Package 'ltsspca'

Title: Sparse Principal Component Based on Least Trimmed Squares
Description: Implementation of robust and sparse PCA algorithm of Wang and Van Aelst (2019) <DOI:10.1080/00401706.2019.1671234>.
Authors: Yixin Wang [aut, cre], Stefan Van Aelst [aut], Holger Cevallos Valdiviezo [ctb] (Original R code for the LTS-PCA algorithm), Tom Reynkens [ctb] (Original R code for angle in the rospca package)
Maintainer: Yixin Wang <[email protected]>
License: GPL (>= 2)
Version: 0.1.0
Built: 2024-10-22 06:48:44 UTC
Source: CRAN

Help Index

Standardized last principal angle

Description

Standardised last principal angle between the subspaces generated by the columns of A and B.

Usage

Angle(A, B)

Arguments

A

numerical matrix of size p by k
B

numerical matrix of size q by l

Value

Standardised last principal angle between A and B.

Author(s)

Tom Reynkens

References

Bjorck, A. and Golub, G. H. (1973), “Numerical Methods for Computing Angles Between Linear Subspaces,” Mathematics of Computation, 27, 579–594.

Hubert, M., Rousseeuw, P. J., and Vanden Branden, K. (2005), “ROBPCA: A New Approach to Robust Principal Component Analysis,” Technometrics, 47, 64–79.

Hubert, M., Reynkens, T., Schmitt, E. and Verdonck, T. (2016), “Sparse PCA for High-Dimensional Data With Outliers,” Technometrics, 58, 424–434.

Simulate data

Description

the function that generates the simulation data set

Usage

dataSim(n = 200, p = 20, bLength = 4, a = c(0.9, 0.5, 0),
  SD = c(10, 5, 2), eps = 0, eta = 25, setting = "3", seed = 123,
  vc = NULL)

Arguments

n

number of observations
p

number of variables
bLength

the number of correlated variables in the first k blocks
a

numveric vector of length k+1 that contains the correlations between the variables in each block (the last block contains uncorrelated variables); by default is (0.9, 0.5, 0)
SD

numveric vector of length k+1 that contains the standard deviation of the variables in each block (the last block contains uncorrelated variables); by default is (10, 5, 2)
eps

proportion of outliers, default is 0
eta

parameter that contols the outlyingness, default is 25
setting

type of outliers: setting="1" generates the outliers which are outlying in the first two variables in the second block; setting="2" generates score outliers; setting="3" generates the orthogonal outliers which are easy to detect (the setting used in Hubert, et al (2016)); default is "3"
seed

random seed used to simulate the data
vc

controls the direction of the score outliers within the PC subspace, default is NULL

Value

a list with components

data

generated data matrix
ind

row indices of outliers
R

Correlation matrix of the data
Sigma

Covariance matrix of the data

Glass data

Description

Glass data of Lemberge et al. (2000) containing Electron Probe X-ray Microanalysis (EPXMA) intensities for different wavelengths of 16–17th century archaeological glass vessels. This dataset was also used in Hubert et al. (2005) and Hubert et al. (2016).

Usage

Glass

Format

A data frame with columns:

A data frame with 180 observations and 750 variables. These variables correspond to EPXMA intensities for different wavelengths and are indicated by V1, V2, ..., V750.

Source

Lemberge, P., De Raedt, I., Janssens, K. H., Wei, F., and Van Espen, P. J. (2000), “Quantitative Z-Analysis of the 16–17th Century Archaelogical Glass Vessels using PLS Regression of EPXMA and μ\mu-XRF Data," Journal of Chemometrics, 14, 751–763.

References

Hubert, M., Rousseeuw, P. J., and Vanden Branden, K. (2005), “ROBPCA: A New Approach to Robust Principal Component Analysis,” Technometrics, 47, 64–79.

Hubert, M., Reynkens, T., Schmitt, E. and Verdonck, T. (2016), “Sparse PCA for High-Dimensional Data With Outliers,” Technometrics, 58, 424–434.

