Non-parametric Bivariate Copulas Estimating p(a, y)=p(a)ply)c(p(a), P(y))(4) Now given a sample ((ai, yi)Jia from p(a, g) From pdf to cdf (pseudo-sample from its copula c) {(u4,v)}=1:={(P(x2),P(v)}=1 Where r.v. (u,v): u=P(r)v= P(y)
Non-parametric Bivariate Copulas • Estimating: Now From pdf to cdf (pseudo-sample from its copula c): Where r.v. (u, v): (4)
Non-parametric Bivariate Copulas (u,v)'s joint density is the copula function c(u; v) Using KDE With Gaussian kernels can approximate c(u; v) but will lead to(u, v),'s support of [0, 1 ]x[0, 1] rather than R2! Instead, performing the density estimation in a transformed space Selecting some continuous distribution with support on R strictly positive density o, cumulative distribution 4 and quantile function重 Letz=重-1(u)and=重-1() their joint pdt p(z,)=(x)()c(重(x),更()(6)
Non-parametric Bivariate Copulas (u,v)’s joint density is the copula function c(u; v)! Using KDE with Gaussian kernels can approximate c(u; v)! but will lead to (u,v)’s support of [0,1]x[0,1] rather than R2 ! Instead, performing the density estimation in a transformed space: Selecting some continuous distribution with support on R, strictly positive density , cumulative distribution and quantile function . Let their joint pdf: (6)
Non-parametric Bivariate Copulas The copula of this new density is identical to the copula of (4), since the performed transformations are marginal wise and the support of (6 )is now R2, Letx;=重1(u)andt;=重-1(v).Then, (2,0)=∑M(,,n,∑) Specially using Gauss density, having Clu.U= (,41)=1N(a,4()中(a),4(), 0((2)04(0)n台 重-1(u)o(4-1() See [Al] for more details of derivation
Non-parametric Bivariate Copulas The copula of this new density is identical to the copula of (4), since the performed transformations are marginalwise and the support of (6) is now R2 ; Specially using Gauss density, having See [A1] for more details of derivation!
Non-parametric Multivariate Copulas From Bivariate(pair copula) to multivariate(copula Extension Trick Introduction of r-vine An R-vine V for a probability density p(a1, . Id) with variable set V=(1,.d is formed y a set of undirected trees T1,..., Td-1, each of them with corresponding set of nodes Vi and set of edges Ei, where Vi= Ei-1 for i E 2, d-1. Any edge e E Ei has associated three sets C(e), D(e), N(e)c v called the conditioned, conditioning and constraint sets of e, respectively F or any edge e g, k)E Ti, i=l,., d-1 with conditioned set C(e)=j, k) and conditioning set D(e) let cik D(e) be the value of the copula density for the conditional distribution of r; and rk when conditioning on ai: iE D(e)), that is, rz:i∈D(e) x)=Ilz)ⅡeD (10) 1e(j,k)∈E
Non-parametric Multivariate Copulas From Bivariate (pair copula) to multivariate (copula): Extension Trick: Introduction of R-vine
Domain adaptation Non-linear regression with continuous data regression (yx)∝p(y ply CjklD(e) i=1e(,k)∈E Given the source pdf: ps(x, y And solving a target task with density pt(x, y)
Domain Adaptation: Non-linear regression with continuous data • regression Given the source pdf: And solving a target task with density: