Package 'FastPCS'

FastPCS estimator of location and scatter.

Description

Uses the FastPCS algorithm to compute the robust PCS estimator of location and scatter.

Details

Package:	FastPCS
Type:	Package
Version:	0.0.9
Date:	2024-03-13
Suggests:	mvtnorm
License:	GPL (>= 2)
LazyLoad:	yes

Index:

FastPCS                 Function to compute the robust FastPCS estimator of location and scatter.
FPCSnumStarts               Internal function used to compute the number of starting points used by FastPCS.
quanf                   Internal function used to compute h, the minimum number of observations expected to be uncontaminated.
plot.FastPCS		Plots the robust distances outputted by a FastPCS model.

Author(s)

Kaveh Vakili [aut, cre], Maintainer: Kaveh Vakili <[email protected]>

References

Vakili, K. and Schmitt, E. (2014). Finding multivariate outliers with FastPCS. Computational Statistics & Data Analysis. Vol. 69, pp 54–66. (http://arxiv.org/abs/1301.2053)

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Computes the FastPCS multivariate outlyingness index.

Description

Computes a fast and robust multivariate outlyingness index for a n by p matrix of multivariate continuous data.

Usage

FastPCS(x,nSamp,alpha=0.5,seed=1)

Arguments

x	A numeric n (n>5*p) by p (p>1) matrix or data frame.
nSamp	A positive integer giving the number of resamples required; "nSamp" may not be reached if too many of the p-subsets, chosen out of the observed vectors, lie on a hyperplane. If "nSamp" is omitted, it is calculated so that the probability of getting at least one uncontaminated starting point is always at least 99 percent when there are n/2 outliers.
alpha	Numeric parameter controlling the size of the active subsets, i.e., "h=quanf(alpha,n,p)". Allowed values are between 0.5 and 1 and the default is 0.5.
seed	Starting value for random generator. A positive integer. Default is seed = 1

Details

The current version of FastPCS includes the use of a C-step procedure to improve efficiency (Rousseeuw and van Driessen (1999)). C-steps are taken after the raw subset (H*) as been chosen (according to the I-index) and before reweighting. In experiments, we found that carrying C-Steps starting from the members of $rawBest improves the speed of convergence without increasing the bias of the final estimates. FastPCS is affine equivariant (Schmitt et al. (2014)) and thus consistent at the elliptical model (Maronna et al., (2006) p. 217).

Value

alpha	The value of alpha used.
nSamp	The value of nSamp used.
obj	The value of the FastPCS objective function of the optimal h subset.
rawBest	The index of the h observation with smallest outlyingness indexes.
best	The index of the observations with outlyingness smaller than the rejection threshold after C-steps are taken.
center	The mean vector of the observations with outlyingness smaller than the rejection threshold after C-steps are taken.
cov	Covariance matrix of the observations with outlyingness smaller than the rejection threshold after C-steps are taken.
distance	The statistical distance of each observation wrt the center vector and cov matrix of the observations with outlyingness smaller than the rejection threshold after C-steps are taken.

Author(s)

Kaveh Vakili

References

Maronna, R. A., Martin R. D. and Yohai V. J. (2006). Robust Statistics: Theory and Methods. Wiley, New York.

P. J. Rousseeuw and K. van Driessen (1999). A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–223.

Eric Schmitt, Viktoria Oellerer, Kaveh Vakili (2014). The finite sample breakdown point of PCS Statistics and Probability Letters, Volume 94, Pages 214-220.

Vakili, K. and Schmitt, E. (2014). Finding multivariate outliers with FastPCS. Computational Statistics & Data Analysis. Vol. 69, pp 54–66. (http://arxiv.org/abs/1301.2053)

Examples

## testing outlier detection 
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
x0[1:30,]<-matrix(rnorm(30*p,4.5,1/100),nc=p)
z<-c(rep(0,30),rep(1,70))
nstart<-FPCSnumStarts(p=p,eps=0.4)
results<-FastPCS(x=x0,nSamp=nstart)
z[results$best]

## testing outlier detection, different value of alpha
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
x0[1:20,]<-matrix(rnorm(20*p,4.5,1/100),nc=p)
z<-c(rep(0,20),rep(1,80))
nstart<-FPCSnumStarts(p=p,eps=0.25)
results<-FastPCS(x=x0,nSamp=nstart,alpha=0.75)
z[results$best]

#testing exact fit
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
x0[1:30,]<-matrix(rnorm(30*p,5,1/100),nc=p)
x0[31:100,3]<-x0[31:100,2]*2+1
z<-c(rep(0,30),rep(1,70))
nstart<-FPCSnumStarts(p=p,eps=0.4)
results<-FastPCS(x=x0,nSamp=nstart)
z[results$rawBest]
results$obj

#testing affine equivariance
n<-100
p<-3
set.seed(123)
x0<-matrix(rnorm(n*p),nc=p)
nstart<-500
results1<-FastPCS(x=x0,nSamp=nstart,seed=1)
a1<-matrix(0.9,p,p)
diag(a1)<-1
x1<-x0%*%a1
results2<-FastPCS(x=x1,nSamp=nstart,seed=1)
results2$center
results2$cov
#should be the same
results1$center%*%a1
a1

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Computes the number of starting p-subsets

Description

Computes the number of starting p-subsets so that the desired probability of selecting at least one clean one is achieved. This is an internal function not intended to be called by the user.

Usage

FPCSnumStarts(p,gamma=0.99,eps=0.5)

Arguments

p	number of dimensions of the data matrix X.
gamma	desired probability of having at least one clean starting p-subset.
eps	suspected contamination rate of the sample.

Value

An integer number of starting p-subsets.

Author(s)

Kaveh Vakili

Examples

FPCSnumStarts(p=3,gamma=0.99,eps=0.4)

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Distance plot for FastPCS

Description

Plots the robust distance values from a FastPCS model fit, and their parametric cut-off.

Usage

## S3 method for class 'FastPCS'
plot(x,col="black",pch=16,...)

Arguments

x	For the plot() method, a FastPCS object, typically resulting as output from FastPCS.
col	A specification for the default plotting color. Vectors of values are recycled.
pch	Either an integer specifying a symbol, or a single character to be used as the default in plotting points. Note that only integers and single-character strings can be set as graphics parameters. Vectors of values are recycled.
...	Further arguments passed to the plot function.

Author(s)

Kaveh Vakili, Eric Schmitt

See Also

FastPCS

Examples

## generate data
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
x0[1:30,]<-matrix(rnorm(30*p,4.5,1/100),nc=p)
z<-c(rep(0,30),rep(1,70))
nstart<-FPCSnumStarts(p=p,eps=0.4)
results<-FastPCS(x=x0,nSamp=nstart)
colvec<-rep("orange",length(z))
colvec[z==1]<-"blue"
plot.FastPCS(results,col=colvec,pch=16)

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Converts alpha values to h-values

Description

FastPCS selects the subset of size h that minimizes the I-index criterion. The function quanf determines the size of h based on the rate of contamination the user expects is present in the data. This is an internal function not intended to be called by the user.

Usage

quanf(n,p,alpha)

Arguments

n	Number of rows of the data matrix.
p	Number of columns of the data matrix.
alpha	Numeric parameter controlling the size of the active subsets, i.e., "h=quanf(alpha,n,p)". Allowed values are between 0.5 and 1 and the default is 0.5.

Value

An integer number of the size of the starting p-subsets.

Author(s)

Kaveh Vakili

Examples

quanf(p=3,n=500,alpha=0.5)

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