Version:	1.7-6
VersionNote:	Released 1.7-5 on 2024-01-30 on CRAN
Title:	Scalable Robust Estimators with High Breakdown Point
Description:	Robust Location and Scatter Estimation and Robust Multivariate Analysis with High Breakdown Point: principal component analysis (Filzmoser and Todorov (2013), <doi:10.1016/j.ins.2012.10.017>), linear and quadratic discriminant analysis (Todorov and Pires (2007)), multivariate tests (Todorov and Filzmoser (2010) <doi:10.1016/j.csda.2009.08.015>), outlier detection (Todorov et al. (2010) <doi:10.1007/s11634-010-0075-2>). See also Todorov and Filzmoser (2009) <urn:isbn:978-3838108148>, Todorov and Filzmoser (2010) <doi:10.18637/jss.v032.i03> and Boudt et al. (2019) <doi:10.1007/s11222-019-09869-x>.
Maintainer:	Valentin Todorov <valentin.todorov@chello.at>
Depends:	R (≥ 2.10), robustbase (≥ 0.92.1), methods
Imports:	stats, stats4, mvtnorm, lattice, pcaPP
Suggests:	grid, MASS
LazyLoad:	yes
License:	GPL (≥ 3)
URL:	https://github.com/valentint/rrcov
BugReports:	https://github.com/valentint/rrcov/issues
Repository:	CRAN
Packaged:	2024-08-19 10:22:55 UTC; valen
NeedsCompilation:	yes
Author:	Valentin Todorov [aut, cre]
Date/Publication:	2024-08-19 11:20:02 UTC
RoxygenNote:	7.2.2

Annual maximum streamflow in central Appalachia

Description

The data on annual maximum streamflow at 104 gaging stations in the central Appalachia region of the United States contains the sample L-moments ratios (L-CV, L-skewness and L-kurtosis) as used by Hosking and Wallis (1997) to illustrate regional freqency analysis (RFA).

Usage

data(Appalachia)

Format

A data frame with 104 observations on the following 3 variables:

L-CV

L-coefficient of variation

L-skewness

L-coefficient of skewness

L-kurtosis

L-coefficient of kurtosis

Details

The sample L-moment ratios (L-CV, L-skewness and L-kurtosis) of a site are regarded as a point in three dimensional space.

Source

Hosking, J. R. M. and J. R. Wallis (1997), Regional Frequency Analysis: An Approach Based on L-moments. Cambridge University Press, p.175–185

References

Neykov, N.M., Neytchev, P.N., Van Gelder, P.H.A.J.M. and Todorov V. (2007), Robust detection of discordant sites in regional frequency analysis, Water Resources Research, 43, W06417, doi:10.1029/2006WR005322

Examples

    data(Appalachia)

    # plot a matrix of scatterplots
    pairs(Appalachia,
          main="Appalachia data set",
          pch=21,
          bg=c("red", "green3", "blue"))

    mcd<-CovMcd(Appalachia)
    mcd
    plot(mcd, which="dist", class=TRUE)
    plot(mcd, which="dd", class=TRUE)

    ##  identify the discordant sites using robust distances and compare 
    ##  to the classical ones
    mcd <- CovMcd(Appalachia)
    rd <- sqrt(getDistance(mcd))
    ccov <- CovClassic(Appalachia)
    cd <- sqrt(getDistance(ccov))
    r.out <- which(rd > sqrt(qchisq(0.975,3)))
    c.out <- which(cd > sqrt(qchisq(0.975,3)))
    cat("Robust: ", length(r.out), " outliers: ", r.out,"\n")
    cat("Classical: ", length(c.out), " outliers: ", c.out,"\n")

Biplot for Principal Components (objects of class 'Pca')

Description

Produces a biplot from an object (derived from) Pca-class.

Usage

    ## S4 method for signature 'Pca'
biplot(x, choices=1L:2L, scale=1, ...)

Arguments

Side Effects

a plot is produced on the current graphics device.

Methods

biplot

signature(x = Pca): Plot a biplot, i.e. represent both the observations and variables of a matrix of multivariate data on the same plot. See also biplot.princomp.

References

Gabriel, K. R. (1971). The biplot graphical display of matrices with applications to principal component analysis. Biometrika, 58, 453–467.

See Also

Pca-class, PcaClassic, PcaRobust-class.

Examples

require(graphics)
biplot(PcaClassic(USArrests, k=2))

Automatic vehicle recognition data

Description

The data set bus (Hettich and Bay, 1999) corresponds to a study in automatic vehicle recognition (see Maronna et al. 2006, page 213, Example 6.3)). This data set from the Turing Institute, Glasgow, Scotland, contains measures of shape features extracted from vehicle silhouettes. The images were acquired by a camera looking downward at the model vehicle from a fixed angle of elevation. Each of the 218 rows corresponds to a view of a bus silhouette, and contains 18 attributes of the image.

Usage

data(bus)

Format

A data frame with 218 observations on the following 18 variables:

V1

compactness

V2

circularity

V3

distance circularity

V4

radius ratio

V5

principal axis aspect ratio

V6

maximum length aspect ratio

V7

scatter ratio

V8

elongatedness

V9

principal axis rectangularity

V10

maximum length rectangularity

V11

scaled variance along major axis

V12

scaled variance along minor axis

V13

scaled radius of gyration

V14

skewness about major axis

V15

skewness about minor axis

V16

kurtosis about minor axis

V17

kurtosis about major axis

V18

hollows ratio

Source

Hettich, S. and Bay, S.D. (1999), The UCI KDD Archive, Irvine, CA:University of California, Department of Information and Computer Science, 'http://kdd.ics.uci.edu'

References

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

Examples

    ## Reproduce Table 6.3 from Maronna et al. (2006), page 213
    data(bus)
    bus <- as.matrix(bus)
    
    ## calculate MADN for each variable
    xmad <- apply(bus, 2, mad)      
    cat("\nMin, Max of MADN: ", min(xmad), max(xmad), "\n")


    ## MADN vary between 0 (for variable 9) and 34. Therefore exclude 
    ##  variable 9 and divide the remaining variables by their MADNs.
    bus1 <- bus[, -9]
    madbus <- apply(bus1, 2, mad)
    bus2 <- sweep(bus1, 2, madbus, "/", check.margin = FALSE)

    ## Compute classical and robust PCA (Spherical/Locantore, Hubert, MCD and OGK)    
    pca  <- PcaClassic(bus2)
    rpca <- PcaLocantore(bus2)
    pcaHubert <- PcaHubert(bus2, k=17, kmax=17, mcd=FALSE)
    pcamcd <- PcaCov(bus2, cov.control=CovControlMcd())
    pcaogk <- PcaCov(bus2, cov.control=CovControlOgk())

    ev    <- getEigenvalues(pca)
    evrob <- getEigenvalues(rpca)
    evhub <- getEigenvalues(pcaHubert)
    evmcd <- getEigenvalues(pcamcd)
    evogk <- getEigenvalues(pcaogk)

    uvar    <- matrix(nrow=6, ncol=6)
    svar    <- sum(ev)
    svarrob <- sum(evrob)
    svarhub <- sum(evhub)
    svarmcd <- sum(evmcd)
    svarogk <- sum(evogk)
    for(i in 1:6){
        uvar[i,1] <- i
        uvar[i,2] <- round((svar - sum(ev[1:i]))/svar, 3)
        uvar[i,3] <- round((svarrob - sum(evrob[1:i]))/svarrob, 3)
        uvar[i,4] <- round((svarhub - sum(evhub[1:i]))/svarhub, 3)
        uvar[i,5] <- round((svarmcd - sum(evmcd[1:i]))/svarmcd, 3)
        uvar[i,6] <- round((svarogk - sum(evogk[1:i]))/svarogk, 3)
    }
    uvar <- as.data.frame(uvar)
    names(uvar) <- c("q", "Classical","Spherical", "Hubert", "MCD", "OGK")
    cat("\nBus data: proportion of unexplained variability for q components\n")
    print(uvar)
 
