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 7 8 |

`X` |
numeric matrix or data frame |

`Y` |
numeric matrix or data frame |

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

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

`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
shrinkage parameters (which uses the formula from Schafer and Strimmer, see the corpcor package `estimate.lambda`

) may
output different results.

Note: when `method = "shrinkage"`

the parameters are estimated using `estimate.lambda`

from the corpcor package. Data are then centered to calculate
the regularised variance-covariance matrices in `rcc`

.

Missing values are handled in the function, except when using `method = "shrinkage"`

.
In that case the estimation of the missing values can be performed by the reconstitution
of the data matrix using the `nipals`

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, Kim-Anh Lê Cao, Florian Rohart, Al J Abadi

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 16 17 18 | ```
## Classic CCA
data(linnerud)
X <- linnerud$exercise
Y <- linnerud$physiological
linn.res <- rcc(X, Y)
## Not run:
## 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
## End(Not run)
``` |

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