The multivariate probability integral transformAB458C9E-20F7-4208-BE75-D340BFF… · (Barbe et al.,...

The multivariate probability integraltransform

Fabrizio Durante

Faculty of Economics and ManagementFree University of Bozen-Bolzano (Italy)fabrizio.durante@unibz.it

http://sites.google.com/site/fbdurante

Austrian Statistical Days, 21–23 October 2015, Vienna

F. Durante (UNIBZ) The multivariate probability integral transform 1 / 58

Outline

1 Univariate Probability Integral Transform

2 Multivariate Probability Integral Transform

3 Extreme Risks and Hazard scenarios

4 Clustering extreme events

5 Conclusions

Probability integral transformLet X be a random variable on the probability space (Ω,F,P) with distributionfunction F.

If F is continuous, then F(X) is uniformly distributed on [0, 1].

The r.v. F(X) is called Probability Integral Transform (shortly, PIT) of X.

Histograms of 1000 points from Z ∼ N(0, 1) (left) and the PIT of Z (right).

Application of PITSuppose that X1, . . . ,Xn, is a random sample from a continuous d.f. F, andsuppose that one is interested to test

H0 : F = F0 vs H1 : F 6= F0

for some completely known continuous d.f. F0.

Under the null hypothesis,

F0(X(1)) < F0(X(2)) < · · · < F0(X(n)),

are distributed like the ordered statistics from a random sample of size n fromthe uniform distribution on [0, 1].Since the i–th smallest ordered value from a sample of size n from the distribu-tion of U(0, 1) has expectation i/(n+1), a plot of the points (i/(n+1),F0(X(i)),i = 1, . . . , n, should lie roughly along a straight line of slope 1, if F = F0.

Application: evaluating forecast accuracy (Diebold et al., 1998).

Quantile function theorem

Let F be a univariate distribution function.

We call quantile inverse of F the function F(−1) : (0, 1)→ (−∞,∞) given by

F(−1)(t) := infx ∈ R : F(x) ≥ t

with the convention inf ∅ = +∞.

If U is a random variable that is uniformly distributed on [0, 1], then F−1(U)has distribution function equal to F.

The previous result gives a procedure for simulating a random sample from agiven d.f. F.

Quantile function

An increasing function (left) and its corresponding quantile inverse (right).

Uniform representation of random vectors

For r.v.’s X1,X2 with continuous d.f.’s F1,F2, U1 := F1(X1) and U2 := F2(X2)are uniformly distributed on [0, 1]. Thus,

F(x1, x2) = P (X1 ≤ x1,X2 ≤ x2)

= P (F1(X1) ≤ F1(x1),F2(X2) ≤ F2(x2))

= P (U1 ≤ F1(x1),U2 ≤ F2(x2))

= C (F1(x1),F2(x2)) ,

where C is the d.f. of (U1,U2) := (F1(X1),F2(X2)).

Since the d.f. C joins or “couples” a multivariate d.f. to its one-dimensionalmarginal d.f.’s, we call it copula.

Sklar’s Theorem

Sklar’s TheoremLet (X1, . . . ,Xd) be a r.v. with joint d.f. F and univariate marginals F1, F2,. . . ,Fd. Then there exists a copula (=joint d.f. with uniform margins) C such that,for all x ∈ Rd,

F(x1, x2, . . . , xd) = C (F1(x1),F2(x2), . . . ,Fd(xd)) .

C is uniquely determined on Range(F1) × · · · × Range(Fd) and, hence, it isunique when F1, . . . ,Fd are continuous.

Every known multivariate d.f. is associated with a copula

The copula C associated with a joint d.f. F is given, for all u ∈ [0, 1]d by

C(u1, . . . , ud) = F(

F(−1)1 (u1), . . . ,F(−1)

d (ud)).

Basic examples of copulas

Basic examples of copulas are:

the independence copula Πd(u) = u1u2 · · · ud associated with a r.v. Xwhose components are independent;

the comonotonicity copula Md(u) = minu1, u2, . . . , ud associated witha r.v. X such that X d

= (T1(Z), . . . ,Td(Z)) for some random variable Zand increasing functions T1, . . . ,Td;

the countermonotonicity copula W2(u1, u2) = maxu1 + u2 − 1, 0 as-sociated with a r.v. (X1,X2) such that X2

d= T(X1) for some decreasing

function T .

