infomaxICA: Estimates independent components via infomax
In steadyICA: ICA and Tests of Independence via Multivariate Distance Covariance
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
Examples
Description
Estimate independent components using the infomax criteria, which is equivalent to maximum likelihood using the logistic density, exp(-S)/(1+exp(-S))^2.
Usage
1
2
infomaxICA(X, n.comp, W.list = NULL, whiten = FALSE, maxit = 500, eps = 1e-08,
alpha.eps = 1e-08, verbose = FALSE, restarts=0)
Arguments
X
the n x p data matrix
n.comp
number of components to be estimated
W.list
list of orthogonal matrices for initialization
whiten
Whitens the data before applying ICA, i.e., X%*%whitener = Z, where Z has mean zero and empirical covariance equal to the identity matrix, and Z is then used as the input.
maxit
maximum number of iterations
eps
algorithm terminates when the norm of the gradient is less than eps
alpha.eps
tolerance controlling the level of annealing: algorithm terminates with a warning if the learning parameter is less than alpha.eps
verbose
if TRUE, prints (1) the value of the infomax objective function at each iteration, (2) the norm of the gradient, and (3) current value of the learning parameter alpha.
restarts
An integer determining the number of initial matrices to use in estimating the ICA model. The objective function has local optima, so multiple starting values are recommended. If whiten=TRUE, then generates random orthogonal matrices. If whiten=FALSE, generate random matrices from rnorm(). See code for details.
Details
This is an R version of ICA using the infomax criteria that provides an alternative to Matlab code (ftp://ftp.cnl.salk.edu/pub/tony/sep96.public), but with a few modifications. First, we use the full data (the so-called offline algorithm) in each iteration rather than an online algorithm with batches. Secondly, we use an adaptive method to choose the step size (based upon Bernaards and Jennrich 2005), which speeds up convergence. We also omitted the bias term (intercept) included in the original formulation because we centered our data.
Value
S
the estimated independent components
W
if whiten=TRUE, returns the orthogonal unmixing matrix; no value is returned when whiten=FALSE
M
Returns the estimated mixing matrix for the model X = S M, where X is not pre-whitened (although X is centered)
f
the value of the objective function at the estimated S
Table
summarizes algorithm status at each iteration
convergence
1 if norm of the gradient is less than eps, 2 if the learning parameter was smaller than alpha.eps, which usually means the gradient is sufficiently small, 0 otherwise
Note
In contrast to most other ICA methods, W is not contrained to be orthogonal.
Author(s)
Benjamin Risk
References
Bell, A. & Sejnowski, T. An information-maximization approach to blind separation and blind deconvolution Neural computation, Neural computation, 1995, 7, 1129-1159.
Bernaards, C. A. and Jennrich, R. I. (2005) Gradient Projection Algorithms and
Software for Arbitrary Rotation Criteria in Factor Analysis, Educational and
Psychological Measurement 65, 676-696. <http://www.stat.ucla.edu/research/gpa>
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
