Copula Constrained SMC Samplers for rare-event …ucaktar/files/Presentations/...Copula Constrained...
Transcript of Copula Constrained SMC Samplers for rare-event …ucaktar/files/Presentations/...Copula Constrained...
Copula Constrained SMC Samplers for rare-event estimationin risk management
Rodrigo S. Targino *
Department of Statistical Science, University College London, UK
* joint work with Gareth W. Peters (UCL)
April 10, 2014
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Table of contents
1 Introduction
2 Overview of Sequential Monte Carlo Methods (SMC)
3 Copulas
4 Copula-Constrained SMC-Sampler
5 Linear constraints in Rd
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Table of contents
1 Introduction
2 Overview of Sequential Monte Carlo Methods (SMC)
3 Copulas
4 Copula-Constrained SMC-Sampler
5 Linear constraints in Rd
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Motivation
• Suppose we have d dependent P&L’s given by Z1, ...,Zd .
• Denote the P&L of a portfolio by Z :=∑d
i=1 Zi .
• Examples: d different positions in a credit portfolio, d risk cells in abank...
(Financial) Problem: Given the total risk of the portfolio, how to ‘coherently’distribute this risk among each of its constituents?
• If ρ( · ) is a risk measure, we want this allocation to satisfy (at least!)
ρ (Z ) =d∑
i=1
ACi ,
where ACi is the risk allocated to the i-th component of the portfolio.
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Capital allocation
• Two common risk measures are the Value at Risk (VaR) and theExpected Shortfall (ES):
VaRα(X ) := qα(X ) := inf{y ∈ R : FX (y) ≥ α};
ESα(X ) = E[X |X ≥ VaRα(X )]
• One of the most popular allocation principles is the Euler allocationprinciple (see McNeil et al. (2010)), where
ρ(Z ) = VaRα(Z ) =⇒ ACi = E[Zi |Z = VaRα(Z )]; (1)
ρ(Z ) = ESα(Z ) =⇒ ACi = E[Zi |Z ≥ VaRα(Z )]. (2)
(Statistical) Problem: How to calculate the expectations in (1) and (2), giventhat P[Z = VaRα(Z )] = 0 and P[Z ≥ VaRα(Z )] = α?
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Capital allocation
(Statistical) Problem: How to calculate the expectations in (1) and (2), giventhat P[Z = VaRα(Z )] = 0 and P[Z ≥ VaRα(Z )] = α?
• If A is either the event [Z = VaRα(Z )] = 0 or [Z ≥ VaRα(Z )] the idea isto create a sequence of nested events {At}T
t=0 shrinking to A, and, foreach t sample from (Z1, ...,Zd) |A.
• We propose to perform the sampling procedure in the “copula space” (ie,[0, 1]d ), where we can efficiently explore the space.
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Table of contents
1 Introduction
2 Overview of Sequential Monte Carlo Methods (SMC)
3 Copulas
4 Copula-Constrained SMC-Sampler
5 Linear constraints in Rd
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Sequential Monte Carlo Methods (SMC)
• We want to approximate a sequence of densities{π̃t}
t≥1 whose supportform an increasing sequence, ie, dim
(Et−1
)< dim
(Et), where
supp(π̃t)= Et .
• We may also assume that π̃t is only known up to a normalizing constant,
π̃t(x1:t) = Z−1t · f̃t(x1:t),
where x1:t := (x1, ..., xt).
• The approximation for π̃t is given by a weighted sum of random samples(also known as “particles”).
• If{
X (i)1:t , W (i)
t
}, with W (i)
t > 0 and∑N
i=1 W (i)t = 1, is a weighted sample
from π̃t we can approximate this distribution by
ˆ̃πt(dx1:t) =N∑
i=1
W (i)t δ
X (i)1:t(dx1:t).
