Canonical correlation

<h2 id="population-cca-definition-via-correlations">Population CCA definition via correlations</h2>
Given two <a href="/facts/Column_vectors/pPuRr9Xk">column vectors</a> 
 
 
 
 X
 =
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 
 )
 
 T
 
 
 
 
 {\displaystyle X=(x_{1},\dots ,x_{n})^{T}}
 
 and 
 
 
 
 Y
 =
 (
 
 y
 
 1
 
 
 ,
 …
 ,
 
 y
 
 m
 
 
 
 )
 
 T
 
 
 
 
 {\displaystyle Y=(y_{1},\dots ,y_{m})^{T}}
 
 of <a href="/facts/Random_variable/TwTBXnLT">random variables</a> with finite <a href="/facts/Second_moments/GL7vJ1nb">second moments</a>, one may define the <a href="/facts/Cross-covariance/nBl7HhGY">cross-covariance</a> 
 
 
 
 
 Σ
 
 X
 Y
 
 
 =
 cov
 ⁡
 (
 X
 ,
 Y
 )
 
 
 {\displaystyle \Sigma _{XY}=\operatorname {cov} (X,Y)}
 
 to be the 
 
 
 
 n
 ×
 m
 
 
 {\displaystyle n\times m}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> whose 
 
 
 
 (
 i
 ,
 j
 )
 
 
 {\displaystyle (i,j)}
 
 entry is the <a href="/facts/Covariance/hILocGNH">covariance</a> 
 
 
 
 cov
 ⁡
 (
 
 x
 
 i
 
 
 ,
 
 y
 
 j
 
 
 )
 
 
 {\displaystyle \operatorname {cov} (x_{i},y_{j})}
 
. In practice, we would estimate the covariance matrix based on sampled data from 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 (i.e. from a pair of data matrices).
Canonical-correlation analysis seeks a sequence of vectors 
 
 
 
 
 a
 
 k
 
 
 
 
 {\displaystyle a_{k}}
 
 (
 
 
 
 
 a
 
 k
 
 
 ∈
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle a_{k}\in \mathbb {R} ^{n}}
 
) and 
 
 
 
 
 b
 
 k
 
 
 
 
 {\displaystyle b_{k}}
 
 (
 
 
 
 
 b
 
 k
 
 
 ∈
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle b_{k}\in \mathbb {R} ^{m}}
 
) such that the random variables 
 
 
 
 
 a
 
 k
 
 
 T
 
 
 X
 
 
 {\displaystyle a_{k}^{T}X}
 
 and 
 
 
 
 
 b
 
 k
 
 
 T
 
 
 Y
 
 
 {\displaystyle b_{k}^{T}Y}
 
 maximize the <a href="/facts/Correlation/egkluAEm">correlation</a> 
 
 
 
 ρ
 =
 corr
 ⁡
 (
 
 a
 
 k
 
 
 T
 
 
 X
 ,
 
 b
 
 k
 
 
 T
 
 
 Y
 )
 
 
 {\displaystyle \rho =\operatorname {corr} (a_{k}^{T}X,b_{k}^{T}Y)}
 
. The (scalar) random variables 
 
 
 
 U
 =
 
 a
 
 1
 
 
 T
 
 
 X
 
 
 {\displaystyle U=a_{1}^{T}X}
 
 and 
 
 
 
 V
 =
 
 b
 
 1
 
 
 T
 
 
 Y
 
 
 {\displaystyle V=b_{1}^{T}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
 }
 
 
 {\displaystyle \min\{m,n\}}
 
 times.

(
        
          a
          
            k
          
        
        ,
        
          b
          
            k
          
        
        )
        =
        
          
            argmax
            
              a
              ,
              b
            
          
        
        corr
        ⁡
        (
        
          a
          
            T
          
        
        X
        ,
        
          b
          
            T
          
        
        Y
        )
        
        
           subject to 
        
        cov
        ⁡
        (
        
          a
          
            T
          
        
        X
        ,
        
          a
          
            j
          
          
            T
          
        
        X
        )
        =
        cov
        ⁡
        (
        
          b
          
            T
          
        
        Y
        ,
        
          b
          
            j
          
          
            T
          
        
        Y
        )
        =
        0
        
           for 
        
        j
        =
        1
        ,
        …
        ,
        k
        −
        1
      
