Compute the correlation matrix for a low-rank structure.
corLevLowRank(par, nlevels, rank, levels,
lowerSQRT = FALSE, compGrad = TRUE,
cov = 0, impl = c("C", "R"))A correlation matrix (or its root) with the optional gradient
attribute.
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.
Number of levels \(m\).
The rank, which must be >1 and < nlevels.
Character representing the levels.
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.
Logical. Should the gradient be computed? This is only possible for the C implementation.
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.
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.
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.
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.
The q1LowRank creator of a corresponding kernel object
with class "covQual", and the similar corLevSymm
function for the full-rank case.