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SMC Algorithm
Inputs: IS density q1, (forward) mutation kernels{
Kt (xt−1, xt )}T
t=1;
Result: Weighted random samples{
X (j)1:t , W (j)
t}T
t=1 approximating{π̃t}T
t=1;for t = 1 do
Choose q1(x1);Sample X1 ∼ q̃1(x1);
Calculate w1(x1) = f̃1(x1)q̃1(x1)
;
Calculate W (j)1 =
w(j)t∑N
j=1 w(j)t
;
endfor t = 2, . . . ,T do
Define q̃t (x1:t ) = q̃1(x1)∏t
j=2 Kj (xj−1, xj );Sample X1:t ∼ q̃t (x1:t ), ie, sample Xt
∣∣Xt−1 ∼ Kt (Xt−1, · );
Calculate wt (x1:t ) =f̃t (x1:t )
q̃t (x1:t )= wt−1(x1:t−1)
f̃t (x1:t )
f̃t−1(x1:t−1)Kt (xt−1, xt )︸ ︷︷ ︸incremental weight: α̃(x1:t )
;
Calculate W (j)t =
w(j)t∑N
j=1 w(j)t
.
end
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Sequential Monte Carlo Samplers
• SMC methods are very useful for dynamic models but can not bedirectly applied for static problems.
• Suppose now that our interest is to approximate a sequence ofprobability distributions {πt}t∈T such that supp(πt) = supp(πt−1) = E .
• The idea presented in Peters (2005) and Del Moral et al. (2006) is totransform this problem in an usual SMC problem, where the sequence oftarget distributions {π̃t}T
t=1 is defined on the product space, i.e.,supp(π̃t) = E × E × ...× E = E t .
• As in the dynamic case, here the distributions of interest may only beknown up to a normalizing constant:
πt(xt) = Z−1t · ft(xt)
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Sequential Monte Carlo Samplers
• The construction of π̃t is carried out as:
π̃1(x1) = π1(x1); π̃t(x1:t) = πt(xt)π̃t(x1:t−1|xt),
where π̃t(x1:t−1|xt) is a pdf on E t−1, for all xt ∈ E .• A wise choice of π̃t(x1:t−1|xt) is given by
π̃t(x1:t−1|xt) =t−1∏s=1
Ls(xs+1, xs),
where the each Ls is the density of an artificial backward Markov kernel.• Similarly, we can define
f̃t(x1:t) = ft(xt)t−1∏s=1
Ls(xs+1, xs), for t = 2, ...,T (3)
q̃t(x1:t) = q1(x1)t∏
j=2
Kj(xj−1, xj), for t = 2, ...,T (4)
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SMC Sampler Algorithm
Inputs: IS density q1, (forward) mutation kernels{
Kt (xt−1, xt )}T
t=1, (artificial)
backward kernels{
Lt−1(xt , xt−1)}T
t=1;
Result: Weighted random samples{
X (j)1:t , W (j)
t}T
t=1 approximating{πt}T
t=1;for t = 1 do
Choose q1(x1);Sample X1 ∼ q1(x1);Calculate w1(x1) = f1(x1)
q1(x1);
Calculate W (j)1 =
w(j)t∑N
j=1 w(j)t
;
endfor t = 2, . . . ,T do
Define q̃t (x1:t ) = q1(x1)∏t
j=2 Kj (xj−1, xj );Sample X1:t ∼ q̃t (x1:t ), ie, sample Xt
∣∣Xt−1 ∼ Kt (Xt−1, · );Using (3) and (4), calculate
wt (x1:t ) =f̃t (x1:t )
q̃t (x1:t )= wt−1(x1:t−1)
ft (xt )Lt−1(xt, xt−1)
ft−1(xt−1)Kt (xt−1, xt )︸ ︷︷ ︸incremental weight: α(xt−1,xt )
;
Calculate W (j)t =
w(j)t∑N
j=1 w(j)t
;
end
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Resample-Move step ( Gilks and Berzuini (2001) )
• In practice a simple SMC will eventually (as t increases) be based onlyin a few distinct particles, in the sense that almost all the other particleswill have negligible weights.
• Liu and Chen (1998) suggested using the Effective Sample Size (ESS)to measure the sample degeneracy, where
ESSt :=
N∑j=1
(W (j)
t
)2
−1
.
• We should, then, resample the particles when ESSt < M, and after thisstep, set W (j)
t = 1/N for all particles.
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Resample-Move step ( Gilks and Berzuini (2001) )
• Although the resample step alleviate the degeneracy problem, itssuccessive usage produces the so-called sample impoverishment,where the number of distinct particles is extremely small.
