Multivariate L-moments based on transports

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14310
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_13You do not have permission to access embedded form.

Résumé

Univariate L-moments are expressed as projections of the quantile function onto an orthogonal basis of univariate polynomials. We present multivariate versions of L-moments expressed as collections of orthogonal projections of a multivariate quantile function on a basis of multivariate polynomials. We propose to consider quantile functions defined as transports from the uniform distribution on [0; 1] d onto the distribution of interest and present some properties of the subsequent L-moments. The properties of estimated L-moments are illustrated for heavy-tailed distributions.

Multivariate L-moments based on transports

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Univariate L-moments are expressed as projections of the quantile function onto an orthogonal basis of univariate polynomials. We present multivariate versions of L-moments expressed as collections of orthogonal projections of a multivariate quantile function on a basis of multivariate polynomials. We propose to consider quantile functions defined as transports from the uniform distribution on [0; 1] d onto the distribution of interest and present some properties of the subsequent L-moments. The properties of estimated L-moments are illustrated for heavy-tailed distributions.

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Multivariate L-Moments Based on Transports Alexis Decurninge Huawei Technologies Geometric Science of Information October 29th, 2015 Outline 1 L-moments Definition of L-moments 2 Quantiles and multivariate L-moments Definitions and properties Rosenblatt quantiles and L-moments Monotone quantiles and L-moments Estimation of L-moments Numerical applications Definition of L-moments L-moments of a distribution : if X1,...,Xr are real random variables with common cumulative distribution function F λr = 1 r r−1 k=0 (−1)k r − 1 k E[Xr−k:r ] with X1:r ≤ X2:r ≤ ... ≤ Xr:r : order statistics λ1 = E[X] : localization λ2 = E[X2:2 − X1:2] : dispersion τ3 = λ3 λ2 = E[X3:3−2X2:3+X1:3] E[X2:2−X1:2] : asymmetry τ4 = λ4 λ2 = E[X4:4−3X3:4+3X2:4−X1:4] E[X2:2−X1:2] : kurtosis Existence if |x|dF(x) < ∞ Characterization of L-moments L-moments are projections of the quantile function on an orthogonal basis λr = 1 0 F−1 (t)Lr (t)dt F−1 generalized inverse of F F−1 (t) = inf {x ∈ R such that F(x) ≥ t} Lr Legendre polynomial (orthogonal basis in L2([0, 1])) Lr (t) = r k=0 (−1)k r k 2 tr−k (1 − t)k L-moments completely characterize a distribution F−1 (t) = ∞ r=1 (2r + 1)λr Lr (t) Definition of L-moments (discrete distributions) L-moments for a multinomial distribution of support x1 ≤ x2 ≤ ... ≤ xn and weights π1, ..., πn ( n i=1 πi = 1) λr = n i=1 w (r) i xi = n i=1 Kr i a=1 πa − Kr i−1 a=1 πa xi with Kr the respective primitive of Lr : Kr = Lr Empirical L-moments U-statistics : mean of all subsequences of size r without replacement ˆλr = 1 n r 1≤i1<··· 1, many multivariate quantiles has been proposed Quantiles coming from depth functions (Tukey, Zuo and Serfling) Spatial Quantiles (Chaudhuri) Generalized quantile processes (Einmahl and Mason) Quantiles as quadratic optimal transports (Galichon and Henry) Multivariate quantiles We define a quantile related to a probability measure ν as a transport from the uniform measure unif on [0; 1]d into ν. Definition Let U and X are random variables with respective measure µ and ν. T is a transport from µ into ν if T(U) = X (we note T#µ = ν). Example of transport families Optimal/monotone transports Rosenblatt transports Moser transports ... Multivariate L-moments X r.v. of interest with related measure ν such that E[ X ] < ∞. Definition Q : [0; 1]d → Rd a transport from unif in [0; 1]d into ν. L-moment λα of multi-index α = (i1, ..., id ) associated to Q : λα := [0;1]d Q(t1, ..., td )Lα(t1, ..., td )dt1...dtd ∈ Rd . with Lα(t1, ..., td ) = d k=1 Lik (tk ). ⇒ Definition compatible with the univariate case : the univariate quantile is a transport from the uniform measure on [0; 1] into the measure of interest (F−1(U) d = X) Multivariate L-moments L-moment of degree 1 λ1(= λ1,1,...,1) = [0;1]d Q(t1, ..., td )dt1...dtd = E[X]. L-moments of degree 2 can be regrouped in a matrix Λ2 = [0;1]d Qi (t1, ..., td )(2tj − 1)dt1...dtd 1≤i,j≤d . with Q(t1, ..., td ) =    Q1(t1, ..., td ) ... Qd (t1, ..., td )    Multivariate L-moments : characterization Proposition Assume that two quantiles Q and Q have same multivariate L-moments (λα)α∈Nd ∗ then Q = Q . Moreover Q(t1, ..., td ) = (i1,...,id )∈Nd ∗ d k=1 (2ik + 1) L(i1,...,id )(t1, ..., td )λ(i1,...,id ) A one-to-one correspondence between quantiles and random vectors is sufficient to guarantee the characteriation of a distribution by its L-moments Monotone transport Proposition Let µ, ν be two probability measures