Sklar’s recipe

Thanks to copulas, we may construct (and fit) parametric statistical models viaa two-stage procedure:

first, choose the univariate marginals Fα11 , . . . ,Fαd

then, choose our favorite copula Cθ,

mix the two ingredients and obtain the model

Fθ,α(x) = Cθ(Fα1

1 (x1), . . . ,Fαdd (xd)

This is potentially very useful for risk management and sensitivity analysissince it allows for testing several scenarios with different kinds of dependencebetween the assets, keeping the marginals fixed.

Tail behaviour under different scenarios

Bivariate sample clouds of 2500 points from the d.f. F = C(F1, F2) where F1, F2 ∼ N(0, 1), the Spearman’s ρ is equal to 0.75, and C is a

Gaussian copula (left) or a Gumbel copula (right).

Outline

5 Conclusions

Kendall d.f.

Let X be a continuous random vector on the probability space (Ω,F,P) whosedistribution function is equal to H. Then we say that:

The PIT of X is the random variable V = H(X).

The distribution function KX of V is called Kendall distribution functionassociated with X.

(Genest and Rivest, 2001)

Notice that, in contrast to the univariate case, it is not generally true that K isuniform on [0, 1].

Kendall d.f.

Empirical Kendall d.f. obtained from a random sample of 1000 points from a Gumbel copula with τ = 0.5.

Calculation of Kendall d.f.

It is neither possible to characterize H(x) = P(X ≤ x) nor reconstruct it fromthe knowledge of K alone. In fact, the calculation of K depends only on thecopula C of X and does not involve the knowledge the marginal distributions.

Specifically, for every t ∈ [0, 1], we have

K(t) = P(H(X) ≤ t)

= µH(x ∈ Rd : H(x) ≤ t)= µC(u ∈ [0, 1]d : C(u) ≤ t)

where µF and µC are, respectively, the measures induced by the d.f. H and thecopula C on Rd.

Properties of Kendall d.f.

K(0−) = 0

t ≤ K(t) ≤ 1 for all t ∈ [0, 1]

K(t) = t if and only if C = Md

In the bivariate case, K(t) = 1 if, and only if, C = W2

The Kendall d.f. is related to the population value of Kendall’s τ rank correla-tion coefficient for a pair of random variables (X,Y) ∼ H via the formula:

τ(X,Y) = 3−∫ 1

0K(t)dt.

Kendall d.f. and associative copula

Notice that two (different) copulas may be associated with the same Kendalld.f.. In particular, Kendall d.f.’s may be used to define an equivalence relation≡K on the class of copulas.

If C1 and C2 are copulas with Kendall d.f.’s K1 and K2, respectively, then

C1 ≡K C2 if and only if K1 = K2

TheoremIn dimension 2, every equivalence class given by the relation ≡K contains oneand only one associative copula.

(Nelsen, Quesada-Molina, Rodríguez-Lallena, Úbeda-Flores, 2009)

Kendall d.f. and Archimedean copula

A copula C is Archimedean if it can be expressed in the form

C(u) = φ−1(φ(u1) + · · ·+ φ(ud))

for a suitable generator φ.

TheoremUnder mild regularity conditions, any Kendall d.f. K is associated with aunique Archimedean copula generated by

φ(t) = exp(∫ t

dww− K(w)

)for some t0 ∈ (0, 1). (Genest and Rivest, 2001; Genest, Neslehova and Ziegel, 2011)

Estimation of the Kendall distribution

Suppose that (X11,X12), . . . , (XT1,XT2) is a random sample from a distributionH with copula C.

The empirical Kendall distribution function KT is given, for all q ∈ [0, 1], by

KT(q) =1T

T∑j=1

1(Wj ≤ q),

where, for each j ∈ 1, . . . ,T,

T∑t=1

1(Xt1 < Xj1,Xt2 < Xj2).

