Canonical correlation

Machine learning and data mining

Problems Classification Clustering Regression Anomaly detection Association rules Reinforcement learning Structured prediction Feature engineering Feature learning Online learning Semi-supervised learning Unsupervised learning Learning to rank Grammar induction
Supervised learning (classification • regression) Decision trees Ensembles (Bagging, Boosting, Random forest) k-NN Linear regression Naive Bayes Neural networks Logistic regression Perceptron Relevance vector machine (RVM) Support vector machine (SVM)
Clustering BIRCH Hierarchical k-means Expectation-maximization (EM) DBSCAN OPTICS Mean-shift
Dimensionality reduction Factor analysis CCA ICA LDA NMF PCA t-SNE
Structured prediction Graphical models (Bayes net, CRF, HMM)
Anomaly detection k-NN Local outlier factor
Neural nets Autoencoder Deep learning Multilayer perceptron RNN Restricted Boltzmann machine SOM Convolutional neural network
Reinforcement Learning Q-Learning SARSA Temporal Difference (TD)
Theory Bias-variance dilemma Computational learning theory Empirical risk minimization Occam learning PAC learning Statistical learning VC theory
Machine learning venues NIPS ICML IJMLC ML JMLR ArXiv:cs.LG
Machine learning portal

In statistics, canonical-correlation analysis (CCA) is a way of making sense of 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 the X_i and Y_j which have maximum correlation with each other.^[1] T. R. Knapp notes "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]

Definition

Given two column vectors $X=(x_{1},\dots ,x_{n})'$ and $Y=(y_{1},\dots ,y_{m})'$ of random variables with finite second moments, one may define the cross-covariance $\Sigma _{XY}=\operatorname {cov} (X,Y)$ to be the $n\times m$ matrix whose $(i,j)$ entry is the covariance $\operatorname {cov} (x_{i},y_{j})$ . In practice, we would estimate the covariance matrix based on sampled data from $X$ and $Y$ (i.e. from a pair of data matrices).

Canonical-correlation analysis seeks vectors $a$ and $b$ such that the random variables $a'X$ and $b'Y$ maximize the correlation $\rho =\operatorname {corr} (a'X,b'Y)$ . The random variables $U=a'X$ and $V=b'Y$ 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 $\min\{m,n\}$ times.

Computation

Derivation

Let $\Sigma _{XX}=\operatorname {cov} (X,X)$ and $\Sigma _{YY}=\operatorname {cov} (Y,Y)$ . The parameter to maximize is

\rho ={\frac {a'\Sigma _{XY}b}{{\sqrt {a'\Sigma _{XX}a}}{\sqrt {b'\Sigma _{YY}b}}}}.

The first step is to define a change of basis and define

c=\Sigma _{XX}^{1/2}a,

d=\Sigma _{YY}^{1/2}b.

And thus we have

\rho ={\frac {c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}{{\sqrt {c'c}}{\sqrt {d'd}}}}.

By the Cauchy-Schwarz inequality, we have

\left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\right)d\leq \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}\left(d'd\right)^{1/2},

\rho \leq {\frac {\left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}}{\left(c'c\right)^{1/2}}}.

There is equality if the vectors $d$ and $\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c$ are collinear. In addition, the maximum of correlation is attained if $c$ is the eigenvector with the maximum eigenvalue for the matrix $\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}$ (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

The solution is therefore:

$c$ is an eigenvector of $\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}$
$d$ is proportional to $\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c$

Reciprocally, there is also:

$d$ is an eigenvector of $\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1/2}$
$c$ is proportional to $\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d$

Reversing the change of coordinates, we have that

$a$ is an eigenvector of $\Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}$
$b$ is an eigenvector of $\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}$
$a$ is proportional to $\Sigma _{XX}^{-1}\Sigma _{XY}b$
$b$ is proportional to $\Sigma _{YY}^{-1}\Sigma _{YX}a$

The canonical variables are defined by:

U=c'\Sigma _{XX}^{-1/2}X=a'X

V=d'\Sigma _{YY}^{-1/2}Y=b'Y

Implementation

CCA can be computed using singular value decomposition on a correlation matrix.^[4] It is available as a function in^[5]

MATLAB as canoncorr
R as cancor or in FactoMineR or in CCP
SAS as proc cancorr
Scikit-Learn, Python as Cross decomposition
SPSS ; as macro CanCorr shipped with the main software

Hypothesis testing

Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row $i$ is zero implies all further correlations are also zero. If we have $p$ independent observations in a sample and ${\widehat {\rho }}_{i}$ is the estimated correlation for $i=1,\dots ,\min\{m,n\}$ . For the $i$ th row, the test statistic is:

\chi ^{2}=-\left(p-1-{\frac {1}{2}}(m+n+1)\right)\ln \prod _{j=i}^{\min\{m,n\}}(1-{\widehat {\rho }}_{j}^{2}),

which is asymptotically distributed as a chi-squared with $(m-i+1)(n-i+1)$ degrees of freedom for large $p$ .^[6] Since all the correlations from $\min\{m,n\}$ to $p$ 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 amongst the two sets. For example, in psychological testing, you 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, you could gain insight into what dimensions were common between the tests and how much variance was shared. For example, you 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.^[7]

