ) where x is the 97% and 99.9% quantiles for global model 190 167.46 North 149.75 East 109.5 69.25 North 29 East

Simulated Fields on Original Scale Exceeds 1000 year level on Danish coast site 7 + North 6.076 5.35 East + North 3.7 East 2.05 + North 0.4 East

Simulated Fields on Original Scale Exceeds 1000 year level on UK coast site 7 North + 6.076 5.35 East North 3.7 + East 2.05 North + 0.4 East

Handling Missing Data for River Flows Partition Y = ( Y M , Y O ) where Y M missing; Y O observed Also Z = ( Z M , Z O ) Then need to model [ Z M | Z O ] Approach is: empty

Handling Missing Data for River Flows Partition Y = ( Y M , Y O ) where Y M missing; Y O observed Also Z = ( Z M , Z O ) Then need to model [ Z M | Z O ] Approach is: • Transform margins empty Z N = T ( Z ) = Φ − 1 (ˆ F ( Z ))

Handling Missing Data for River Flows Partition Y = ( Y M , Y O ) where Y M missing; Y O observed Also Z = ( Z M , Z O ) Then need to model [ Z M | Z O ] Approach is: • Transform margins empty Z N = T ( Z ) = Φ − 1 (ˆ F ( Z )) • Model dependence by MVN copula � � �� � � �� Z N 0 Σ 11 Σ 12 M ∼ MVN , Z N 0 Σ 21 Σ 22 O

Handling Missing Data for River Flows Partition Y = ( Y M , Y O ) where Y M missing; Y O observed Also Z = ( Z M , Z O ) Then need to model [ Z M | Z O ] Approach is: • Transform margins empty Z N = T ( Z ) = Φ − 1 (ˆ F ( Z )) • Model dependence by MVN copula � � �� � � �� Z N 0 Σ 11 Σ 12 M ∼ MVN , Z N 0 Σ 21 Σ 22 O • Take a sample from this conditional distribution [ˆ Z N M | Z N O ]