## Example when p > d. The MD function and amari measures
# are not defined for M. We can compare the
# "true W inverse", which is the mixing matrix multiplied
# by the whitening matrix; alternatively, we can use
# multidcov::frobICA. These two approaches are
# demonstrated below:
set.seed(999)
nObs <- 1024
nComp <- 3
# simulate from gamma distributions with
# varying amounts of skewness:
simS<-cbind(rgamma(nObs, shape = 1, scale = 2),
rgamma(nObs, shape = 3, scale = 2),
rgamma(nObs, shape = 9, scale = 0.5))
#standardize by expected value and variance:
simS[,1] = (simS[,1] - 1*2)/sqrt(1*2^2)
simS[,2] = (simS[,2] - 3*2)/sqrt(3*2^2)
simS[,3] = (simS[,3] - 9*0.5)/sqrt(9*0.5^2)
# slightly revised 'mixmat' function (from ProDenICA)
# for p>=d: uses fastICA and ProDenICA parameterization:
myMixmat <- function (p = 2, d = NULL) {
if(is.null(d)) d = p
a <- matrix(rnorm(d * p), d, p)
sa <- La.svd(a)
dL <- sort(runif(d) + 1)
mat <- sa$u%*%(sa$vt*dL)
attr(mat, "condition") <- dL[d]/dL[1]
mat
}
simM <- myMixmat(p = 6, d = nComp)
xData <- simS%*%simM
xWhitened <- whitener(xData, n.comp = nComp)
#Define a 'true' W (uses the estimated whitening matrix):
W.true <- solve(simM%*%xWhitened$whitener)
estInfomax <- infomaxICA(X = xData, n.comp = nComp, whiten = TRUE, verbose = TRUE)
frobICA(estInfomax$M,simM)
library(JADE)
MD(t(estInfomax$W),t(solve(W.true)))
amari.error(t(estInfomax$W),t(solve(W.true)))
Example output
Loading required package: Rcpp
Loading required package: MASS
Loading required package: clue
Loading required package: combinat
Attaching package: 'combinat'
The following object is masked from 'package:utils':
combn
iter: 1; newF: -4416.845894; log10||grad||: -1.059842; alpha: 2
iter: 2; newF: -4349.538901; log10||grad||: -1.193744; alpha: 4
iter: 3; newF: -4316.57333; log10||grad||: -1.866519; alpha: 2
iter: 4; newF: -4315.058557; log10||grad||: -1.757356; alpha: 4
iter: 5; newF: -4314.075226; log10||grad||: -1.639013; alpha: 4
iter: 6; newF: -4309.199036; log10||grad||: -1.998738; alpha: 2
iter: 7; newF: -4306.161831; log10||grad||: -1.954943; alpha: 4
iter: 8; newF: -4303.018115; log10||grad||: -1.614786; alpha: 8
iter: 9; newF: -4297.895403; log10||grad||: -1.978995; alpha: 2
iter: 10; newF: -4295.200099; log10||grad||: -1.940444; alpha: 4
iter: 11; newF: -4293.312979; log10||grad||: -1.855048; alpha: 4
iter: 12; newF: -4292.667846; log10||grad||: -1.722532; alpha: 4
iter: 13; newF: -4289.74446; log10||grad||: -2.119811; alpha: 2
iter: 14; newF: -4288.70826; log10||grad||: -2.061477; alpha: 4
iter: 15; newF: -4288.331644; log10||grad||: -1.933059; alpha: 4
iter: 16; newF: -4287.229329; log10||grad||: -2.310636; alpha: 2
iter: 17; newF: -4286.823324; log10||grad||: -2.247909; alpha: 4
iter: 18; newF: -4286.713287; log10||grad||: -2.11228; alpha: 4
iter: 19; newF: -4286.242831; log10||grad||: -2.509627; alpha: 2
iter: 20; newF: -4286.102464; log10||grad||: -2.432625; alpha: 4
iter: 21; newF: -4286.087117; log10||grad||: -2.283365; alpha: 4
iter: 22; newF: -4285.882887; log10||grad||: -2.699351; alpha: 2
iter: 23; newF: -4285.839592; log10||grad||: -2.598032; alpha: 4
iter: 24; newF: -4285.783824; log10||grad||: -2.915212; alpha: 2
iter: 25; newF: -4285.75343; log10||grad||: -2.876714; alpha: 4
iter: 26; newF: -4285.741996; log10||grad||: -2.750683; alpha: 4