    
    {\displaystyle (a_{k},b_{k})={\underset {a,b}{\operatorname {argmax} }}\operatorname {corr} (a^{T}X,b^{T}Y)\quad {\text{ subject to }}\operatorname {cov} (a^{T}X,a_{j}^{T}X)=\operatorname {cov} (b^{T}Y,b_{j}^{T}Y)=0{\text{ for }}j=1,\dots ,k-1}

The sets of vectors 
 
 
 
 
 a
 
 k
 
 
 ,
 
 b
 
 k
 
 
 
 
 {\displaystyle a_{k},b_{k}}
 
 are called canonical directions or weight vectors or simply weights. The 'dual' sets of vectors 
 
 
 
 
 Σ
 
 X
 X
 
 
 
 a
 
 k
 
 
 ,
 
 Σ
 
 Y
 Y
 
 
 
 b
 
 k
 
 
 
 
 {\displaystyle \Sigma _{XX}a_{k},\Sigma _{YY}b_{k}}
 
 are called canonical loading vectors or simply loadings; these are often more straightforward to interpret than the weights.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="computation">Computation</h2>
<h3>Derivation</h3>
Let 
 
 
 
 
 Σ
 
 X
 Y
 
 
 
 
 {\displaystyle \Sigma _{XY}}
 
 be the <a href="/facts/Cross-covariance_matrix/kcfhptRE">cross-covariance matrix</a> for any pair of (vector-shaped) random variables 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. The target function to maximize is

ρ
        =
        
          
            
              
                a
                
                  T
                
              
              
                Σ
                
                  X
                  Y
                
              
              b
            
            
              
                
                  
                    a
                    
                      T
                    
                  
                  
                    Σ
                    
                      X
                      X
                    
                  
                  a
                
              
              
                
                  
                    b
                    
                      T
                    
                  
                  
                    Σ
                    
                      Y
                      Y
                    
                  
                  b
                
              
            
          
        
        .
      
    
    {\displaystyle \rho ={\frac {a^{T}\Sigma _{XY}b}{{\sqrt {a^{T}\Sigma _{XX}a}}{\sqrt {b^{T}\Sigma _{YY}b}}}}.}

The first step is to define a <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a> and define

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

d
        =
        
          Σ
          
            Y
            Y
          
          
            1
            
              /
            
            2
          
        
        b
        ,
      
    
    {\displaystyle d=\Sigma _{YY}^{1/2}b,}

where 
 
 
 
 
 Σ
 
 X
 X
 
 
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle \Sigma _{XX}^{1/2}}
 
 and 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle \Sigma _{YY}^{1/2}}
 
 can be obtained from the eigen-decomposition (or by <a href="/facts/Square_root_of_a_matrix/uWGkQQqN">diagonalization</a>):

Σ
          
            X
            X
          
          
            1
            
              /
            
            2
          
        
        =
        
          V
          
            X
          
        
        
          D
          
            X
          
          
            1
            
              /
            
            2
          
        
        
          V
          
            X
          
          
            ⊤
          
        
        ,
        
        
          V
          
            X
          
        
        
          D
          
            X
          
        
        
          V
          
            X
          
          
            ⊤
          
        
        =
        
          Σ
          
            X
            X
          
        
        ,
      
    
    {\displaystyle \Sigma _{XX}^{1/2}=V_{X}D_{X}^{1/2}V_{X}^{\top },\qquad V_{X}D_{X}V_{X}^{\top }=\Sigma _{XX},}

and

Σ
          
            Y
            Y
          
          
            1
            
              /
            
            2
          
        
        =
        
          V
          
            Y
          
        
        
          D
          
            Y
          
          
            1
            
              /
            
            2
          
        
        
          V
          
            Y
          
          
            ⊤
          
        
        ,
        
        
          V
          
            Y
          
        
        
          D
          
            Y
          
        
        
          V
          
            Y
          
          
            ⊤
          
        
        =
        
          Σ
          
            Y
            Y
          
        
        .
      
    
    {\displaystyle \Sigma _{YY}^{1/2}=V_{Y}D_{Y}^{1/2}V_{Y}^{\top },\qquad V_{Y}D_{Y}V_{Y}^{\top }=\Sigma _{YY}.}

Thus

ρ
        =
        
          
            
              
                c
                
                  T
                
              
              
                Σ
                
                  X
                  X
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
              
                Σ
                
                  X
                  Y
                
              
              
                Σ
                
                  Y
                  Y
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
              d
            
            
              
                
                  
                    c
                    
                      T
                    
                  
                  c
                
              
              
                
                  
                    d
                    
                      T
                    
                  
                  d
                
              
            
          
        
        .
      
