rmgarch (version 1.2-9)

radical: The Robust Accurate, Direct ICA aLgorithm (RADICAL).

Description

An ICA algorithm based on an efficient entropy estimator (due to Vasicek) which is robust to outliers.

Usage

radical(X, n.comp = dim(X)[2], demean = TRUE, pca.cov = c("ML", "LW", "ROB", "EWMA"), 
k = 150, augment = FALSE, replications = 30, sd = 0.175, firstEig = 1, 
lastEig = dim(X)[1], pcaE = NULL, pcaD = NULL, whiteSig = NULL, whiteMat = NULL, 
dewhiteMat = NULL, rseed = NULL, trace = FALSE, ...)

Arguments

X

The multidimensional signal matrix, where each column of matrix represents one observed signal.

n.comp

Number of independent components to estimate, defaults to the dimension of the data (rows). Is overwritten by firstEig and lastEig.

demean

(Logical) Whether the data should be centered.

pca.cov

The method to use for the calculation of the covariance matrix during the PCA whitening phase. ML is the standard maximum likelihood method, LW is the Ledoit-Wolf method, ROB is the robust method from

k

The number of angles at which to evaluate the contrast function. The ICA contrast function will be evaluated at K evenly spaced rotations from -Pi/4 to Pi/4

augment

Whether to augment the data (as explained in paper). For large datasets of >10,000 points this should be set to FALSE.

replications

This is the number of replicated points for each original point. The default value is 30. The larger the number of points in the data set, the smaller this value can be. For data sets of 10,000 points or more, point replication should be de-activate

sd

This is the standard deviation (noise) of the replicated points when using the augmentation option.

firstEig

This and lastEig specify the range for eigenvalues that are retained, firstEig is the index of largest eigenvalue to be retained. Making use of this option overwrites n.comp.

lastEig

This is the index of the last (smallest) eigenvalue to be retained and overwrites n.comp argument.

pcaE

Optionally provided eigenvector (must also supply pcaD).

pcaD

Optionally provided eigenvalues (must also supply pcaE).

whiteSig

Optionally provided Whitened signal.

whiteMat

Optionally provided Whitening matrix (no.factors by no.signals).

dewhiteMat

Optionally provided dewhitening matrix (no.signals by no.factors).

rseed

Optionally provided seed to initialize the augmented data matrix.

trace

To report progress in the console, set this to TRUE.

...

Optional arguments passed to the pca.cov methods.

Value

• A list containing the following values:
• A
  Estimated Mixing Matrix (no.signals by no.factors).
• W
  Estimated UnMixing Matrix (no.factors by no.signals).
• U
  Estimated rotation Matrix (no.factors by no.factors).
• S
  The column vectors of estimated independent components (no.obs by no.factors).
• C
  Estimated Covariance Matrix (no.signals by no.signals).
• whiteningMatrix
  The Whitening matrix (no.factors by no.signals).
• dewhiteningMatrix
  The de-Whitening matrix (no.signals by no.factors).
• rseed
  The random seed used (if any) for initializing the mixing matrix A.
• elapsed
  The elapsed time.

Details

The interested reader should consult the paper in the references section for details on the properties of the algorithm. The algorithm is quite slow, despite partial implementation in C++, and should only be used on small to medium sized sets.

References

Learned-Miller, A.G and Fisher III, J.W., 2003, ICA Using Spacings Estimates of Entropy, Journal of Machine Learning Research, 4, 1271-1295. http://www.cs.umass.edu/~elm/ICA/

Examples

# create a set of independent signals S, glued together by a mixing matrix A
# (note the notation and matrix multiplication direction as we are dealing with
# row rather than column vectors)
set.seed(100)
S <- matrix(runif(10000), 5000, 2)
A <- matrix(c(1, 1, -1, 2), 2, 2, byrow = TRUE)
# the mixed signal X
X = S %*% t(A)
# The function centers and whitens (by the eigenvalue decomposition of the 
# unconditional covariance matrix) the data before applying the theICA algorithm.
IC <- radical(X, n.comp = 2)

# demeaned data:
X_bar = scale(X, scale = FALSE)

# whitened data:
X_white = X_bar %*% t(IC$whiteningMatrix)

# check whitening:
# check correlations are zero
cor(X_white)
# check diagonals are 1 in covariance
cov(X_white)

# check that the estimated signals(S) multiplied by the
# estimated mxing matrix (A) are the same as the original dataset (X)
round(head(IC$S %*% t(IC$A)), 12) == round(head(X), 12)

# do some plots:
par(mfrow = c(1, 3))
plot(IC$S %*% t(IC$A), main = "Pre-processed data")
plot(X_white, main = "Whitened and Centered components")
plot(IC$S, main = "ICA components")