Pseudo-Observations
Compute the pseudo-observations for the given data matrix.
pobs(x, na.last = "keep",
ties.method = eval(formals(rank)$ties.method), lower.tail = TRUE)x |
n x d-matrix (or d-vector) of random variates to be converted to pseudo-observations. |
na.last |
string passed to |
ties.method |
string specifying how ranks should be computed if
there are ties in any of the coordinate samples of |
lower.tail |
|
Given n realizations
x_i=(x_{i1},...,x_{id}),
i in {1,...,n} of a random vector X,
the pseudo-observations are defined via u_{ij}=r_{ij}/(n+1) for
i in {1,...,n} and j in
{1,...,d}, where r_{ij} denotes the rank of x_{ij} among all
x_{kj}, k in {1,...,n}. When there are
no ties in any of the coordinate samples of x, the
pseudo-observations can thus also be computed by component-wise applying the
marginal empirical distribution functions to the data and scaling the result by
n/(n+1). This asymptotically negligible scaling factor is used to
force the variates to fall inside the open unit hypercube, for example, to
avoid problems with density evaluation at the boundaries. Note that
pobs(, lower.tail=FALSE) simply returns 1-pobs().
## Simple definition of the function:
pobs
## Draw from a multivariate normal distribution
d <- 10
set.seed(1)
P <- Matrix::nearPD(matrix(pmin(pmax(runif(d*d), 0.3), 0.99), ncol=d))$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]))
## All components: pairwise plot
pairs(u, gap=0, pch=".", labels =
as.expression( lapply(1:d, function(j) bquote(italic(U[.(j)]))) ))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.