iter: 27; newF: -4285.716439; log10||grad||: -3.109438; alpha: 2
iter: 28; newF: -4285.706535; log10||grad||: -3.039898; alpha: 4
iter: 29; newF: -4285.704729; log10||grad||: -2.891422; alpha: 4
iter: 30; newF: -4285.692235; log10||grad||: -3.288818; alpha: 2
iter: 31; newF: -4285.689197; log10||grad||: -3.187474; alpha: 4
iter: 32; newF: -4285.685479; log10||grad||: -3.49347; alpha: 2
iter: 33; newF: -4285.683344; log10||grad||: -3.451837; alpha: 4
iter: 34; newF: -4285.682526; log10||grad||: -3.322549; alpha: 4
iter: 35; newF: -4285.680694; log10||grad||: -3.673572; alpha: 2
iter: 36; newF: -4285.679957; log10||grad||: -3.599313; alpha: 4
iter: 37; newF: -4285.679816; log10||grad||: -3.44813; alpha: 4
iter: 38; newF: -4285.678853; log10||grad||: -3.836976; alpha: 2
iter: 39; newF: -4285.6786; log10||grad||: -3.733675; alpha: 4
iter: 40; newF: -4285.678296; log10||grad||: -4.027392; alpha: 2
iter: 41; newF: -4285.678105; log10||grad||: -3.984566; alpha: 4
iter: 42; newF: -4285.67802; log10||grad||: -3.858417; alpha: 4
iter: 43; newF: -4285.677859; log10||grad||: -4.186181; alpha: 2
iter: 44; newF: -4285.67778; log10||grad||: -4.119141; alpha: 4
iter: 45; newF: -4285.677752; log10||grad||: -3.976623; alpha: 4
iter: 46; newF: -4285.677664; log10||grad||: -4.330149; alpha: 2
iter: 47; newF: -4285.677628; log10||grad||: -4.244121; alpha: 4
iter: 48; newF: -4285.67762; log10||grad||: -4.090614; alpha: 4
iter: 49; newF: -4285.677569; log10||grad||: -4.462373; alpha: 2
iter: 50; newF: -4285.677552; log10||grad||: -4.362654; alpha: 4
iter: 51; newF: -4285.67755; log10||grad||: -4.202009; alpha: 4
iter: 52; newF: -4285.677521; log10||grad||: -4.586212; alpha: 2
iter: 53; newF: -4285.677512; log10||grad||: -4.477068; alpha: 4
iter: 54; newF: -4285.677512; log10||grad||: -4.311819; alpha: 4
iter: 55; newF: -4285.677494; log10||grad||: -4.704495; alpha: 2
iter: 56; newF: -4285.67749; log10||grad||: -4.588957; alpha: 4
iter: 57; newF: -4285.677484; log10||grad||: -4.868585; alpha: 2
iter: 58; newF: -4285.67748; log10||grad||: -4.819269; alpha: 4
iter: 59; newF: -4285.677477; log10||grad||: -4.699317; alpha: 4
iter: 60; newF: -4285.677474; log10||grad||: -4.985367; alpha: 2
iter: 61; newF: -4285.677471; log10||grad||: -4.931881; alpha: 4
iter: 62; newF: -4285.67747; log10||grad||: -4.808755; alpha: 4
iter: 63; newF: -4285.677468; log10||grad||: -5.099789; alpha: 2
iter: 64; newF: -4285.677466; log10||grad||: -5.04318; alpha: 4
iter: 65; newF: -4285.677465; log10||grad||: -4.917637; alpha: 4
iter: 66; newF: -4285.677464; log10||grad||: -5.212764; alpha: 2
iter: 67; newF: -4285.677463; log10||grad||: -5.153677; alpha: 4
iter: 68; newF: -4285.677463; log10||grad||: -5.026179; alpha: 4
iter: 69; newF: -4285.677462; log10||grad||: -5.324849; alpha: 2
iter: 70; newF: -4285.677462; log10||grad||: -5.263681; alpha: 4
iter: 71; newF: -4285.677461; log10||grad||: -5.134508; alpha: 4
iter: 72; newF: -4285.677461; log10||grad||: -5.436384; alpha: 2
iter: 73; newF: -4285.677461; log10||grad||: -5.373374; alpha: 4
iter: 74; newF: -4285.67746; log10||grad||: -5.242699; alpha: 4
iter: 75; newF: -4285.67746; log10||grad||: -5.54757; alpha: 2
iter: 76; newF: -4285.67746; log10||grad||: -5.482864; alpha: 4
iter: 77; newF: -4285.67746; log10||grad||: -5.350799; alpha: 4