The empirical process√

T(KT −K) convergence in law to a centered Gaussianlimit under mild regularity conditions.

(Barbe et al., 1996; Genest, Neslehova and Ziegel, 2011)

Approximation of copulas via Kendall distribution

Given the correspondence between Kendall d.f.’s and Archimedean copulas,we can consider a general approximation of the dependence structure as fol-lows:

Consider a sequence of iid observations X1,X2, . . . from a continuousX ∼ F = C(F1, . . . ,Fd).

Choose an estimator Kn of the corresponding Kendall d.f. KC.

Provided that Kn is a bona fide Kendall d.f., construct the related Archi-medean copula.

The procedure works well even if we restrict to consider Kn from specificclasses (e.g. piecewise linear).

Example: Gumbel copula

Comparison between Kendall’s d.f. of the reference Gumbel copula, its empirical estimate, and the corresponding piecewise linear approxima-tion. See (Salvadori, Durante and Perrone, 2013).

Application: structural riskWe consider an application in coastal engineering related to the preliminarydesign of a rubble mound breakwater.

The target is to compute the quantiles associated to the weight W of a concretecube element forming the breakwater structure, assuming that the environmen-tal load is given by

H, the significant wave height (in meters),

D, the sea storm duration (in hours),and the existence of a structure function Ψ given by

W = Ψ(H,D) = ρS ·

g [4.597 · H0.328]2

)0.13/

ρW− 1)·

6.7 · N0.4d

(3600 D/ [4.597 · H0.328])0.3

For more details, see (Salvadori, Tomasicchio and D’Alessandro, 2014).

Application: structural risk

Behavior of the approximations of the cube weight for four different design quantiles and three different Sea Storm Sample Sizes (SSSS).Reference copula: Gumbel with τ = 0.5. See (Pappadà et al., 2015).

Outline

5 Conclusions

Global economic losses from extreme weather

The global insured and uninsured economic losses from the two biggest categories of weather related extreme events (category 6 “great natural

catastrophes‘” and category 5 “devastating catastrophes”) over the last 30 years from the Munich Re NatCatSE RVICE database.

Motivation

Climate Extreme (extreme weather or climate event): The occurrence of avalue of a weather or climate variable above (or below) a threshold value nearthe upper (or lower) ends of the range of observed values of the variable.

Much of the analysis of changes of extremes has, up to now, focused on in-dividual extremes of a single variable. However, recent literature in climateresearch is starting to consider compound events and explore appropriate meth-ods for their analysis.

(IPCC Report, 2012)

Motivation

The flood risk management should require the implementation of suitableflood hazard maps covering the geographical areas which could be floodedaccording to the following scenarios:

(a) floods with a low probability, or extreme event scenarios;

(b) floods with a medium probability (likely return period ≥ 100 years);

(c) floods with a high probability, where appropriate.

For each flood scenario, the following quantities should be considered:

(a) the flood extent;

(b) water depths or water level, as appropriate;

(c) where appropriate, the flow velocity or the relevant water flow.

Directive 2007/60/EC of The European Parliament and of The Council

Hazard scenario

Let X be a random vector describing the phenomenon of interest.A Hazard Scenario (hereinafter, HS) of level α ∈ (0, 1) is any Borel upper setS ⊆ Rd such that P(X ∈ S) = α.

If S is an upper set, it follows that, for all x ∈ S, S also contains all the occur-rences y ≥ x componentwise.

As will be shown below, given a realization x ∈ Rd, there exist several waysto associate x with a suitable HS, occasionally denoted by Sx ⊆ Rd.

Hazard scenario: OR

For every x ∈ Rd, OR HS is given by

S∨x =

d⋃i=1

(R× · · · × R× [xi,+∞[ ×R× · · · × R) .

For the realization of the event X ∈ S∨x it is enough that one of the variablesXi’s exceeds the corresponding threshold xi.

In turn,

α∨x = P(X ∈ S∨x

)= 1− C(F1(x1), . . . ,Fd(xd))

= 1− C(u1, . . . , ud),

where (u1, . . . , ud) = (F1(x1), . . . ,Fd(xd)).

