Uncertainty Analysis Applications of GDB Copula with TSPGenerating Densities

Leon Adams

May 10, 2011

Dissertation Defense

Uncertainty Analysis Applications of GDB Copula with TSP Generating Densities

Dissertation Committee:

Johan R. van Dorp Dissertation DirectorEnrique Campos-Nanez Committee MemberJonathan Pierce Deason Committee MemberRobert A. Roncace Committee MemberThomas Andrew Mazzuchi Committee Member

Overview

Introduction

• Dissertation background• Preview dissertation contributions

Multivariate Models

• Details of the single common risk factor multivariate model• Details of the multiple common risk factor multivariate model

Application Examples

• Hydrological frequency analysis• Stock returns

Concluding remarks

• Review model success• Future work

Introduction

Definitions – Sklar’s Theorem

Theorem (Sklar). Given the CDF H with continuous margins F and G. Then there exists a uniquecopula represented by:

H(x, y) = C{F (x), G(y)}

Corollary. Given the CDF H with continuous margins F and G and the copula C. Then foru, v ∈ [0, 1]:

C(u, v) = H{F−1(u), G−1(v)},

where F−1 and G−1 are quantile functions.

We also have the density representation of copulas:

h(x, y) = c{F (x), G(y)} · f(x)g(y)

where h, c, f, and g are densities.

Graphical representation of the diagonal band copula

Support region and copula PDF representation of the original diagonal band copula [4].

There are two extension to the diagonal band copula–both using generating densities

• Ferguson (1995) and Bojarski (2002)• Lewandowski (2005) showed that Ferguson and Bojarski extension were equivalent

c(u, v) =1

2{g(|u− v|) + g(1− |1− u− v|)} 0 < u, v < 1 [5]

g(z) =

{

(gθ(−z) + gθ(z) for z ∈ [0, 1− θ] [10]0 elsewhere

GDB TSP copula [8]

c {x, y|p (·|Ψ)} = 12×

p (1− x− y|Ψ) + p (1 + x− y|Ψ) , (x, y) ∈ A1,

p (1− x− y|Ψ) + p (1− x+ y|Ψ) , (x, y) ∈ A2,

p (x+ y − 1|Ψ) + p (1 + x− y|Ψ) , (x, y) ∈ A3,

p (x+ y − 1|Ψ) + p (1− x+ y|Ψ) , (x, y) ∈ A4.

If we consider the TSP generating density (Kotz, Van Dorp 2010)

p(·|Ψ) = nzn−1 → P (·|Ψ) = zn

c { x, y | p (·|Ψ) } = 12×

n (1− x− y)n−1 + n (1 + x− y)n−1 , (x, y) ∈ A1,

n (1− x− y)n−1 + n (1− x+ y)n−1 , (x, y) ∈ A2,

n (x+ y − 1)n−1 + n (1 + x− y)n−1 , (x, y) ∈ A3,

n (x+ y − 1)n−1 + n (1− x+ y)n−1 , (x, y) ∈ A4.

• For nomenclature we use: single parameter GDB TSP copula• This single parameter copula is utilized in the current research model• Could be extended to other copula models

Preview research contributions

Contributions:

• Specification of estimation procedure for a multivariate copula framework

• Bivariate model with single common risk factor• Multivariate model with single common risk factor• Multivariate model with multiple common risk factors

• Derivation of novel relationships between model parameters and the Spearman’s ρ andBlomqvist’s β dependency measures

• Specification of efficient joint sampling procedures

Motivation:

• Literature review reveals focus on bivariate copulas• Applications tend to use select few family

• Elliptical - Gaussian, Student t• Archimedean - Gumbel, Clayton, Frank

• Increase availability of multivariate copula models

Multivariate Models

Single common risk factor multivariate GDB TSP copula - Parameter estimation

Copula model assumptions

• Single common risk factor• Uniformly distributed observations & common risk factor• Conditional independence of observed random variables given U

• Dependence on common risk factor defined by GDB TSP copula• Model parameters scale with dimension of the problem

ρij = 1− 12

(ni + 2)(ni + 3)− 12

(nj + 2)(nj + 3)+

24

(ni + 1)(nj + 1)(ni + nj + 3)

