FastICA algorithm
This is an R and C code implementation of the FastICA algorithm of Aapo Hyvarinen et al. (http://www.cs.helsinki.fi/u/ahyvarin/) to perform Independent Component Analysis (ICA) and Projection Pursuit.
fastICA(X, n.comp, alg.typ = c("parallel","deflation"),
fun = c("logcosh","exp"), alpha = 1.0, method = c("R","C"),
row.norm = FALSE, maxit = 200, tol = 1e-04, verbose = FALSE,
w.init = NULL)X |
a data matrix with |
n.comp |
number of components to be extracted |
alg.typ |
if |
fun |
the functional form of the G function used in the approximation to neg-entropy (see ‘details’). |
alpha |
constant in range [1, 2] used in approximation to
neg-entropy when |
method |
if |
row.norm |
a logical value indicating whether rows of the data
matrix |
maxit |
maximum number of iterations to perform. |
tol |
a positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged. |
verbose |
a logical value indicating the level of output as the algorithm runs. |
w.init |
Initial un-mixing matrix of dimension
|
Independent Component Analysis (ICA)
The data matrix X is considered to be a linear combination of non-Gaussian (independent) components i.e. X = SA where columns of S contain the independent components and A is a linear mixing matrix. In short ICA attempts to ‘un-mix’ the data by estimating an un-mixing matrix W where XW = S.
Under this generative model the measured ‘signals’ in X will tend to be ‘more Gaussian’ than the source components (in S) due to the Central Limit Theorem. Thus, in order to extract the independent components/sources we search for an un-mixing matrix W that maximizes the non-gaussianity of the sources.
In FastICA, non-gaussianity is measured using approximations to neg-entropy (J) which are more robust than kurtosis-based measures and fast to compute.
The approximation takes the form
J(y) = [E G(y) - E G(v)]^2
where v is a N(0,1) r.v.
The following choices of G are included as options G(u) = 1/alpha log cosh (alpha u) and G(u) = -exp(-u^2/2).
Algorithm
First, the data are centered by subtracting the mean of each column of the data matrix X.
The data matrix is then ‘whitened’ by projecting the data onto its principal component directions i.e. X -> XK where K is a pre-whitening matrix. The number of components can be specified by the user.
The ICA algorithm then estimates a matrix W s.t XKW = S . W is chosen to maximize the neg-entropy approximation under the constraints that W is an orthonormal matrix. This constraint ensures that the estimated components are uncorrelated. The algorithm is based on a fixed-point iteration scheme for maximizing the neg-entropy.
Projection Pursuit
In the absence of a generative model for the data the algorithm can be used to find the projection pursuit directions. Projection pursuit is a technique for finding ‘interesting’ directions in multi-dimensional datasets. These projections and are useful for visualizing the dataset and in density estimation and regression. Interesting directions are those which show the least Gaussian distribution, which is what the FastICA algorithm does.
A list containing the following components
X |
pre-processed data matrix |
K |
pre-whitening matrix that projects data onto the first |
W |
estimated un-mixing matrix (see definition in details) |
A |
estimated mixing matrix |
S |
estimated source matrix |
J L Marchini and C Heaton
A. Hyvarinen and E. Oja (2000) Independent Component Analysis: Algorithms and Applications, Neural Networks, 13(4-5):411-430
#---------------------------------------------------
#Example 1: un-mixing two mixed independent uniforms
#---------------------------------------------------
S <- matrix(runif(10000), 5000, 2)
A <- matrix(c(1, 1, -1, 3), 2, 2, byrow = TRUE)
X <- S %*% A
a <- fastICA(X, 2, alg.typ = "parallel", fun = "logcosh", alpha = 1,
method = "C", row.norm = FALSE, maxit = 200,
tol = 0.0001, verbose = TRUE)
par(mfrow = c(1, 3))
plot(a$X, main = "Pre-processed data")
plot(a$X %*% a$K, main = "PCA components")
plot(a$S, main = "ICA components")
#--------------------------------------------
#Example 2: un-mixing two independent signals
#--------------------------------------------
S <- cbind(sin((1:1000)/20), rep((((1:200)-100)/100), 5))
A <- matrix(c(0.291, 0.6557, -0.5439, 0.5572), 2, 2)
X <- S %*% A
a <- fastICA(X, 2, alg.typ = "parallel", fun = "logcosh", alpha = 1,
method = "R", row.norm = FALSE, maxit = 200,
tol = 0.0001, verbose = TRUE)
par(mfcol = c(2, 3))
plot(1:1000, S[,1 ], type = "l", main = "Original Signals",
xlab = "", ylab = "")
plot(1:1000, S[,2 ], type = "l", xlab = "", ylab = "")
plot(1:1000, X[,1 ], type = "l", main = "Mixed Signals",
xlab = "", ylab = "")
plot(1:1000, X[,2 ], type = "l", xlab = "", ylab = "")
plot(1:1000, a$S[,1 ], type = "l", main = "ICA source estimates",
xlab = "", ylab = "")
plot(1:1000, a$S[, 2], type = "l", xlab = "", ylab = "")
#-----------------------------------------------------------
#Example 3: using FastICA to perform projection pursuit on a
# mixture of bivariate normal distributions
#-----------------------------------------------------------
if(require(MASS)){
x <- mvrnorm(n = 1000, mu = c(0, 0), Sigma = matrix(c(10, 3, 3, 1), 2, 2))
x1 <- mvrnorm(n = 1000, mu = c(-1, 2), Sigma = matrix(c(10, 3, 3, 1), 2, 2))
X <- rbind(x, x1)
a <- fastICA(X, 2, alg.typ = "deflation", fun = "logcosh", alpha = 1,
method = "R", row.norm = FALSE, maxit = 200,
tol = 0.0001, verbose = TRUE)
par(mfrow = c(1, 3))
plot(a$X, main = "Pre-processed data")
plot(a$X %*% a$K, main = "PCA components")
plot(a$S, main = "ICA components")
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