Hazard scenario: OR

OR HS: illustration

Evaluation of dangerous flood events at the confluence of two rivers, where the threatening occurrence can be due to the contribution of oneriver, or the other, or both (Favre et al., 2004)

Hazard scenario: AND

For every x ∈ Rd AND HS is given by

S∧x =

d⋂i=1

(R× · · · × R× [xi,+∞[ ×R× · · · × R) .

For the realization of the event X ∈ S∧x it is necessary that all the variablesXi’s exceed the corresponding thresholds xi’s.

In turn,

α∧x = P(X ∈ S∧x

)= C(F1(x1), . . . ,Fd(xd)),

where F1, . . . ,Fd are the survival functions and C is the survival copula asso-ciated with C.

Hazard scenario: AND

AND HS: illustration

The Mekong Delta is at risk if bothflood peak and flood volume are high.The hydraulic system in the Viet-namese Mekong Delta is character-ized by a large number of channels,dikes and control structures such assluice gates in the dikes connectingchannels and floodplains. The pres-ence of the dike system requires cer-tain water levels [...], but as socio-economic and agricultural systems arewell adapted to the annual floods,large discharge values exceeding dikelevels do not automatically imply adisaster. For a flood event to become adisaster, it needs also a high flood vol-ume. (Dung et al., 2015)

Hazard scenario: Kendall

Given a d–dimensional continuous d.f. F with strictly increasing margins andt ∈ (0, 1) we define:

the critical layerLF

t = x ∈ Rd : F(x) = t.

the sub–critical region (lower level set)

R<t = x ∈ Rd : F(x) < t.

the super–critical region (upper level set)

R>t = x ∈ Rd : F(x) > t.

At any occurrence of the phenomenon, with probability 1, either a realizationof X lies in R>t or in R<t .

Kendall scenarioFor every x ∈ Rd with F(x) = t the Kendall HS is defined as

SKt = x ∈ Rd : F(x) > t = x ∈ Rd : C(F1(x1), . . . ,Fd(xd)) > t,

whose level α is given by

αKu = αK

x = αKt = P

(X ∈ SK

= 1− K(t),

where t = C(u) = F(x).(Salvadori, De Michele and Durante, 2011)

Kendall HS: illustration

For more details, see S. Corbella and D. D. Stretch (2012).

Hazard scenario: Survival Kendall

Survival Kendall scenarioFor every x ∈ Rd with F(x) = t, the Survival Kendall HS is defined as

SKt = x ∈ Rd : F(x) < t = x ∈ Rd : C(F1(x1), . . . ,Fd(xd)) < t,

whose level α is given by

αKx = αK

t = P(

X ∈ SKt

)= 1− K(t) = K(t),

where t = C(1− u) = F(x) and

K(t) = P(F(X1, . . . ,Xd) ≤ t

(C(F1(X1), . . . ,Fd(Xd)) ≤ t

(Salvadori, De Michele and Durante, 2014)

Hazard scenario: Survival Kendall

Survival Kendall HS: illustration

Concurrent temperature and precipitation extremes return period based on November-April data from 1896 to 2014. The blue dots represent

historical observations, and the isolines show the return periods. For more details, see AgaKouchak et al. (2014).

Survival Kendall HS: illustration

Univariate empirical return period of extreme droughts in California and their corresponding concurrent extreme (red text) return periods. The

latter includes November-April 1896-2014 precipitation and temperature data, whereas the former is solely based on precipitation in the same

period. For more details, see AgaKouchak et al. (2014).

Example: Ceppo Morelli dam (Italy)

Variables of interest: Q, maximum annual flood peak; V maximum annual volume. See (Salvadori et al., 2015).

Example: Ceppo Morelli dam (Italy)

Comparison of the Failure Probabilities (probability of occurrence of at least one extreme event under a given time period considering different

Hazard Scenarios and samples. See (Salvadori et al., 2015).

Outline

5 Conclusions

Motivation

The requirement for projections of extreme events has motivated the de-velopment of regionalization techniques to simulate weather and climateat finer spatial resolutions.

