PRCCA: Canonical Correlation analysis with partial L2 penalty

In ElenaTuzhilina/RCCA: Regularized Canonical Correlation Analysis in high dimensions

Description

PRCCA function performs Canonical Correlation Analysis with partial L2 regularization and allows to conduct Canonical Correlation Analysis in high dimensions. It imposes L2 penalty only on a subset of α and β coefficients. Specifically, if

x = (x_1, ..., x_p) and y = (y_1, ..., y_q)

are random vectors and

I = {i_1, ..., i_m} is a subset of {1, ..., p} and J = {j_1, ..., i_r} is a subset of {1, ..., q}

then PRCCA seeks for such vectors

α = (α_1, ..., α_p) and β = (β_1, ..., β_q)

that satisfy partial L2 constraints

||α_I|| <= t_1 and ||β_J|| <= t_2

and that maximize the correlation cor(u, v) between the linear combnations

u = <x , α> and v = <y , β>.

Here <a , b> refers to the inner product between two vectors and

α_I and β_J

are corresponding subvectors of α and β with indices belonging to I and J, respectively. Again, the above optimization problem is equivalet to maximizing the modified correlation coefficient

cov(<x , α>, <y , β>) / ( cov(<x , α>) + λ_1 ||α_I||^2 )^1/2 ( var(<y , β>) + λ_2 ||β_J||^2 )^1/2,

where

λ_1 and λ_2

control the resulting sparsity of the canonical coefficients within

α_I and β_J

parts of the coefficient vectors.

Usage

PRCCA(X, Y, index1 = 1:ncol(X), index2 = 1:ncol(Y), lambda1 = 0, lambda2 = 0)

Arguments

X	a rectangular n x p matrix containing n observations of random vector x.
Y	a rectangular n x q matrix containing n observations of random vector y.
index1	a subset of indices the penalty is imposed on while regularizing the X side. By default index1 = 1:ncol(X), i.e. we include all α coefficients in the relularization term.
index2	a subset of indices the penalty is imposed on while regularizing the Y side. By default index2 = 1:ncol(Y), i.e. we include all β coefficients in the relularization term.
lambda1	a non-negative penalty factor used for regularizing X side coefficients α. By default lambda1 = 0, i.e. no regularization is imposed. Increasing lambda1 incourages sparsity of the resulting canonical coefficients.
lambda2	a non-negative penalty factor used for regularizing Y side coefficients β. By default lambda2 = 0, i.e. no regularization is imposed. Increasing lambda2 incourages sparsity of the resulting canonical coefficients.

Value

A list containing the PCMS problem solution:

• n.comp – the number of computed canonical components, i.e. k = min(p, q).

• cors – the resulting k canonical correlations.

• mod.cors – the resulting k values of modified canonical correlation.

• x.coefs – p x k matrix representing k canonical coefficient vectors α[1], ..., α[k].

• x.vars – n x k matrix representing k canonical variates u[1], ..., u[k].

• y.coefs – q x k matrix representing k canonical coefficient vectors β[1], ..., β[k].

• y.vars – n x k matrix representing k canonical variates v[1], ..., v[k].

Examples

data(X)
data(Y)
#run RCCA 
prcca = PRCCA(X, Y, lambda1 = 100, lambda2 = 0, index1 = 1:(ncol(X) - 10))
#check the modified canonical correlations 
plot(1:prcca$n.comp, prcca$mod.cors, pch = 16, xlab = 'component', 'ylab' = 'correlation', ylim = c(0, 1))
#check the canonical correlations
points(1:prcca$n.comp, prcca$cors, pch = 16, col = 'purple')
#compare them to cor(x*alpha, y*beta)
points(1:prcca$n.comp, diag(cor(X %*% prcca$x.coefs, Y %*% prcca$y.coefs)), col = 'cyan', pch = 16, cex = 0.7)
#check the canonical coefficients for the first canonical variates
barplot(prcca$x.coefs[,'can.comp1'], col = 'orange', 'xlab' = 'X feature', ylab = 'value')
barplot(prcca$y.coefs[,'can.comp1'], col = 'darkgreen', 'xlab' = 'Y feature', ylab = 'value')

ElenaTuzhilina/RCCA documentation built on July 11, 2021, 6:09 p.m.