Replies: 1 comment
-
---- Second part Source for ideas : https://vinecopulib.github.io/ graphical structure : an undirected graph with a set of nodes and edges. Tree 1 is just on nodes 1 to d special vines tree sequences: Dvines (only path-like tree) and canonical C-nives (only star-like tree). Formule des vines régulières bedford and cooke 2001 slide 11. joli exemple slide 12. A lotmore of stuff on the fitting side in the second part of the slides that is more or less interesting. basically what we need is the dissmann algorithm. |
Beta Was this translation helpful? Give feedback.
-
Taking notes from the tutorial by C. Czado. Rememebr that goal is to implement vines and vines estimators in the package.
Remember that if f(x,y) = C(F_X(x), F_Y(y)), then f_{X [ Y}(x | y) = c(..) * f_Y(y) so nothing more is needed to compute it.
Plots of copulas should be doable on x-scale, u-scale and z-scale.
Interesting class of extreme value copulas : the TAWN copula, a non-symetric EVT class. Implementation with 3 parameters is not done anywhere yet.
Guassian kendall tau = 2/pi arcsin(\rho), student idem. These are still missingin the package we could add them.
Upper and lower tail dependence coefficients = lim{t -> 1-} (1 - 2t - C(t,t))/(1-t) and lim{t -> 0+} C(t,t)/t. Could be implemented generically and added as a feature ? Is there a multivariate equivalent ?
up = low = 0 for gaussian
up and low have formula for student and clayton and gumbel (slide 20).
Bivariate rotations and reflections are already covered by the
SurvivalCopula
class so no need to include them.Estimation mthods : difference between IFM (margins estimated first to comput ranks and estimate the copula after) and semiparametric (estimate the copula on empirical ranks).
Nonparametric bivariate: you may use the mepirical copula, but also some kernel density for margins and a bivariate kernel for the dependence structure; see Nagler & Czado 2016. -> do the bivariate kernel on z-scale and not on u-scale !
Rappel : AIC = 2llh + 2p, BIC = 2ll + log(n)*p.
ESTIMATION OF A BIVARIATE MODEL :
They do it for all bivariate families availiale in the rvinecopulib package.
-> function "bicop" in rvinecopulib makes the estimation.
-> hbicop gives conditional cdf, usefull later.
pairs-plot parfait pour les copules multivarié : z-scale en bas, u-scale en haut et histograme d'uniformité sur la diagonale ! bonus : pour Sklardist, metre les marginals au milieux.
----------------- No go to higher dims with vines.
pair copula construction through conditionning : harry Joe in 1996 alrdeay had the D-vine. bedford and cooke 2002 introduced the graph structure to organize them.
-> send the result to vine-copulas.org !
Idea : look at bivariate conditional distributions and their copulas, denoted C_{i,j; D}. while C_{i,j | D} (given) is not the same. slide 34 à reprendre.
Impotant thing: at the end of the conditioning extantions, conditional copulas have conditionals cdf in them.
Simplifying assumption : the conditional copula those not depend on the values of conditioning variables (the the conditional cdf might still !)
matrix storage of the graph : a bit complicated (slide 38) but maybe we can do better ?
Sequential two-step estimation :
!!! this depend on the simplifying assumption !
Jointly maximisation :
Then compare structures w.r.t AIC, BIC, LLH...
so the dificulty seems ot be the conditionning stuff, apart from which it is mostly MLE issue so not that difficult to do.
------------ end of firs tpart.
Beta Was this translation helpful? Give feedback.
All reactions