    
    {\displaystyle \rho ={\frac {c^{T}\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}{{\sqrt {c^{T}c}}{\sqrt {d^{T}d}}}}.}

By the <a href="/facts/Cauchy%25E2%2580%2593Schwarz_inequality/poMzQUIY">Cauchy–Schwarz inequality</a>, ...can someone check the this, particularly the term to the right of "(d) leq"?

(
          
            
              c
              
                T
              
            
            
              Σ
              
                X
                X
              
              
                −
                1
                
                  /
                
                2
              
            
            
              Σ
              
                X
                Y
              
            
            
              Σ
              
                Y
                Y
              
              
                −
                1
                
                  /
                
                2
              
            
          
          )
        
        (
        d
        )
        ≤
        
          
            (
            
              
                c
                
                  T
                
              
              
                Σ
                
                  X
                  X
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
              
                Σ
                
                  X
                  Y
                
              
              
                Σ
                
                  Y
                  Y
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
              
                Σ
                
                  Y
                  Y
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
              
                Σ
                
                  Y
                  X
                
              
              
                Σ
                
                  X
                  X
                
                
                  −
                  1
                  
                    /
                  
                  2
                
              
              c
            
            )
          
          
            1
            
              /
            
            2
          
        
        
          
            (
            
              
                d
                
                  T
                
              
              d
            
            )
          
          
            1
            
              /
            
            2
          
        
        ,
      
    
    {\displaystyle \left(c^{T}\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\right)(d)\leq \left(c^{T}\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^{T}d\right)^{1/2},}

ρ
        ≤
        
          
            
              
                (
                
                  
                    c
                    
                      T
                    
                  
                  
                    Σ
                    
                      X
                      X
                    
                    
                      −
                      1
                      
                        /
                      
                      2
                    
                  
                  
                    Σ
                    
                      X
                      Y
                    
                  
                  
                    Σ
                    
                      Y
                      Y
                    
                    
                      −
                      1
                    
                  
                  
                    Σ
                    
                      Y
                      X
                    
                  
                  
                    Σ
                    
                      X
                      X
                    
                    
                      −
                      1
                      
                        /
                      
                      2
                    
                  
                  c
                
                )
              
              
                1
                
                  /
                
                2
              
            
            
              
                (
                
                  
                    c
                    
                      T
                    
                  
                  c
                
                )
              
              
                1
                
                  /
                
                2
              
            
          
        
        .
      
    
    {\displaystyle \rho \leq {\frac {\left(c^{T}\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}}{\left(c^{T}c\right)^{1/2}}}.}

There is equality if the vectors 
 
 
 
 d
 
 
 {\displaystyle d}
 
 and 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 /
 
 2
 
 
 
 Σ
 
 Y
 X
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 c
 
 
 {\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}
 
 are collinear. In addition, the maximum of correlation is attained if 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is the <a href="/facts/Eigenvector/8TjEoT8u">eigenvector</a> with the maximum eigenvalue for the matrix 
 
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 
 Σ
 
 X
 Y
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 
 
 Σ
 
 Y
 X
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}
 
 (see <a href="/facts/Rayleigh_quotient/yH7sD1K4">Rayleigh quotient</a>). The subsequent pairs are found by using <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> of decreasing magnitudes. Orthogonality is guaranteed by the symmetry of the correlation matrices.
Another way of viewing this computation is that 
 
 
 
 c
 
 
 {\displaystyle c}
 
 and 
 
 
 
 d
 
 
 {\displaystyle d}
 
 are the left and right <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular vectors</a> of the correlation matrix of X and Y corresponding to the highest singular value.

