Type:	Package
Title:	Tools for Exploring Multivariate Data via ICS/ICA
Version:	1.4-2
Date:	2025-03-17
Author:	Klaus Nordhausen [aut, cre], Andreas Alfons [aut], Aurore Archimbaud [aut], Hannu Oja [aut], Anne Ruiz-Gazen [aut], David E. Tyler [aut]
Maintainer:	Klaus Nordhausen <klausnordhausenR@gmail.com>
Depends:	R (≥ 2.5.0), methods, mvtnorm
Imports:	survey, graphics
Suggests:	pixmap, robustbase, MASS, ICSNP, testthat (≥ 3.0.0), ICSOutlier
Description:	Implementation of Tyler, Critchley, Duembgen and Oja's (JRSS B, 2009, <doi:10.1111/j.1467-9868.2009.00706.x>) and Oja, Sirkia and Eriksson's (AJS, 2006, <https://www.ajs.or.at/index.php/ajs/article/view/vol35,%20no2%263%20-%207>) method of two different scatter matrices to obtain an invariant coordinate system or independent components, depending on the underlying assumptions.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
LazyLoad:	yes
Encoding:	UTF-8
NeedsCompilation:	no
RoxygenNote:	7.2.3
Config/testthat/edition:	3
Packaged:	2025-03-18 14:43:10 UTC; nordklau
Repository:	CRAN
Date/Publication:	2025-03-18 21:30:02 UTC

Tools for Exploring Multivariate Data via ICS/ICA

Description

Implementation of Tyler, Critchley, Duembgen and Oja's (JRSS B, 2009, <doi:10.1111/j.1467-9868.2009.00706.x>) and Oja, Sirkia and Eriksson's (AJS, 2006, <https://www.ajs.or.at/index.php/ajs/article/view/vol35,%20no2%263%20-%207>) method of two different scatter matrices to obtain an invariant coordinate system or independent components, depending on the underlying assumptions.

Details

Some multivariate tests and estimates are not affine equivariant by nature. A possible remedy for the lack of that property is to transform the data points to an invariant coordinate system, construct tests and estimates from the transformed data, and if needed, retransform the estimates back. The use of two different scatter matrices to obtain invariant coordinates is implemeted in this package by the function ICS. For an invariant coordinate selection no assumptions are made about the data or the scatter matrices and it can be seen as a data transformation method. If the data come, however, from a so called independent component model the ICS function can recover the independent components and estimate the mixing matrix under general assumptions. Besides, the function ICS provides these package tools to work with objects of this class, and some scatter matrices which can be used in the ICS function. Furthermore, there are also two tests for multinormality. Note that starting with version 1.4-0 the functions ics and ics2 are not recommended anymore and everything can be done in a more efficient way using the function ICS which combines the functionality of the original two functions and also provides an improved algorithm for certain scatter combinations. Furthermore, does ICS return an S3 object and not anymore S4 objects as ics and ics2 did. In the long run functions ics and ics2 will be removed from the package.

Index of help topics:

ICS-S3                  Two Scatter Matrices ICS Transformation
ICS-package             Tools for Exploring Multivariate Data via
                        ICS/ICA
ICS_scatter             Location and Scatter Estimates for ICS
Mean3Cov4               Location Vector Based on 3rd Moments and
                        Scatter Matrix Based on 4th Moments
MeanCov                 Mean Vector and Covariance Matrix
coef.ICS-S3             To extract the Coefficient Matrix of the ICS
                        Transformation
coef.ics                To extract the Unmixing Matrix
components              To extract the Component Scores of the ICS
                        Transformation
cov4                    Scatter Matrix based on Fourth Moments
cov4.wt                 Weighted Scatter Matrix based on Fourth Moments
covAxis                 One step Tyler Shape Matrix
covOrigin               Covariance Matrix with Respect to the Origin
covW                    One-step M-estimator
fitted.ICS-S3           Fitted Values of the ICS Transformation
fitted.ics              Fitted Values of an ICS Object
gen_kurtosis            To extract the Generalized Kurtosis Values of
                        the ICS Transformation
ics                     Two Scatter Matrices ICS Transformation
ics-class               Class ICS
ics.components          Extracting ICS Components
ics2                    Two Scatter Matrices ICS Transformation
                        Augmented by Two Location Estimates
ics2-class              Class ICS2
mean3                   Location Estimate based on Third Moments
mvnorm.kur.test         Test of Multivariate Normality Based on
                        Kurtosis
mvnorm.skew.test        Test of Multivariate Normality Based on
                        Skewness
plot.ICS-S3             Scatterplot Matrix of Component Scores from the
                        ICS Transformation
plot.ics                Scatterplot for a ICS Object
print.ICS-S3            Basic information of ICS Object
print.ics               Basic information of ICS Object
print.ics2              Basic information of ICS2 Object
scovq                   Supervised scatter matrix based on quantiles
screeplot.ICS-S3        Screeplot for an 'ICS' Object
screeplot.ics           Screeplot for an ICS Object
summary.ICS-S3          To summarize an 'ICS' object
summary.ics             To summarize an ICS object
summary.ics2            To summarize an ICS2 object
tM                      Joint M-estimation of Location and Scatter for
                        a Multivariate t-distribution

Further information is available in the following vignettes:

Author(s)

Klaus Nordhausen [aut, cre] (<https://orcid.org/0000-0002-3758-8501>), Andreas Alfons [aut] (<https://orcid.org/0000-0002-2513-3788>), Aurore Archimbaud [aut] (<https://orcid.org/0000-0002-6511-9091>), Hannu Oja [aut] (<https://orcid.org/0000-0002-4945-5976>), Anne Ruiz-Gazen [aut] (<https://orcid.org/0000-0001-8970-8061>), David E. Tyler [aut]

Maintainer: Klaus Nordhausen <klausnordhausenR@gmail.com>

References

Tyler, D.E., Critchley, F., Dümbgen, L. and Oja, H. (2009), Invariant co-ordinate selecetion, Journal of the Royal Statistical Society,Series B, 71, 549–592. <doi:10.1111/j.1467-9868.2009.00706.x>.

Oja, H., Sirkiä, S. and Eriksson, J. (2006), Scatter matrices and independent component analysis, Austrian Journal of Statistics, 35, 175–189.

Nordhausen, K., Oja, H. and Tyler, D.E. (2008), Tools for exploring multivariate data: The package ICS, Journal of Statistical Software, 28, 1–31. <doi:10.18637/jss.v028.i06>.

Archimbaud, A., Drmac, Z., Nordhausen, K., Radojicic, U. and Ruiz-Gazen, A. (2023), Numerical considerations and a new implementation for ICS, SIAM Journal on Mathematics of Data Science, 5, 97–121. <doi:10.1137/22M1498759>.

Two Scatter Matrices ICS Transformation

Description

Transforms the data via two scatter matrices to an invariant coordinate system or independent components, depending on the underlying assumptions. Function ICS() is intended as a replacement for ics() and ics2(), and it combines their functionality into a single function. Importantly, the results are returned as an S3 object rather than an S4 object. Furthermore, ICS() implements recent improvements, such as a numerically stable algorithm based on the QR algorithm for a common family of scatter pairs.