    ## Reproduce Table 6.4 from Maronna et al. (2006), page 214
    ##
    ## Compute classical and robust PCA extracting only the first 3 components
    ## and take the squared orthogonal distances to the 3-dimensional hyperplane
    ##
    pca3 <- PcaClassic(bus2, k=3)               # classical
    rpca3 <- PcaLocantore(bus2, k=3)            # spherical (Locantore, 1999)
    hpca3 <- PcaHubert(bus2, k=3)               # Hubert
    dist <- pca3@od^2
    rdist <- rpca3@od^2
    hdist <- hpca3@od^2

    ## calculate the quantiles of the distances to the 3-dimensional hyperplane
    qclass  <- round(quantile(dist, probs = seq(0, 1, 0.1)[-c(1,11)]), 1)
    qspc <- round(quantile(rdist, probs = seq(0, 1, 0.1)[-c(1,11)]), 1)
    qhubert <- round(quantile(hdist, probs = seq(0, 1, 0.1)[-c(1,11)]), 1)
    qq <- cbind(rbind(qclass, qspc, qhubert), round(c(max(dist), max(rdist), max(hdist)), 0))
    colnames(qq)[10] <- "Max"
    rownames(qq) <- c("Classical", "Spherical", "Hubert")
    cat("\nBus data: quantiles of distances to hiperplane\n")
    print(qq)

    ## 
    ## Reproduce Fig 6.1 from Maronna et al. (2006), page 214
    ## 
    cat("\nBus data: Q-Q plot of logs of distances to hyperplane (k=3) 
    \nfrom classical and robust estimates. The line is the identity diagonal\n")
    plot(sort(log(dist)), sort(log(rdist)), xlab="classical", ylab="robust")
    lines(sort(log(dist)), sort(log(dist)))
   
    

Campbell Bushfire Data with added missing data items

Description

This data set is based on the bushfire data set which was used by Campbell (1984) to locate bushfire scars - see bushfire in package robustbase. The original dataset contains satelite measurements on five frequency bands, corresponding to each of 38 pixels.

Usage

data(bushmiss)

Format

A data frame with 190 observations on 6 variables.

The original data set consists of 38 observations in 5 variables. Based on it four new data sets are created in which some of the data items are replaced by missing values with a simple "missing completely at random " mechanism. For this purpose independent Bernoulli trials are realized for each data item with a probability of success 0.1, 0.2, 0.3, 0.4, where success means that the corresponding item is set to missing. The obtained five data sets, including the original one (each with probability of a data item to be missing equal to 0, 0.1, 0.2, 0.3 and 0.4 which is reflected in the new variable MPROB) are merged. (See also Beguin and Hulliger (2004).)

Source

Maronna, R.A. and Yohai, V.J. (1995) The Behavoiur of the Stahel-Donoho Robust Multivariate Estimator. Journal of the American Statistical Association 90, 330–341.

Beguin, C. and Hulliger, B. (2004) Multivariate outlier detection in incomplete survey data: the epidemic algorithm and transformed rank correlations. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 127, 2, 275–294.

Examples

## The following code will result in exactly the same output
##  as the one obtained from the original data set
data(bushmiss)
bf <- bushmiss[bushmiss$MPROB==0,1:5]
plot(bf)
covMcd(bf)


## Not run: 
##  This is the code with which the missing data were created:
##
##  Creates a data set with missing values (for testing purposes)
##  from a complete data set 'x'. The probability of
##  each item being missing is 'pr' (Bernoulli trials).
##
getmiss <- function(x, pr=0.1)
{
    n <- nrow(x)
    p <- ncol(x)
    done <- FALSE
    iter <- 0
    while(iter <= 50){
        bt <- rbinom(n*p, 1, pr)
        btmat <- matrix(bt, nrow=n)
        btmiss <- ifelse(btmat==1, NA, 0)
        y <- x+btmiss
        if(length(which(rowSums(nanmap(y)) == p)) == 0)
            return (y)
        iter <- iter + 1
    }
    y
}

## End(Not run)

Consumer reports car data: dimensions

Description

A data frame containing 11 variables with different dimensions of 111 cars

Usage

data(Cars)

Format

A data frame with 111 observations on the following 11 variables.

length

a numeric vector

wheelbase

a numeric vector

width

a numeric vector

height

a numeric vector

front.hd

a numeric vector

rear.hd

a numeric vector

front.leg

a numeric vector

rear.seating

a numeric vector

front.shoulder

a numeric vector

rear.shoulder

a numeric vector

luggage

a numeric vector

Source

Consumer reports. (April 1990). http://backissues.com/issue/Consumer-Reports-April-1990, pp. 235–288.

References

Chambers, J. M. and Hastie, T. J. (1992). Statistical models in S. Cole, Pacific Grove, CA: Wadsworth and Brooks, pp. 46–47.

M. Hubert, P. J. Rousseeuw, K. Vanden Branden (2005), ROBPCA: A new approach to robust principal components analysis, Technometrics, 47, 64–79.

Examples

    data(Cars)

## Plot a pairwise scaterplot matrix
    pairs(Cars[,1:6])

    mcd <- CovMcd(Cars[,1:6])    
    plot(mcd, which="pairs")
    
## Start with robust PCA
    pca <- PcaHubert(Cars, k=ncol(Cars), kmax=ncol(Cars))
    pca

## Compare with the classical PCA
    prcomp(Cars)

## or  
    PcaClassic(Cars, k=ncol(Cars), kmax=ncol(Cars))
    
## If you want to print the scores too, use
    print(pca, print.x=TRUE)

## Using the formula interface
    PcaHubert(~., data=Cars, k=ncol(Cars), kmax=ncol(Cars))

## To plot the results:

    plot(pca)                    # distance plot
    pca2 <- PcaHubert(Cars, k=4)  
    plot(pca2)                   # PCA diagnostic plot (or outlier map)
    
## Use the standard plots available for prcomp and princomp
    screeplot(pca)    # it is interesting with all variables    
    biplot(pca)       # for biplot we need more than one PCs
    
## Restore the covraiance matrix     
    py <- PcaHubert(Cars, k=ncol(Cars), kmax=ncol(Cars))
    cov.1 <- py@loadings %*% diag(py@eigenvalues) %*% t(py@loadings)
    cov.1      

Annual precipitation totals for the North Cascades region

Description

The data on annual precipitation totals for the North Cascades region contains the sample L-moments ratios (L-CV, L-skewness and L-kurtosis) for 19 sites as used by Hosking and Wallis (1997), page 53, Table 3.4, to illustrate screening tools for regional freqency analysis (RFA).

Usage

data(Cascades)

Format

A data frame with 19 observations on the following 3 variables.

L-CV

L-coefficient of variation

L-skewness

L-coefficient of skewness

L-kurtosis

L-coefficient of kurtosis

Details

The sample L-moment ratios (L-CV, L-skewness and L-kurtosis) of a site are regarded as a point in three dimensional space.