on Rd , such that µ does not give mass to "small sets". Then, there is exactly one measurable map T such that T#µ = ν and T = ϕ for some convex function ϕ. These transports, gradient of convex functions, are called monotone transports by analogy with the univariate case If defined, the transport is solution to the quadratic optimal transport ϕ∗ = arg inf T:T#µ=ν Rd u − T(u) 2 dµ(u) Example : monotone quantile for a random vector with independent marginals X = (X1, ..., Xd ) random vector with independent marginals. The monotone quantile of X is the collection of its marginals quantiles Q(t1, ..., td ) =    Q1(t1) ... Qd (td )    =    φ1(t1) ... φd (td )    Indeed, if φ(t1, ..., td ) = φ1(t1) + · · · + φd (td ) φ = Q The associated L-moments are then    λ1,...,1 = E[X] λ1...1,r,1,...,1 = (0, . . . , 0, λr (Xi ), 0, . . . , 0)T λα = 0 otherwise Monotone transport from the standard Gaussian distribution QN the monotone distribution from unif onto the standard Gaussian distribution N(0, Id ) defined by QN (t1, .., td ) =    N−1(t1) ... N−1(td )    T0 the monotone transport from the standard Gaussian distribution from ν (rotation equivariant) ([0; 1]d , du) QN → (Rd , dN) T0 → (Rd , dν) ⇒ Q = T0 ◦ QN is then a quantile. Monotone transport from the standard Gaussian distribution : Gaussian distribution with a random covariance For x ∈ Rd , A positive symmetric matrix ϕ(x) = m.x + 1 2 xT Ax T0(x) = ϕ(x) = m + Ax ⇒ T0(Nd (0, Id )) d = Nd (m, AAT ). The L-moments of a Gaussian with mean m and covariance AAT are : λα = m if α = (1, ..., 1) Aλα(Nd (0, Id )) otherwise In particular, the L-moments of degree 2 : Λ2 = (λ2,1...,1 . . . λ1,...,1,2) = 1 √ π A. Monotone transport from the standard Gaussian distribution : quasi-elliptic distribution For x ∈ Rd , u convex ϕ(x) = m.x + 1 2u(xT Ax) T0(x) = m + u (xT Ax)Ax. The L-moments of this distribution are then λα = m if α = (1, ..., 1) A Rd u (xT Ax)Lα(N(x))xdN(x) otherwise Si A = Id , T0(X) follows a spherical distribution Monotone transport from the standard Gaussian distribution : quasi-elliptic distribution Figure: Samples with T0(x) = − Ax xT Ax and A = Id (left) or A = 1 0.8 0.8 1 (right) Estimation : general case x1, ..., xn ∈ Rd an iid sample issued from a same r.v. X with measure ν of quantile Q. Empirical measure : νn = n i=1 δxi Estimation of Q : Qn corresponding transport from unif onto νn Empirical L-moment ˆλα = [0;1]d Qn(t)Lα(t)dt Estimation of a monotone transport Monotone transport of an absolutely continuous measure µ (of support Ω) onto the discrete measure νn Power diagrams of (x1, w1), ..., (xn, wn) 1≤i≤n u ∈ Ω s.t. u − xi 2 + wi ≤ u − xj 2 + wj ∀j = i Piecewise linear functions (PL) for any u ∈ Ω, φh(u) = max 1≤i≤n {u.xi + hi } . Areas of gradient of a PL function = power diagrams with weights wi = xi 2 + 2hi Wi (h) = {u ∈ Ω s.t. φh(u) = xi }. Estimation of a monotone transport Gradient of PL functions ⇒ Monotone transport Theorem φh is a monotone transport from µ onto νn for some h = h∗ , unique up to constant (b, ..., b), verifying h∗ = arg min h∈Rn Ω φh(u)dµ − 1 n n i=1 hi . For any 1 ≤ i ≤ n Wi (h∗ ) dµ(x) = 1 n Estimation of a monotone transport : Newton descent Computation of h∗ : minimization of E(h) = Ω φh(u)dµ − 1 n n i=1 hi Gradient descent : while | E(ht)| > η ht+1 = ht − γ( 2 E(ht))−1 E(ht) t ← t + 1 end while However : delicate Hessian computation Estimation of a monotone transport : sample in [0; 1]2 Voronoi cells of the sample Optimal power diagram Figure: Monotone transport for a sample of size 10 onto the uniform distribution on [0; 1]2 Estimation of a monotone transport : Gaussian sample Voronoi cells of the sample Optimal power diagram Figure: Monotone transport for a sample of size 100 onto the standard Gaussian Estimation of a monotone transport : consistency T transport from µ onto ν Tn transport from µ onto νn Theorem If ν verifies x dν(x) < +∞, T − Tn 1 = Rd T(x) − Tn(x) dµ(x) a.s. → 0. Q, Qn monotone quantiles having µ as a reference measure Theorem For α ∈ Nd ∗ . ˆλα = [0;1]d Qn(u)Lα(u)du a.s. → λα = [0;1]d Q(u)Lα(u)du Numerical applications We simulate a linear combination of independent vectors in R2 X = P σ1Z1 σ2Z2 with P a rotation matrix P = 1 √ 2 −1 1 1 1 Z1, Z2 are drawn from a symmetrical Weibull distribution Wν of scale parameter equal to 1 and shape parameter ν = 0.5. Numerical applications We perform N = 100 estimations of the second L-moment matrix Λ2 and the covariance matrix Σ for a sample of size n = 30 or 100. The mean of the different estimates The median of the different estimates The coefficient of variation of the estimates ˆθ1, ..., ˆθN (for an arbitrary parameter θ) CV = N i=1 ˆθi − 1 N N i=1 ˆθi 2 1/2 1 N N i=1 ˆθi n = 30 n = 100 Parameter True Value Mean Median CV Mean Median CV Λ2,11 0.38 0.28 0.27 0.30 0.38 0.37 0.18 Λ2,12 0.19 0.14 0.13 0.65 0.20 0.20 0.33 Σ11 0.69 0.70 0.48 1.23 0.69 0.59 0.55 Σ12 0.55 0.55 0.29 1.62 0.55 0.47 0.67 Thank you for your attention !