Regularized canonical correlation analysis

<p>Regularized canonical correlation analysis is a way of using <a href="/facts/Ridge_regression/eq6kH8PX">ridge regression</a> to solve the <a href="/facts/Singularity_theory/MgLkyrhl">singularity</a> problem in the <a href="/facts/Cross-covariance_matrix/kcfhptRE">cross-covariance matrices</a> of <a href="/facts/Canonical_correlation_analysis/LuX8FltG">canonical correlation analysis</a>. By converting 
  
    
      
        cov
        ⁡
        (
        X
        ,
        X
        )
      
    
    {\displaystyle \operatorname {cov} (X,X)}
  
 and 
  
    
      
        cov
        ⁡
        (
        Y
        ,
        Y
        )
      
    
    {\displaystyle \operatorname {cov} (Y,Y)}
  
 into 
  
    
      
        cov
        ⁡
        (
        X
        ,
        X
        )
        +
        λ
        
          I
          
            X
          
        
      
    
    {\displaystyle \operatorname {cov} (X,X)+\lambda I_{X}}
  
 and 
  
    
      
        cov
        ⁡
        (
        Y
        ,
        Y
        )
        +
        λ
        
          I
          
            Y
          
        
      
    
    {\displaystyle \operatorname {cov} (Y,Y)+\lambda I_{Y}}
  
, it ensures that the above matrices will have reliable <a href="/facts/Matrix_inverse/fPqXk3V8">inverses</a>.
</p><p>The idea probably dates back to <a href="/facts/Hrishikesh_Vinod/XcyEzWG7">Hrishikesh D. Vinod</a>'s publication in 1976 where he called it "Canonical ridge".
It has been suggested for use in the analysis of <a href="/facts/Functional_neuroimaging/r0k8tKGR">functional neuroimaging</a> data as such data are often singular.
It is possible to compute the regularized canonical vectors in the lower-dimensional space.
</p>

<h3>Further reading</h3>
<ul><li><a href="/facts/Sue_Leurgans/5I5dVoT5">Leurgans, S.E.</a>; Moyeed, R.A.; Silverman, B.W. (1993). "Canonical correlation analysis when the data are curves". <i><a href="/facts/Journal_of_the_Royal_Statistical_Society/14Twyyg1">Journal of the Royal Statistical Society</a></i>. Series B (Methodological). 55 (3): 725–740. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1111%2Fj.2517-6161.1993.tb01936.x">10.1111/j.2517-6161.1993.tb01936.x</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2345883">2345883</a>.</li>
<li>Elena Tuzhilina; Leonardo Tozzi; <a href="/facts/Trevor_Hastie/9bSQQGo7">Trevor Hastie</a> (3 October 2021). "Canonical correlation analysis in high dimensions with structured regularization". <i><a href="/facts/Statistical_Modelling/jKkT5ftm">Statistical Modelling</a></i>. 23 (3): 203–227. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1177%2F1471082X211041033">10.1177/1471082X211041033</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1471-082X">1471-082X</a>. <a href="/facts/WDQ_(identifier)/5Myy74K5">Wikidata</a> Q132973388.</li></ul>

Regularized canonical correlation analysis open-in-new

Regularized canonical correlation analysis