hsicCCA (version 1.0)

ktaCCA: Canonical Correlation Analysis based on the Centered Kernel Target Alignment.

Description

Given two multi-dimensional data sets, find pairs of canonical projection pairs that maximize the Centered Kernel Target Alignment Algorithm.

Usage

ktaCCA(x, y, M, sigmax = NULL, sigmay = NULL, numrepeat = 5, numiter = 100, reltolstop = 1e-04)

Arguments

x

The x-variable data matrix. One row per observation.

y

The y-variable data matrix. One row per observation.

M

Number of canonical projection pairs to extract.

sigmax

The bandwidth parameter for the Gaussian kernel on the x-variable set. A positive value. The smaller the smoother. If NULL, set to median(dist(x)), and will be updated automatically for extracting different pairs of canonical projection.

sigmay

The bandwidth parameter for the Gaussian kernel on the y-variable set. A positive value. The smaller the smoother. If NULL, set to median(dist(y)), and will be updated automatically for extracting different pairs of canonical projection.

numrepeat

Number of random restarts.

numiter

Maximum number of iterations for extracting each pair of canonical projections.

reltolstop

Convergence threshold. Algorithm stops when relative change in cost from consecutive iterations is less than the threshold and will then move on to find the next pair of canonical vectors.

Value

• A list containing:
• Wx
  The M canoncial projection vectors for the x-variable set. Each column corresponds to a projection vector.
• Wy
  The M canoncial projection vectors for the y-variable set. Each column corresponds to a projection vector.

Details

Optimization is done by gradient descent, where Nelder-Mead is used for step-size selection. Nelder Mead may fail to increase the cost at times (when stuck at local minima). User may consider restarting the algorithm when this happens.

References

Chang et. al. (2013) Canonical Correlation Analysis based on Hilbert-Schmidt Independence Criterion and Centered Kernel Target Alignment. ICML 2013.

Cortes et. al. (2012) Algorithms for learning kernels based on centered alignments. JMLR 13:795-828.

See Also

hsicCCA, ktaCCAfunc

Examples

set.seed(1)
numData <- 100
numDim <- 3
x <- matrix(rnorm(numData*numDim),numData,numDim)
y <- matrix(rnorm(numData*numDim),numData,numDim)
z <- runif(numData,-pi,pi)
y[,1] <- cos(z)+rnorm(numData,sd=0.1); x[,1] <- sin(z)+rnorm(numData,sd=0.1)
y[,2] <- x[,2]+rnorm(numData,sd=0.5)
x <- scale(x)
y <- scale(y)

fit <- ktaCCA(x,y,2,numrepeat=2,numiter=10)
par(mfrow=c(1,2))
for (K in 1:2) plot(x%*%fit$Wx[,K],y%*%fit$Wy[,K])