Source

Hosking, J. R. M. and J. R. Wallis (1997), Regional Frequency Analysis: An Approach Based on L-moments. Cambridge University Press, p. 52–53

References

Neykov, N.M., Neytchev, P.N., Van Gelder, P.H.A.J.M. and Todorov V. (2007), Robust detection of discordant sites in regional frequency analysis, Water Resources Research, 43, W06417, doi:10.1029/2006WR005322

Examples

    data(Cascades)

    # plot a matrix of scatterplots
    pairs(Cascades,
          main="Cascades data set",
          pch=21,
          bg=c("red", "green3", "blue"))

    mcd<-CovMcd(Cascades)
    mcd
    plot(mcd, which="dist", class=TRUE)
    plot(mcd, which="dd", class=TRUE)

    ##  identify the discordant sites using robust distances and compare 
    ##  to the classical ones
    rd <- sqrt(getDistance(mcd))
    ccov <- CovClassic(Cascades)
    cd <- sqrt(getDistance(ccov))
    r.out <- which(rd > sqrt(qchisq(0.975,3)))
    c.out <- which(cd > sqrt(qchisq(0.975,3)))
    cat("Robust: ", length(r.out), " outliers: ", r.out,"\n")
    cat("Classical: ", length(c.out), " outliers: ", c.out,"\n")

Class "Cov" – a base class for estimates of multivariate location and scatter

Description

The class Cov represents an estimate of the multivariate location and scatter of a data set. The objects of class Cov contain the classical estimates and serve as base for deriving other estimates, i.e. different types of robust estimates.

Objects from the Class

Objects can be created by calls of the form new("Cov", ...), but the usual way of creating Cov objects is a call to the function Cov which serves as a constructor.

Slots

call:

Object of class "language"

cov:

covariance matrix

center:

location

n.obs:

number of observations used for the computation of the estimates

mah:

mahalanobis distances

det:

determinant

flag:

flags (FALSE if suspected an outlier)

method:

a character string describing the method used to compute the estimate: "Classic"

singularity:

a list with singularity information for the covariance matrix (or NULL of not singular)

X:

data

Methods

getCenter

signature(obj = "Cov"): location vector

getCov

signature(obj = "Cov"): covariance matrix

getCorr

signature(obj = "Cov"): correlation matrix

getData

signature(obj = "Cov"): data frame

getDistance

signature(obj = "Cov"): distances

getEvals

signature(obj = "Cov"): Computes and returns the eigenvalues of the covariance matrix

getDet

signature(obj = "Cov"): Computes and returns the determinant of the covariance matrix (or 0 if the covariance matrix is singular)

getShape

signature(obj = "Cov"): Computes and returns the shape matrix corresponding to the covariance matrix (i.e. the covariance matrix scaled to have determinant =1)

getFlag

signature(obj = "Cov"): Flags observations as outliers if the corresponding mahalanobis distance is larger then qchisq(prob, p) where prob defaults to 0.975.

isClassic

signature(obj = "Cov"): returns TRUE by default. If necessary, the robust classes will override

plot

signature(x = "Cov"): plot the object

show

signature(object = "Cov"): display the object

summary

signature(object = "Cov"): calculate summary information

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

showClass("Cov")

Classical Estimates of Multivariate Location and Scatter

Description

Computes the classical estimates of multivariate location and scatter. Returns an S4 class CovClassic with the estimated center, cov, Mahalanobis distances and weights based on these distances.

Usage

    CovClassic(x, unbiased=TRUE)
    Cov(x, unbiased=TRUE)

Arguments

Value

An object of class "CovClassic".

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Cov-class, CovClassic-class

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
cv <- CovClassic(hbk.x)
cv
summary(cv)
plot(cv)

Class "CovClassic" - classical estimates of multivariate location and scatter

Description

The class CovClassic represents an estimate of the multivariate location and scatter of a data set. The objects of class CovClassic contain the classical estimates.

Objects from the Class

Objects can be created by calls of the form new("CovClassic", ...), but the usual way of creating CovClassic objects is a call to the function CovClassic which serves as a constructor.

Slots

call:

Object of class "language"

cov:

covariance matrix

center:

location

n.obs:

number of observations used for the computation of the estimates

mah:

mahalanobis distances

method:

a character string describing the method used to compute the estimate: "Classic"

singularity:

a list with singularity information for the ocvariance matrix (or NULL of not singular)

X:

data

Methods

getCenter

signature(obj = "CovClassic"): location vector

getCov

signature(obj = "CovClassic"): covariance matrix

getCorr

signature(obj = "CovClassic"): correlation matrix

getData

signature(obj = "CovClassic"): data frame

getDistance

signature(obj = "CovClassic"): distances

getEvals

signature(obj = "CovClassic"): Computes and returns the eigenvalues of the covariance matrix

plot

signature(x = "CovClassic"): plot the object

show

signature(object = "CovClassic"): display the object

summary

signature(object = "CovClassic"): calculate summary information

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
cv <- CovClassic(hbk.x)
cv
summary(cv)
plot(cv)

Class "CovControl" is a VIRTUAL base control class

Description

The class "CovControl" is a VIRTUAL base control class for the derived classes representing the control parameters for the different robust methods

Arguments

Objects from the Class

A virtual Class: No objects may be created from it.

Methods

No methods defined with class "CovControl" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Constructor function for objects of class "CovControlMcd"

Description

This function will create a control object CovControlMcd containing the control parameters for CovMcd

Usage

CovControlMcd(alpha = 0.5, nsamp = 500, scalefn=NULL, maxcsteps=200, 
seed = NULL, trace= FALSE, use.correction = TRUE)

Arguments

Value

A CovControlMcd object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMcd", alpha=0.75)
    ctrl2 <- CovControlMcd(alpha=0.75)

    data(hbk)
    CovMcd(hbk, control=ctrl1)

Class 'CovControlMcd' - contains control parameters for CovMcd

Description

This class extends the CovControl class and contains the control parameters for "CovMcd"

Objects from the Class

Objects can be created by calls of the form new("CovControlMcd", ...) or by calling the constructor-function CovControlMcd.

Slots

alpha:

numeric parameter controlling the size of the subsets over which the determinant is minimized, i.e., alpha*n observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.5.

nsamp

number of subsets used for initial estimates or "best", "exact" or "deterministic". Default is nsamp = 500. For nsamp="best" exhaustive enumeration is done, as long as the number of trials does not exceed 5000. For "exact", exhaustive enumeration will be attempted however many samples are needed. In this case a warning message will be displayed saying that the computation can take a very long time.

For "deterministic", the deterministic MCD is computed; as proposed by Hubert et al. (2012) it starts from the h most central observations of six (deterministic) estimators.

scalefn

function to compute a robust scale estimate or character string specifying a rule determining such a function.

maxcsteps

maximal number of concentration steps in the deterministic MCD; should not be reached.

seed:

starting value for random generator. Default is seed = NULL

use.correction:

whether to use finite sample correction factors. Default is use.correction=TRUE.

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlMcd"): the generic function restimate allows the different methods for robust estimation to be used polymorphically - this function will call CovMcd passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMcd", alpha=0.75)
    ctrl2 <- CovControlMcd(alpha=0.75)

    data(hbk)
    CovMcd(hbk, control=ctrl1)

Constructor function for objects of class "CovControlMest"

Description

This function will create a control object CovControlMest containing the control parameters for CovMest

Usage

CovControlMest(r = 0.45, arp = 0.05, eps = 0.001, maxiter = 120)

Arguments

Value

A CovControlMest object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMest", r=0.4)
    ctrl2 <- CovControlMest(r=0.4)

    data(hbk)
    CovMest(hbk, control=ctrl1)

Class 'CovControlMest' - contains control parameters for "CovMest"

Description

This class extends the CovControl class and contains the control parameters for CovMest

Objects from the Class

Objects can be created by calls of the form new("CovControlMest", ...) or by calling the constructor-function CovControlMest.

