copula (version 1.1-3)

pobs: Pseudo-Observations

Description

Compute the pseudo-observations for the given data matrix.

Usage

pobs(x, na.last = "keep",
     ties.method = eval(formals(rank)$ties.method), lower.tail = TRUE)

Value

matrix (or vector) of the same dimensions as x containing the pseudo-observations.

Arguments

x

\(n\times d\)-matrix (or \(d\)-vector) of random variates to be converted to pseudo-observations.

na.last

string passed to rank; see there.

ties.method

character string specifying how ranks should be computed if there are ties in any of the coordinate samples of x; passed to rank.

lower.tail

logical which, if FALSE, returns the pseudo-observations when applying the empirical marginal survival functions.

Details

Given \(n\) realizations \(\bm{x}_i=(x_{i1},\dots,x_{id})^T\), \(i\in\{1,\dots,n\}\) of a random vector \(\bm{X}\), the pseudo-observations are defined via \(u_{ij}=r_{ij}/(n+1)\) for \(i\in\{1,\dots,n\}\) and \(j\in\{1,\dots,d\}\), where \(r_{ij}\) denotes the rank of \(x_{ij}\) among all \(x_{kj}\), \(k\in\{1,\dots,n\}\). When there are no ties in any of the coordinate samples of x, the pseudo-observations can thus also be computed by component-wise applying the marginal empirical distribution functions to the data and scaling the result by \(n/(n+1)\). This asymptotically negligible scaling factor is used to force the variates to fall inside the open unit hypercube, for example, to avoid problems with density evaluation at the boundaries. Note that pobs(, lower.tail=FALSE) simply returns 1-pobs().

Examples

## Simple definition of the function:
pobs

## Draw from a multivariate normal distribution
d <- 10
set.seed(1)
P <- Matrix::nearPD(matrix(pmin(pmax(runif(d*d), 0.3), 0.99), ncol=d))$mat
diag(P) <- rep(1, d)
n <- 500
x <- MASS::mvrnorm(n, mu = rep(0, d), Sigma = P)

## Compute pseudo-observations (should roughly follow a Gauss
## copula with correlation matrix P)
u <- pobs(x)
plot(u[,5],u[,10], xlab=quote(italic(U)[1]), ylab=quote(italic(U)[2]))
if(requireNamespace("zoo")) {
 z <- zoo::zoo(x)
 uz <- pobs(z)
 stopifnot( uz == u, identical(dim(uz), dim(u)), inherits(uz, "zoo"))
}
## All components: pairwise plot
pairs(u, gap=0, pch=".", labels =
      as.expression( lapply(1:d, function(j) bquote(italic(U[.(j)]))) ))