<h3>Solution</h3>
The solution is therefore:

<ul><li>
 
 
 
 c
 
 
 {\displaystyle c}
 
 is an eigenvector of 
 
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 
 Σ
 
 X
 Y
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 
 
 Σ
 
 Y
 X
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}
 
</li>
<li>
 
 
 
 d
 
 
 {\displaystyle d}
 
 is proportional to 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 /
 
 2
 
 
 
 Σ
 
 Y
 X
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 c
 
 
 {\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}
 
</li></ul>
Reciprocally, there is also:

<ul><li>
 
 
 
 d
 
 
 {\displaystyle d}
 
 is an eigenvector of 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 /
 
 2
 
 
 
 Σ
 
 Y
 X
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 
 
 Σ
 
 X
 Y
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1/2}}
 
</li>
<li>
 
 
 
 c
 
 
 {\displaystyle c}
 
 is proportional to 
 
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 /
 
 2
 
 
 
 Σ
 
 X
 Y
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 /
 
 2
 
 
 d
 
 
 {\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}
 
</li></ul>
Reversing the change of coordinates, we have that

<ul><li>
 
 
 
 a
 
 
 {\displaystyle a}
 
 is an eigenvector of 
 
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 
 
 Σ
 
 X
 Y
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 
 
 Σ
 
 Y
 X
 
 
 
 
 {\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}}
 
,</li>
<li>
 
 
 
 b
 
 
 {\displaystyle b}
 
 is proportional to 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 
 
 Σ
 
 Y
 X
 
 
 a
 ;
 
 
 {\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}a;}
 
</li>
<li>
 
 
 
 b
 
 
 {\displaystyle b}
 
 is an eigenvector of 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 −
 1
 
 
 
 Σ
 
 Y
 X
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 
 
 Σ
 
 X
 Y
 
 
 ,
 
 
 {\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY},}
 
</li>
<li>
 
 
 
 a
 
 
 {\displaystyle a}
 
 is proportional to 
 
 
 
 
 Σ
 
 X
 X
 
 
 −
 1
 
 
 
 Σ
 
 X
 Y
 
 
 b
 
 
 {\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}b}
 
.</li></ul>
The canonical variables are defined by:

U
        =
        
          c
          
            T
          
        
        
          Σ
          
            X
            X
          
          
            −
            1
            
              /
            
            2
          
        
        X
        =
        
          a
          
            T
          
        
        X
      
    
    {\displaystyle U=c^{T}\Sigma _{XX}^{-1/2}X=a^{T}X}

V
        =
        
          d
          
            T
          
        
        
          Σ
          
            Y
            Y
          
          
            −
            1
            
              /
            
            2
          
        
        Y
        =
        
          b
          
            T
          
        
        Y
      
    
    {\displaystyle V=d^{T}\Sigma _{YY}^{-1/2}Y=b^{T}Y}

<h3>Implementation</h3>
CCA can be computed using <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value decomposition</a> on a correlation matrix.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> It is available as a function in<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<ul><li><a href="/facts/MATLAB/qPjLISCk">MATLAB</a> as <a href="http://www.mathworks.co.uk/help/stats/canoncorr.html">canoncorr</a> (<a href="https://sourceforge.net/p/octave/statistics/ci/default/tree/inst/canoncorr.m">also</a> in <a href="/facts/GNU_Octave/vVVq5evM">Octave</a>)</li>
<li><a href="/facts/R_(programming_language)/LSrkr8K8">R</a> as the standard function <a href="http://stat.ethz.ch/R-manual/R-devel/library/stats/html/cancor.html">cancor</a> and several other packages, including <a href="https://cran.r-project.org/web/packages/CCA/index.html">CCA</a> and <a href="https://cran.r-project.org/web/packages/vegan/index.html">vegan</a>. <a href="https://cran.r-project.org/web/packages/CCP/index.html">CCP</a> for statistical hypothesis testing in canonical correlation analysis.</li>
<li><a href="/facts/SAS_language/pV7hwEJV">SAS</a> as <a href="https://go.documentation.sas.com/doc/en/pgmsascdc/9.4_3.5/statug/statug_cancorr_toc.htm">proc cancorr</a></li>
<li><a href="/facts/Python_(programming_language)/YbuGqofa">Python</a> in the library <a href="/facts/Scikit-learn/HqxDHRMJ">scikit-learn</a>, as <a href="http://scikit-learn.org/stable/modules/cross_decomposition.html">cross decomposition</a> and in statsmodels, as <a href="https://devdocs.io/statsmodels/generated/statsmodels.multivariate.cancorr.cancorr">CanCorr</a>. The CCA-Zoo library <a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> implements CCA extensions, such as probabilistic CCA, sparse CCA, multi-view CCA, and deep CCA.</li>
<li><a href="/facts/SPSS/6PGBVUXF">SPSS</a> as macro CanCorr shipped with the main software</li>
<li><a href="/facts/Julia_(programming_language)/AoB0PJ9C">Julia (programming language)</a> in the <a href="https://github.com/JuliaStats/MultivariateStats.jl">MultivariateStats.jl</a> package.</li></ul>
CCA computation using <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value decomposition</a> on a correlation matrix is related to the <a href="/facts/Cosine/czej7ur0">cosine</a> of the <a href="/facts/Angles_between_flats/JMrfQQfL">angles between flats</a>. The <a href="/facts/Cosine/czej7ur0">cosine</a> function is <a href="/facts/Ill-conditioned/9Tjtk6wh">ill-conditioned</a> for small angles, leading to very inaccurate computation of highly correlated principal vectors in finite <a href="/facts/Precision_(computer_science)/6rgoFI6d">precision</a> <a href="/facts/Computer_arithmetic/SgIjPzSc">computer arithmetic</a>. To <a href="/facts/Angles_between_flats/JMrfQQfL">fix this trouble</a>, alternative algorithms<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> are available in

