Description Usage Arguments Details Value Author(s) References See Also Examples

The function performs the regularized extension of the Canonical Correlation Analysis to seek correlations between two data matrices.

1 2 3 4 5 6 | ```
rcc(X,
Y,
ncomp = 2,
method = "ridge", #choose between c("ridge", "shrinkage")
lambda1 = 0,
lambda2 = 0)
``` |

`X` |
numeric matrix or data frame |

`Y` |
numeric matrix or data frame |

`method` |
One of "ridge" or "shrinkage". If "ridge", |

`ncomp` |
the number of components to include in the model. Default to 2. |

`lambda1, lambda2` |
a non-negative real. The regularization parameter for the |

The main purpose of Canonical Correlations Analysis (CCA) is the exploration of sample
correlations between two sets of variables *X* and *Y*
observed on the same individuals (experimental units)
whose roles in the analysis are strictly symmetric.

The `cancor`

function performs the core of computations
but additional tools are required to deal with data sets highly
correlated (nearly collinear), data sets with more variables
than units by example.

The `rcc`

function, the regularized version of CCA,
is one way to deal with this problem by
including a regularization step in the computations of CCA.
Such a regularization in this context
was first proposed by Vinod (1976), then developped by Leurgans *et al.* (1993).
It consists in the regularization of the empirical covariances matrices of
*X* and *Y* by adding a multiple of the matrix identity, that is,
Cov*(X)+ λ_1 I* and Cov*(Y)+ λ_2 I*.

When `lambda1=0`

and `lambda2=0`

, `rcc`

performs a classical
CCA, if possible (i.e. when *n > p+q*.

The shrinkage estimates `method = "shrinkage"`

can be used to bypass `tune.rcc`

to choose the shrinkage parameters - which can be long and costly to compute with very large data sets. Note that both functions `tune.rcc`

(which uses cross-validation) and the whrinkage parameters (which uses the formula from Schafer and Strimmer) may output different results.

Note: when `method = "shrinkage"`

the input data are centered and scaled for the estimation of the shrinkage parameters and the calculation of the regularised variance-covariance matrices in `rcc`

.

The estimation of the missing values can be performed
by the reconstitution of the data matrix using the `nipals`

function. Otherwise, missing
values are handled by casewise deletion in the `rcc`

function.

`rcc`

returns a object of class `"rcc"`

, a list that
contains the following components:

`X` |
the original |

`Y` |
the original |

`cor` |
a vector containing the canonical correlations. |

`lambda` |
a vector containing the regularization parameters whether those were input if ridge method or directly estimated with the shrinkage method. |

`loadings` |
list containing the estimated coefficients used to calculate the canonical variates in |

`variates` |
list containing the canonical variates. |

`names` |
list containing the names to be used for individuals and variables. |

Sébastien Déjean, Ignacio González, Francois Bartolo.

González, I., Déjean, S., Martin, P. G., and Baccini, A. (2008). CCA: An R package to extend canonical correlation analysis. Journal of Statistical Software, 23(12), 1-14.

González, I., Déjean, S., Martin, P., Goncalves, O., Besse, P., and Baccini, A. (2009). Highlighting relationships between heterogeneous biological data through graphical displays based on regularized canonical correlation analysis. Journal of Biological Systems, 17(02), 173-199.

Leurgans, S. E., Moyeed, R. A. and Silverman, B. W. (1993).
Canonical correlation analysis when the data are curves.
*Journal of the Royal Statistical Society. Series B* **55**, 725-740.

Vinod, H. D. (1976). Canonical ridge and econometrics of joint production.
*Journal of Econometrics* **6**, 129-137.

Opgen-Rhein, R., and K. Strimmer. 2007. Accurate ranking of differentially expressed genes by a distribution-free shrinkage approach. Statist. emphAppl. Genet. Mol. Biol. **6**:9. (http://www.bepress.com/sagmb/vol6/iss1/art9/)

Sch"afer, J., and K. Strimmer. 2005. A shrinkage approach to large-scale covariance estimation and implications for functional genomics. Statist. emphAppl. Genet. Mol. Biol. **4**:32. (http://www.bepress.com/sagmb/vol4/iss1/art32/)

`summary`

, `tune.rcc`

,
`plot.rcc`

, `plotIndiv`

,
`plotVar`

, `cim`

, `network`

and http://www.mixOmics.org for more details.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
## Classic CCA
data(linnerud)
X <- linnerud$exercise
Y <- linnerud$physiological
linn.res <- rcc(X, Y)
## Regularized CCA
data(nutrimouse)
X <- nutrimouse$lipid
Y <- nutrimouse$gene
nutri.res1 <- rcc(X, Y, ncomp = 3, lambda1 = 0.064, lambda2 = 0.008)
## using shrinkage parameters
nutri.res2 <- rcc(X, Y, ncomp = 3, method = 'shrinkage')
nutri.res2$lambda # the shrinkage parameters
``` |

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