Pseudo-Observations

Compute the pseudo-observations for the given data matrix.

Usage

pseudo_obs(x, ties_method = "average", lower_tail = TRUE)

Arguments

x

vector or matrix random variates to be converted (column wise) to pseudo-observations.

ties_method

similar to ties.method of rank() (only "average", "first" and "random" currently available).

lower_tail

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

Value

a vector of matrix of the same dimension as the input containing the pseudo-observations.

Details

Given n realizations \(x_i=(x_{i1}, \ldots,x_{id})\), \(i \in \left\lbrace 1, \ldots,n \right\rbrace \) of a random vector X, the pseudo-observations are defined via \(u_{ij}=r_{ij}/(n+1)\) for \(i \in \left\lbrace 1, \ldots,n \right\rbrace\) and \(j \in \left\lbrace 1, \ldots,d \right\rbrace \), where \(r_{ij}\) denotes the rank of \(x_{ij}\) among all \(x_{kj}\), \(k \in \left\lbrace 1, \ldots,n \right\rbrace \).

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.

When lower_tail = FALSE, then pseudo_obs() simply returns 1 - pseudo_obs().

Examples

# pseudo-observations for a vector
pseudo_obs(rnorm(10))
#>  [1] 0.63636364 0.72727273 0.81818182 0.27272727 0.45454545 0.54545455
#>  [7] 0.09090909 0.18181818 0.90909091 0.36363636

# pseudo-observations for a matrix
pseudo_obs(cbind(rnorm(10), rnorm(10)))
#>             [,1]       [,2]
#>  [1,] 0.36363636 0.54545455
#>  [2,] 0.18181818 0.18181818
#>  [3,] 0.09090909 0.09090909
#>  [4,] 0.54545455 0.63636364
#>  [5,] 0.90909091 0.90909091
#>  [6,] 0.63636364 0.72727273
#>  [7,] 0.27272727 0.36363636
#>  [8,] 0.81818182 0.81818182
#>  [9,] 0.45454545 0.45454545
#> [10,] 0.72727273 0.27272727