<ul><li><a href="/facts/SciPy/bNcMQGc0">SciPy</a> as <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.subspace_angles.html">linear-algebra function subspace_angles</a></li>
<li><a href="/facts/MATLAB/qPjLISCk">MATLAB</a> as <a href="https://www.mathworks.com/matlabcentral/fileexchange/55-subspacea-m">FileExchange function subspacea</a></li></ul>
<h2 id="hypothesis-testing">Hypothesis testing</h2>
Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is zero implies all further correlations are also zero. If we have 
 
 
 
 p
 
 
 {\displaystyle p}
 
 independent observations in a sample and 
 
 
 
 
 
 
 
 ρ
 ^
 
 
 
 
 i
 
 
 
 
 {\displaystyle {\widehat {\rho }}_{i}}
 
 is the estimated correlation for 
 
 
 
 i
 =
 1
 ,
 …
 ,
 min
 {
 m
 ,
 n
 }
 
 
 {\displaystyle i=1,\dots ,\min\{m,n\}}
 
. For the 
 
 
 
 i
 
 
 {\displaystyle i}
 
th row, the test statistic is:

χ
          
            2
          
        
        =
        −
        
          (
          
            p
            −
            1
            −
            
              
                1
                2
              
            
            (
            m
            +
            n
            +
            1
            )
          
          )
        
        ln
        ⁡
        
          ∏
          
            j
            =
            i
          
          
            min
            {
            m
            ,
            n
            }
          
        
        (
        1
        −
        
          
            
              
                ρ
                ^
              
            
          
          
            j
          
          
            2
          
        
        )
        ,
      
    
    {\displaystyle \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 <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared</a> with 
 
 
 
 (
 m
 −
 i
 +
 1
 )
 (
 n
 −
 i
 +
 1
 )
 
 
 {\displaystyle (m-i+1)(n-i+1)}
 
 <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86">degrees of freedom</a> for large 
 
 
 
 p
 
 
 {\displaystyle p}
 
.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> Since all the correlations from 
 
 
 
 min
 {
 m
 ,
 n
 }
 
 
 {\displaystyle \min\{m,n\}}
 
 to 
 
 
 
 p
 
 
 {\displaystyle p}
 
 are logically zero (and estimated that way also) the product for the terms after this point is irrelevant.
Note that in the small sample size limit with 
 
 
 
 p
 <
 n
 +
 m
 
 
 {\displaystyle p<n+m}
 
 then we are guaranteed that the top 
 
 
 
 m
 +
 n
 −
 p
 
 
 {\displaystyle m+n-p}
 
 correlations will be identically 1 and hence the test is meaningless.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h2 id="practical-uses">Practical uses</h2>
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.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> For example, in psychological testing, one could take two well established multidimensional <a href="/facts/Personality_tests/elzLWUmX">personality tests</a> such as the <a href="/facts/Minnesota_Multiphasic_Personality_Inventory/bCTlY94q">Minnesota Multiphasic Personality Inventory</a> (MMPI-2) and the <a href="/facts/Neuroticism_Extraversion_Openness_Personality_Inventory/37pYRth9">NEO</a>. 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 <a href="/facts/Extraversion_and_introversion/MUQ9pRhe">extraversion</a> or <a href="/facts/Neuroticism/tmhrDpUS">neuroticism</a> 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.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>
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.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="examples">Examples</h2>
Let 
 
 
 
 X
 =
 
 x
 
 1
 
 
 
 
 {\displaystyle X=x_{1}}
 
 with zero <a href="/facts/Expected_value/1XV0JKL8">expected value</a>, i.e., 
 
 
 
 E
 ⁡
 (
 X
 )
 =
 0
 
 
 {\displaystyle \operatorname {E} (X)=0}
 
. 