iter: 78; newF: -4285.67746; log10||grad||: -5.65853; alpha: 2
iter: 79; newF: -4285.67746; log10||grad||: -5.592215; alpha: 4
iter: 80; newF: -4285.67746; log10||grad||: -5.458833; alpha: 4
iter: 81; newF: -4285.67746; log10||grad||: -5.769335; alpha: 2
iter: 82; newF: -4285.677459; log10||grad||: -5.701469; alpha: 4
iter: 83; newF: -4285.677459; log10||grad||: -5.56682; alpha: 4
iter: 84; newF: -4285.677459; log10||grad||: -5.880031; alpha: 2
iter: 85; newF: -4285.677459; log10||grad||: -5.810647; alpha: 4
iter: 86; newF: -4285.677459; log10||grad||: -5.674769; alpha: 4
iter: 87; newF: -4285.677459; log10||grad||: -5.990645; alpha: 2
iter: 88; newF: -4285.677459; log10||grad||: -5.919765; alpha: 4
iter: 89; newF: -4285.677459; log10||grad||: -5.782687; alpha: 4
iter: 90; newF: -4285.677459; log10||grad||: -6.101193; alpha: 2
iter: 91; newF: -4285.677459; log10||grad||: -6.028833; alpha: 4
iter: 92; newF: -4285.677459; log10||grad||: -5.890578; alpha: 4
iter: 93; newF: -4285.677459; log10||grad||: -6.211689; alpha: 2
iter: 94; newF: -4285.677459; log10||grad||: -6.137858; alpha: 4
iter: 95; newF: -4285.677459; log10||grad||: -5.998446; alpha: 4
iter: 96; newF: -4285.677459; log10||grad||: -6.322137; alpha: 2
iter: 97; newF: -4285.677459; log10||grad||: -6.246843; alpha: 4
iter: 98; newF: -4285.677459; log10||grad||: -6.106292; alpha: 4
iter: 99; newF: -4285.677459; log10||grad||: -6.432544; alpha: 2
iter: 100; newF: -4285.677459; log10||grad||: -6.355791; alpha: 4
iter: 101; newF: -4285.677459; log10||grad||: -6.214118; alpha: 4
iter: 102; newF: -4285.677459; log10||grad||: -6.542913; alpha: 2
iter: 103; newF: -4285.677459; log10||grad||: -6.464705; alpha: 4
iter: 104; newF: -4285.677459; log10||grad||: -6.321926; alpha: 4
iter: 105; newF: -4285.677459; log10||grad||: -6.653246; alpha: 2
iter: 106; newF: -4285.677459; log10||grad||: -6.573587; alpha: 4
iter: 107; newF: -4285.677459; log10||grad||: -6.429716; alpha: 4
iter: 108; newF: -4285.677459; log10||grad||: -6.763545; alpha: 2
iter: 109; newF: -4285.677459; log10||grad||: -6.682437; alpha: 4
iter: 110; newF: -4285.677459; log10||grad||: -6.537488; alpha: 4
iter: 111; newF: -4285.677459; log10||grad||: -6.873811; alpha: 2
iter: 112; newF: -4285.677459; log10||grad||: -6.791257; alpha: 4
iter: 113; newF: -4285.677459; log10||grad||: -6.645245; alpha: 4
iter: 114; newF: -4285.677459; log10||grad||: -6.984045; alpha: 2
iter: 115; newF: -4285.677459; log10||grad||: -6.900047; alpha: 4
iter: 116; newF: -4285.677459; log10||grad||: -6.752985; alpha: 4
iter: 117; newF: -4285.677459; log10||grad||: -7.094247; alpha: 2
iter: 118; newF: -4285.677459; log10||grad||: -7.008807; alpha: 4
iter: 119; newF: -4285.677459; log10||grad||: -6.86071; alpha: 4
iter: 120; newF: -4285.677459; log10||grad||: -7.204419; alpha: 2
iter: 121; newF: -4285.677459; log10||grad||: -7.11754; alpha: 4
iter: 122; newF: -4285.677459; log10||grad||: -6.96842; alpha: 4
iter: 123; newF: -4285.677459; log10||grad||: -7.31456; alpha: 2
iter: 124; newF: -4285.677459; log10||grad||: -7.226245; alpha: 4
iter: 125; newF: -4285.677459; log10||grad||: -7.076115; alpha: 4
iter: 126; newF: -4285.677459; log10||grad||: -7.424671; alpha: 2
iter: 127; newF: -4285.677459; log10||grad||: -7.334922; alpha: 4