Examples

## Not run: 
 data(Glass)

## End(Not run)

Principal Component Analysis Based on Least Trimmed Squaers (LTS-PCA)

Description

the function that computes LTS-PCA

Usage

ltspca(x, q, alpha = 0.5, b.choice = NULL, tol = 1e-06, N1 = 3,
  N2 = 2, N2bis = 10, Npc = 10)

Arguments

x

the input data matrix
q

the dimension of the PC subspace
alpha

the robust parameter which takes value between 0 to 0.5, default is 0.5
b.choice

intial loading matrix; by default is NULL and the deterministic starting values will be computed by the algorithm
tol

convergence criterion
N1

the number controls the updates for a without updating b in the concentration step
N2

the number controls outer loop in the concentration step
N2bis

the number controls the outer loop for the selected b
Npc

the number controls the inner loop

Value

the object of class "ltspca" is returned

b

the unnormalized loading matrix
mu

the center estimate
ws

if the observation in included in the h-subset ws=1; otherwise ws=0
best.cand

the method which computes the best deterministic starting value in the concentration step

Author(s)

Cevallos Valdiviezo

References

Cevallos Valdiviezo, H., Van Aelst, S. (2019), “ Fast computation of robust subspace estimators”, Computational Statistics & Data Analysis, 134, 171–185.

Examples

## Not run: 
ltspcaM <- ltspca(x = x, q = 2, alpha = 0.5)

## End(Not run)

Sparse Principal Component Analysis Based on Least Trimmed Squaers (LTS-SPCA)

Description

the function that computes the initial LTS-SPCA

Usage

ltsspca(x, kmax, alpha = 0.5, mu.choice = NULL, l.search = NULL,
  ls.min = 1, tol = 1e-06, N1 = 3, N2 = 2, N2bis = 10,
  Npc = 10)

Arguments

x

the input data matrix
kmax

the maximal number of PCs searched by the intial LTS-SPCA
alpha

the robust parameter which takes value between 0 to 0.5, default is 0.5
mu.choice

the center estimate fixed by the user; by default, the center will be estimated automatically by the algorithm
l.search

a list of length kmax which contains the search grids chosen by the user; default is NULL
ls.min

the smallest grid step when searching for the sparsity of each PC; default is 1
tol

convergence criterion
N1

the number controls the updates for a without updating b in the concentration step for LTS-PCA
N2

the number controls outer loop in the concentration step for LTS-PCA
N2bis

the number controls the outer loop for the selected b for both LTS-PCA and LTS-SPCA
Npc

the number controls the inner loop for both LTS-PCA and LTS-SPCA

Value

the object of class "ltsspca" is returned

loadings

the initially estimated loading matrix by LTS-SPCA
mu

the center estimates associated with each PC
spca.it

the list that contains the results of LTS-SPCA when searching for the individual PCs
ls

the list that contains the final search grid for each PC direction

Author(s)

Yixin Wang

References

Wang, Y., Van Aelst, S. (2019), “ Sparse Principal Component Based On Least Trimmed Squares”, Technometrics, accepted.

Examples

library(mvtnorm)
dataM <- dataSim(n = 200, p = 20, bLength = 4, a = c(0.9, 0.5, 0),
                SD = c(10, 5, 2), eps = 0, seed = 123)
x <- dataM$data
ltsspcaMI <- ltsspca(x = x, kmax = 5, alpha = 0.5)
ltsspcaMR <- ltsspcaRw(x = x, obj = ltsspcaMI, k = 2, alpha = 0.5)
matplot(ltsspcaMR$loadings,type="b",ylab="Loadings")

Reweighted LTS-SPCA

Description

the function that computes the reweighted LTS-SPCA

Usage

ltsspcaRw(x, obj, k = NULL, alpha = 0.5, co.sd = 0.25)