− 24Γ(ni + 2)Γ(nj + 2)

(ni + 1)(nj + 1)Γ(ni + nj + 4)

min (ρn − ρ̂)2 s.t. n > 0

Estimate model parameters

U

Multiple common risk factor multivariate GDB TSP copula - Parameter estimation

G(Y ) = vol (H(γ) ∩ C) =∑

v∈C(sign v)(γ−α·v)n+n!(Πn

i=1αi)

vi = Gi{yi(u|W,n)} = Pr(Yi ≤ yi)

Yi =∑k

l=1 wilUl , yi =∑k

l=1 wilul

ρωn ←∫ 1

0. . .

∫ 1

0

{

vi +(1− vi)

ni+1

ni + 1− v

ni+1i

ni + 1

}

×

vj +(1− vj)

nj+1

nj + 1−

vnj+1

j

nj + 1

du

min (ρωn − ρ̂)2 s.t. n > 0;k

∑

i,j=1

ωi,j = 1; ωi,j ≥ 0

Multiple common risk factor multivariate GDB TSP copula

Extension to multiple common risk factor multivariate model

• Main addition is the incorporation of GDB-TSP copula in common risk factors model• With multiple common risk factors, model flexibility and complexity increase

• Must know account for the Yi ↔ G(Yi)• Must relate model parameters to global dependence measure

• Increases the challenge of the optimization procedure in the parameter estimation process• Two classes of model parameters

• Copula parameters constrained by ni > 0• Weights of the common risk factors with following constraints:

0 ≤ wji ≤ 1∑m

i=1 ωji = 1 ∀j → 1..k

Multivariate sampling algorithm

Application Examples

Application Examples

• Hydrological frequency analysis

• Objective: Study relationship between rainfall duration and amounts• Demonstrated importance of correctly modeling dependence structure• GDB-TSP model outperforms traditional distribution approach• Differences in model outputs have practical implications to flood mitigation strategies

• Flood example

• Objective: Study relationship between locations upstream and downstream• Demonstrated importance of correctly modeling marginals• Gamma distribution selected as the univariate model for both marginals

• Sediment composition

• Objective: Spatial study of relationship between composition of Cerium and Scandium• Demonstrated importance of correctly modeling marginals• GEV distribution selected as univariate model for Cerium marginal• Logistic distribution selected as univariate model for Scandium marginal

• Salmonid risk assessment

• Objective: Monte Carlo Salmonid risk assessment(

dsDg

, hL1

,L2L1

)

• Demonstrated importance of correctly modeling dependence structure• In parts of modeled space correlated model had a higher estimation of risk in achieving target survival rates

• Stock returns analysis

• Objective: Study effectiveness of research model on 7-dim model. Data matrix ↔ Sample matrix• Use law of parsimony to select 3 common risk factors model over 2 common risk factors model• Demonstrated high degree of fidelity of Sample Correlation matrix with Data Correlation matrix

Hydrological frequency analysis

The task is to better understand thebehavior of extreme rainfall events:

• Magnitude• Duration• Frequency• Taking a distributional approach

Motivation:

• Potential for great loss• Inform mitigation strategies• Insurance underwriting• Input into rainfall runoff models

An aerial view of the submerged runway at Rockhampton airport in Australia. Ge�y Images / Jonathan Wood

Source: h�p://blogs.sacbee.com/photos/2011/01/new-storms-soak-flood-weary-au.html

Korean rainfall example

Data:

• Seoul Korean dataset [9]• Bivariate dataset of rainfall

maximums of amount andduration

• Study period from 1965-2005

Approach:

• Comparative investigationbetween GDB-TSP modeland Gumbel mixed model

• Maintain assumption ofGumbel marginals

• Relies on the calculations ofreturn periods

• Examine differences in modelpredictions of returns period

Estimated parameters for the GDB-TSP model with Gumbel marginals

Marginal Mean Std. dev Scale Location Correlation GDB-TSPµ σ λ u ρ n

Duration 56.25 30.55 23.82 42.5 0.55 2.689

Amount 225.23 111.9 87.2 174.9

Returns definitions

• T (x, y) is defined as the joint returns period of amount and duration.• T (x|y) is defined as the conditional returns period of amount given duration.• T ′(x, y) is the non-standard joint return of amount and duration.• T ′(x|y) is defined as the non-standard conditional return of amount given duration.