Slots

r:

a numeric value specifying the required breakdown point. Allowed values are between (n - p)/(2 * n) and 1 and the default is 0.45

arp:

a numeric value specifying the asympthotic rejection point, i.e. the fraction of points receiving zero weight (see Rocke (1996)). Default is 0.05

eps:

a numeric value specifying the relative precision of the solution of the M-estimate. Defaults to 1e-3

maxiter:

maximum number of iterations allowed in the computation of the M-estimate. Defaults to 120

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlMest"): the generic function restimate allowes the different methods for robust estimation to be used polymorphically - this function will call CovMest passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMest", r=0.4)
    ctrl2 <- CovControlMest(r=0.4)

    data(hbk)
    CovMest(hbk, control=ctrl1)

Constructor function for objects of class "CovControlMMest"

Description

This function will create a control object CovControlMMest containing the control parameters for CovMMest

Usage

    CovControlMMest(bdp = 0.5, eff=0.95, maxiter = 50, sest=CovControlSest(),
        trace = FALSE, tolSolve = 1e-7)

Arguments

Value

A CovControlSest object.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

   ## the following two statements are equivalent
    ctrl1 <- new("CovControlMMest", bdp=0.25)
    ctrl2 <- CovControlMMest(bdp=0.25)

    data(hbk)
    CovMMest(hbk, control=ctrl1)
    
    
    

Class 'CovControlMMest' - contains control parameters for "CovMMest"

Description

This class extends the CovControl class and contains the control parameters for CovMMest

Objects from the Class

Objects can be created by calls of the form new("CovControlMMest", ...) or by calling the constructor-function CovControlMMest.

Slots

bdp

a numeric value specifying the required breakdown point. Allowed values are between 0.5 and 1 and the default is bdp=0.5.

eff

a numeric value specifying the required efficiency for the MM estimates. Default is eff=0.95.

sest

an CovControlSest object containing control parameters for the initial S-estimate.

maxiter

maximum number of iterations allowed in the computation of the MM-estimate. Default is maxiter=50.

trace, tolSolve:

from the "CovControl" class. tolSolve is used as a convergence tolerance for the MM-iteration.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlMMest"): the generic function restimate allowes the different methods for robust estimation to be used polymorphically - this function will call CovMMest passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMMest", bdp=0.25)
    ctrl2 <- CovControlMMest(bdp=0.25)

    data(hbk)
    CovMMest(hbk, control=ctrl1)

Constructor function for objects of class "CovControlMrcd"

Description

This function will create a control object CovControlMrcd containing the control parameters for CovMrcd

Usage

CovControlMrcd(alpha = 0.5, h=NULL, maxcsteps=200, rho=NULL, 
    target=c("identity", "equicorrelation"), maxcond=50,
    trace= FALSE)

Arguments

Value

A CovControlMrcd object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMrcd", alpha=0.75)
    ctrl2 <- CovControlMrcd(alpha=0.75)

    data(hbk)
    CovMrcd(hbk, control=ctrl1)

Class 'CovControlMrcd' - contains control parameters for CovMrcd()

Description

This class extends the CovControl class and contains the control parameters for "CovMrcd"

Objects from the Class

Objects can be created by calls of the form new("CovControlMrcd", ...) or by calling the constructor-function CovControlMrcd.

Slots

alpha:

numeric parameter controlling the size of the subsets over which the determinant is minimized, i.e., alpha*n observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.5.

h

the size of the subset (can be between ceiling(n/2) and n). Normally NULL and then it h will be calculated as h=ceiling(alpha*n). If h is provided, alpha will be calculated as alpha=h/n.

maxcsteps

maximal number of concentration steps in the deterministic MCD; should not be reached.

rho

regularization parameter. Normally NULL and will be estimated from the data.

target

structure of the robust positive definite target matrix: a) "identity": target matrix is diagonal matrix with robustly estimated univariate scales on the diagonal or b) "equicorrelation": non-diagonal target matrix that incorporates an equicorrelation structure (see (17) in paper).

maxcond

maximum condition number allowed (see step 3.4 in algorithm 1).

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlMrcd"): the generic function restimate allows the different methods for robust estimation to be used polymorphically - this function will call CovMrcd passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

"CovControlMcd"

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMrcd", alpha=0.75)
    ctrl2 <- CovControlMrcd(alpha=0.75)

    data(hbk)
    CovMrcd(hbk, control=ctrl1)

Constructor function for objects of class "CovControlMve"

Description

This function will create a control object CovControlMve containing the control parameters for CovMve

Usage

CovControlMve(alpha = 0.5, nsamp = 500, seed = NULL, trace= FALSE)

Arguments

Value

A CovControlMve object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMve", alpha=0.75)
    ctrl2 <- CovControlMve(alpha=0.75)

    data(hbk)
    CovMve(hbk, control=ctrl1)

Class 'CovControlMve' - contains control parameters for CovMve

Description

This class extends the CovControl class and contains the control parameters for "CovMve"

Objects from the Class

Objects can be created by calls of the form new("CovControlMve", ...) or by calling the constructor-function CovControlMve.

Slots

alpha:

numeric parameter controlling the size of the subsets over which the determinant is minimized, i.e., alpha*n observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.5.

nsamp:

number of subsets used for initial estimates or "best" or "exact". Default is nsamp = 500. For nsamp="best" exhaustive enumeration is done, as long as the number of trials does not exceed 5000. For "exact", exhaustive enumeration will be attempted however many samples are needed. In this case a warning message will be displayed saying that the computation can take a very long time.

seed:

starting value for random generator. Default is seed = NULL

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlMve"): the generic function restimate allowes the different methods for robust estimation to be used polymorphically - this function will call CovMve passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlMve", alpha=0.75)
    ctrl2 <- CovControlMve(alpha=0.75)

    data(hbk)
    CovMve(hbk, control=ctrl1)

Constructor function for objects of class "CovControlOgk"

Description

This function will create a control object CovControlOgk containing the control parameters for CovOgk

Usage

CovControlOgk(niter = 2, beta = 0.9, mrob = NULL, 
vrob = .vrobGK, smrob = "scaleTau2", svrob = "gk")

Arguments

Details

If the user does not specify a scale and covariance function to be used in the computations or specifies one by using the arguments smrob and svrob (i.e. the names of the functions as strings), a native code written in C will be called which is by far faster than the R version.

If the arguments mrob and vrob are not NULL, the specified functions will be used via the pure R implementation of the algorithm. This could be quite slow.

Value

A CovControlOgk object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Maronna, R.A. and Zamar, R.H. (2002) Robust estimates of location and dispersion of high-dimensional datasets; Technometrics 44(4), 307–317.

Yohai, R.A. and Zamar, R.H. (1998) High breakdown point estimates of regression by means of the minimization of efficient scale JASA 86, 403–413.

Gnanadesikan, R. and John R. Kettenring (1972) Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics 28, 81–124.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlOgk", beta=0.95)
    ctrl2 <- CovControlOgk(beta=0.95)

    data(hbk)
    CovOgk(hbk, control=ctrl1)

Class 'CovControlOgk' - contains control parameters for CovOgk

Description

This class extends the CovControl class and contains the control parameters for "CovOgk"

Objects from the Class

Objects can be created by calls of the form new("CovControlOgk", ...) or by calling the constructor-function CovControlOgk.

Slots

niter

number of iterations, usually 1 or 2 since iterations beyond the second do not lead to improvement.

beta

coverage parameter for the final reweighted estimate

mrob

function for computing the robust univariate location and dispersion - defaults to the tau scale defined in Yohai and Zamar (1998)

vrob

function for computing robust estimate of covariance between two random vectors - defaults the one proposed by Gnanadesikan and Kettenring (1972)

smrob

A string indicating the name of the function for computing the robust univariate location and dispersion - defaults to scaleTau2 - the scale 'tau' function defined in Yohai and Zamar (1998)

svrob

A string indicating the name of the function for computing robust estimate of covariance between two random vectors - defaults to gk, the one proposed by Gnanadesikan and Kettenring (1972).