Usage

ICS(
  X,
  S1 = ICS_cov,
  S2 = ICS_cov4,
  S1_args = list(),
  S2_args = list(),
  algorithm = c("whiten", "standard", "QR"),
  center = FALSE,
  fix_signs = c("scores", "W"),
  na.action = na.fail
)

Arguments

Details

For a given scatter pair S_{1} and S_{2}, a matrix Z (in which the columns contain the scores of the respective invariant coordinates) and a matrix W (in which the rows contain the coefficients of the linear transformation to the respective invariant coordinates) are found such that:

• The columns of Z are whitened with respect to S_{1}. That is, S_{1}(Z) = I, where I denotes the identity matrix.

• The columns of Z are uncorrelated with respect to S_{2}. That is, S_{2}(Z) = D, where D is a diagonal matrix.

• The columns of Z are ordered according to their generalized kurtosis.

Given those criteria, W is unique up to sign changes in its rows. The argument fix_signs provides two ways to ensure uniqueness of W:

• If argument fix_signs is set to "scores", the signs in W are fixed such that the generalized skewness values of all components are positive. If S1 and S2 provide location components, which are denoted by T_{1} and T_{2}, the generalized skewness values are computed as T_{1}(Z) - T_{2}(Z). Otherwise, the skewness is computed by subtracting the column medians of Z from the corresponding column means so that all components are right-skewed. This way of fixing the signs is preferred in an invariant coordinate selection framework.

• If argument fix_signs is set to "W", the signs in W are fixed independently of Z such that the maximum element in each row of W is positive and that each row has norm 1. This is the usual way of fixing the signs in an independent component analysis framework.

In principal, the order of S_{1} and S_{2} does not matter if both are true scatter matrices. Changing their order will just reverse the order of the components and invert the corresponding generalized kurtosis values.

The same does not hold when at least one of them is a shape matrix rather than a true scatter matrix. In that case, changing their order will also reverse the order of the components, but the ratio of the generalized kurtosis values is no longer 1 but only a constant. This is due to the fact that when shape matrices are used, the generalized kurtosis values are only relative ones.

Different algorithms are available to compute the invariant coordinate system of a data frame X_n with n observations:

• "whiten": whitens the data X_n with respect to the first scatter matrix before computing the second scatter matrix. If S2 is not a function, whitening is not applicable.

  • whiten the data X_n with respect to the first scatter matrix: Y_n = X_n S_1(X_n)^{-1/2}

  • compute S_2 for the uncorrelated data: S_2(Y_n)

  • perform the eigendecomposition of S_2(Y_n): S_2(Y_n) = UDU'

  • compute W: W = U' S_1(X_n)^{-1/2}

• "standard": performs the spectral decomposition of the symmetric matrix M(X_n)

  • compute M(X_n) = S_1(X_n)^{-1/2} S_2(X_n) S_1(X_n)^{-1/2}

  • perform the eigendecomposition of M(X_n): M(X_n) = UDU'

  • compute W: W = U' S_1(X_n)^{-1/2}

• "QR": numerically stable algorithm based on the QR algorithm for a common family of scatter pairs: if S1 is ICS_cov() or cov(), and if S2 is one of ICS_cov4(), ICS_covW() , ICS_covAxis(), cov4(), covW(), or covAxis(). For other scatter pairs, the QR algorithm is not applicable. See Archimbaud et al. (2023) for details. Note that the QR algorithm requires LAPACK version \geq 3.11.0 to attain better numerical stability.

The "whiten" algorithm is the most natural version and therefore the default. The option "standard" should be only used if the scatters provided are not functions but precomputed matrices. The option "QR" is mainly of interest when there are numerical issues when "whiten" is used and the scatter combination allows its usage.

Note that when the purpose of ICS is outlier detection the package ICSOutlier provides additional functionalities as does the package ICSClust in case the goal of ICS is dimension reduction prior clustering.

Value

An object of class "ICS" with the following components:

Author(s)

Andreas Alfons and Aurore Archimbaud, based on code for ics() and ics2() by Klaus Nordhausen

References

Tyler, D.E., Critchley, F., Duembgen, L. and Oja, H. (2009) Invariant Co-ordinate Selection. Journal of the Royal Statistical Society, Series B, 71(3), 549–592. doi:10.1111/j.1467-9868.2009.00706.x.

Archimbaud, A., Drmac, Z., Nordhausen, K., Radojcic, U. and Ruiz-Gazen, A. (2023) Numerical Considerations and a New Implementation for Invariant Coordinate Selection. SIAM Journal on Mathematics of Data Science, 5(1), 97–121. doi:10.1137/22M1498759.

See Also

gen_kurtosis(), coef(), components(), fitted(), and plot() methods

Examples

# import data
data("iris")
X <- iris[,-5]

# run ICS
out_ICS <- ICS(X)
out_ICS
summary(out_ICS)

# extract generalized eigenvalues
gen_kurtosis(out_ICS)
# Plot
screeplot(out_ICS)

# extract the components
components(out_ICS)
components(out_ICS, select = 1:2)

# Plot
plot(out_ICS)

# equivalence with previous functions
out_ics <- ics(X, S1 = cov, S2 = cov4, stdKurt = FALSE)
out_ics
out_ics2 <- ics2(X, S1 = MeanCov, S2 = Mean3Cov4)
out_ics2
out_ICS


# example using two functions
X1 <- rmvnorm(250, rep(0,8), diag(c(rep(1,6),0.04,0.04)))
X2 <- rmvnorm(50, c(rep(0,6),2,0), diag(c(rep(1,6),0.04,0.04)))
X3 <- rmvnorm(200, c(rep(0,7),2), diag(c(rep(1,6),0.04,0.04)))
X.comps <- rbind(X1,X2,X3)
A <- matrix(rnorm(64),nrow=8)
X <- X.comps %*% t(A)
ics.X.1 <- ICS(X)
summary(ics.X.1)
plot(ics.X.1)
# compare to
pairs(X)
pairs(princomp(X,cor=TRUE)$scores)