Bivariate returns periods

Return Period Event

TX,Y (x, y) {(X > x or Y > y) or (X > x & Y > y) }T ′X,Y (x, y) {X > x and Y > y }

TX|Y (x|y) {X > x given Y = y }T ′X|Y

(x|y) {X > x given Y ≤ y }

T (x, y) =1

PE(x, y)Where PE(x, y) = 1− F (x, y)

Candidate models

Gumbel mixed model F (x, y) = Fx(x)Fy(y)× exp

{

−θ[

1lnFx(x)

+ 1lnFy(y)

]−1}

GDB-TSP model F (x, y) = C {Fx(x), Fy(y)} Where C is the GDB-TSP copula

For both models marginals are assumed to be Gumbel marginals:

Fz(z) = exp[

− exp(

− z−uzλz

)]

Distribution and density for GDB-TSP model

Goodness of fit comparison

Goodness of fit details

• Distance from empirical CDF [6]

• Sn =n∑

i=1{Fn − Fθn}2

• Tn = sup√n |Fn − Fθn |

Model predictions comparison for the T(x,y) study

D=12hrs

Amount (mm)

Re

turn

Pe

rio

d (

yr)

Image: hp://www.abbey-associates.com/splash-splash/storm_water_management.html

Storm water runoff system

Findings of the comparative study

• Study compared GDB-TSP model to the Gumbel mixed model• Based on Goodness of fit results, we select the GDB-TSP model• For T (x, y) the comparison found:

• Good agreement in the low amount-duration regime• GDB-TSP model predicted shorter joint returns elsewhere

• For T ′(x, y) the comparison found:

• Good agreement in most of the modeled space• GDB-TSP model predicted smaller rainfall amounts in the higher duration events

• For both T (x|y) and T ′(x|y) the comparison found:

• GDB-TSP model predicted smaller rainfall amounts in the shorter duration events• GDB-TSP model predicted larger rainfall amounts in the higher duration events

Higher dimensional example: stock returns

• Data

• Weekly returns: January 1st 1990 – January 3rd 2011 Rt =Pt−Pt−1

Pt−1

• Stocks: XOM APA CVX SLB SU IMO NBL• Indices: NDX DJA GSPC

• Goal

• Reproduce data correlation matrix• Simulate from resulting distribution• data correlation↔ fit correlation↔ sampled correlation

Stocks estimated parameters

Objec�ve func�on = 0.00036

Parameter Vector Weight Vector

Objec�ve func�on = 0.035

Parameter Vector Weight Vector

Correlation matrix comparison

Data Correlaon Matrix

Fi�ed Correlaon Matrix

Sampled Correlaon Matrix

XOM APA CVX SLB SU NDX DJA

XOM APA CVX SLB SU NDX DJA

XOM APA CVX SLB SU NDX DJA

Canonical Correlation

YX

XOMAPACVXSLBSUIMONBL

NDX

DJA

GSPC

COMP1

COMP2

COMP3

U1

U2

U3

Linear

Model 1 Model 2 Model 3

R2M R1 R2 R3

Model 1 1.000 1.000 1.000 1.000Model 2 6.56e−6 0.550 0.097 0.048Model 3 0.232 0.950 0.774 0.655

R2M

=

∣

∣

∣

∣

S−1yy SyxS

−1xx Sxy

∣

∣

∣

∣

=s∏

i=1

r2i

Research Summary

Research contributions

• Novel relations linking copula parameters to traditional dependence measures• Copula models

• A two parameter bivariate copula model based on a common risk factor• A multivariate copula model based on a common risk factor• A multivariate copula model based on multiple common risk factor

• Estimation procedures and sampling routines

Application examples

• A flood example demonstrating improvement over traditional distribution approach• A geochemical sediment composition example leveraging the flexibility of arbitrary marginals• A hydrology example of returns period• A Monte Carlo simulation for risk assessment• A multivariate example of stock market returns