iter: 128; newF: -4285.677459; log10||grad||: -7.183796; alpha: 4
iter: 129; newF: -4285.677459; log10||grad||: -7.534752; alpha: 2
iter: 130; newF: -4285.677459; log10||grad||: -7.443572; alpha: 4
iter: 131; newF: -4285.677459; log10||grad||: -7.291463; alpha: 4
iter: 132; newF: -4285.677459; log10||grad||: -7.644803; alpha: 2
iter: 133; newF: -4285.677459; log10||grad||: -7.552195; alpha: 4
iter: 134; newF: -4285.677459; log10||grad||: -7.399116; alpha: 4
iter: 135; newF: -4285.677459; log10||grad||: -7.754824; alpha: 2
iter: 136; newF: -4285.677459; log10||grad||: -7.660792; alpha: 4
iter: 137; newF: -4285.677459; log10||grad||: -7.506755; alpha: 4
iter: 138; newF: -4285.677459; log10||grad||: -7.864815; alpha: 2
iter: 139; newF: -4285.677459; log10||grad||: -7.769363; alpha: 4
iter: 140; newF: -4285.677459; log10||grad||: -7.614381; alpha: 4
iter: 141; newF: -4285.677459; log10||grad||: -7.974776; alpha: 2
iter: 142; newF: -4285.677459; log10||grad||: -7.877908; alpha: 4
iter: 143; newF: -4285.677459; log10||grad||: -7.721995; alpha: 4
iter: 144; newF: -4285.677459; log10||grad||: -8.084708; alpha: 2
[1] 0.0445615
[1] 0.06572981
[1] 0.03725169
steadyICA documentation built on May 2, 2019, 7:30 a.m.
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Description Usage Arguments Details Value Note Author(s) References Examples
Description
Estimate independent components using the infomax criteria, which is equivalent to maximum likelihood using the logistic density, exp(-S)/(1+exp(-S))^2.
Usage
1 2 | infomaxICA(X, n.comp, W.list = NULL, whiten = FALSE, maxit = 500, eps = 1e-08,
alpha.eps = 1e-08, verbose = FALSE, restarts=0)
|
Arguments
X |
the n x p data matrix |
n.comp |
number of components to be estimated |
W.list |
list of orthogonal matrices for initialization |
whiten |
Whitens the data before applying ICA, i.e., X%*%whitener = Z, where Z has mean zero and empirical covariance equal to the identity matrix, and Z is then used as the input. |
maxit |
maximum number of iterations |
eps |
algorithm terminates when the norm of the gradient is less than eps |
alpha.eps |
tolerance controlling the level of annealing: algorithm terminates with a warning if the learning parameter is less than alpha.eps |
verbose |
if TRUE, prints (1) the value of the infomax objective function at each iteration, (2) the norm of the gradient, and (3) current value of the learning parameter alpha. |
restarts |
An integer determining the number of initial matrices to use in estimating the ICA model. The objective function has local optima, so multiple starting values are recommended. If whiten=TRUE, then generates random orthogonal matrices. If whiten=FALSE, generate random matrices from rnorm(). See code for details. |
Details
This is an R version of ICA using the infomax criteria that provides an alternative to Matlab code (ftp://ftp.cnl.salk.edu/pub/tony/sep96.public), but with a few modifications. First, we use the full data (the so-called offline algorithm) in each iteration rather than an online algorithm with batches. Secondly, we use an adaptive method to choose the step size (based upon Bernaards and Jennrich 2005), which speeds up convergence. We also omitted the bias term (intercept) included in the original formulation because we centered our data.