# slow:
if (require("ICSNP")) {
  ics.X.2 <- ICS(X, S1 = tyler.shape, S2 = duembgen.shape,
  S1_args = list(location=0))
  summary(ics.X.2)
  plot(ics.X.2)
  # example using three pictures
  library(pixmap)
  fig1 <- read.pnm(system.file("pictures/cat.pgm", package = "ICS")[1],
                   cellres = 1)
  fig2 <- read.pnm(system.file("pictures/road.pgm", package = "ICS")[1],
                   cellres = 1)
  fig3 <- read.pnm(system.file("pictures/sheep.pgm", package = "ICS")[1],
                   cellres = 1)
  p <- dim(fig1@grey)[2]
  fig1.v <- as.vector(fig1@grey)
  fig2.v <- as.vector(fig2@grey)
  fig3.v <- as.vector(fig3@grey)
  X <- cbind(fig1.v, fig2.v, fig3.v)
  A <- matrix(rnorm(9), ncol = 3)
  X.mixed <- X %*% t(A)
  ICA.fig <- ICS(X.mixed)
  par.old <- par()
  par(mfrow=c(3,3), omi = c(0.1,0.1,0.1,0.1), mai = c(0.1,0.1,0.1,0.1))
  plot(fig1)
  plot(fig2)
  plot(fig3)
  plot(pixmapGrey(X.mixed[,1], ncol = p, cellres = 1))
  plot(pixmapGrey(X.mixed[,2], ncol = p, cellres = 1))
  plot(pixmapGrey(X.mixed[,3], ncol = p, cellres = 1))
  plot(pixmapGrey(components(ICA.fig)[,1], ncol = p, cellres = 1))
  plot(pixmapGrey(components(ICA.fig)[,2], ncol = p, cellres = 1))
  plot(pixmapGrey(components(ICA.fig)[,3], ncol = p, cellres = 1))
}

Location and Scatter Estimates for ICS

Description

Computes a scatter matrix and an optional location vector to be used in transforming the data to an invariant coordinate system or independent components.

Usage

ICS_cov(x, location = TRUE)

ICS_cov4(x, location = c("mean", "mean3", "none"))

ICS_covW(x, location = TRUE, alpha = 1, cf = 1)

ICS_covAxis(x, location = TRUE)

ICS_tM(x, location = TRUE, df = 1, ...)

ICS_scovq(x, y, ...)

Arguments

Details

ICS_cov() is a wrapper for the sample covariance matrix as computed by cov().

ICS_cov4() is a wrapper for the scatter matrix based on fourth moments as computed by cov4(). Note that the scatter matrix is always computed with respect to the sample mean, even though the returned location component can be specified to be based on third moments as computed by mean3(). Setting a location component other than the sample mean can be used to fix the signs of the invariant coordinates in ICS() based on generalized skewness values, for instance when using the scatter pair ICS_cov() and ICS_cov4().

ICS_covW() is a wrapper for the one-step M-estimator of scatter as computed by covW().

ICS_covAxis() is a wrapper for the one-step Tyler shape matrix as computed by covAxis(), which is can be used to perform Principal Axis Analysis.

ICS_tM() is a wrapper for the M-estimator of location and scatter for a multivariate t-distribution, as computed by tM().

ICS_scovq() is a wrapper for the supervised scatter matrix based on quantiles scatter, as computed by scovq().

Value

An object of class "ICS_scatter" with the following components:

Author(s)

Andreas Alfons and Aurore Archimbaud

References

Arslan, O., Constable, P.D.L. and Kent, J.T. (1995) Convergence behaviour of the EM algorithm for the multivariate t-distribution, Communications in Statistics, Theory and Methods, 24(12), 2981–3000. doi:10.1080/03610929508831664.

Critchley, F., Pires, A. and Amado, C. (2006) Principal Axis Analysis. Technical Report, 06/14. The Open University, Milton Keynes.

Kent, J.T., Tyler, D.E. and Vardi, Y. (1994) A curious likelihood identity for the multivariate t-distribution, Communications in Statistics, Simulation and Computation, 23(2), 441–453. doi:10.1080/03610919408813180.

Oja, H., Sirkia, S. and Eriksson, J. (2006) Scatter Matrices and Independent Component Analysis. Austrian Journal of Statistics, 35(2&3), 175-189.

Tyler, D.E., Critchley, F., Duembgen, L. and Oja, H. (2009) Invariant Co-ordinate Selection. Journal of the Royal Statistical Society, Series B, 71(3), 549–592. doi:10.1111/j.1467-9868.2009.00706.x.

See Also

ICS()

colMeans(), mean3()

cov(), cov4(), covW(), covAxis(), tM(), scovq()

Examples

data("iris")
X <- iris[,-5]
ICS_cov(X)
ICS_cov4(X)
ICS_covW(X, alpha = 1, cf = 1/(ncol(X)+2))
ICS_covAxis(X)
ICS_tM(X)


# The number of explaining variables
p <- 10
# The number of observations
n <- 400
# The error variance
sigma <- 0.5
# The explaining variables
X <- matrix(rnorm(p*n),n,p)
# The error term
epsilon <- rnorm(n, sd = sigma)
# The response
y <- X[,1]^2 + X[,2]^2*epsilon
ICS_scovq(X, y = y)

Location Vector Based on 3rd Moments and Scatter Matrix Based on 4th Moments

Description

Returns, for some multivariate data, the location vector based on 3rd moments and the scatter matrix based on 4th moments.

Usage

Mean3Cov4(x)

Arguments

Details

Note that the scatter matrix of 4th moments is computed with respect to the mean vector and not with respect to the location vector based on 3rd moments.

Value

A list containing:

Author(s)

Klaus Nordhausen

See Also

mean3, cov4

Examples

X <- rmvnorm(200, 1:3, diag(2:4))
Mean3Cov4(X)

Mean Vector and Covariance Matrix

Description

Returns, for some multivariate data, the mean vector and covariance matrix.

Usage

MeanCov(x)

Arguments

Value

A list containing:

Author(s)

Klaus Nordhausen

See Also

colMeans, cov

Examples

X <- rmvnorm(200, 1:3, diag(2:4))
MeanCov(X)

To extract the Coefficient Matrix of the ICS Transformation

Description

Extracts the coefficient matrix of a linear transformation to an invariant coordinate system. Each row of the matrix contains the coefficients of the transformation to the corresponding component.

Usage

## S3 method for class 'ICS'
coef(object, select = NULL, drop = FALSE, index = NULL, ...)

Arguments

Value

A numeric matrix or vector containing the coefficients for the requested components.

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

gen_kurtosis(), components(), fitted(), and plot() methods

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
coef(out)
coef(out, select = c(1,4))
coef(out, select = 1, drop = FALSE)

To extract the Unmixing Matrix

Description

Extracts the unmixing matrix of a class ics object.

Usage

## S4 method for signature 'ics'
coef(object)

Arguments

Value

The unmixing matrix of a class ics object.

Author(s)

Klaus Nordhausen

See Also

ics-class and ics

To extract the Component Scores of the ICS Transformation

Description

Extracts the components scores of an invariant coordinate system obtained via an ICS transformation.

Usage

components(x, ...)

## S3 method for class 'ICS'
components(x, select = NULL, drop = FALSE, index = NULL, ...)

Arguments

Value

A numeric matrix or vector containing the requested components.

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

gen_kurtosis(), coef(), fitted(), and plot() methods

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
components(out)
components(out, select = c(1,4))
components(out, select = 1, drop = FALSE)

Scatter Matrix based on Fourth Moments

Description

Estimates the scatter matrix based on the 4th moments of the data.