Future work

• Investigate alternatives to numerical integration• Investigate the interpretation of the common risk factors

Uncertainty Analysis Applications of GDB Copula with TSPGenerating Densities

Leon Adams

May 10, 2011

References

[1] D. L. Barrow and P. W. Smith. Spline notation applied to a volume problem. The AmericanMathematical Monthly, 86:50–51, Jan. 1979.[2] R. W. Carter. Floods in Georgia. Geological Survey Circular, 1951. No. 100. [24.3-1].[3] R. Dennis Cook and Mark E. Johnson. A family of distributions for modelling non-ellipticallysymmetric multivariate data. Journal of the Royal Statistical Society. Series B (Methodological),43(2):210–218, 1981.[4] Roger M. Cooke and Rudi Waij. Monte carlo sampling for generalized knowledge dependencewith application to human reliability. Risk Analysis, 6(3):335–343, 1986.[5] T.F. Ferguson. A class of symmetric bivariate uniform distributions. Statistical Papers,36(1):31–40, 1995.[7] Christian Genest and Louis-Paul Rivest. Statistical inference procedures for bivariate archimedeancopulas. Journal of the American statistical association, 88(423):1034–1043, September 1993.[6] Christian Genest, Bruno Remillard, and David Beaudoin. Goodness-of-fit tests for copulas: Areview and a power study. Insurance: Mathematics and Economics, 44:199–213, 2009.[8] Samuel Kotz and Johan Rene Van Dorp. Generalized diagonal band copulas with two-sidedgenerating densities. Decison Analysis, 7(2):196–214, 2010.[9] Chang Lee, Tae-Woong Kim, Gunhui Chung, Minha Choi, and Chulsang Yoo. Application ofbivariate frequency analysis to the derivation of rainfall–frequency curves. Stochastic EnvironmentalResearch and Risk Assessment, 24:389–397, 2010. 10.1007/s00477-009-0328-9.[10] Daniel Lewandowski. Generalized diagonal band copulas. Insurance: Mathematics andEconomics, 37:49–67, 2005.[11] Fu-Chun Wu and Yin-Phan Tsang. Second-order monte carlo uncertainty/variability analysisusing correlated model parameters: application to salmonid embryo survival risk assesment.Ecological Modelling, 177:393–414, 2004.

GDB-TSP Canonical Correlation

Zero covariance

X correlation Y correlation

Cross−correlation

−1.0 −0.5 0.0 0.5 1.0

• Reminiscent of factor rotation

• Independent X’s• Significant dependence Y↔ X

• Structural differences

• Factor rotation is a linear model• GDB-TSP model provides full distribution approach

Multiple common risk factor multivariate GDB TSP copula

Model correlation matrix detail calculations

Dependence Parameters: W : k × m-matrix, n : m-vector

k = 3,m = 7 ⇒ 21 dependence parameters as opposed to(7

2

)

= 21

k = 3,m = 10 ⇒ 30 dependence parameters as opposed to(10

2

)

= 45

E[XiXj |W,n] =

∫

1

u1=0

. . .

∫

1

uk=0

E[XiXj |U = u,W, n]duk . . . du1

=

∫

1

u1=0

. . .

∫

1

uk=0

E[Xi|U = u,W, n]E[Xj |U = u,W, n]duk . . . du1

=

∫

1

u1=0

. . .

∫

1

uk=0

E[Xi|Yi = yi(u|W ), n]E[Xj |Yj = yj(u|W ), n]duk . . . du1

=

∫

1

u1=0

. . .

∫

1

uk=0

E[Xi|Vi = Gi{yi(u|W ), n}]E[Xj |Vj = Gj{yj(u|W ), n}]duk . . . du1

=

1∫

u1=0

. . .

1∫

uk=0

{

vi +(1 − vi)

ni+1

ni + 1−

vni+1

i

ni + 1

}{

vj +(1 − vj)

nj+1

nj + 1−

vnj+1

j

nj + 1

}

duk . . . du1

where vi = Gi{yi(u|W,n)} = Pr(Yi ≤ yi), Yi =∑k

l=1wilUl , yi =

∑kl=1

wilul.

Validation of estimation procedures