FastPCS estimator of location and scatter.

Description

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

Details

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 <vakili.kaveh.email@gmail.com>

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)

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

Value

An integer number of starting p-subsets.

Author(s)

Kaveh Vakili

Examples

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

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

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

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

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

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)

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

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)