The identification of different groups in a set of climate time series isrelevant to identify subgroups characterized by similar behavior in orderto adopt specific risk management strategies.

Moreover, management of environmental risk often requires the analysisof spatial rainfall extremes which typically exhibit joint tail dependence.

The goal

We aim at creating clusters of climate time series that are homogeneous in thesense that the elements of each cluster tend to comove, while elements fromdifferent clusters are characterized by some weak dependence.

The procedure is based on the introduction of a suitable dissimilarity matrix∆ = (δij) that describes the pairwise association and it is tailored to some riskmeasure currently adopted in environmental science.

Once the dissimilarity matrix is obtained, we may apply standard clusteringtechniques like hierarchical clustering (R: hclust) or “fuzzy” clustering (R:fanny).

The clustering procedureConsider an iid sample Xt

1, . . . ,Xtn from a given r.v. X corresponding to n dif-

ferent measurements collected at time t ∈ 1, . . . ,T.

The procedure to calculate the dissimilarity matrix ∆ = (δij) consists of thefollowing steps:

Calculate the Kendall d.f. Kij for each pair (Xi,Xj).

Define a dissimilarity matrix among the time series such that the dissimi-larity between Xi and Xj is given by:

δij =

0(q− Kij(q))2dq,

where K(t) = t is the Kendall d.f. of the comonotone case.

Apply a suitable cluster algorithm.(Durante and Pappadà, 2015)

The data

Gauge stations in South–Tyrol. Total area: 7399.97 km2. Population: 511750.

The data

We consider daily rainfall measurements recorded at 18 gauge stations spreadacross the province of Bolzano-Bozen in the North-Eastern Italy.

From these time series, we extracted annual maxima at each spatial locationresulting in a 50× 18 matrix of time series observations

The selection of annual maxima transforms strong seasonal time series intodata that are assumed to be iid.

The data

Box plots of annual maxima (in mm) at each station from 1961 to 2010.

The cluster dendrogram

Dendrogram for the 18 rainfall measurement stations based on the complete linkage method.

The regionalization

Map of the rainfall measurement stations marked according the the 4-clusters solution in the province of Bolzano–Bozen (North-Eastern, Italy).

See (Durante and Pappadà, 2015).

Outline

5 Conclusions

Conclusions

We have presented the univariate PIT and shown how it is can be used tointroduce the concept of copula.

We have considered a multivariate PIT and outlined how it can be used toaddress the problem of identifying hazard scenarios, when several depen-dent variables are involved.

Several studies are hence illustrated about their use in hydrology.

Bibliography

G. Salvadori, C. De Michele, and F. Durante. On the return period and design in a multi-variate framework. Hydrol. Earth Syst. Sci., 15:3293–3305, 2011.

G. Salvadori, F. Durante, and C. De Michele. Multivariate return period calculation viasurvival functions. Water Resour. Res., 49:2308–2311, 2013.

G. Salvadori, F. Durante, and E. Perrone. Semi–parametric approximation of the Kendall’sdistribution and multivariate return periods. J. SFdS, 154(1):151–173, 2013.

F. Durante and R. Pappadà. Cluster analysis of time series via Kendall distribution. InP. Grzegorzewski, M. Gagolewski, O. Hryniewicz, and M.A. Gil, editors, StrengtheningLinks Between Data Analysis and Soft Computing, volume 315 of Advances in IntelligentSystems and Computing, pages 209–216. Springer International Publishing, 2015.

R. Pappadà, E. Perrone, F. Durante, and G. Salvadori. Spin-off extreme value and Archi-medean copulas for estimating the bivariate structural risk. Stoch. Environ. Res RiskAssess., in press, 2015.

G. Salvadori, F. Durante, C. De Michele, M. Bernardi, and L. Petrella. Hazard scenarios

and threatening assessment: guidelines for understanding the multivariate risk via copulas

and failure probabilities. Water Resour. Res., under review, 2015.

Questions? Comments?

Thanks for your attention!

More information about this talk:

visit my home-pagehttp://sites.google.com/site/fbdurante

The multivariate probability integral transformAB458C9E-20F7-4208-BE75-D340BFF… · (Barbe et al.,...

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