• In Gilks and Berzuini (2001) propose to add a move with any kernel suchthat the target distribution is invariant with respect to it to rejuvenate thesystem (for example, a MCMC kernel).
• In the MCMC case, for a proposal density q(xt , x∗t ) that moves theparticle xt to x∗t we would accept the move to x∗t with probability
α = min{
1,π̃(x∗t )qt(x∗t , xt)
π̃(xt)qt(xt , x∗t )
}.
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Table of contents
1 Introduction
2 Overview of Sequential Monte Carlo Methods (SMC)
3 Copulas
4 Copula-Constrained SMC-Sampler
5 Linear constraints in Rd
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Introduction to Copulas
Definition (Copula):
A d-dimensional copula is a distribution function on [0, 1]d with uniformmarginal distributions.
Theorem (Sklar’s):Let FZ be a joint distributions with margins FZ1 , ...,FZd . Then there exists acopula C : [0, 1]d → [0, 1] such that
FZ(z) = C(FZ1(z1), ...,FZd (zd)
), ∀z = (z1, ..., zd) ∈ R
d. (5)
If the margins are continuous then C is unique, given by
C(u1, ..., ud) = FZ(F−1Z1
(u1), ...,F−1Zd
(ud))
Conversely, if C is a copula and FZ1 , ...,FZd are univariate distributions, thenthe F defined in (5) is a joint distribution function with margins FZ1 , ...,FZd .
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Introduction to Copulas
• Copulas are a useful way to describe dependence structures in amultivariate random variable.
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0 20 40 60 80 100
020
4060
8010
0
X1
X2
0 20 40 60 80 100
0.00
0.01
0.02
0.03
0.04
0.05
U
f(U
)
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8010
0
X1
X2
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f(U
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0.0 0.2 0.4 0.6 0.8 1.0
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U1
U2
Figure: Two different joint distributions: (top) X1,X2 ∼ log-Norm(µ = 1, σ = 2.5),(bottom) X1,X2 ∼ Γ(α = 60, β = 61/60). (Right) Underlying copula: the same for bothjoint distributions.
17 / 33
Table of contents
1 Introduction
2 Overview of Sequential Monte Carlo Methods (SMC)
3 Copulas
4 Copula-Constrained SMC-Sampler
5 Linear constraints in Rd
18 / 33
Copula-Constrained SMC-Sampler
Main objective:To develop a procedure to sample from the joint distribution of Z conditionalon g(Z) ∈ A, where P[g(Z) ∈ A] is “small” (in some sense).
• For a fixed function g and set A, let {At}Tt=0 be a sequence of nested
sets shrinking to A, ie, At ⊂ At−1 and At ↓ A.• This sequence of sets define a sequence of regions:
GZt := {zt ∈ Rd : g(zt) ∈ At},
GUt :={
ut ∈ [0, 1]d :(F−1
1 (ut,1), ...,F−1d (ut,d)
)∈ GZt
}.
• These regions present a duality, in the sense that zt ∈ GZt ⇔ ut ∈ GUt ,with ut = (ut,1, ..., ut,d) :=
(F1(zt,1), ...,Fd(zt,d)
).
• The idea is to use SMC Samplers to iteratively move particles from A0 toA1 and then from A1 to A2, and then all the way to AT = A.
19 / 33
Copula-Constrained SMC-Sampler
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050
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Constraint in the original space
z1
z 2
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0.0 0.2 0.4 0.6 0.8 1.0
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Constraint in the Copula space
u1u 2
Figure: Gumbel copula with parameter 3, Log-Normal margins, both with same µ = 3,and σ1 = 0.4, σ2 = 0.6. (Left) Constraint in the original space, (Right) Constraint in theCopula space, [0, 1]2.
20 / 33
Copula-Constrained SMC-Sampler
The Copula-Constrained SMC-SamplerIn order to sample from fZ(z |g(Z) ∈ A) we can perform the samplingprocedure in the copula space!
• Let fZ(z) = c(F1(z1), ...,Fd(zd)
)∏di=1 fi(zi).