Usage

cov4(X, location = "Mean", na.action = na.fail)

Arguments

Details

If location is Mean the scatter matrix of 4th moments is computed wrt to the sample mean. For location = Origin it is the scatter matrix of 4th moments wrt to the origin. The scatter matrix is standardized in such a way to be consistent for the regular covariance matrix at the multinormal model. It is given for n \times p matrix X by

\frac{1}{p+2} ave_{i}\{[(x_{i}-\bar{x})S^{-1}(x_{i}-\bar{x})'](x_{i}-\bar{x})'(x_{i}-\bar{x})\},

where \bar{x} is the mean vector and S the regular covariance matrix.

Value

A matrix containing the estimated fourth moments scatter.

Author(s)

Klaus Nordhausen

References

Cardoso, J.F. (1989), Source separation using higher order moments, in Proc. IEEE Conf. on Acoustics, Speech and Signal Processing (ICASSP'89), 2109–2112. <doi:10.1109/ICASSP.1989.266878>.

Oja, H., Sirki?, S. and Eriksson, J. (2006), Scatter matrices and independent component analysis, Austrian Journal of Statistics, 35, 175–189.

Examples

set.seed(654321)
cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
X <- rmvnorm(100, c(0,0,0), cov.matrix)
cov4(X)
cov4(X, location="Origin")
rm(.Random.seed)

Weighted Scatter Matrix based on Fourth Moments

Description

Estimates the weighted scatter matrix based on the 4th moments of the data.

Usage

cov4.wt(x, wt = rep(1/nrow(x), nrow(x)), location = TRUE,
        method = "ML", na.action = na.fail)

Arguments

Details

If location = TRUE, then the scatter matrix is given for a n \times p data matrix X by

\frac{1}{p+2} ave_{i}\{w_i[(x_{i}-\bar{x}_w)S_w^{-1}(x_{i}-\bar{x}_w)'](x_{i}-\bar{x}_w)'(x_{i}-\bar{x}_w)\},

where w_i are the weights standardized such that \sum{w_i}=1, \bar{x}_w is the weighted mean vector and S_w the weighted covariance matrix. For details about the weighted mean vector and weighted covariance matrix see cov.wt.

Value

A matrix containing the estimated weighted fourth moments scatter.

Author(s)

Klaus Nordhausen

See Also

cov4, cov.wt

Examples

cov.matrix.1 <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
X.1 <- rmvnorm(100, c(0,0,0), cov.matrix.1)
cov.matrix.2 <- diag(1,3)
X.2 <- rmvnorm(50, c(1,1,1), cov.matrix.2)
X <- rbind(X.1, X.2)

cov4.wt(X, rep(c(0,1), c(100,50)))
cov4.wt(X, rep(c(1,0), c(100,50)))

One step Tyler Shape Matrix

Description

This matrix can be used to get the principal axes from ics, which is then known as principal axis analysis.

Usage

covAxis(X, na.action = na.fail)

Arguments

Details

The covAxis matrix V is a given for a n \times p data matrix X as

p \ ave_{i}\{[(x_{i}-\bar{x})S^{-1}(x_{i}-\bar{x})']^{-1}(x_{i}-\bar{x})'(x_{i}-\bar{x})\},

where \bar{x} is the mean vector and S the regular covariance matrix.

covAxis can be used to perform a Prinzipal Axis Analysis (Critchley et al. 2006) using the function ics. In that case, for a centered data matrix X, covAxis can be used as S2 in ics, where S1 should be in that case the regular covariance matrix.

Value

A matrix containing the estimated one step Tyler shape matrix.

Author(s)

Klaus Nordhausen

References

Critchley , F., Pires, A. and Amado, C. (2006), Principal axis analysis, Technical Report, 06/14, The Open University Milton Keynes.

Tyler, D.E., Critchley, F., D?mbgen, L. and Oja, H. (2009), Invariant co-ordinate selecetion, Journal of the Royal Statistical Society,Series B, 71, 549–592. <doi:10.1111/j.1467-9868.2009.00706.x>.

See Also

ics

Examples

data(iris)
iris.centered <- sweep(iris[,1:4], 2, colMeans(iris[,1:4]), "-")
iris.paa <- ics(iris.centered, cov, covAxis, stdKurt = FALSE)
summary(iris.paa)
plot(iris.paa, col=as.numeric(iris[,5]))
mean(iris.paa@gKurt)
emp.align <- iris.paa@gKurt
emp.align

screeplot(iris.paa)
abline(h = 1)

Covariance Matrix with Respect to the Origin

Description

Estimates the covariance matrix with respect to the origin.

Usage

covOrigin(X, location = NULL, na.action = na.fail)

Arguments

Details

The covariance matrix S_{0} with respect to origin is given for a matrix X with n observations by

S_{0}= \frac{1}{n}X'X.

Value

A matrix containing the estimated covariance matrix with respect to the origin.

Author(s)

Klaus Nordhausen

See Also

cov

Examples

set.seed(654321)
cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
X <- rmvnorm(100,c(0,0,0),cov.matrix)
covOrigin(X)
rm(.Random.seed)

One-step M-estimator

Description

Estimates the scatter matrix based on one-step M-estimator using mean and covariance matrix as starting point.

Usage

covW(X, na.action = na.fail, alpha = 1, cf = 1)

Arguments

Details

It is given for n \times p matrix X by

COV_{w}(X)=\frac{1}{n} {cf} \sum_{i=1}^n w(D^2(x_i)) (x_i - \bar{ x})^\top(x_i - \bar{ x}),

where \bar{x} is the mean vector, D^2(x_i) is the squared Mahalanobis distance, w(d)=d^\alpha is a non-negative and continuous weight function and {cf} is a consistency factor. Note that the consistency factor, which makes the estimator consistent at the multivariate normal distribution, is in most case unknown and therefore the default is to use simply cf = 1.

• If w(d)=1, we get the covariance matrix cov() (up to the factor 1/(n-1) instead of 1/n).

• If \alpha=-1, we get the covAxis().

• If \alpha=1, we get the cov4() with {cf} = \frac{1}{p+2}.

Value

A matrix containing the one-step M-scatter.

Author(s)

Aurore Archimbaud and Klaus Nordhausen

References

Archimbaud, A., Drmac, Z., Nordhausen, K., Radojicic, U. and Ruiz-Gazen, A. (2023). SIAM Journal on Mathematics of Data Science (SIMODS), Vol.5(1):97–121. doi:10.1137/22M1498759.

See Also

cov(), cov4(), covAxis()

Examples

data(iris)
X <- iris[,1:4]

# Equivalence with covAxis
covW(X, alpha = -1, cf = ncol(X))
covAxis(X)

# Equivalence with cov4
covW(X, alpha = 1, cf = 1/(ncol(X)+2))
cov4(X)

# covW with alpha = 0.5
covW(X, alpha = 0.5)

Fitted Values of the ICS Transformation

Description

Computes the fitted values based on an invariant coordinate system obtained via an ICS transformation. When using all components, computing the fitted values constitutes a backtransformation to the observed data. When using fewer components, the fitted values can often be viewed as reconstructions of the observed data with noise removed.

