Generalized canonical correlation
Encyclopedia
In statistics
, the generalized canonical correlation
analysis (gCCA), is a way of making sense of cross-correlation
matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes Principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.
(HWB) method of estimating linear regression
parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the generalized canonical correlation
Canonical correlation
In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross-covariance matrices. If we have two sets of variables, x_1, \dots, x_n and y_1, \dots, y_m, and there are correlations among the variables, then canonical correlation analysis will...
analysis (gCCA), is a way of making sense of cross-correlation
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes Principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.
Applications
The Helmert-Wolf blockingHelmert-Wolf blocking
The Helmert–Wolf blocking is a least squares solution for a sparse system of linear equations. Friedrich Robert Helmert reported on the use of such systems for geodesy in his book "Die mathematischen und physikalischen Theorieen der höheren Geodäsie, 1. Teil" published in Leipzig, 1880...
(HWB) method of estimating linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.
External links
- FactoMineR (free exploratory multivariate data analysis software linked to R)