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlOgk"): the generic function restimate allowes the different methods for robust estimation to be used polymorphically - this function will call CovOgk passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlOgk", beta=0.95)
    ctrl2 <- CovControlOgk(beta=0.95)

    data(hbk)
    CovOgk(hbk, control=ctrl1)

Constructor function for objects of class "CovControlSde"

Description

This function will create a control object CovControlSde containing the control parameters for CovSde

Usage

CovControlSde(nsamp = 0, maxres = 0, tune = 0.95, eps = 0.5, prob = 0.99,
    seed = NULL, trace = FALSE, tolSolve = 1e-14)

Arguments

Value

A CovControlSde object.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlSde", nsamp=2000)
    ctrl2 <- CovControlSde(nsamp=2000)

    data(hbk)
    CovSde(hbk, control=ctrl1)

Class 'CovControlSde' - contains control parameters for "CovSde"

Description

This class extends the CovControl class and contains the control parameters for CovSde

Objects from the Class

Objects can be created by calls of the form new("CovControlSde", ...) or by calling the constructor-function CovControlSde.

Slots

nsamp

a positive integer giving the number of resamples required

maxres

a positive integer specifying the maximum number of resamples to be performed including those that are discarded due to linearly dependent subsamples.

tune

a numeric value between 0 and 1 giving the fraction of the data to receive non-zero weight. Default is tune = 0.95.

prob

a numeric value between 0 and 1 specifying the probability of high breakdown point; used to compute nsamp when nsamp is omitted. Default is prob = 0.99.

eps

a numeric value between 0 and 0.5 specifying the breakdown point; used to compute nsamp when nresamp is omitted. Default is eps = 0.5.

seed

starting value for random generator. Default is seed = NULL.

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlSde"): ...

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlSde", nsamp=2000)
    ctrl2 <- CovControlSde(nsamp=2000)

    data(hbk)
    CovSde(hbk, control=ctrl1)

Constructor function for objects of class "CovControlSest"

Description

This function will create a control object CovControlSest containing the control parameters for CovSest

Usage

    CovControlSest(bdp = 0.5, arp = 0.1, eps = 1e-5, maxiter = 120,
        nsamp = 500, seed = NULL, trace = FALSE, tolSolve = 1e-14, method= "sfast")

Arguments

Value

A CovControlSest object.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlSest", bdp=0.4)
    ctrl2 <- CovControlSest(bdp=0.4)

    data(hbk)
    CovSest(hbk, control=ctrl1)

Class 'CovControlSest' - contains control parameters for "CovSest"

Description

This class extends the CovControl class and contains the control parameters for CovSest

Objects from the Class

Objects can be created by calls of the form new("CovControlSest", ...) or by calling the constructor-function CovControlSest.

Slots

bdp

a numeric value specifying the required breakdown point. Allowed values are between (n - p)/(2 * n) and 1 and the default is bdp=0.45.

arp

a numeric value specifying the asympthotic rejection point (for the Rocke type S estimates), i.e. the fraction of points receiving zero weight (see Rocke (1996)). Default is arp=0.1.

eps

a numeric value specifying the relative precision of the solution of the S-estimate (bisquare and Rocke type). Default is to eps=1e-5.

maxiter

maximum number of iterations allowed in the computation of the S-estimate (bisquare and Rocke type). Default is maxiter=120.

nsamp

the number of random subsets considered. Default is nsamp = 500.

seed

starting value for random generator. Default is seed = NULL.

method

Which algorithm to use: 'sfast'=FAST-S, 'surreal'=Ruppert's SURREAL algorithm, 'bisquare'=Bisquare S-estimation with HBDP start or 'rocke' for Rocke type S-estimates

trace, tolSolve:

from the "CovControl" class.

Extends

Class "CovControl", directly.

Methods

restimate

signature(obj = "CovControlSest"): the generic function restimate allowes the different methods for robust estimation to be used polymorphically - this function will call CovSest passing it the control object and will return the obtained CovRobust object

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    ## the following two statements are equivalent
    ctrl1 <- new("CovControlSest", bdp=0.4)
    ctrl2 <- CovControlSest(bdp=0.4)

    data(hbk)
    CovSest(hbk, control=ctrl1)

Robust Location and Scatter Estimation via MCD

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the ‘Fast MCD’ (Minimum Covariance Determinant) estimator.

Usage

CovMcd(x,
       raw.only=FALSE, alpha=control@alpha, nsamp=control@nsamp,
       scalefn=control@scalefn, maxcsteps=control@maxcsteps,
       initHsets=NULL, save.hsets=FALSE,
       seed=control@seed, trace=control@trace,
       use.correction=control@use.correction,
       control=CovControlMcd(), ...)

Arguments

Details

This function computes the minimum covariance determinant estimator of location and scatter and returns an S4 object of class CovMcd-class containing the estimates. The implementation of the function is similar to the existing R function covMcd() which returns an S3 object. The MCD method looks for the h (> n/2) observations (out of n) whose classical covariance matrix has the lowest possible determinant. The raw MCD estimate of location is then the average of these h points, whereas the raw MCD estimate of scatter is their covariance matrix, multiplied by a consistency factor and a finite sample correction factor (to make it consistent at the normal model and unbiased at small samples). Both rescaling factors are returned also in the vector raw.cnp2 of length 2. Based on these raw MCD estimates, a reweighting step is performed which increases the finite-sample efficiency considerably - see Pison et al. (2002). The rescaling factors for the reweighted estimates are returned in the vector cnp2 of length 2. Details for the computation of the finite sample correction factors can be found in Pison et al. (2002). The finite sample corrections can be suppressed by setting use.correction=FALSE. The implementation in rrcov uses the Fast MCD algorithm of Rousseeuw and Van Driessen (1999) to approximate the minimum covariance determinant estimator.

Value

An S4 object of class CovMcd-class which is a subclass of the virtual class CovRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and Outlier Detection. Wiley.

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

M. Hubert, P. Rousseeuw and T. Verdonck (2012) A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics 21(3), 618–637.

Pison, G., Van Aelst, S., and Willems, G. (2002), Small Sample Corrections for LTS and MCD, Metrika, 55, 111-123.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

cov.rob from package MASS

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovMcd(hbk.x)
cD <- CovMcd(hbk.x, nsamp = "deterministic")
summary(cD)

## the following three statements are equivalent
c1 <- CovMcd(hbk.x, alpha = 0.75)
c2 <- CovMcd(hbk.x, control = CovControlMcd(alpha = 0.75))
## direct specification overrides control one:
c3 <- CovMcd(hbk.x, alpha = 0.75,
             control = CovControlMcd(alpha=0.95))
c1

MCD Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates MCD Estimates of multivariate location and scatter computed by the ‘Fast MCD’ algorithm.

Objects from the Class

Objects can be created by calls of the form new("CovMcd", ...), but the usual way of creating CovMcd objects is a call to the function CovMcd which serves as a constructor.

Slots

alpha:

Object of class "numeric" - the size of the subsets over which the determinant is minimized (the default is (n+p+1)/2)

quan:

Object of class "numeric" - the number of observations on which the MCD is based. If quan equals n.obs, the MCD is the classical covariance matrix.

best:

Object of class "Uvector" - the best subset found and used for computing the raw estimates. The size of best is equal to quan

raw.cov:

Object of class "matrix" the raw (not reweighted) estimate of location

raw.center:

Object of class "vector" - the raw (not reweighted) estimate of scatter

raw.mah:

Object of class "Uvector" - mahalanobis distances of the observations based on the raw estimate of the location and scatter

raw.wt:

Object of class "Uvector" - weights of the observations based on the raw estimate of the location and scatter

raw.cnp2:

Object of class "numeric" - a vector of length two containing the consistency correction factor and the finite sample correction factor of the raw estimate of the covariance matrix

cnp2:

Object of class "numeric" - a vector of length two containing the consistency correction factor and the finite sample correction factor of the final estimate of the covariance matrix.

iter, crit, wt:

from the "CovRobust" class.