• From the duality between GZt and GUt ,
(z1, ..., zd) ∼ f (z) (z1, ..., zd) ∼ fZ(z | g(Z) ∈ A)
1. (u1, ..., ud ) ∼ c(u1, ..., ud ) 1. (u1, ..., ud ) ∼ c(u1, ..., ud |u ∈ GU)
2. z1 = F−11 (u1), ..., zd = F−1
d (ud ) 2. z1 = F−11 (u1), ..., zd = F−1
d (ud )
SMC Sampler in the bounded [0, 1]d space!
21 / 33
Copula-Constrained SMC-Sampler
The Copula-Constrained SMC-SamplerIn order to sample from fZ(z |g(Z) ∈ A) we can perform the samplingprocedure in the copula space!
• Let fZ(z) = c(F1(z1), ...,Fd(zd)
)∏di=1 fi(zi).
• From the duality between GZt and GUt ,
(z1, ..., zd) ∼ f (z) (z1, ..., zd) ∼ fZ(z | g(Z) ∈ A)
1. (u1, ..., ud ) ∼ c(u1, ..., ud ) 1. (u1, ..., ud ) ∼ c(u1, ..., ud |u ∈ GU)
2. z1 = F−11 (u1), ..., zd = F−1
d (ud ) 2. z1 = F−11 (u1), ..., zd = F−1
d (ud )
• SMC Sampler in the bounded [0, 1]d space!
22 / 33
Table of contents
1 Introduction
2 Overview of Sequential Monte Carlo Methods (SMC)
3 Copulas
4 Copula-Constrained SMC-Sampler
5 Linear constraints in Rd
23 / 33
Linear constraint
• Our constraint region is linear in the original Rd space, ie,g(Z) =
∑di=1 Zi , A = [B,+∞).
• Using the SMC Sampler’s notation, we construct the following sequenceof distributions in the copula space:
πt(ut) =c(ut)1{ut∈GUt }(ut)
P[Ut ∈ GUt ]
with
GZt :=
{zt ∈ Rd :
d∑i=1
zi ≥ Bt
},
GUt :={
ut ∈ [0, 1]d :(F−1
1 (ut,1), ...,F−1d (ut,d)
)∈ GZt
}.
• To be designed: (1) the forward kernels Kt(xt−1, xt), (2) the backwardkernels Lt−1(xt , xt−1) and (3) a proposal density q(ut , u∗t ) for theMCMC move.
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26 / 33
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Figure: Frank(2) copula, Log-Normal marginals, with parameters (µ = 3, σ = 0.4) and(µ = 3, σ = 0.6). On the left and on top, the Beta distributions used in the mutationstep.
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Example
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Figure: Frank(2) copula, Log-Normal marginals, with parameters (µ = 3, σ = 0.4) and(µ = 3, σ = 0.6). On the left and on top, the Beta distributions used in the mutationstep.
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Backward kernel Lt and MCMC proposal qt
• Since the mutation kernel at time t is given by a slice-sampler thatguarantees new particles are already in the next level set, we propose touse the same kernel as the MCMC proposal.
• For the backward kernel Lt we use an approximation of the optimalkernel (see Del Moral et al. (2006)):
Loptt (ut+1,ut) ≈
ft(ut)Kt+1(ut ,ut+1)∑Nj=1 W (j)
t Kt+1(u(j)t ,ut+1)
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Example
Figure: Frank(2) copula, Log-Normal marginals, with parameters (µ = 3, σ = 0.4) and(µ = 3, σ = 0.6).
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References I
Del Moral, P., Doucet, A., and Jasra, A. (2006). Sequential monte carlosamplers. Journal of the Royal Statistical Society: Series B (StatisticalMethodology), 68(3):411–436.
Gilks, W. R. and Berzuini, C. (2001). Following a moving targetmonte carloinference for dynamic bayesian models. Journal of the Royal StatisticalSociety: Series B (Statistical Methodology), 63(1):127–146.
Liu, J. S. and Chen, R. (1998). Sequential monte carlo methods for dynamicsystems. Journal of the American statistical association,93(443):1032–1044.
McNeil, A. J., Frey, R., and Embrechts, P. (2010). Quantitative riskmanagement: concepts, techniques, and tools. Princeton university press.
Peters, G. (2005). Topics in sequential monte carlo samplers. M. Sc.,University of Cambridge, Department of Engineering, 6.
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