FastPCS (version 0.1.4)

FastPCS: 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)

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.

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

Author

Kaveh Vakili

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).

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