FastRCS {FastRCS}

Computes the FastRCS outlyingness index for regression.

Description

Computes a fast and robust regression model for a n by p matrix of multivariate continuous regressors and a single dependent variable.

Usage

  FastRCS(x,y,nSamp,alpha=0.5,seed=1,intercept=1)

Arguments

x

A numeric n (n>5*p) by p (p>1) matrix or data frame. Should not contain an intercept.

y

A numeric nvector.

nSamp

a positive integer giving the number of resamples required; "nSamp" may not be reached if too many of the p-subsamples, chosen out of the observed vectors, are in 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

intercept

If true, a model with constant term will be estimated; otherwise no constant term will be included. Default is intercept=TRUE.

Details

The current version of FastRCS includes the use of a C-step procedure to improve efficiency (Rousseeuw and van Driessen (1999)). C-steps are taken after the raw subset is found 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. FastRCS is regression and affine equivariant and thus consistent at the elliptical model (Grubel and Rock (1990)).

Value

nSamp

The value of nSamp used.

alpha

The value of alpha used.

obj

The value of the FastRCS objective function (the I-index) obtained for H*.

rawBest

The index of the h observation with smallest outlyingness indexes.

rawDist

The distances of the observations to the model defined by rawBest.

best

The index of the J observation with outlyingness smaller than the rejection threshold.

coefficients

The vector of coefficients of the hyperplane fitted to the members of $rew$best.

fitted.values

the fitted mean values: cbind(1,x)%*%rew$coefficients.

residuals

the residuals, that is response minus fitted values.

rank

the numeric rank of the fitted linear model.

weights

(only for weighted fits) the specified weights.

df.residual

the residual degrees of freedom.

scale

(robust) scale estimate of the reweighted residuals.

Author(s)

Kaveh Vakili

References

Grubel, R. and Rocke, D. M. (1990). On the cumulants of affine equivariant estimators in elliptical families. Journal of Multivariate Analysis, Vol. 35, p. 203–222. Journal of Multivariate Analysis

Rousseeuw, P. J., and van Driessen, K. (2006). Computing lts regression for large data sets. Data mining and Knowledge Discovery, 12, 29–45

Vakili, K. and Schmitt, E. (2014). Finding Regression Outliers With FastRCS. (http://arxiv.org/abs/1307.4834)

Examples

## testing outlier detection
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
y0<-rnorm(n)
z<-c(rep(0,30),rep(1,70))
x0[1:30,]<-matrix(rnorm(30*p,5,1/100),nc=p)
y0[1:30]<-rnorm(30,10,1/100)
ns<-FRCSnumStarts(p=p,eps=0.4);
results<-FastRCS(x=x0,y=y0,alpha=0.5,nSamp=ns)
z[results$best]

## testing outlier detection, different value of alpha
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
y0<-rnorm(n)
z<-c(rep(0,20),rep(1,80))
x0[1:20,]<-matrix(rnorm(20*p,5,1/100),nc=p)
y0[1:20]<-rnorm(20,10,1/100)
ns<-FRCSnumStarts(p=p,eps=0.25);
results<-FastRCS(x=x0,y=y0,alpha=0.75,nSamp=ns)
z[results$best]



#testing exact fit
set.seed(123)
n<-100
p<-3
x0<-matrix(rnorm(n*p),nc=p)
y0<-rep(1,n)
z<-c(rep(0,30),rep(1,70))
x0[1:30,]<-matrix(rnorm(30*p,5,1/100),nc=p)
y0[1:30]<-rnorm(30,10,1/100)
ns<-FRCSnumStarts(p=p,eps=0.4);
results<-FastRCS(x=x0,y=y0,alpha=0.5,nSamp=ns,seed=1)
z[results$rawBest]
results$obj

#testing regression equivariance
n<-100
p<-3
x0<-matrix(rnorm(n*(p-1)),nc=p-1)
y0<-rnorm(n)
ns<-FRCSnumStarts(p=p,eps=0.4);
y1<-y0+cbind(1,x0)%*%rep(-1,p)
results1<-FastRCS(y=y0,x=x0,nSamp=ns,seed=1)$coefficients
results2<-FastRCS(y=y1,x=x0,nSamp=ns,seed=1)$coefficients
results1+rep(-1,p)
#should be the same:
results2