canonical correlation analysis cca
Download
Skip this Video
Download Presentation
Canonical Correlation Analysis (CCA)

Loading in 2 Seconds...

play fullscreen
1 / 20

Canonical Correlation Analysis (CCA) - PowerPoint PPT Presentation


  • 117 Views
  • Uploaded on

Canonical Correlation Analysis (CCA). CCA. This is it! The mother of all linear statistical analysis. When ? We want to find a structural relation between a set of independent variables and a set of dependent variables. CCA. When ? (part 2)

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Canonical Correlation Analysis (CCA)' - constance-morse


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide2
CCA
  • This is it!
    • The mother of all linear statistical analysis
  • When ?
    • We want to find a structural relation between a set of independent variables and a set of dependent variables.
slide3
CCA
  • When ? (part 2)
    • To what extend can one set of two or more variables be predicted or “explained” by another set of two or more variables?
    • What contribution does a single variable make to the explanatory power to the set of variables to which the variable belongs?
    • What contribution does a single variable contribute to predicting or “explaining” the composite of the variables in the variable set to which the variable does not belong?
    • What different dynamics are involved in the ability of one variable set to “explain” in different ways different portions of other variable set?
    • What relative power do different canonical functions have to predict or explain relationships?
    • How stable are canonical results across samples or sample subgroups?
    • How closely do obtained canonical results conform to expected canonical results?
slide4
CCA
  • Assumptions
    • Linearity: if not, nonlinear canonical correlation analysis.
    • Absence of multicollinearity: If not, Partial Least Squares (PLS) regression to reduce the space.
    • Homoscedasticity: If not, data transformation.
    • Normality: If not, re-sampling.
    • A lot of data: Max(p, q)20nb of pairs.
    • Absence of outliers.
slide5
CCA
  • Toy example

IVs

DVs

=X

slide6
CCA
  • Z score transformation

IV1

IV1

DV2

DV2

=Z

slide7
CCA
  • Canonical Correlation Matrix
slide8
CCA
  • Relations with other subspace methods
slide9
CCA
  • Eigenvalues and eigenvectors decomposition

R =

PCA

slide10
CCA
  • Eigenvalues and eigenvectors decomposition
  • The roots of the eigenvalues are the canonical correlation values
slide11
CCA
  • Significance test for the canonical correlation
  • A significant output indicates that there is a variance share between IV and DV sets
  • Procedure:
    • We test for all the variables (m=1,…,min(p,q))
    • If significant, we removed the first variable (canonical correlate) and test for the remaining ones (m=2,…, min(p,q)
    • Repeat
slide12
CCA
  • Significance test for the canonical correlation

Since all canonical variables are significant, we will keep them all.

slide13
CCA
  • Canonical Coefficients
    • Analogous to regression coefficients

BY=

Eigenvectors

Correlation matrix of the dependant variables

Bx=

slide14
CCA
  • Canonical Variates
    • Analogous to regression coefficients
slide15
CCA
  • Loading matrices
    • Matrices of correlations between the variables and the canonical coefficients

Ax

Ay

slide16
CCA
  • Loadings and canonical correlations for both canonical variate pairs
    • Only coefficient higher than |0.3| are interpreted.

Loading

Canonical correlation

slide17
CCA
  • Proportion of variance extracted
    • How much variance does each of the canonical variates extract form the variables on its own side of the equation?

First

First

Second

Second

slide18
CCA
  • Redundancy
    • How much variance the canonical variates form the IVs extract from the DVs, and vice versa.

rdyx

Eigenvalues

slide19
CCA
  • Redundancy
    • How much variance the canonical variates form the IVs extract from the DVs, and vice versa.

Summary

The first canonical variate from IVs extract 40% of the variance in the y variable.

The second canonical variate form IVs extract 30% of the variance in the y variable.

Together they extract 70% of the variance in the DVs.

The first canonical variate from DVs extract 49% of the variance in the x variable.

The second canonical variate form DVs extract 24% of the variance in the x variable.

Together they extract 73% of the variance in the IVs.

slide20
CCA
  • Rotation
    • A rotation does not influence the variance proportion or the redundancy.

= Loading matrix =

ad