Arguments

x

the input data matrix
obj

initial LTS-SPCA object given by ltsspca function
k

dimension of the PC subspace; by default is NULL then k takes the value of kmax in the initial LTS-SPCA
alpha

the robust parameter which takes value between 0 to 0.5, default is 0.5
co.sd

cutoff value for score outlier weight, default is 0.25

Value

the object of class "ltsspcaRw" is returned

loadings

the sparse loading matrix estimated with reweighted LTS-SPCA
scores

the estimated score matrix
eigenvalues

the estimated eigenvalues
mu

the center estimate
rw.obj

the list that contains the results of sPCA_rSVD on the reduced data
od

the orthonal distances with respect to the initially estimated PC subspace with all the noisy variables removed
co.od

the cutoff value for the orthogonal distances
ws.od

if the observation is outlying in the orthgonal complement of the initially estimated PC subspace ws.od=0; otherwise ws.od=1
sc.wt

the score outlier weight, which is compared with 0.25 (by default) to flag score outliers
co.sd

the cutoff value for score outlier weight, default is 0.25
ws.sd

if the observation is outlying with the PC subspace ws.sd=0; otherwise ws.sd=1
sc.out

the retruned object when computing the score outlier weights

Make diagnostic plot using the estimated PC subspace

Description

Make diagnostic plot using the estimated PC subspace

Usage

mydiagPlot(x, obj, k, alpha = 0.5, co.sd = 0.25)

Arguments

x

the input data matrix
obj

the returned output from rwltsspca
k

dimension of the PC subspace
alpha

the robust parameter which takes value between 0 to 0.5, default is 0.5
co.sd

cutoff value for score outlier weight, default is 0.25

Value

the diagnostics of outliers

od

the orthgonal distances with respect to the k-dimensional PC subspace
ws.od

if the observation is outlying in the orthgonal complement of the PC subspace ws.od=0; otherwise ws.sd=1
co.od

the cutoff value for orthogonal distances
sc.wt

the score outlier weight, which is compared with 0.25 (by default) to flag score outliers
ws.sd

if the observation is outlying with the PC subspace ws.sd=0; otherwise ws.sd=1
co.sd

the cutoff value for score outlier weight, default is 0.25
sc.out

the retruned object when computing the score outlier weights

Sparse Principal Component Analysis via Regularized Singular Value Decompsition (sPCA-rSVD)

Description

the function that computes sPCA_rSVD

Usage

sPCA_rSVD(x, k, method = "hard", center = FALSE, scale = FALSE,
  l.search = NULL, ls.min = 1)

Arguments

x

the input data matrix
k

the maximal number of PC's to seach for in the initial stage
method

threshold method used in the algorithm; If method = "hard" (defauls), the hard threshold function is used; if method = "soft", the soft threshold function is used; if method = "scad", the scad threshold function is used
center

if center = TRUE the data will be centered by the columnwise means; default is center = FALSE
scale

if scale = TRUE the data will be scaled by the columnwise standard deviations; default is scaled = FALSE
l.search

a list of length kmax which contains the search grids chosen by the user; default is NULL
ls.min

the smallest grid step when searching for the sparsity of each PC; default is 1

Value

an object of class "sPCA_rSVD" is returned

loadings

the sparse loading matrix estimated with sPCA_rSVD
scores

the estimated score matrix
eigenvalues

the estimated eigenvalues
spca.it

the list that contains the results of sPCA_rSVD when searching for the individual PCs
ls

the list that contains the final search grid for each PC direction

References

Shen, H. and Huang, J. (2008), “Sparse principal component anlysis via regularized low rank matrix decomposition”, Journal of Multivariate Analysis, 99, 1015–1034.

Shen, D., Shen, H., and Marron, J. (2013). “Consistency of sparse PCA in high dimensional low sample size context”, Journal of Multivariate Analysis, 115, 315–333.

Examples

## Not run: 
nonrobM <- sPCA_rSVD(x = x, k = 2, center =  T, scale = F)

## End(Not run)