FastPCS {FastPCS} | R Documentation |
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;
|
alpha |
Numeric parameter controlling the size of the active subsets,
i.e., |
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