Usage

## S3 method for class 'ICS'
fitted(object, select = NULL, index = NULL, ...)

Arguments

Value

A numeric matrix containing the fitted values.

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

gen_kurtosis(), coef(), components(), and plot() methods

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
fitted(out)
fitted(out, select = 4)

Fitted Values of an ICS Object

Description

Computes the fitted values of an ics object.

Usage

## S4 method for signature 'ics'
fitted(object,index=NULL)

Arguments

Value

Returns a dataframe with the fitted values.

Author(s)

Klaus Nordhausen

See Also

ics-class and ics

Examples

    set.seed(123456)
    X1 <- rmvnorm(250, rep(0,8), diag(c(rep(1,6),0.04,0.04)))
    X2 <- rmvnorm(50, c(rep(0,6),2,0), diag(c(rep(1,6),0.04,0.04)))
    X3 <- rmvnorm(200, c(rep(0,7),2), diag(c(rep(1,6),0.04,0.04)))

    X.comps <- rbind(X1,X2,X3)
    A <- matrix(rnorm(64),nrow=8)
    X <- X.comps %*% t(A)

    ics.X.1 <- ics(X)
    fitted(ics.X.1)
    fitted(ics.X.1,index=c(1,2,3,6,7,8))

    rm(.Random.seed)

To extract the Generalized Kurtosis Values of the ICS Transformation

Description

Extracts the generalized kurtosis values of the components obtained via an ICS transformation.

Usage

gen_kurtosis(object, ...)

## S3 method for class 'ICS'
gen_kurtosis(object, select = NULL, scale = FALSE, index = NULL, ...)

Arguments

Details

The argument scale is useful when ICS is performed with shape matrices rather than true scatter matrices. Let S_{1} and S_{2} denote the scatter or shape matrices used in ICS.

If both S_{1} and S_{2} are true scatter matrices, their order in principal does not matter. Changing their order will just reverse the order of the components and invert the corresponding generalized kurtosis values.

The same does not hold when at least one of them is a shape matrix rather than a true scatter matrix. In that case, changing their order will also reverse the order of the components, but the ratio of the generalized kurtosis values is no longer 1 but only a constant. This is due to the fact that when shape matrices are used, the generalized kurtosis values are only relative ones. It is then useful to scale the generalized kurtosis values such that their product is 1.

Value

A numeric vector containing the generalized kurtosis values of the requested components.

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

coef(), components(), fitted(), and plot() methods

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
gen_kurtosis(out)
gen_kurtosis(out, scale = TRUE)
gen_kurtosis(out, select = c(1,4))

Two Scatter Matrices ICS Transformation

Description

Implements the two scatter matrices transformation to obtain an invariant coordinate sytem or independent components, depending on the underlying assumptions.

Usage

ics(X, S1 = cov, S2 = cov4, S1args = list(), S2args = list(),
    stdB = "Z", stdKurt = TRUE, na.action = na.fail)

Arguments

Details

Seeing this function as a tool for data transformation the result is an invariant coordinate selection which can be used for test and estimation. And if needed the results can be easily retransformed to the original scale. It is possible to use it also for dimension reduction, in order to find outliers or clusters in the data. The function can, also be used in a modelling framework. In this case it is assumed that the data were created by mixing independent components which have different kurtosis values. If the two scatter matrices used have the so-called independence property the function can recover the independent components by estimating the unmixing matrix.

By default S1 is the regular covariance matrix cov and S2 the matrix of fourth moments cov4. However those can be replaced with any other scatter matrix the user prefers. The package ICS offers for example also cov4.wt, covAxis, covOrigin, covW or tM and the ICSNP offers further scatters as duembgen.shape, tyler.shape, HR.Mest or HP1.shape. But of course also scatters from any other package can be used.

Note that when function names are submitted, the function should return only a scatter matrix. If the function returns more, the scatter should be computed in advance or a wrapper written that yields the required output. For example tM returns a list with four elements where the scatter estimate is called V. A simple wrapper would then be my.tm <- function(x, ...) tM(x, ...)$V.

For a given choice of S1 and S2, the general idea of the ics function is to find the unmixing matrix B and the invariant coordinates (independent coordinates) Z in such a way, that:

(i)

The elements of Z are standardized with respect to S1 (S1(Z)=I).

(ii)

The elements of Z are uncorrelated with respect to S2. (S2(Z)=D, where D is a diagonal matrix).

(iii)

The elements of Z are ordered according to their generalized kurtosis.

Given those criteria, B is unique up to sign changes of its rows. The function provides two options to decide the exact form of B.

(i)

Method 'Z' standardizes B such, that all components are right skewed. The criterion used is the sign of each componentwise difference of mean vector and transformation retransformation median. This standardization is prefered in an invariant coordinate framework.

(ii)

Method 'B' standardizes B independent of Z such that the maximum element per row is positive and each row has norm 1. Usual way in an independent component analysis framework.

In principal, if S1 and S2 are true scatter matrices the order does not matter. It will just reverse and invert the kurtosis value vector. This is however not true when one or both are shape matrices (and not both of them are scatter matrices). In this case the order of the kurtosis values is also reversed, the ratio however then is not 1 but only constant. This is due to the fact that when shape matrices are used, the kurtosis values are only relative ones. Therefore by the default the kurtosis values are standardized such that their product is 1. If no standardization is wanted, the 'stdKurt' argument should be used.

Value

an object of class ics.

Note

Function ics() reached the end of its lifecycle, please use ICS() instead. In future versions, ics() will be deprecated and eventually removed.

Author(s)

Klaus Nordhausen

References

Tyler, D.E., Critchley, F., D?mbgen, L. and Oja, H. (2009), Invariant co-ordinate selecetion, Journal of the Royal Statistical Society,Series B, 71, 549–592. <doi:10.1111/j.1467-9868.2009.00706.x>.

Oja, H., Sirki?, S. and Eriksson, J. (2006), Scatter matrices and independent component analysis, Austrian Journal of Statistics, 35, 175–189.

Nordhausen, K., Oja, H. and Tyler, D.E. (2008), Tools for exploring multivariate data: The package ICS, Journal of Statistical Software, 28, 1–31. <doi:10.18637/jss.v028.i06>.

