## Documentation Center |

Canonical correlation

`[A,B] = canoncorr(X,Y)[A,B,r] = canoncorr(X,Y)[A,B,r,U,V] = canoncorr(X,Y)[A,B,r,U,V,stats] = canoncorr(X,Y) `

`[A,B] = canoncorr(X,Y)` computes
the sample canonical coefficients for the `n`-by-`d1` and `n`-by-`d2` data
matrices `X` and `Y`. `X` and `Y` must
have the same number of observations (rows) but can have different
numbers of variables (columns). `A` and `B` are `d1`-by-`d` and `d2`-by-`d` matrices,
where `d = min(rank(X),rank(Y))`. The `j`th
columns of `A` and `B` contain the
canonical coefficients, i.e., the linear combination of variables
making up the `j`th canonical variable for `X` and `Y`,
respectively. Columns of `A` and `B` are
scaled to make the covariance matrices of the canonical variables
the identity matrix (see `U` and `V` below).
If `X` or `Y` is less than full
rank, `canoncorr` gives a warning and returns zeros
in the rows of `A` or `B` corresponding
to dependent columns of `X` or `Y`.

`[A,B,r] = canoncorr(X,Y)` also
returns a 1-by-`d` vector containing the sample canonical
correlations. The `j`th element of `r` is
the correlation between the *j*th columns of `U` and `V` (see
below).

`[A,B,r,U,V] = canoncorr(X,Y)` also
returns the canonical variables, scores. `U` and `V` are `n`-by-`d` matrices
computed as

U = (X-repmat(mean(X),N,1))*A V = (Y-repmat(mean(Y),N,1))*B

`[A,B,r,U,V,stats] = canoncorr(X,Y) `
also returns a structure `stats` containing information
relating to the sequence of `d` null hypotheses
, that the (`k+1`)st
through `d`th correlations are all zero, for `k
= 0:(d-1)`. `stats` contains seven fields,
each a `1`-by-`d` vector with elements
corresponding to the values of `k`, as described
in the following table:

Field | Description |
---|---|

Wilks | Wilks' lambda (likelihood ratio) statistic |

df1 | Degrees of freedom for the chi-squared statistic, and
the numerator degrees of freedom for the |

df2 | Denominator degrees of freedom for the |

F | Rao's approximate |

pF | Right-tail significance level for |

chisq | Bartlett's approximate chi-squared statistic for with Lawley's modification |

pChisq | Right-tail significance level for |

`stats` has two other fields (`dfe` and `p`)
which are equal to `df1` and `pChisq`,
respectively, and exist for historical reasons.

[1] Krzanowski, W. J. *Principles
of Multivariate Analysis: A User's Perspective*. New York:
Oxford University Press, 1988.

[2] Seber, G. A. F. *Multivariate
Observations*. Hoboken, NJ: John Wiley & Sons, Inc.,
1984.

Was this topic helpful?