pobs: Pseudo-Observations

Description

Compute the pseudo-observations for the given data matrix.

Usage

pobs(x, na.last = "keep",
     ties.method = c("average", "first", "random", "max", "min"))

Arguments

$n\times d$-matrix of random variates to be converted to pseudo-observations.

na.last, ties.method

are passed to rank; see there.

Value

matrix of the same dimensions as x containing the pseudo-observations.

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 $x_{kj}$, $k\in{1,\dots,n}$. The pseudo-observations can thus also be computed by component-wise applying the 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.

Examples

Run this code

## 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 Gaussian
## copula with correlation matrix P)
u <- pobs(x)
plot(u[,5],u[,10], xlab=expression(italic(U)[1]), ylab=expression(italic(U)[2]))

## All components: pairwise plot
pairs(u, gap=0, pch=".", labels=as.expression( sapply(1:d, function(j) bquote(italic(U[.(j)]))) ))