corLevLowRank: Correlation Matrix for a Low-Rank Structure

In kergp: Gaussian Process Laboratory

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.

See Also

The q1LowRank creator of a corresponding kernel object with class "covQual", and the similar corLevSymm function for the full-rank case.

kergp documentation built on May 29, 2024, 10:25 a.m.