See Also

ICS-package, ICS

Examples

    # example using two functions
    set.seed(123456)
    X1 <- rmvnorm(250, rep(0,8), diag(c(rep(1,6),0.04,0.04)))
    X2 <- rmvnorm(50, c(rep(0,6),2,0), diag(c(rep(1,6),0.04,0.04)))
    X3 <- rmvnorm(200, c(rep(0,7),2), diag(c(rep(1,6),0.04,0.04)))

    X.comps <- rbind(X1,X2,X3)
    A <- matrix(rnorm(64),nrow=8)
    X <- X.comps %*% t(A)

    ics.X.1 <- ics(X)
    summary(ics.X.1)
    plot(ics.X.1)

    # compare to
    pairs(X)
    pairs(princomp(X,cor=TRUE)$scores)

    # slow:

    # library(ICSNP)
    # ics.X.2 <- ics(X, tyler.shape, duembgen.shape, S1args=list(location=0))
    # summary(ics.X.2)
    # plot(ics.X.2)

    rm(.Random.seed)

    # example using two computed scatter matrices for outlier detection

    library(robustbase)
    ics.wood<-ics(wood,tM(wood)$V,tM(wood,2)$V)
    plot(ics.wood)

    # example using three pictures
    library(pixmap)

    fig1 <- read.pnm(system.file("pictures/cat.pgm", package = "ICS")[1])
    fig2 <- read.pnm(system.file("pictures/road.pgm", package = "ICS")[1])
    fig3 <- read.pnm(system.file("pictures/sheep.pgm", package = "ICS")[1])

    p <- dim(fig1@grey)[2]

    fig1.v <- as.vector(fig1@grey)
    fig2.v <- as.vector(fig2@grey)
    fig3.v <- as.vector(fig3@grey)
    X <- cbind(fig1.v,fig2.v,fig3.v)

    set.seed(4321)
    A <- matrix(rnorm(9), ncol = 3)
    X.mixed <- X %*% t(A)

    ICA.fig <- ics(X.mixed)

    par.old <- par()
    par(mfrow=c(3,3), omi = c(0.1,0.1,0.1,0.1), mai = c(0.1,0.1,0.1,0.1))

    plot(fig1)
    plot(fig2)
    plot(fig3)

    plot(pixmapGrey(X.mixed[,1],ncol=p))
    plot(pixmapGrey(X.mixed[,2],ncol=p))
    plot(pixmapGrey(X.mixed[,3],ncol=p))

    plot(pixmapGrey(ics.components(ICA.fig)[,1],ncol=p))
    plot(pixmapGrey(ics.components(ICA.fig)[,2],ncol=p))
    plot(pixmapGrey(ics.components(ICA.fig)[,3],ncol=p))

    par(par.old)
    rm(.Random.seed)
    

Class ICS

Description

A S4 class to store results from an invariant coordinate system transformation or independent component computation based on two scatter matrices.

Objects from the Class

Objects can be created by calls of the form new("ics", ...). But usually objects are created by the function ics.

Slots

gKurt:

Object of class "numeric". Gives the generalized kurtosis measures of the components

UnMix:

Object of class "matrix". The unmixing matrix.

S1:

Object of class "matrix". The first scatter matrix.

S2:

Object of class "matrix". The second scatter matrix.

S1name:

Object of class "character". Name of the first scatter matrix.

S2name:

Object of class "character". Name of the second scatter matrix.

Scores:

Object of class "data.frame". The underlying components in the invariant coordinate system.

DataNames:

Object of class "character". Names of the original variables.

StandardizeB:

Object of class "character". Names standardization method for UnMix.

StandardizegKurt:

Object of class "logical". States wether the generalized kurtosis is standardized or not.

Methods

For this class the following generic functions are available: print.ics, summary.ics, coef.ics, fitted.ics and plot.ics

Note

In case no extractor function for the slots exists, the component can be extracted the usual way using '@'.

Author(s)

Klaus Nordhausen

See Also

ics

Extracting ICS Components

Description

Function to extract the ICS components of a ics object.

Usage

ics.components(object)

Arguments

Value

Dataframe that contains the components.

Author(s)

Klaus Nordhausen

See Also

ics-class and ics

Two Scatter Matrices ICS Transformation Augmented by Two Location Estimates

Description

This function implements the two scatter matrices transformation to obtain an invariant coordinate sytem or independent components, depending on the underlying assumptions. Differently to ics here, there are also two location functionals used to fix the signs of the components and to get a measure of skewness.

Usage

ics2(X, S1 = MeanCov, S2 = Mean3Cov4, S1args = list(), S2args = list(),
     na.action = na.fail)

Arguments

Details

For a general discussion about ICS see the help for ics. The difference to ics is that S1 and S2 are either functions which return a list containing a multivariate location and scatter computed on X or lists containing these measures computed in advance. Of importance for the resulting lists is that in both cases the location vector is the first element of the list and the scatter matrix the second element. This means most multivariate location - scatter functions can be used directly without the need to write a wrapper.

The invariant coordinates Z are then computed such that (i) T1(Z) = 0, the origin. (ii) S1(Z) = I_p, the identity matrix. (iii) T2(Z) = S, where S is a vector having positive elements which can be seen as a generalized skewness measure (gSkew). (iv) S2(Z) = D, a diagonal matrix with descending elements which can be seen as a generalized kurtosis measure (gKurt).

Hence in this function there are no options to standardize Z or the transformation matrix B as everything is specified by S1 and S2.

Note also that ics2 makes hardly any input checks.

Value

an object of class ics2 inheriting from class ics.

Note

Function ics2() reached the end of its lifecycle, please use ICS() instead. In future versions, ics2() will be deprecated and eventually removed.

Author(s)

Klaus Nordhausen

References

Tyler, D.E., Critchley, F., Dümbgen, L. and Oja, H. (2009), Invariant co-ordinate selecetion, Journal of the Royal Statistical Society,Series B, 71, 549–592. <doi:10.1111/j.1467-9868.2009.00706.x>.

Nordhausen, K., Oja, H. and Ollila, E. (2011), Multivariate Models and the First Four Moments, In Hunter, D.R., Richards, D.S.R. and Rosenberger, J.L. (editors) "Nonparametric Statistics and Mixture Models: A Festschrift in Honor of Thomas P. Hettmansperger", 267–287, World Scientific, Singapore. <doi:10.1142/9789814340564_0016>.

See Also

ICS

Examples

 set.seed(123456)
 X1 <- rmvnorm(250, rep(0,8), diag(c(rep(1,6),0.04,0.04)))
 X2 <- rmvnorm(50, c(rep(0,6),2,0), diag(c(rep(1,6),0.04,0.04)))
 X3 <- rmvnorm(200, c(rep(0,7),2), diag(c(rep(1,6),0.04,0.04)))

 X.comps <- rbind(X1,X2,X3)
 A <- matrix(rnorm(64),nrow=8)
 X <- X.comps %*% t(A)

 # the default
 ics2.X.1 <- ics2(X2)
 summary(ics2.X.1)

 # using another function as S2 not with its default
 ics2.X.2 <- ics2(X2, S2 = tM, S2args = list(df = 2))
 summary(ics2.X.2)

 # computing in advance S2 and using another S1
 Scauchy <- tM(X)
 ics2.X.2 <- ics2(X2, S1 = tM, S2 = Scauchy, S1args = list(df = 5))
 summary(ics2.X.2)
 plot(ics2.X.2)

Class ICS2

Description

A S4 class to store results from an invariant coordinate system transformation or independent component computation based on two scatter matrices and two location vectors.

Objects from the Class

Objects can be created by calls of the form new("ics2", ...). But usually objects are created by the function ics2. The Class inherits from the ics class.

