GRCCA: Canonical Correlation analysis with group L2 penalty

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

Description

GRCCA function performs Canonical Correlation Analysis with group L2 regularization and allows to conduct Canonical Correlation Analysis in high dimensions. It imposes group L2 penalty on the coefficient vectors α and β coefficients. Specifically, if

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

are random vectors and

I_1, ..., I_K is a partition of {1, ..., p} and J_1, ..., J_L is a partition of {1, ..., q}

then GRCCA seeks for such vectors

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

that satisfy two within group constraints

||α||_w = var(α_I_1) + ... + var(α_I_K) <= t_1

and

||β||_w = var(β_J_1) + ... + var(β_J_L) <= t_2

as well as two between group constraints

||α||_b = |I_1| mean(α_I_1)^2 + ... + |I_K| mean(α_I_K)^2 <= s_1

and

||β||_b = |J_1| mean(β_J_1)^2 + ... + |J_L| mean(β_J_L)^2 <= s_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;

α_I_k and β_J_l

are corresponding subvectors of α and β with indices belonging to I_k and J_l, respectively; and |A| referes to the set candinality. The above optimization problem is equivalet to maximizing the modified correlation coefficient

cov(<x , α>, <y , β>) / ( cov(<x , α>) + λ_1 ||α||_w + μ_1 ||α||_b )^1/2 ( var(<y , β>) + λ_2 ||β||_w + μ_2 ||β||_b )^1/2,

where

λ_1 and λ_2

control the resulting within group variation of the coefficiens and

μ_1 and μ_2

control the sparsity on a group level of the canonical coefficients α and β.

Usage

GRCCA(
  X,
  Y,
  group1 = rep(1, ncol(X)),
  group2 = rep(1, ncol(Y)),
  lambda1 = 0,
  lambda2 = 0,
  mu1 = 0,
  mu2 = 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.
group1	an integer valued vector representing the group assignment of α coefficients. By default group1 = rep(1, ncol(X)), i.e. we include all α coefficients in the same group.
group2	an integer valued vector representing the group assignment of β coefficients. By default group2 = rep(1, ncol(Y)), i.e. we include all β coefficients in the same group.
lambda1	a non-negative penalty factor used for controlling the within variation of α coefficients. By default lambda1 = 0, i.e. no regularization is imposed. Increasing lambda1 shrinks each coefficient toward it's group mean.
lambda2	a non-negative penalty factor used for controlling the within variation of β coefficients. By default lambda2 = 0, i.e. no regularization is imposed. Increasing lambda2 shrinks each coefficient toward it's group mean.
mu1	a non-negative penalty factor used for controlling the between variation of α coefficients. By default mu1 = 0, i.e. no regularization is imposed. Increasing mu1 shrinks each group mean toward zero.
mu2	a non-negative penalty factor used for controlling the between variation of β coefficients. By default mu2 = 0, i.e. no regularization is imposed. Increasing mu2 shrinks each group mean toward zero.

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
n.groups = 5
#run GRCCA with no sparsity on a group level
group1 = rep(1:n.groups, rep(ncol(X)/n.groups, n.groups))
grcca = GRCCA(X, Y, group1, group2 = NULL, lambda1 = 1000, lambda2 = 0, mu1 = 0, mu2 = 0)
#check the modified canonical correlations 
plot(1:grcca$n.comp, grcca$mod.cors, pch = 16, xlab = 'component', 'ylab' = 'correlation', ylim = c(0, 1))
#check the canonical coefficients for the first canonical variates
barplot(grcca$x.coefs[,'can.comp1'], col = 'orange', 'xlab' = 'X feature', ylab = 'value')
n.groups = 50
#run GRCCA with sparsity on a group level
group1 = rep(1:n.groups, rep(ncol(X)/n.groups, n.groups))
grcca = GRCCA(X, Y, group1, group2 = NULL, lambda1 = 10000, lambda2 = 0, mu1 = 100, mu2 = 0)
#check the modified canonical correlations 
plot(1:grcca$n.comp, grcca$mod.cors, pch = 16, xlab = 'component', 'ylab' = 'correlation', ylim = c(0, 1))
#check the canonical coefficients for the first canonical variates
barplot(grcca$x.coefs[,'can.comp1'], col = 'orange', 'xlab' = 'X feature', ylab = 'value')

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