R: Correlation Matrix for a Low-Rank Structure

corLevLowRank {kergp}

R Documentation

Correlation Matrix for a Low-Rank Structure

Description

Compute the correlation matrix for a low-rank structure.

Usage


corLevLowRank(par, nlevels, rank, levels,
              lowerSQRT = FALSE, compGrad = TRUE,
              cov = 0, impl = c("C", "R"))

Arguments

`par`	A numeric vector with length `npCor + npVar` where `npCor = (rank - 1) *` `(nlevels - rank / 2)` is the number of correlation parameters, and `npVar` is the number of variance parameters, which depends on the value of `cov`. The value of `npVar` is `0`, `1` or `nlevels` corresponding to the values of `cov`: `0`, `1` and `2`. The correlation parameters are assumed to be located at the head of `par` i.e. at indices `1` to `npCor`. The variance parameter(s) are assumed to be at the tail, i.e. at indices `npCor +1` to `npCor + npVar`.
`nlevels`	Number of levels `m`.
`rank`	The rank, which must be `>1` and `< nlevels`.
`levels`	Character representing the levels.
`lowerSQRT`	Logical. When `TRUE` a lower-triangular root `\mathbf{L}` of the correlation or covariance matrix `\mathbf{C}` is returned instead of the correlation matrix. Note that this matrix can have negative diagonal elements hence is not a (pivoted) Cholesky root.
`compGrad`	Logical. Should the gradient be computed? This is only possible for the C implementation.
`cov`	Integer `0`, `1` or `2`. If `cov` is `0`, the matrix is a correlation matrix (or its root). If `cov` is `1` or `2`, the matrix is a covariance (or its root) with constant variance vector for `code = 1` and with arbitrary variance for `code = 2`. The variance parameters `par` are located at the tail of the `par` vector, so at locations `npCor + 1` to `npCor + nlevels` when `code = 2` where `npCor` is the number of correlation parameters.
`impl`	A character telling which of the C and R implementations should be chosen. The R implementation is only for checks and should not be used.

Details

The correlation matrix with size m is the general symmetric correlation matrix with rank \leq r where r is given, as described by Rapisarda et al. It depends on (r - 1) \times (m - r / 2) / 2 parameters \theta_{ij} where the indices i and j are such that 1 \leq j < i for i \leq r or such that 1 \leq j < r for r < i \leq n. The parameters \theta_{ij} are angles and are to be taken to be in [0, 2\pi) if j = 1 and in [0, \pi) otherwise.

Value

A correlation matrix (or its root) with the optional gradient attribute.

Note

This function is essentially for internal use and the corresponding correlation or covariance kernels are created as covQual objects by using the q1LowRank creator.

Here the parameters \theta_{ij} are used in row order rather than in the column order. This order simplifies the computation of the gradient.

References

Francesco Rapisarda, Damanio Brigo, Fabio Mercurio (2007). "Parameterizing Correlations a Geometric Interpretation". IMA Journal of Management Mathematics, 18(1): 55-73.

Igor Grubišić, Raoul Pietersz (2007). "Efficient Rank Reduction of Correlation Matrices". Linear Algebra and its Applications, 422: 629-653.