# Pseudo-Observations of Radial and Uniform Part of Elliptical and Archimedean Copulas Given a matrix of iid multivariate data from a meta-elliptical or meta-Archimedean model, `RSpobs()` computes pseudo-observations of the radial part $R$ and the vector $S$ which follows a uniform distribution on the unit sphere (for elliptical copulas) or the unit simplex (for Archimedean copulas). These quantities can be used for (graphical) goodness-of-fit tests, for example. ```r RSpobs(x, do.pobs = TRUE, method = c("ellip", "archm"), ...) ``` ## Arguments - `x`: an $(n, d)$-`matrix` of data; if `do.pobs=FALSE`, the rows of `x` are assumed to lie in the $d$-dimensional unit hypercube (if they do not, this leads to an error). - `do.pobs`: `logical` indicating whether `pobs()` is applied to `x` for transforming the data to the $d$-dimensional unit hypercube. - `method`: `character` string indicating the assumed underlying model, being meta-elliptical if `method="ellip"` (in which case `S` should be approximately uniform on the $d$-dimensional unit sphere) or meta-Archimedean if `method="archm"` (in which case `S` should be approximately uniform on the $d$-dimensional unit simplex). - ``...``: additional arguments passed to the implemented methods. These can be - **`method="ellip"`**: `qQg()` (the quantile `function` of the (assumed) distribution function $G[g]$ as given in Genest, Hofert, G. (2014)); if provided, `qQg()` is used in the transformation for obtaining pseudo-observations of $R$ and $S$ (see the code for more details). - **`method="archm"`**: `iPsi()` (the assumed underlying generator inverse); if provided, `iPsi()` is used in the transformation for obtaining pseudo-observations of $R$ and $S$ (see the code for more details). ## Returns A `list` with components `R` (an $n$-`vector` containing the pseudo-observations of the radial part) and `S` (an c("$(n,\n$", "$ d)$")-`matrix` containing the pseudo-observations of the uniform distribution (on the unit sphere/simplex)). ## Details The construction of the pseudo-obersvations of the radial part and the uniform distribution on the unit sphere/simplex is described in Genest, Hofert, G. (2014). ## References Genest, C., Hofert, M., G. , J., (2014). Is the dependence Archimedean, elliptical, or what? **To be submitted**. ## See Also `pobs()` for computing the classical pseudo-observations. ## Examples ```r set.seed(100) n <- 250 # sample size d <- 5 # dimension nu <- 3 # degrees of freedom ## Build a mean vector and a dispersion matrix, ## and generate multivariate t_nu data: mu <- rev(seq_len(d)) # d, d-1, .., 1 L <- diag(d) # identity in dim d L[lower.tri(L)] <- 1:(d*(d-1)/2)/d # Cholesky factor (diagonal > 0) Sigma <- crossprod(L) # pos.-def. dispersion matrix (*not* covariance of X) X <- rep(mu, each=n) + mvtnorm::rmvt(n, sigma=Sigma, df=nu) # multiv. t_nu data ## note: this is *wrong*: mvtnorm::rmvt(n, mean=mu, sigma=Sigma, df=nu) ## compute pseudo-observations of the radial part and uniform distribution ## once for 1a), once for 1b) below RS.t <- RSpobs(X, method="ellip", qQg=function(p) qt(p, df=nu)) # 'correct' RS.norm <- RSpobs(X, method="ellip", qQg=qnorm) # for testing 'wrong' distribution stopifnot(length(RS.norm$R) == n, length(RS.t$R) == n, dim(RS.norm$S) == c(n,d), dim(RS.t$S) == c(n,d)) ## 1) Graphically testing the radial part ## 1a) Q-Q plot of R against the correct quantiles qqplot2(RS.t$R, qF=function(p) sqrt(d*qf(p, df1=d, df2=nu)), main.args=list(text= substitute(bold(italic(F[list(d.,nu.)](r^2/d.))~~"Q-Q Plot"), list(d.=d, nu.=nu)), side=3, cex=1.3, line=1.1, xpd=NA)) ## 1b) Q-Q plot of R against the quantiles of F_R for a multivariate normal ## distribution qqplot2(RS.norm$R, qF=function(p) sqrt(qchisq(p, df=d)), main.args=list(text= substitute(bold(italic(chi[D_]) ~~ "Q-Q Plot"), list(D_=d)), side=3, cex=1.3, line=1.1, xpd=NA)) ## 2) Graphically testing the angular distribution ## auxiliary function qqp <- function(k, Bmat) { d <- ncol(Bmat) + 1 qqplot2(Bmat[,k], qF = function(p) qbeta(p, shape1=k/2, shape2=(d-k)/2), main.args=list(text= substitute(plain("Beta")(s1,s2) ~~ bold("Q-Q Plot"), list(s1 = k/2, s2 = (d-k)/2)), side=3, cex=1.3, line=1.1, xpd=NA)) } ## 2a) Q-Q plot of the 'correct' angular distribution ## (Bmat[,k] should follow a Beta(k/2, (d-k)/2) distribution) Bmat.t <- gofBTstat(RS.t$S) qqp(1, Bmat=Bmat.t) # k=1 qqp(3, Bmat=Bmat.t) # k=3 ## 2b) Q-Q plot of the 'wrong' angular distribution Bmat.norm <- gofBTstat(RS.norm$S) qqp(1, Bmat=Bmat.norm) # k=1 qqp(3, Bmat=Bmat.norm) # k=3 ## 3) Graphically check independence between radial part and B_1 and B_3 ## 'correct' distributions (multivariate t) plot(pobs(cbind(RS.t$R, Bmat.t[,1])), # k = 1 xlab=quote(italic(R)), ylab=quote(italic(B)[1]), main=quote(bold("Rank plot between"~~italic(R)~~"and"~~italic(B)[1]))) plot(pobs(cbind(RS.t$R, Bmat.t[,3])), # k = 3 xlab=quote(italic(R)), ylab=quote(italic(B)[3]), main=quote(bold("Rank plot between"~~italic(R)~~"and"~~italic(B)[3]))) ## 'wrong' distributions (multivariate normal) plot(pobs(cbind(RS.norm$R, Bmat.norm[,1])), # k = 1 xlab=quote(italic(R)), ylab=quote(italic(B)[1]), main=quote(bold("Rank plot between"~~italic(R)~~"and"~~italic(B)[1]))) plot(pobs(cbind(RS.norm$R, Bmat.norm[,3])), # k = 3 xlab=quote(italic(R)), ylab=quote(italic(B)[3]), main=quote(bold("Rank plot between"~~italic(R)~~"and"~~italic(B)[3]))) ```

RSpobs function