Before the advent of high speed computers, considerable effort was devoted to finding approximate solutions to the problem, particularly in estimating the communalities by other means, which then simplifies the problem considerably by yielding a known reduced correlation matrix.
32.
If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the positive definite if no variable can have all its values exactly generated as a linear combination of the others.
33.
Correlation coefficients are calculated between the six time-series and the principal components are the eigenvectors of the resulting correlation matrix . ( Empirical orthogonal function analysis proceeds instead from the variance-covariance matrix which again is biased to the eastern longitudes .)
34.
In practice, typically a single correlation parameter ( not a correlation matrix ) is used to model the default correlation between any two entities in a collateralized debt obligation, CDO . Conceptually this correlation parameter should be the same for the entire CDO portfolio.
35.
Note that the node-node correlations ( or for simplicity the node correlations ) for all pairs of nodes define a symmetric correlation matrix whose ( i, j ) element is the correlation between nodes " i " and " j ".
36.
In probability and statistics, the term "'cross-correlations "'is used for referring to the correlations between the entries of " X " itself, those forming the correlation matrix ( matrix of correlations ) of " X ".
37.
For example, the unique optimal whitening transformation achieving maximal component-wise correlation between original X and whitened Y is produced by the whitening matrix W = P ^ {-1 / 2 } V ^ {-1 / 2 } where P is the correlation matrix and V the variance matrix.
38.
A special case of generalized least squares called "'weighted least squares "'occurs when all the off-diagonal entries of " ? " ( the correlation matrix of the residuals ) are null; the variances of the observations ( along the covariance matrix diagonal ) may still be unequal ( heteroscedasticity ).
39.
For the cases where has full row or column rank, and the inverse of the correlation matrix ( AA ^ * for with full row rank or A ^ * A for full column rank ) is already known, the pseudoinverse for matrices related to A can be computed by applying the Sherman Morrison Woodbury formula to update the inverse of the correlation matrix, which may need less work.
40.
For the cases where has full row or column rank, and the inverse of the correlation matrix ( AA ^ * for with full row rank or A ^ * A for full column rank ) is already known, the pseudoinverse for matrices related to A can be computed by applying the Sherman Morrison Woodbury formula to update the inverse of the correlation matrix, which may need less work.
