matrix_tools: Tools to Work with Matrices

Description

p2P() creates a matrix from a given vector of parameters. P2p() creates a numeric vector from a given matrix, currently useful for elliptical copulas.

getSigma() returns the \(d \times d\) symmetric matrix \(\Sigma\) which is called “Rho” as well, written (capital Greek \(\rho\) !) as \(P\) (and hence sometimes erronously pronounced "Pee"). Note that getSigma() works for all elliptical copulas and uses p2P() for the “unstuctured” case, dispstr = "un".

extremePairs() identifies pairs with the largest (or smallest or both) entries in a symmetric matrix.

Usage

p2P(param, d = floor(1 + sqrt(2*length(param))))
P2p(P)
getSigma(copula)
extremePairs(x, n = 6, method = c("largest", "smallest", "both"),
             use.names = FALSE)

Value

p2P:: a symmetric matrix with ones on the diagonal and the values of param filled column-wise below the diagonal (which corresponds to row-wise filling above the diagonal).
P2p:: vector of column-wise below-diagonal entries of P (equal to the row-wise above-diagonal entries in case of a symmetric matrix).
getSigma:: matrix as from p2P() for all cases of elliptical copulas.
extremePairs:: a data.frame consisting of three columns (row (index or name), col (index or name), value).

Arguments

param: a parameter vector.
d: dimension of the resulting matrix. The default is correct under the assumption (of p2P() in general!) that param is the lower-triangular part of a correlation matrix P and hence corresponds to ellipCopula(.., dispstr = "un").
P: a matrix which should be converted to a vector.
copula: an elliptical copula, i.e., an object (extending) class ellipCopula; typically resulting from tCopula() or normalCopula().
x: a symmetric matrix.
n: the number of pairs with smallest (or largest) values to be displayed.
method: a character string indicating the method to be used (with "largest" to comute the n pairs with largest entries in x (sorted in decreasing order); with "smallest" to compute the n pairs with smallest entries in x (sorted in increasing order); and with "both" to comute the 2n pairs with n largest entries and n smallest entries (sorted in decreasing order)).
use.names: A logical indicating whether colnames(x) are used as labels (if !is.null(colnames(x))).

Details

These auxiliary functions are often used when working with elliptical copulas.

Examples

Run this code

## display the two simple definitions:
p2P
P2p

param <- (2:7)/10
tC <- tCopula(param, dim = 4, dispstr = "un", df = 3)
## consistency of the three functions :
P <- p2P(param) # (using the default 'd')
stopifnot(identical(param, P2p(P)),
	  identical(P, getSigma(tC)))

## Toeplitz case:
(tCt <- tCopula((2:6)/10, dim = 6, disp = "toep"))
(rhoP <- tCt@getRho(tCt))
stopifnot(identical(getSigma (tCt),
		    toeplitz (c(1, rhoP))))

## "AR1" case:
nC.7 <- normalCopula(0.8, dim = 7, dispstr = "ar1")
(Sar1.7 <- getSigma(nC.7))
0.8^(0:(7-1)) #  1  0.8  0.64  0.512 ..
stopifnot(all.equal(Sar1.7, toeplitz(0.8^(0:(7-1)))))