<ol><li>If 
 
 
 
 Y
 =
 X
 
 
 {\displaystyle Y=X}
 
, i.e., 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are perfectly correlated, then, e.g., 
 
 
 
 a
 =
 1
 
 
 {\displaystyle a=1}
 
 and 
 
 
 
 b
 =
 1
 
 
 {\displaystyle b=1}
 
, so that the first (and only in this example) pair of canonical variables is 
 
 
 
 U
 =
 X
 
 
 {\displaystyle U=X}
 
 and 
 
 
 
 V
 =
 Y
 =
 X
 
 
 {\displaystyle V=Y=X}
 
.</li>
<li>If 
 
 
 
 Y
 =
 −
 X
 
 
 {\displaystyle Y=-X}
 
, i.e., 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are perfectly anticorrelated, then, e.g., 
 
 
 
 a
 =
 1
 
 
 {\displaystyle a=1}
 
 and 
 
 
 
 b
 =
 −
 1
 
 
 {\displaystyle b=-1}
 
, so that the first (and only in this example) pair of canonical variables is 
 
 
 
 U
 =
 X
 
 
 {\displaystyle U=X}
 
 and 
 
 
 
 V
 =
 −
 Y
 =
 X
 
 
 {\displaystyle V=-Y=X}
 
.</li></ol>
We notice that in both cases 
 
 
 
 U
 =
 V
 
 
 {\displaystyle U=V}
 
, which illustrates that the canonical-correlation analysis treats correlated and anticorrelated variables similarly.

<h2 id="connection-to-principal-angles">Connection to principal angles</h2>
Assuming that 
 
 
 
 X
 =
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 
 )
 
 T
 
 
 
 
 {\displaystyle X=(x_{1},\dots ,x_{n})^{T}}
 
 and 
 
 
 
 Y
 =
 (
 
 y
 
 1
 
 
 ,
 …
 ,
 
 y
 
 m
 
 
 
 )
 
 T
 
 
 
 
 {\displaystyle Y=(y_{1},\dots ,y_{m})^{T}}
 
 have zero <a href="/facts/Expected_value/1XV0JKL8">expected values</a>, i.e., 
 
 
 
 E
 ⁡
 (
 X
 )
 =
 E
 ⁡
 (
 Y
 )
 =
 0
 
 
 {\displaystyle \operatorname {E} (X)=\operatorname {E} (Y)=0}
 
, their <a href="/facts/Covariance/hILocGNH">covariance</a> matrices 
 
 
 
 
 Σ
 
 X
 X
 
 
 =
 Cov
 ⁡
 (
 X
 ,
 X
 )
 =
 E
 ⁡
 [
 X
 
 X
 
 T
 
 
 ]
 
 
 {\displaystyle \Sigma _{XX}=\operatorname {Cov} (X,X)=\operatorname {E} [XX^{T}]}
 
 and 
 
 
 
 
 Σ
 
 Y
 Y
 
 
 =
 Cov
 ⁡
 (
 Y
 ,
 Y
 )
 =
 E
 ⁡
 [
 Y
 
 Y
 
 T
 
 
 ]
 
 
 {\displaystyle \Sigma _{YY}=\operatorname {Cov} (Y,Y)=\operatorname {E} [YY^{T}]}
 
 can be viewed as <a href="/facts/Gram_matrix/SGb2VGTU">Gram matrices</a> in an <a href="/facts/Inner_product/HoyjElEy">inner product</a> for the entries of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, correspondingly. In this interpretation, the random variables, entries 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 
 y
 
 j
 
 
 
 
 {\displaystyle y_{j}}
 
 of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are treated as elements of a vector space with an inner product given by the <a href="/facts/Covariance/hILocGNH">covariance</a> 
 
 
 
 cov
 ⁡
 (
 
 x
 
 i
 
 
 ,
 
 y
 
 j
 
 
 )
 
 
 {\displaystyle \operatorname {cov} (x_{i},y_{j})}
 
; see <a href="/facts/Covariance/hILocGNH">Covariance#Relationship to inner products</a>.
The definition of the canonical variables 
 
 
 
 U
 
 
 {\displaystyle U}
 
 and 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is then equivalent to the definition of <a href="/facts/Principal_angles/JMrfQQfL">principal vectors</a> for the pair of subspaces spanned by the entries of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 with respect to this <a href="/facts/Inner_product/HoyjElEy">inner product</a>. The canonical correlations 
 
 
 
 corr
 ⁡
 (
 U
 ,
 V
 )
 
 
 {\displaystyle \operatorname {corr} (U,V)}
 
 is equal to the <a href="/facts/Cosine/czej7ur0">cosine</a> of <a href="/facts/Principal_angles/JMrfQQfL">principal angles</a>.