Slots

gSkew:

Object of class "numeric". Gives the generalized skewness measures of the components

gKurt:

Object of class "numeric". Gives the generalized kurtosis measures of the components

UnMix:

Object of class "matrix". The unmixing matrix.

S1:

Object of class "matrix". The first scatter matrix.

S2:

Object of class "matrix". The second scatter matrix.

T1:

Object of class "numeric". The first location vector.

T2:

Object of class "numeric". The second location vector.

S1name:

Object of class "character". Name of the first scatter matrix.

S2name:

Object of class "character". Name of the second scatter matrix.

S1args:

Object of class "list". Additional arguments needed when calling function S1.

S2args:

Object of class "list". Additional arguments needed when calling function S2.

Scores:

Object of class "data.frame". The underlying components in the invariant coordinate system.

DataNames:

Object of class "character". Names of the original variables.

StandardizeB:

Object of class "character". Names standardization method for UnMix.

StandardizegKurt:

Object of class "logical". States wether the generalized kurtosis is standardized or not.

Methods

For this class the following generic functions are available: print.ics2, summary.ics2 But naturally the other methods like plot, coef, fitted and so from class ics work via inheritance.

Note

In case no extractor function for the slots exists, the component can be extracted the usual way using '@'.

Author(s)

Klaus Nordhausen

See Also

ics2

Location Estimate based on Third Moments

Description

Estimates the location based on third moments.

Usage

mean3(X, na.action = na.fail)

Arguments

Details

This location estimate is defined for a n \times p matrix X as

\frac{1}{p} ave_{i}\{[(x_{i}-\bar{x})S^{-1}(x_{i}-\bar{x})'] x_{i}\},

where \bar{x} is the mean vector and S the regular covariance matrix.

Value

A vector.

Author(s)

Klaus Nordhausen

References

Oja, H., Sirki?, S. and Eriksson, J. (2006), Scatter matrices and independent component analysis, Austrian Journal of Statistics, 35, 175–189.

Examples

set.seed(654321)
cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
X <- rmvnorm(100, c(0,0,0), cov.matrix)
mean3(X)
rm(.Random.seed)

Test of Multivariate Normality Based on Kurtosis

Description

Test for multivariate normality which uses as criterion the kurtosis measured by the ratio of regular covariance matrix and matrix of fourth moments.

Usage

mvnorm.kur.test(X, method = "integration", n.simu = 1000, 
                na.action = na.fail)

Arguments

Details

This test implements the multivariate normality test based on kurtosis measured by two different scatter estimates as described in Kankainen, Taskinen and Oja. The choice here is based on the regular covariance matrix and matrix of fourth moments (cov4). The limiting distribution of the test statistic W is a linear combination of independent chi-square variables with different degrees of freedom. Exact limiting p-values or approximated p-values are obtained by using the function pchisqsum. However Kankainen et al. mention that even for n = 200 the convergence can be poor, therefore also p-values simulated under the NULL can be obtained.

Note that the test statistic used is a symmetric version of the one in the paper to guarantee affine invariance.

Value

A list with class 'htest' containing the following components:

Author(s)

Klaus Nordhausen

References

Kankainen, A., Taskinen, S. and Oja, H. (2007), Tests of multinormality based on location vectors and scatter matrices, Statistical Methods and Applications, 16, 357–379. <doi:10.1007/s10260-007-0045-9>.

See Also

mvnorm.skew.test

Examples

X<-rmvnorm(100, c(2, 4, 5))
mvnorm.kur.test(X)
mvnorm.kur.test(X, method = "satt")
mvnorm.kur.test(X, method = "simu")

Test of Multivariate Normality Based on Skewness

Description

Test for multivariate normality that uses as criterion the skewness measured as the difference between location estimates based on first respectively third moments

Usage

mvnorm.skew.test(X, na.action = na.fail)

Arguments

Details

This test implements the multivariate normality test based on skewness measured by two different location estimates as described in Kankainen, Taskinen and Oja. The choice here is based on the regular mean vector and the location estimate based on third moments (mean3). The scatter matrix used is the regular covariance matrix.

Value

A list with class 'htest' containing the following components:

Author(s)

Klaus Nordhausen

References

Kankainen, A., Taskinen, S. and Oja, H. (2007),Tests of multinormality based on location vectors and scatter matrices, Statistical Methods and Applications, 16, 357–379. <doi:10.1007/s10260-007-0045-9>.

See Also

mvnorm.kur.test

Examples

X<-rmvnorm(100,c(2,4,5))
mvnorm.skew.test(X)

Scatterplot Matrix of Component Scores from the ICS Transformation

Description

Produces a scatterplot matrix of the component scores of an invariant coordinate system obtained via an ICS transformation.

Usage

## S3 method for class 'ICS'
plot(x, select = NULL, index = NULL, ...)

Arguments

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

gen_kurtosis(), coef(), components(), and fitted() methods

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
plot(out)
plot(out, select = c(1,4))

Scatterplot for a ICS Object

Description

Scatterplot matrix for an ics object.

Usage

## S4 method for signature 'ics,missing'
plot(x, index = NULL, ...)

Arguments

Details

If no index vector is given the function plots the full scatterplots matrix only if there are less than seven components. Otherwise the three first and three last components will be plotted. This is because the components with extreme kurtosis are the most interesting ones.

Author(s)

Klaus Nordhausen

See Also

screeplot.ics, ics-class and ics

Examples

    set.seed(123456)
    X1 <- rmvnorm(250, rep(0,8), diag(c(rep(1,6),0.04,0.04)))
    X2 <- rmvnorm(50, c(rep(0,6),2,0), diag(c(rep(1,6),0.04,0.04)))
    X3 <- rmvnorm(200, c(rep(0,7),2), diag(c(rep(1,6),0.04,0.04)))

    X.comps <- rbind(X1,X2,X3)
    A <- matrix(rnorm(64),nrow=8)
    X <- X.comps %*% t(A)

    ics.X.1 <- ics(X)
    plot(ics.X.1)
    plot(ics.X.1,index=1:8)
    rm(.Random.seed)

Basic information of ICS Object

Description

Prints information of an ICS object.

Usage

## S3 method for class 'ICS'
print(x, info = FALSE, digits = 4L, ...)

Arguments

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
print(out)
print(out, info = TRUE)

Basic information of ICS Object

Description

Prints the minimal information of an ics object.

Usage

## S4 method for signature 'ics'
show(object)

Arguments

Author(s)

Klaus Nordhausen

See Also

ics-class and ics

Basic information of ICS2 Object

Description

Prints the minimal information of an ics2 object.

Usage

## S4 method for signature 'ics2'
show(object)

Arguments

Author(s)

Klaus Nordhausen

See Also

ics2-class and ics2

Supervised scatter matrix based on quantiles

Description

Function for a supervised scatter matrix that is the weighted covariance matrix of x with weights 1/(q2-q1) if y is between the lower (q1) and upper (q2) quantile and 0 otherwise (or vice versa).