Value
S |
the estimated independent components |
W |
if whiten=TRUE, returns the orthogonal unmixing matrix; no value is returned when whiten=FALSE |
M |
Returns the estimated mixing matrix for the model X = S M, where X is not pre-whitened (although X is centered) |
f |
the value of the objective function at the estimated S |
Table |
summarizes algorithm status at each iteration |
convergence |
1 if norm of the gradient is less than eps, 2 if the learning parameter was smaller than alpha.eps, which usually means the gradient is sufficiently small, 0 otherwise |
Note
In contrast to most other ICA methods, W is not contrained to be orthogonal.
Author(s)
Benjamin Risk
References
Bell, A. & Sejnowski, T. An information-maximization approach to blind separation and blind deconvolution Neural computation, Neural computation, 1995, 7, 1129-1159.
Bernaards, C. A. and Jennrich, R. I. (2005) Gradient Projection Algorithms and Software for Arbitrary Rotation Criteria in Factor Analysis, Educational and Psychological Measurement 65, 676-696. <http://www.stat.ucla.edu/research/gpa>
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | ## Example when p > d. The MD function and amari measures
# are not defined for M. We can compare the
# "true W inverse", which is the mixing matrix multiplied
# by the whitening matrix; alternatively, we can use
# multidcov::frobICA. These two approaches are
# demonstrated below:
set.seed(999)
nObs <- 1024
nComp <- 3
# simulate from gamma distributions with
# varying amounts of skewness:
simS<-cbind(rgamma(nObs, shape = 1, scale = 2),
rgamma(nObs, shape = 3, scale = 2),
rgamma(nObs, shape = 9, scale = 0.5))
#standardize by expected value and variance:
simS[,1] = (simS[,1] - 1*2)/sqrt(1*2^2)
simS[,2] = (simS[,2] - 3*2)/sqrt(3*2^2)
simS[,3] = (simS[,3] - 9*0.5)/sqrt(9*0.5^2)
# slightly revised 'mixmat' function (from ProDenICA)
# for p>=d: uses fastICA and ProDenICA parameterization:
myMixmat <- function (p = 2, d = NULL) {
if(is.null(d)) d = p
a <- matrix(rnorm(d * p), d, p)
sa <- La.svd(a)
dL <- sort(runif(d) + 1)
mat <- sa$u%*%(sa$vt*dL)
attr(mat, "condition") <- dL[d]/dL[1]
mat
}
simM <- myMixmat(p = 6, d = nComp)
xData <- simS%*%simM
xWhitened <- whitener(xData, n.comp = nComp)
#Define a 'true' W (uses the estimated whitening matrix):
W.true <- solve(simM%*%xWhitened$whitener)
estInfomax <- infomaxICA(X = xData, n.comp = nComp, whiten = TRUE, verbose = TRUE)
frobICA(estInfomax$M,simM)
library(JADE)
MD(t(estInfomax$W),t(solve(W.true)))
amari.error(t(estInfomax$W),t(solve(W.true)))
|
Example output
Loading required package: Rcpp
Loading required package: MASS
Loading required package: clue
Loading required package: combinat
Attaching package: 'combinat'
The following object is masked from 'package:utils':
combn
iter: 1; newF: -4416.845894; log10||grad||: -1.059842; alpha: 2
iter: 2; newF: -4349.538901; log10||grad||: -1.193744; alpha: 4
iter: 3; newF: -4316.57333; log10||grad||: -1.866519; alpha: 2
iter: 4; newF: -4315.058557; log10||grad||: -1.757356; alpha: 4
iter: 5; newF: -4314.075226; log10||grad||: -1.639013; alpha: 4
iter: 6; newF: -4309.199036; log10||grad||: -1.998738; alpha: 2
iter: 7; newF: -4306.161831; log10||grad||: -1.954943; alpha: 4
iter: 8; newF: -4303.018115; log10||grad||: -1.614786; alpha: 8
iter: 9; newF: -4297.895403; log10||grad||: -1.978995; alpha: 2
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[1] 0.0445615
[1] 0.06572981
[1] 0.03725169
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