<h2 id="whitening-and-probabilistic-canonical-correlation-analysis">Whitening and probabilistic canonical correlation analysis</h2>
CCA can also be viewed as a special <a href="/facts/Whitening_transformation/q3MhJVIY">whitening transformation</a> where the random vectors 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are simultaneously transformed in such a way that the cross-correlation between the whitened vectors 
 
 
 
 
 X
 
 C
 C
 A
 
 
 
 
 {\displaystyle X^{CCA}}
 
 and 
 
 
 
 
 Y
 
 C
 C
 A
 
 
 
 
 {\displaystyle Y^{CCA}}
 
 is diagonal.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
The canonical correlations are then interpreted as regression coefficients linking 
 
 
 
 
 X
 
 C
 C
 A
 
 
 
 
 {\displaystyle X^{CCA}}
 
 and 
 
 
 
 
 Y
 
 C
 C
 A
 
 
 
 
 {\displaystyle Y^{CCA}}
 
 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.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Generalized_canonical_correlation/GjLEnT8M">Generalized canonical correlation</a></li>
<li><a href="/facts/RV_coefficient/CwovuFMv">RV coefficient</a></li>
<li><a href="/facts/Angles_between_flats/JMrfQQfL">Angles between flats</a></li>
<li><a href="/facts/Principal_component_analysis/pCbVTqdR">Principal component analysis</a></li>
<li><a href="/facts/Linear_discriminant_analysis/Mj6KQhRN">Linear discriminant analysis</a></li>
<li><a href="/facts/Regularized_canonical_correlation_analysis/9uxpmFh8">Regularized canonical correlation analysis</a></li>
<li><a href="/facts/Singular_value_decomposition/8t3v6wk3">Singular value decomposition</a></li>
<li><a href="/facts/Partial_least_squares_regression/jYIcInPw">Partial least squares regression</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://github.com/mhaghighat/dcaFuse">Discriminant Correlation Analysis (DCA)</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> (<a href="/facts/MATLAB/qPjLISCk">MATLAB</a>)</li>
<li>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. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.6452">10.1.1.14.6452</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1162%2F0899766042321814">10.1162/0899766042321814</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/15516276">15516276</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:202473">202473</a>.</li>
<li><a href="http://mpra.ub.uni-muenchen.de/12796/">A note on the ordinal canonical-correlation analysis of two sets of ranking scores</a> (Also provides a <a href="/facts/FORTRAN/m1KqjcMU">FORTRAN</a> program)- in Journal of Quantitative Economics 7(2), 2009, pp. 173–199</li>
<li><a href="http://ssrn.com/abstract=1331886">Representation-Constrained Canonical Correlation Analysis: A Hybridization of Canonical Correlation and Principal Component Analyses</a> (Also provides a <a href="/facts/FORTRAN/m1KqjcMU">FORTRAN</a> program)- in Journal of Applied Economic Sciences 4(1), 2009, pp. 115–124</li></ul>

<h2 id="references">References</h2>

<ol>
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<li id="fn:2">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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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. <a href="/wiki/Harold_Hotelling" target="_blank">/wiki/Harold_Hotelling</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Jordan, C. (1875). "Essai sur la géométrie à 
 