Usage

scovq(x, y, q1 = 0, q2 = 0.5, pos = TRUE, type = 7, 
      method = "unbiased", na.action = na.fail, 
      check = TRUE)

Arguments

Details

The weights for this supervised scatter matrix for pos=TRUE are w(y) = I(q1-quantile < y < q2-quantile)/(q2-q1). Then scovq is calculated as

scovq = \sum w(y) (x-\bar{x}_w)'(x-\bar{x}_w).

where \bar{x}_w = \sum w(y) x.

To see how this function can be used in the context of supervised invariant coordinate selection see the example below.

Value

a matrix.

Author(s)

Klaus Nordhausen

References

Liski, E., Nordhausen, K. and Oja, H. (2014), Supervised invariant coordinate selection, Statistics: A Journal of Theoretical and Applied Statistics, 48, 711–731. <doi:10.1080/02331888.2013.800067>.

See Also

cov.wt and ics

Examples

# Creating some data

# The number of explaining variables
p <- 10
# The number of observations
n <- 400
# The error variance
sigma <- 0.5
# The explaining variables 
X <- matrix(rnorm(p*n),n,p)
# The error term
epsilon <- rnorm(n, sd = sigma)
# The response
y <- X[,1]^2 + X[,2]^2*epsilon


# SICS with ics

X.centered <- sweep(X,2,colMeans(X),"-")
SICS <- ics(X.centered, S1=cov, S2=scovq, S2args=list(y=y, q1=0.25, 
        q2=0.75, pos=FALSE), stdKurt=FALSE, stdB="Z")

# Assuming it is known that k=2, then the two directions 
# of interest are choosen as:

k <- 2
KURTS <- SICS@gKurt 
KURTS.max <- ifelse(KURTS >= 1, KURTS, 1/KURTS)
ordKM <- order(KURTS.max, decreasing = TRUE)

indKM <- ordKM[1:k]

# The two variables of interest
Zk <- ics.components(SICS)[,indKM]

# The correspondings transformation matrix
Bk <- coef(SICS)[indKM,]

# The corresponding projection matrix
Pk <- t(Bk) %*% solve(Bk %*% t(Bk)) %*% Bk

# Visualization
pairs(cbind(y,Zk))

# checking the subspace difference

# true projection

B0 <- rbind(rep(c(1,0),c(1,p-1)),rep(c(0,1,0),c(1,1,p-2)))
P0 <- t(B0) %*% solve(B0 %*% t(B0)) %*% B0

# crone and crosby subspace distance measure, should be small
k - sum(diag(P0 %*% Pk))

Screeplot for an ICS Object

Description

Plots the kurtosis measures of an ICS object against its index number. Two versions of this screeplot are available.

Usage

## S3 method for class 'ICS'
screeplot(
  x,
  index = NULL,
  type = "barplot",
  main = deparse(substitute(x)),
  ylab = "generalized kurtosis",
  xlab = "component",
  names.arg = index,
  labels = TRUE,
  ...
)

Arguments

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

gen_kurtosis() method

Examples

X <- iris[,-5]
out <- ICS(X)
screeplot(out)
screeplot(out, type = "lines")

Screeplot for an ICS Object

Description

Plots the kurtosis measures of an ics object against its index number. Two versions of this screeplot are available.

Usage

## S3 method for class 'ics'
screeplot(x, index = NULL, type = "barplot", 
          main = deparse(substitute(x)), ylab = "generalized kurtosis", 
          xlab = "component", names.arg = index, labels = TRUE, ...)

Arguments

Author(s)

Klaus Nordhausen

See Also

plot.ics, ics-class and ics

Examples

set.seed(654321)
A <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2),ncol=3)
eigen.A <- eigen(A)
sqrt.A <- eigen.A$vectors %*% (diag(eigen.A$values))^0.5 %*% t(eigen.A$vectors)
normal.ic <- cbind(rnorm(800), rnorm(800), rnorm(800))
mix.ic <- cbind(rt(800,4), rnorm(800), runif(800,-2,2))

data.normal <- normal.ic %*% t(sqrt.A)
data.mix <- mix.ic %*% t(sqrt.A)

par(mfrow=c(1,2))
screeplot(ics(data.normal))
screeplot(ics(data.mix), type="lines")
par(mfrow=c(1,1))
rm(.Random.seed)

screeplot(ics(data.normal), names.arg=paste("IC", 1:ncol(A), sep=""), xlab="")

To summarize an ICS object

Description

Summarizes and prints an ICS object in an informative way.

Usage

## S3 method for class 'ICS'
summary(object, ...)

Arguments

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS()

print.ICS()

Examples

data("iris")
X <- iris[,-5]
out <- ICS(X)
summary(out)

To summarize an ICS object

Description

Summarizes and prints a ics object in an informative way.

Usage

## S4 method for signature 'ics'
summary(object, digits = 4)

Arguments

Author(s)

Klaus Nordhausen

See Also

ics-class and ics

To summarize an ICS2 object

Description

Summarizes and prints a ics2 object in an informative way.

Usage

## S4 method for signature 'ics2'
summary(object, digits = 4)

Arguments

Author(s)

Klaus Nordhausen

See Also

ics2-class and ics2

Joint M-estimation of Location and Scatter for a Multivariate t-distribution

Description

Implements three EM algorithms to M-estimate the location vector and scatter matrix of a multivariate t-distribution.

Usage

tM(X, df = 1, alg = "alg3", mu.init = NULL, V.init = NULL,
        gamma.init = NULL, eps = 1e-06, maxiter = 100,
        na.action = na.fail)

Arguments

Details

This function implements the EM algorithms described in Kent et al. (1994). The norm used to define convergence is as in Arslan et al. (1995).

Algorithm 1 is valid for all degrees of freedom df > 0. Algorithm 2 is well defined only for degrees of freedom df > 1. Algorithm 3 is the limiting case of Algorithm 2 with degrees of freedom df = 1.

The performance of the algorithms are compared in Arslan et al. (1995).

Note that cov.trob in the MASS package implements also a covariance estimate for a multivariate t-distribution. That function provides for example also the possibility to fix the location. It requires however that the degrees of freedom exceeds 2.

Value

A list containing:

Author(s)

Klaus Nordhausen

References

Kent, J.T., Tyler, D.E. and Vardi, Y. (1994), A curious likelihood identity for the multivariate t-distribution, Communications in Statistics, Simulation and Computation, 23, 441–453. <doi:10.1080/03610919408813180>.

Arslan, O., Constable, P.D.L. and Kent, J.T. (1995), Convergence behaviour of the EM algorithm for the multivariate t-distribution, Communications in Statistics, Theory and Methods, 24, 2981–3000. <doi:10.1080/03610929508831664>.

See Also

cov.trob

Examples

set.seed(654321)
cov.matrix <- matrix(c(3,2,1,2,4,-0.5,1,-0.5,2), ncol=3)
X <- rmvt(100, cov.matrix, 1)
tM(X)
rm(.Random.seed)