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovMcd" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMcd, Cov-class, CovRobust-class

Examples

showClass("CovMcd")

Constrained M-Estimates of Location and Scatter

Description

Computes constrained M-Estimates of multivariate location and scatter based on the translated biweight function (‘t-biweight’) using a High breakdown point initial estimate. The default initial estimate is the Minimum Volume Ellipsoid computed with CovMve. The raw (not reweighted) estimates are taken and the covariance matrix is standardized to determinant 1.

Usage

covMest(x, cor=FALSE, r = 0.45, arp = 0.05, eps=1e-3,
    maxiter=120, control, t0, S0)

Arguments

.

Details

Rocke (1996) has shown that the S-estimates of multivariate location and scatter in high dimensions can be sensitive to outliers even if the breakdown point is set to be near 0.5. To mitigate this problem he proposed to utilize the translated biweight (or t-biweight) method with a standardization step consisting of equating the median of rho(d) with the median under normality. This is then not an S-estimate, but is instead a constrained M-estimate. In order to make the smooth estimators to work, a reasonable starting point is necessary, which will lead reliably to a good solution of the estimator. In covMest the MVE computed by CovMve is used, but the user has the possibility to give her own initial estimates.

Value

An object of class "mest" which is basically a list with the following components. This class is "derived" from "mcd" so that the same generic functions - print, plot, summary - can be used. NOTE: this is going to change - in one of the next revisions covMest will return an S4 class "mest" which is derived (i.e. contains) form class "cov".

Note

The psi, rho and weight functions for the M estimation are encapsulated in a virtual S4 class PsiFun from which a PsiBwt class, implementing the translated biweight (t-biweight), is dervied. The base class PsiFun contains also the M-iteration itself. Although not documented and not accessibale directly by the user these classes will form the bases for adding other functions (biweight, LWS, etc.) as well as S-estimates.

Author(s)

Valentin Todorov valentin.todorov@chello.at,

(some code from C. Becker - http://www.sfb475.uni-dortmund.de/dienst/de/content/struk-d/bereicha-d/tpa1softw-d.html)

References

D.L.Woodruff and D.M.Rocke (1994) Computable robust estimation of multivariate location and shape on high dimension using compound estimators, Journal of the American Statistical Association, 89, 888–896.

D.M.Rocke (1996) Robustness properties of S-estimates of multivariate location and shape in high dimension, Annals of Statistics, 24, 1327-1345.

D.M.Rocke and D.L.Woodruff (1996) Identification of outliers in multivariate data Journal of the American Statistical Association, 91, 1047–1061.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

covMcd

Constrained M-Estimates of Location and Scatter

Description

Computes constrained M-Estimates of multivariate location and scatter based on the translated biweight function (‘t-biweight’) using a High breakdown point initial estimate as defined by Rocke (1996). The default initial estimate is the Minimum Volume Ellipsoid computed with CovMve. The raw (not reweighted) estimates are taken and the covariance matrix is standardized to determinant 1.

Usage

    CovMest(x, r = 0.45, arp = 0.05, eps=1e-3,
        maxiter=120, control, t0, S0, initcontrol)

Arguments

Details

Rocke (1996) has shown that the S-estimates of multivariate location and scatter in high dimensions can be sensitive to outliers even if the breakdown point is set to be near 0.5. To mitigate this problem he proposed to utilize the translated biweight (or t-biweight) method with a standardization step consisting of equating the median of rho(d) with the median under normality. This is then not an S-estimate, but is instead a constrained M-estimate. In order to make the smooth estimators to work, a reasonable starting point is necessary, which will lead reliably to a good solution of the estimator. In CovMest the MVE computed by CovMve is used, but the user has the possibility to give her own initial estimates.

Value

An object of class CovMest-class which is a subclass of the virtual class CovRobust-class.

Note

The psi, rho and weight functions for the M estimation are encapsulated in a virtual S4 class PsiFun from which a PsiBwt class, implementing the translated biweight (t-biweight), is dervied. The base class PsiFun contains also the M-iteration itself. Although not documented and not accessibale directly by the user these classes will form the bases for adding other functions (biweight, LWS, etc.) as well as S-estimates.

Author(s)

Valentin Todorov valentin.todorov@chello.at,

(some code from C. Becker - http://www.sfb475.uni-dortmund.de/dienst/de/content/struk-d/bereicha-d/tpa1softw-d.html)

References

D.L.Woodruff and D.M.Rocke (1994) Computable robust estimation of multivariate location and shape on high dimension using compound estimators, Journal of the American Statistical Association, 89, 888–896.

D.M.Rocke (1996) Robustness properties of S-estimates of multivariate location and shape in high dimension, Annals of Statistics, 24, 1327-1345.

D.M.Rocke and D.L.Woodruff (1996) Identification of outliers in multivariate data Journal of the American Statistical Association, 91, 1047–1061.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

covMcd, Cov-class, CovMve, CovRobust-class, CovMest-class

Examples

library(rrcov)
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovMest(hbk.x)

## the following four statements are equivalent
c0 <- CovMest(hbk.x)
c1 <- CovMest(hbk.x, r = 0.45)
c2 <- CovMest(hbk.x, control = CovControlMest(r = 0.45))
c3 <- CovMest(hbk.x, control = new("CovControlMest", r = 0.45))

## direct specification overrides control one:
c4 <- CovMest(hbk.x, r = 0.40,
             control = CovControlMest(r = 0.25))
c1
summary(c1)
plot(c1)

Constrained M-estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates constrained M-Estimates of multivariate location and scatter based on the translated biweight function (‘t-biweight’) using a High breakdown point initial estimate (Minimum Covariance Determinant - ‘Fast MCD’)

Objects from the Class

Objects can be created by calls of the form new("CovMest", ...), but the usual way of creating CovMest objects is a call to the function CovMest which serves as a constructor.

Slots

vt:

Object of class "vector" - vector of weights (v)

iter, crit, wt:

from the "CovRobust" class.

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovMest" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMest, Cov-class, CovRobust-class

Examples

showClass("CovMest")

MM Estimates of Multivariate Location and Scatter

Description

Computes MM-Estimates of multivariate location and scatter starting from an initial S-estimate

Usage

    CovMMest(x, bdp = 0.5, eff = 0.95, eff.shape=TRUE, maxiter = 50, 
        trace = FALSE, tolSolve = 1e-7, control)

Arguments

Details

Computes MM-estimates of multivariate location and scatter starting from an initial S-estimate.

Value

An S4 object of class CovMMest-class which is a subclass of the virtual class CovRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Tatsuoka, K.S. and Tyler, D.E. (2000). The uniqueness of S and M-functionals under non-elliptical distributions. Annals of Statistics 28, 1219–1243

M. Salibian-Barrera, S. Van Aelstt and G. Willems (2006). Principal components analysis based on multivariate MM-estimators with fast and robust bootstrap. Journal of the American Statistical Association 101, 1198–1211.

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

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

library(rrcov)
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovMMest(hbk.x)

## the following four statements are equivalent
c0 <- CovMMest(hbk.x)
c1 <- CovMMest(hbk.x, bdp = 0.25)
c2 <- CovMMest(hbk.x, control = CovControlMMest(bdp = 0.25))
c3 <- CovMMest(hbk.x, control = new("CovControlMMest", bdp = 0.25))

## direct specification overrides control one:
c4 <- CovMMest(hbk.x, bdp = 0.40,
             control = CovControlMMest(bdp = 0.25))
c1
summary(c1)
plot(c1)

## Deterministic MM-estmates
CovMMest(hbk.x, control=CovControlMMest(sest=CovControlSest(method="sdet")))

MM Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates MM Estimates of multivariate location and scatter.