 
 
 n
 
 
 {\displaystyle n}
 
 dimensions". Bull. Soc. Math. France. 3: 103. <a href="/wiki/Camille_Jordan" target="_blank">/wiki/Camille_Jordan</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Andrew, Galen; Arora, Raman; Bilmes, Jeff; Livescu, Karen (2013-05-26). "Deep Canonical Correlation Analysis". Proceedings of the 30th International Conference on Machine Learning. PMLR: 1247–1255. <a href="https://proceedings.mlr.press/v28/andrew13.html" target="_blank">https://proceedings.mlr.press/v28/andrew13.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Ju, Ce; Kobler, Reinmar J; Tang, Liyao; Guan, Cuntai; Kawanabe, Motoaki (2024). Deep Geodesic Canonical Correlation Analysis for Covariance-Based Neuroimaging Data. The Twelfth International Conference on Learning Representations (ICLR 2024, spotlight). <a href="https://openreview.net/pdf?id=PnR1MNen7u" target="_blank">https://openreview.net/pdf?id=PnR1MNen7u</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Statistical Learning with Sparsity: the Lasso and Generalizations". hastie.su.domains. Retrieved 2023-09-12. <a href="https://hastie.su.domains/StatLearnSparsity/" target="_blank">https://hastie.su.domains/StatLearnSparsity/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Gu, Fei; Wu, Hao (2018-04-01). "Simultaneous canonical correlation analysis with invariant canonical loadings". Behaviormetrika. 45 (1): 111–132. doi:10.1007/s41237-017-0042-8. ISSN 1349-6964. <a href="https://doi.org/10.1007/s41237-017-0042-8" target="_blank">https://doi.org/10.1007/s41237-017-0042-8</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">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. S2CID 220740158. <a href="http://www.cs.mcgill.ca/~colt2009/papers/011.pdf" target="_blank">http://www.cs.mcgill.ca/~colt2009/papers/011.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">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. Archived from the original (PDF) on 2017-03-13. Retrieved 2015-09-04. <a href="https://web.archive.org/web/20170313203427/http://www.stat.sinica.edu.tw/syhuang/papersdownload/KCCA-080906.pdf" target="_blank">https://web.archive.org/web/20170313203427/http://www.stat.sinica.edu.tw/syhuang/papersdownload/KCCA-080906.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Chapman, James; Wang, Hao-Ting (2021-12-18). "CCA-Zoo: A collection of Regularized, Deep Learning based, Kernel, and Probabilistic CCA methods in a scikit-learn style framework". Journal of Open Source Software. 6 (68): 3823. Bibcode:2021JOSS....6.3823C. doi:10.21105/joss.03823. ISSN 2475-9066. <a href="https://doi.org/10.21105%2Fjoss.03823" target="_blank">https://doi.org/10.21105%2Fjoss.03823</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Knyazev, A.V.; Argentati, M.E. (2002), "Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates", SIAM Journal on Scientific Computing, 23 (6): 2009–2041, Bibcode:2002SJSC...23.2008K, CiteSeerX 10.1.1.73.2914, doi:10.1137/S1064827500377332 <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press. <a href="/wiki/Kanti_V._Mardia" target="_blank">/wiki/Kanti_V._Mardia</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Yang Song, Peter J. Schreier, David Ram´ırez, and Tanuj Hasija Canonical correlation analysis of high-dimensional data with very small sample support arXiv:1604.02047 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Sieranoja, S.; Sahidullah, Md; Kinnunen, T.; Komulainen, J.; Hadid, A. (July 2018). "Audiovisual Synchrony Detection with Optimized Audio Features" (PDF). 2018 IEEE 3rd International Conference on Signal and Image Processing (ICSIP). pp. 377–381. doi:10.1109/SIPROCESS.2018.8600424. ISBN 978-1-5386-6396-7. S2CID 51682024. <a href="978-1-5386-6396-7" target="_blank">978-1-5386-6396-7</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Tofallis, C. (1999). "Model Building with Multiple Dependent Variables and Constraints". Journal of the Royal Statistical Society, Series D. 48 (3): 371–378. arXiv:1109.0725. doi:10.1111/1467-9884.00195. S2CID 8942357. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
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<li id="fn:18">Jendoubi, T.; Strimmer, K. (2018). "A whitening approach to probabilistic canonical correlation analysis for omics data integration". BMC Bioinformatics. 20 (1): 15. arXiv:1802.03490. doi:10.1186/s12859-018-2572-9. PMC 6327589. PMID 30626338. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6327589" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6327589</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Jendoubi, Takoua; Strimmer, Korbinian (9 January 2019). "A whitening approach to probabilistic canonical correlation analysis for omics data integration". BMC Bioinformatics. 20 (1): 15. doi:10.1186/s12859-018-2572-9. ISSN 1471-2105. PMC 6327589. PMID 30626338. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6327589" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6327589</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
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</ol>

Canonical correlation open-in-new

Canonical correlation