Valores Extremos Multivariados mediante R-vines
Leonardo Moreno
II Jornadas LPE-MAREN
La Paloma, Rocha, 2014
Estaciones seleccionadas
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11020
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• Se decide trabajar con 20 estaciones que presentan datossimultaneamente en 37 anos.
Copulas: Teorema de Sklar
Para copulas bidimensionales
Sea H la funcion de distribucion conjunta de una v.a vectorial(X ,Y ) con F y G las distribuciones marginales (supongamoscontinuas). Entonces existe y es unica la copula C tal que∀(x , y) ∈ R
2cumple que,
H(x , y) = C(F (x),G(y)). (1)
Propiedades
• Sean M(u, v) ≡ min(u, v) y W (u, v) ≡ max(u + v − 1,0).Entonces para toda copula C se cumple que,
W (u, v) ≤ C(u, v) ≤ M(u, v). (2)
W y M son llamadas las cotas inferior y superior deFrechet-Hoeffding para Copulas
• Sean X e Y v.a absolutamente continuas. X e Y sonindependientes si y solo si CXY (u, v) = Π(u, v) ≡ u.v .
Densidad de la Funcion de Copula
Se llama densidad asociada a la copula C(u1,u2) a,
c(u1,u2) =∂2C(u1,u2)
∂u1∂u2. (3)
En v.a absolutamente continuas se cumple que,
f (x1, x2) = c(F1(x1),F2(x2))f1(x1)f2(x2), (4)
Copulas parametricas
AlgunasFamilias• Farlie-Gumbel-
Morgenstern
• Gaussiana
• Student
• Clayton
x
y
C(u,v)
Denisdad Cópula FGM
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
θ = 0,7
Copulas parametricas
AlgunasFamilias• Farlie-Gumbel-
Morgenstern
• Gaussiana
• Student
• Clayton
x
y
C(u,v)
Densidad Cópula Normal
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
θ = 0,25
Copulas parametricas
AlgunasFamilias• Farlie-Gumbel-
Morgenstern
• Gaussiana
• Student
• Clayton
x
y
C(u,v)
Densidad Cópula Student
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
ρ = 0,7 y 4 g.l
Copulas parametricas
AlgunasFamilias• Farlie-Gumbel-
Morgenstern
• Gaussiana
• Student
• Clayton
x
y
C(u,v)
Densidad Cópula Clayton
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
θ = 1,5
Copulas Extremas
AlgunasFamilias• Gumbel-Hougaard
• Galambos
• Husler-Reiss
• t-Extremal
x
y
C(u,v)
Denisdad Cópula Gumbel
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
α = 2/3
Copulas Extremas
AlgunasFamilias• Gumbel-Hougaard
• Galambos
• Husler-Reiss
• t-Extremal
x
y
C(u,v)
Denisdad Cópula Galambos
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
α = 2/3
Copulas Extremas
AlgunasFamilias• Gumbel-Hougaard
• Galambos
• Husler-Reiss
• t-Extremal
x
y
C(u,v)
Densidad Cópula Husler−Reiss
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0.00 0.25 0.50 0.75 1.00x
y
Simulación de 200 valores
a = 3/2
Copulas Extremas
AlgunasFamilias• Gumbel-Hougaard
• Galambos
• Husler-Reiss
• t-Extremal
x
y
C(u,v)
Densidad Cópula t−extremal
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y
Simulación de 200 valores
ρ = 2/3 y 4 g.l
Estructura de R-vines
f1,2,3,4,5 = f1f2f3f4f5c12c13c34c15c23/1c14/3c35/1c24/13c45/13c25/134,
1
2
12
5
15
3
434
13
13
12
23/1
34
14/3
15
35/1
23/1 14/3 35/145/1324/13
24/13 45/1325/134
El numero posible de R-vines en Rd es d!2 2
(d−2)(d−3)2 .
Ejemplo. C-vines
f1,2,3,4,5 = f1f2f3f4f5c12c13c14c15c23/1c24/1c25/1c34/12c35/12c45/123
A1
1
2
12
3
13
4
14
5
15
A2
12
13
23/1
14
24/1
15
25/1
A3
23/1
24/1
34/1
2
25/135/12
A4
34/12
35/12
45/123
Ejemplos: D-vines
f1,2,3,4,5 = f1f2f3f4f5c12c23c34c45c13/2c24/3c35/4c14/23c25/34c15/234,
A1
1 2 3 4 545342312
A2
12 23 34 4535/424/313/2
A3
13/2 24/3 35/425/3414/23
A4
14/23 25/3415/234
Seleccion del Modelo
La seleccion de la copula R-vines consta de 3 pasos.