Objects from the Class

Objects can be created by calls of the form new("CovMMest", ...), but the usual way of creating CovSest objects is a call to the function CovMMest which serves as a constructor.

Slots

det, flag, iter, crit:

from the "CovRobust" class.

c1

tuning parameter of the loss function for MM-estimation (depend on control parameters eff and eff.shape). Can be computed by the internal function .csolve.bw.MM(p, eff, eff.shape=TRUE). For the tuning parameters of the underlying S-estimate see the slot sest and "CovSest".

sest

an CovSest object containing the initial S-estimate.

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovMMest" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMMest, Cov-class, CovRobust-class

Examples

showClass("CovMMest")

Robust Location and Scatter Estimation via Minimum Regularized Covariance Determonant (MRCD)

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the Minimum Regularized Covariance Determonant (MRCD) estimator.

Usage

CovMrcd(x,
       alpha=control@alpha, 
       h=control@h,
       maxcsteps=control@maxcsteps,
       initHsets=NULL, save.hsets=FALSE,
       rho=control@rho,
       target=control@target,
       maxcond=control@maxcond,
       trace=control@trace,
       control=CovControlMrcd())

Arguments

Details

This function computes the minimum regularized covariance determinant estimator (MRCD) of location and scatter and returns an S4 object of class CovMrcd-class containing the estimates. Similarly like the MCD method, MRCD looks for the h (> n/2) observations (out of n) whose classical covariance matrix has the lowest possible determinant, but replaces the subset-based covariance by a regularized covariance estimate, defined as a weighted average of the sample covariance of the h-subset and a predetermined positive definite target matrix. The Minimum Regularized Covariance Determinant (MRCD) estimator is then the regularized covariance based on the h-subset which makes the overall determinant the smallest. A data-driven procedure sets the weight of the target matrix (rho), so that the regularization is only used when needed.

Value

An S4 object of class CovMrcd-class which is a subclass of the virtual class CovRobust-class.

Author(s)

Kris Boudt, Peter Rousseeuw, Steven Vanduffel and Tim Verdonk. Improved by Joachim Schreurs and Iwein Vranckx. Adapted for rrcov by Valentin Todorov valentin.todorov@chello.at

References

Kris Boudt, Peter Rousseeuw, Steven Vanduffel and Tim Verdonck (2020) The Minimum Regularized Covariance Determinant estimator, Statistics and Computing, 30, pp 113–128 doi:10.1007/s11222-019-09869-x.

Mia Hubert, Peter Rousseeuw and Tim Verdonck (2012) A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics 21(3), 618–637.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMcd

Examples

## The result will be (almost) identical to the raw MCD
##  (since we do not do reweighting of MRCD)
##
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
c0 <- CovMcd(hbk.x, alpha=0.75, use.correction=FALSE)
cc <- CovMrcd(hbk.x, alpha=0.75)
cc$rho
all.equal(c0$best, cc$best)
all.equal(c0$raw.center, cc$center)
all.equal(c0$raw.cov/c0$raw.cnp2[1], cc$cov/cc$cnp2)

summary(cc)

## the following three statements are equivalent
c1 <- CovMrcd(hbk.x, alpha = 0.75)
c2 <- CovMrcd(hbk.x, control = CovControlMrcd(alpha = 0.75))
## direct specification overrides control one:
c3 <- CovMrcd(hbk.x, alpha = 0.75,
             control = CovControlMrcd(alpha=0.95))
c1

## Not run: 

##  This is the first example from Boudt et al. (2020). The first variable is 
##  the dependent one, which we remove and remain with p=226 NIR absorbance spectra 

data(octane)

octane <- octane[, -1]    # remove the dependent variable y

n <- nrow(octane)
p <- ncol(octane)

##  Compute MRCD with h=33, which gives approximately 15 percent breakdown point.
##  This value of h was found by Boudt et al. (2020) using a data driven approach, 
##  similar to the Forward Search of Atkinson et al. (2004). 
##  The default value of h would be 20 (i.e. alpha=0.5) 

out <- CovMrcd(octane, h=33) 
out$rho

## Please note that in the paper is indicated that the obtained rho=0.1149, however,
##  this value of rho is obtained if the parameter maxcond is set equal to 999 (this was 
##  the default in an earlier version of the software, now the default is maxcond=50). 
##  To reproduce the result from the paper, change the call to CovMrcd() as follows 
##  (this will not influence the results shown further):

##  out <- CovMrcd(octane, h=33, maxcond=999) 
##  out$rho

robpca = PcaHubert(octane, k=2, alpha=0.75, mcd=FALSE)
(outl.robpca = which(robpca@flag==FALSE))

# Observations flagged as outliers by ROBPCA:
# 25, 26, 36, 37, 38, 39

# Plot the orthogonal distances versus the score distances:
pch = rep(20,n); pch[robpca@flag==FALSE] = 17
col = rep('black',n); col[robpca@flag==FALSE] = 'red'
plot(robpca, pch=pch, col=col, id.n.sd=6, id.n.od=6)

## Plot now the MRCD mahalanobis distances
pch = rep(20,n); pch[!getFlag(out)] = 17
col = rep('black',n); col[!getFlag(out)] = 'red'
plot(out, pch=pch, col=col, id.n=6)

## End(Not run)

MRCD Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates MRCD Estimates of multivariate location and scatter computed by a variant of the ‘Fast MCD’ algorithm.

Objects from the Class

Objects can be created by calls of the form new("CovMrcd", ...), but the usual way of creating CovMrcd objects is a call to the function CovMrcd which serves as a constructor.

Slots

alpha:

Object of class "numeric" - the size of the subsets over which the determinant is minimized (the default is (n+p+1)/2)

quan:

Object of class "numeric" - the number of observations on which the MCD is based. If quan equals n.obs, the MCD is the classical covariance matrix.

best:

Object of class "Uvector" - the best subset found and used for computing the raw estimates. The size of best is equal to quan

cnp2:

Object of class "numeric" - containing the consistency correction factor of the estimate of the covariance matrix.

icov:

The inverse of the covariance matrix.

rho:

The estimated regularization parameter.

target:

The estimated target matrix.

crit:

from the "CovRobust" class.

call, cov, center, n.obs, mah, method, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovMrcd" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMrcd, Cov-class, CovRobust-class, CovMcd-class

Examples

showClass("CovMrcd")

Robust Location and Scatter Estimation via MVE

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the ‘MVE’ (Minimum Volume Ellipsoid) estimator.

Usage

CovMve(x, alpha = 1/2, nsamp = 500, seed = NULL, trace = FALSE, control)

Arguments

Details

This function computes the minimum volume ellipsoid estimator of location and scatter and returns an S4 object of class CovMve-class containing the estimates.

The approximate estimate is based on a subset of size alpha*n with an enclosing ellipsoid of smallest volume. The mean of the best found subset provides the raw estimate of the location, and the rescaled covariance matrix is the raw estimate of scatter. The rescaling of the raw covariance matrix is by median(dist)/qchisq(0.5, p) and this scale factor is returned in the slot raw.cnp2. Currently no finite sample corrction factor is applied. The Mahalanobis distances of all observations from the location estimate for the raw covariance matrix are calculated, and those points within the 97.5 under Gaussian assumptions are declared to be good. The final (reweightd) estimates are the mean and rescaled covariance of the good points. The reweighted covariance matrix is rescaled by 1/pgamma(qchisq(alpha, p)/2, p/2 + 1)/alpha (see Croux and Haesbroeck, 1999) and this scale factor is returned in the slot cnp2.

