From Sparse Regression to Sparse Multiple Correspondence Analysis
Résumé
High dimensional data means that the number of variables p if far larger
than the number of observations n.
This talk starts from a survey of various solutions in linear regression .
When p n the OLS estimator does not exist . Since it is a case of forced multi-
collinearity, one may use regularized techniques such as ridge regression, principal
component regression or PLS regression which keep all the predictors.
However if p n combinations of all variables cannot be interpreted. Sparse so-
lutions, ie with a large number of zero coe cients, are preferred. Lasso, elastic net,
sparse PLS perform simultaneously regularization and variable selection thanks to
non quadratic penalties: L1, SCAD etc.
In PCA, the singular value decomposition shows that if we regress principal com-
ponents onto the input variables, the vector of regression coe cients is equal to the
factor loadings. It su ces to adapt sparse regression techniques to get sparse ver-
sions of PCA and of PCA with groups of variables. We conclude by a presentation of
a sparse version of Multiple Correspondence Analysis and give several applications.
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