“Canonical correlation”的意思、由来-开放百科全书

In statistics, canonical-correlation analysis (CCA, also called 'Canonical Variates Analysis') is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X₁, ..., X_n) and Y = (Y₁, ..., Y_m) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y which have maximum correlation with each other.^[1] T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables."^[2] The method was first introduced by Harold Hotelling in 1936,^[3] although in the context of angles between flats the mathematical concept was published by Jordan in 1875.^[4]

Definition

Given two column vectors

and

of random variables with finite second moments, one may define the cross-covariance

to be the

matrix whose

entry is the covariance

. In practice, we would estimate the covariance matrix based on sampled data from

and

(i.e. from a pair of data matrices).

Canonical-correlation analysis seeks vectors

(

) and

(

) such that the random variables

and

maximize the correlation

. The random variables

and

are the first pair of canonical variables. Then one seeks vectors maximizing the same correlation subject to the constraint that they are to be uncorrelated with the first pair of canonical variables; this gives the second pair of canonical variables. This procedure may be continued up to

times.

Computation

Derivation

There is equality if the vectors

and

are collinear. In addition, the maximum of correlation is attained if

is the eigenvector with the maximum eigenvalue for the matrix

(see Rayleigh quotient). The subsequent pairs are found by using eigenvalues of decreasing magnitudes. Orthogonality is guaranteed by the symmetry of the correlation matrices.

Solution

Implementation

CCA can be computed using singular value decomposition on a correlation matrix.^[5] It is available as a function in^[6]

CCA computation using singular value decomposition on a correlation matrix is related to the cosine of the angles between flats. The cosine function is ill-conditioned for small angles, leading to very inaccurate computation of highly correlated principal vectors in finite precision computer arithmetic. To fix this trouble, alternative algorithms^[7] are available in

Hypothesis testing

Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row

is zero implies all further correlations are also zero. If we have

independent observations in a sample and

is the estimated correlation for

. For the

th row, the test statistic is:

which is asymptotically distributed as a chi-squared with

degrees of freedom for large

.^[8] Since all the correlations from

are logically zero (and estimated that way also) the product for the terms after this point is irrelevant.

Practical uses

A typical use for canonical correlation in the experimental context is to take two sets of variables and see what is common among the two sets.^[9] For example, in psychological testing, one could take two well established multidimensional personality tests such as the Minnesota Multiphasic Personality Inventory (MMPI-2) and the NEO. By seeing how the MMPI-2 factors relate to the NEO factors, one could gain insight into what dimensions were common between the tests and how much variance was shared. For example, one might find that an extraversion or neuroticism dimension accounted for a substantial amount of shared variance between the two tests.

One can also use canonical-correlation analysis to produce a model equation which relates two sets of variables, for example a set of performance measures and a set of explanatory variables, or a set of outputs and set of inputs. Constraint restrictions can be imposed on such a model to ensure it reflects theoretical requirements or intuitively obvious conditions. This type of model is known as a maximum correlation model.^[10]

Visualization of the results of canonical correlation is usually through bar plots of the coefficients of the two sets of variables for the pairs of canonical variates showing significant correlation. Some authors suggest that they are best visualized by plotting them as heliographs, a circular format with ray like bars, with each half representing the two sets of variables.^[11]

Examples

Let

with zero expected value, i.e.,

. If

, i.e.,

and

are perfectly correlated, then, e.g.,

and

, so that the first (and only in this example) pair of canonical variables is

and

. If

, i.e.,

and

are perfectly anticorrelated, then, e.g.,

and

, so that the first (and only in this example) pair of canonical variables is

and

. We notice that in both cases

, which illustrates that the canonical-correlation analysis treats correlated and anticorrelated variables similarly.

Connection to principal angles

Assuming that

and

have zero expected values, i.e.,

, their covariance matrices

and

can be viewed as Gram matrices in an inner product for the entries of

and

, correspondingly. In this interpretation, the random variables, entries

and

are treated as elements of a vector space with an inner product given by the covariance

; see Covariance#Relationship to inner products.

The definition of the canonical variables

and

is then equivalent to the definition of principal vectors for the pair of subspaces spanned by the entries of

and

with respect to this inner product. The canonical correlations

is equal to the cosine of principal angles.