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.^[8]

Examples

Let $X=x_{1}$ with zero expected value, i.e., $\operatorname {E} (X)=0$ . If $Y=X$ , i.e., $X$ and $Y$ are perfectly correlated, then, e.g., $a=1$ and $b=1$ , so that the first (and only in this example) pair of canonical variables is $U=X$ and $V=Y=X$ . If $Y=-X$ , i.e., $X$ and $Y$ are perfectly anticorrelated, then, e.g., $a=1$ and $b=-1$ , so that the first (and only in this example) pair of canonical variables is $U=X$ and $V=-Y=X$ . We notice that in both cases $U=V$ , which illustrates that the canonical-correlation analysis treats correlated and anticorrelated variables similarly.

Connection to principal angles

Assuming that $X=(x_{1},\dots ,x_{n})'$ and $Y=(y_{1},\dots ,y_{m})'$ have zero expected values, i.e., $\operatorname {E} (X)=\operatorname {E} (Y)=0$ , their covariance matrices $\Sigma _{XX}=\operatorname {Cov} (X,X)=\operatorname {E} [XX']$ and $\Sigma _{YY}=\operatorname {Cov} (Y,Y)=\operatorname {E} [YY']$ can be viewed as Gram matrices in an inner product for the entries of $X$ and $Y$ , correspondingly. In this interpretation, the random variables, entries $x_{i}$ of $X$ and $y_{j}$ of $Y$ are treated as elements of a vector space with an inner product given by the covariance $\operatorname {cov} (x_{i},y_{j})$ , see Covariance#Relationship to inner products.

The definition of the canonical variables $U$ and $V$ is then equivalent to the definition of principal vectors for the pair of subspaces spanned by the entries of $X$ and $Y$ with respect to this inner product. The canonical correlations $\operatorname {corr} (U,V)$ is equal to the cosine of principal angles.

References

↑ Härdle, Wolfgang; Simar, Léopold (2007). "Canonical Correlation Analysis". Applied Multivariate Statistical Analysis. pp. 321–330. doi:10.1007/978-3-540-72244-1_14. ISBN 978-3-540-72243-4.
↑ Knapp, T. R. (1978). "Canonical correlation analysis: A general parametric significance-testing system". Psychological Bulletin. 85 (2): 410–416. doi:10.1037/0033-2909.85.2.410.
↑ Hotelling, H. (1936). "Relations Between Two Sets of Variates". Biometrika. 28 (3–4): 321–377. doi:10.1093/biomet/28.3-4.321. JSTOR 2333955.
↑ Hsu, D.; Kakade, S. M.; Zhang, T. (2012). "A spectral algorithm for learning Hidden Markov Models" (PDF). Journal of Computer and System Sciences. 78 (5): 1460. arXiv:0811.4413. doi:10.1016/j.jcss.2011.12.025.
↑ Huang, S. Y.; Lee, M. H.; Hsiao, C. K. (2009). "Nonlinear measures of association with kernel canonical correlation analysis and applications" (PDF). Journal of Statistical Planning and Inference. 139 (7): 2162. doi:10.1016/j.jspi.2008.10.011.
↑ Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press.
↑ Tofallis, C. (1999). "Model Building with Multiple Dependent Variables and Constraints". Journal of the Royal Statistical Society: Series D (The Statistician). 48 (3): 371–378. arXiv:1109.0725. doi:10.1111/1467-9884.00195.
↑ Degani, A.; Shafto, M.; Olson, L. (2006). "Canonical Correlation Analysis: Use of Composite Heliographs for Representing Multiple Patterns". Diagrammatic Representation and Inference (PDF). Lecture Notes in Computer Science. 4045. p. 93. doi:10.1007/11783183_11. ISBN 978-3-540-35623-3.

External links

Hardoon, D. R.; Szedmak, S.; Shawe-Taylor, J. (2004). "Canonical Correlation Analysis: An Overview with Application to Learning Methods". Neural Computation. 16 (12): 2639–2664. doi:10.1162/0899766042321814. PMID 15516276.
A note on the ordinal canonical-correlation analysis of two sets of ranking scores (Also provides a FORTRAN program)- in J. of Quantitative Economics 7(2), 2009, pp. 173–199
Representation-Constrained Canonical Correlation Analysis: A Hybridization of Canonical Correlation and Principal Component Analyses (Also provides a FORTRAN program)- in J. of Applied Economic Sciences 4(1), 2009, pp. 115–124

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic Median Mode

Dispersion	Variance Standard deviation Coefficient of variation Percentile Range Interquartile range

Shape	Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Sample size determination Missing data

Survey methodology	Sampling Standard error stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in

Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife

Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F

Goodness of fit	Chi-squared Kolmogorov–Smirnov Anderson–Darling Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC

Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

Category
Portal
Commons
WikiProject

This article is issued from Wikipedia - version of the 12/2/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.