1. Seleccionar la estructura R-vines entre todas la posibles.2. Elegir las d(d − 1)/2 familias parametricas de copulas de a
pares.3. Estimar los parametros de las copulas seleccionadas.
Seleccion de la estructura R-vines
• Procedimiento secuencial introducido por Diımann,Brechmann, Czado y Kurowicka (2013), [5].
• Se elige una medida de dependencia δi,j que le asigne pesos alas aristas.
• Bajo un algoritmo (usamos Prim) adecuado se selecciona unarbol, el primero del conjunto, que maximice,∑
arsitas e={i,j}
|δi,j |
Otras medidas de Dependencia
• Rango Correlacion de Kendall, (“Kendall’s tau”).
ρτ (X ,Y ) = P[(X1 − X2)(Y1 − Y2) > 0]− P[(X1 − X2)(Y1 − Y2) < 0],
donde (X1,Y1) y (X2,Y2) pares de v.a iid con distribucion F .• Coeficiente Dependencia en las Colas.
λu = lımu→1−
P(Y > F−12 (u)|X > F−1
1 (u)),
• El F -madograma (para un campo aleatorio).
νF (h) =12
E∣∣F(Z (x + h)
)− F
(Z (x)
)∣∣ , (5)
• Distancia covarianza (Szekely 2007) y MIC (Reshef 2011).
Seleccion de Familias.
• Luego de elegida la estructura se seleccionan las familias decopulas bidimensionales y se estiman los parametros de cadacopula por maxima verosimilitud.
• El metodo de simplificacion, que consiste en considerarcopulas de una sola familia, y truncamiento, suponer que apartir de cierto nivel las copulas son independientes, fueronintroducidos por Brechmann, Czado y Aas en el 2012, [1].
• Por ultimo se calculan las distribuciones condicionalesempıricas de forma recursiva pues. A partir de ellas se reiterael algoritmo.
Algunas Predicciones
perıodo (anos) Junio − Agosto
Mediana Media Desvıo P∗(supere 100 mm) P∗(supere 150 mm)
10 105.2 110.6 23.4 0.625 0.06320 116.0 122.5 25.3 0.87 0.12130 123.4 129.1 25.4 0.948 0.16140 127.2 133.5 27.4 0.98 0.19450 133.0 139.2 28.7 0.994 0.268
Referencias (1)
E.Brechmann; C. Czado; K. Aas.Truncated regular vines in high dimensions with application to financial data.Canadian Journal of Statistics, 40(1):68–85, 2012.
K. Aas; C. Czado; A. Frigessi; H. Bakken.Pair-copula constructions of multiple dependence.Insurance, Mathematics and Economics, 44:182–198, 2009.
J.F. Dissmann.Statistical inference for regular vines and application.Master’s thesis, Technische Universit at Munchen., 2010.
D. Kurowicka and R. Cooke.Uncertainty analysis with high dimensional dependence modelling.Wiley, 2006.
J. DiıMann; E. C. Brechmann; C. Czado; D. Kurowicka.Selecting and estimating regular vine copulae and application to financial returns.Computational Statistics & Data Analysis, 59:52–69, 2013.
O. Morales-Napoles.Bayesian belief nets and vines in aviation safety and other applications.PhD thesis, Technische Universiteit Delft, 2008.
Referencias (2)
R. Nelsen.An introduction to copulas.Springer-Verlag, 2006.
A. Sklar.Fonctions de repartition a n dimensions et leurs marges.Publ. Inst. Stat. Univ. Paris, 8:229–231, 1959.
R.M Cooke T. Bedford.Probability density decomposition for conditionally dependent random variables modeled by vines.Annals of Mathematics and Artificial Intelligence, 32:245–268, 2001.
H.Joe; J.J. Xu.The estimation method of inference functions for margins for multivariate models.Technical report, Department of Statistics, University of British Columbia, 1996.