Whitening and probabilistic canonical correlation analysis

CCA can also be viewed as special whitening transformation where random vectors

and

are simultaneously transformed in such a way that the cross-correlation between the whitened vectors

and

is diagonal.^[12]

The canonical correlations are then interpreted as regression coefficients linking

and

and may also be negative. The regression view of CCA also provides a way to construct a latent variable probabilistic generative model for CCA, with uncorrelated hidden variables representing shared and non-shared variability.

See also

References

1. ^{{Cite book | doi = 10.1007/978-3-540-72244-1_14 | chapter = Canonical Correlation Analysis | title = Applied Multivariate Statistical Analysis | pages = 321–330 | year = 2007 | isbn = 978-3-540-72243-4 | first1 = Wolfgang | last1 = Härdle| first2 = Léopold | last2 = Simar| pmid = | pmc = | citeseerx = 10.1.1.324.403 }}
2. ^{{Cite journal | last1 = Knapp | first1 = T. R. | title = Canonical correlation analysis: A general parametric significance-testing system | doi = 10.1037/0033-2909.85.2.410 | journal = Psychological Bulletin | volume = 85 | issue = 2 | pages = 410–416 | year = 1978 | pmid = | pmc = }}
3. ^{{Cite journal | last1 = Hotelling | first1 = H. | authorlink1 = Harold Hotelling| title = Relations Between Two Sets of Variates | doi = 10.1093/biomet/28.3-4.321 | journal = Biometrika | volume = 28 | issue = 3–4 | pages = 321–377 | year = 1936 | jstor = 2333955| pmid = | pmc = }}
4. ^{{cite journal |last=Jordan |first=C. |authorlink=Camille Jordan |date=1875 |title=Essai sur la géométrie à

dimensions |journal=Bull. Soc. Math. France |volume=3 |pages=103 |url=http://www.numdam.org/item?id=BSMF_1875__3__103_2}}
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6. ^{{Cite journal | last1 = Huang | first1 = S. Y. | last2 = Lee | first2 = M. H. | last3 = Hsiao | first3 = C. K. | doi = 10.1016/j.jspi.2008.10.011 | title = Nonlinear measures of association with kernel canonical correlation analysis and applications | journal = Journal of Statistical Planning and Inference | volume = 139 | issue = 7 | pages = 2162 | year = 2009 | url = http://www.stat.sinica.edu.tw/syhuang/papersdownload/KCCA-080906.pdf| pmid = | pmc = }}
7. ^{{Citation | last = Knyazev | first = A.V. | last2 = Argentati | first2 = M.E. | title = Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates | journal = SIAM Journal on Scientific Computing | volume = 23 | issue = 6 | pages = 2009–2041 | year = 2002 | doi = 10.1137/S1064827500377332 | id = | citeseerx = 10.1.1.73.2914 }}
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11. ^{{Cite book | last1 = Degani | first1 = A. | last2 = Shafto | first2 = M. | last3 = Olson | first3 = L. | doi = 10.1007/11783183_11 | chapter = Canonical Correlation Analysis: Use of Composite Heliographs for Representing Multiple Patterns | title = Diagrammatic Representation and Inference | series = Lecture Notes in Computer Science | volume = 4045 | pages = 93 | year = 2006 | isbn = 978-3-540-35623-3 | chapter-url = http://ti.arc.nasa.gov/m/profile/adegani/Composite_Heliographs.pdf| pmid = | pmc = | citeseerx = 10.1.1.538.5217 }}
12. ^{{cite journal | last1 = Jendoubi | first1 = T. | last2 = Strimmer | first2 = K. | title = A whitening approach to probabilistic canonical correlation analysis for omics data integration | journal = BMC Bioinformatics | volume = 20 | issue = 1 | pages = 15 | year = 2018 | arxiv = 1802.03490 | doi = 10.1186/s12859-018-2572-9 | pmid = 30626338 | pmc = 6327589 }}
13. ^{{Cite journal |doi = 10.1109/TIFS.2016.2569061|title = Discriminant Correlation Analysis: Real-Time Feature Level Fusion for Multimodal Biometric Recognition|journal = IEEE Transactions on Information Forensics and Security|volume = 11|issue = 9|pages = 1984–1996|year = 2016|last1 = Haghighat|first1 = Mohammad|last2 = Abdel-Mottaleb|first2 = Mohamed|last3 = Alhalabi|first3 = Wadee}}