ccipca {onlinePCA}

Candid Covariance-Free Incremental PCA

Description

Stochastic gradient ascent algorithm CCIPCA of Weng et al. (2003).

Usage

ccipca(lambda, U, x, n, q = length(lambda), l=2, center, tol = 1e-8, sort = TRUE)

Arguments

lambda

vector of eigenvalues.

U

matrix of eigenvectors (PC) stored in columns.

x

new data vector.

n

sample size before observing x.

q

number of eigenvectors to compute.

l

'amnesic' parameter.

center

optional centering vector for x.

tol

numerical tolerance.

sort

Should the new eigenpairs be sorted?

Details

The 'amnesic' parameter l determines the weight of past observations in the PCA update. If l=0, all observations have equal weight, which is appropriate for stationary processes. Otherwise, typical values of l range between 2 and 4. As l increases, more weight is placed on new observations and less on older ones. For meaningful results, the condition 0<=l<n should hold.
The CCIPCA algorithm iteratively updates the PCs while deflating x. If at some point the Euclidean norm of x becomes less than tol, the algorithm stops to prevent numerical overflow.
If sort is TRUE, the updated eigenpairs are sorted by decreasing eigenvalue. If FALSE, they are not sorted.

Value

A list with components

values

updated eigenvalues.

vectors

updated eigenvectors.

References

Weng et al. (2003). Candid Covariance-free Incremental Principal Component Analysis. IEEE Trans. Pattern Analysis and Machine Intelligence.

Examples

## Simulation of Brownian motion
n <- 100 # number of paths
d <- 50	 # number of observation points
q <- 10	 # number of PCs to compute
x <- matrix(rnorm(n*d,sd=1/sqrt(d)), n, d)
x <- t(apply(x,1,cumsum))	

## Initial PCA
n0 <- 50
pca <- princomp(x[1:n0,])
xbar <- pca$center
pca <- list(values=pca$sdev^2, vectors=pca$loadings)

## Incremental PCA
for (i in n0:(n-1))
{	xbar <- updateMean(xbar, x[i+1,], i)
  	pca <- ccipca(pca$values, pca$vectors, x[i+1,], i, q = q, center = xbar) }

# Uncentered PCA
nx1 <- sqrt(sum(x[1,]^2))
pca <- list(values=nx1^2, vectors=as.matrix(x[1,]/nx1))
for (i in n0:(n-1))
  	pca <- ccipca(pca$values, pca$vectors, x[i+1,], i, q = q)