The search for the approximate solution is made over ellipsoids determined by the covariance matrix of p+1 of the data points and applying a simple but effective improvement of the subsampling procedure as described in Maronna et al. (2006), p. 198. Although there exists no formal proof of this improvement (as for MCD and LTS), simulations show that it can be recommended as an approximation of the MVE.

Value

An S4 object of class CovMve-class which is a subclass of the virtual class CovRobust-class.

Note

Main reason for implementing the MVE estimate was that it is the recommended initial estimate for S estimation (see Maronna et al. (2006), p. 199) and will be used by default in CovMest (after removing the correction factors from the covariance matrix and rescaling to determinant 1).

Author(s)

Valentin Todorov valentin.todorov@chello.at and Matias Salibian-Barrera matias@stat.ubc.ca

References

P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and Outlier Detection. Wiley.

C. Croux and G. Haesbroeck (1999). Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. Journal of Multivariate Analysis, 71, 161–190.

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

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

cov.rob from package MASS

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovMve(hbk.x)

## the following three statements are equivalent
c1 <- CovMve(hbk.x, alpha = 0.75)
c2 <- CovMve(hbk.x, control = CovControlMve(alpha = 0.75))
## direct specification overrides control one:
c3 <- CovMve(hbk.x, alpha = 0.75,
             control = CovControlMve(alpha=0.95))
c1

MVE Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates MVE Estimates of multivariate location and scatter computed by the ‘Fast MVE’ algorithm.

Objects from the Class

Objects can be created by calls of the form new("CovMve", ...), but the usual way of creating CovMve objects is a call to the function CovMve which serves as a constructor.

Slots

alpha:

Object of class "numeric" - the size of the subsets over which the volume of the ellipsoid is minimized (the default is (n+p+1)/2)

quan:

Object of class "numeric" - the number of observations on which the MVE is based. If quan equals n.obs, the MVE is the classical covariance matrix.

best:

Object of class "Uvector" - the best subset found and used for computing the raw estimates. The size of best is equal to quan

raw.cov:

Object of class "matrix" the raw (not reweighted) estimate of location

raw.center:

Object of class "vector" - the raw (not reweighted) estimate of scatter

raw.mah:

Object of class "Uvector" - mahalanobis distances of the observations based on the raw estimate of the location and scatter

raw.wt:

Object of class "Uvector" - weights of the observations based on the raw estimate of the location and scatter

raw.cnp2:

Object of class "numeric" - a vector of length two containing the consistency correction factor and the finite sample correction factor of the raw estimate of the covariance matrix

cnp2:

Object of class "numeric" - a vector of length two containing the consistency correction factor and the finite sample correction factor of the final estimate of the covariance matrix.

iter, crit, wt:

from the "CovRobust" class.

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovMve" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMve, Cov-class, CovRobust-class

Examples

showClass("CovMve")

Robust Location and Scatter Estimation - Ortogonalized Gnanadesikan-Kettenring (OGK)

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the pairwise algorithm proposed by Marona and Zamar (2002) which in turn is based on the pairwise robust estimator proposed by Gnanadesikan-Kettenring (1972).

Usage

CovOgk(x, niter = 2, beta = 0.9, control)

Arguments

Details

The method proposed by Marona and Zamar (2002) allowes to obtain positive-definite and almost affine equivariant robust scatter matrices starting from any pairwise robust scatter matrix. The default robust estimate of covariance between two random vectors used is the one proposed by Gnanadesikan and Kettenring (1972) but the user can choose any other method by redefining the function in slot vrob of the control object CovControlOgk. Similarly, the function for computing the robust univariate location and dispersion used is the tau scale defined in Yohai and Zamar (1998) but it can be redefined in the control object.

The estimates obtained by the OGK method, similarly as in CovMcd are returned as 'raw' estimates. To improve the estimates a reweighting step is performed using the coverage parameter beta and these reweighted estimates are returned as 'final' estimates.

Value

An S4 object of class CovOgk-class which is a subclass of the virtual class CovRobust-class.

Note

If the user does not specify a scale and covariance function to be used in the computations or specifies one by using the arguments smrob and svrob (i.e. the names of the functions as strings), a native code written in C will be called which is by far faster than the R version.

If the arguments mrob and vrob are not NULL, the specified functions will be used via the pure R implementation of the algorithm. This could be quite slow.

See CovControlOgk for details.

Author(s)

Valentin Todorov valentin.todorov@chello.at and Kjell Konis kjell.konis@epfl.ch

References

Maronna, R.A. and Zamar, R.H. (2002) Robust estimates of location and dispersion of high-dimensional datasets; Technometrics 44(4), 307–317.

Yohai, R.A. and Zamar, R.H. (1998) High breakdown point estimates of regression by means of the minimization of efficient scale JASA 86, 403–413.

Gnanadesikan, R. and John R. Kettenring (1972) Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics 28, 81–124.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMcd, CovMest

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovOgk(hbk.x)

## the following three statements are equivalent
c1 <- CovOgk(hbk.x, niter=1)
c2 <- CovOgk(hbk.x, control = CovControlOgk(niter=1))

## direct specification overrides control one:
c3 <- CovOgk(hbk.x, beta=0.95,
             control = CovControlOgk(beta=0.99))
c1

x<-matrix(c(1,2,3,7,1,2,3,7), ncol=2)
##  CovOgk(x)   - this would fail because the two columns of x are exactly collinear.
##              In order to fix it, redefine the default 'vrob' function for example
##              in the following way and pass it as a parameter in the control
##              object.
cc <- CovOgk(x, control=new("CovControlOgk",
                            vrob=function(x1, x2, ...)
                            {
                                r <- .vrobGK(x1, x2, ...)
                                if(is.na(r))
                                    r <- 0
                                r
                            })
)
cc

OGK Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates OGK Estimates of multivariate location and scatter computed by the algorithm proposed by Marona and Zamar (2002).

Objects from the Class

Objects can be created by calls of the form new("CovOgk", ...), but the usual way of creating CovOgk objects is a call to the function CovOgk which serves as a constructor.

Slots

raw.cov:

Object of class "matrix" the raw (not reweighted) estimate of covariance matrix

raw.center:

Object of class "vector" - the raw (not reweighted) estimate of the location vector

raw.mah:

Object of class "Uvector" - mahalanobis distances of the observations based on the raw estimate of the location and scatter

raw.wt:

Object of class "Uvector" - weights of the observations based on the raw estimate of the location and scatter

iter, crit, wt:

from the "CovRobust" class.

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovOgk" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMcd-class, CovMest-class

Examples

showClass("CovOgk")

Robust Location and Scatter Estimation

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using one of the available estimators.

Usage

CovRobust(x, control, na.action = na.fail)

Arguments

Details

This function simply calls the restimate method of the control object control. If a character string naming an estimator is specified, a new control object will be created and used (with default estimation options). If this argument is missing or a character string 'auto' is specified, the function will select the robust estimator according to the size of the dataset. If there are less than 1000 observations and less than 10 variables or less than 5000 observations and less than 5 variables, Stahel-Donoho estimator will be used. Otherwise, if there are less than 50000 observations either bisquare S-estimates (for less than 10 variables) or Rocke type S-estimates (for 10 to 20 variables) will be used. In both cases the S iteration starts at the initial MVE estimate. And finally, if there are more than 50000 observations and/or more than 20 variables the Orthogonalized Quadrant Correlation estimator (CovOgk with the corresponding parameters) is used.

Value

An object derived from a CovRobust object, depending on the selected estimator.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovRobust(hbk.x)
CovRobust(hbk.x, CovControlSest(method="bisquare"))

Class "CovRobust" - virtual base class for robust estimates of multivariate location and scatter

Description

CovRobust is a virtual base class used for deriving the concrete classes representing different robust estimates of multivariate location and scatter. Here are implemeted the standard methods common for all robust estimates like show, summary and plot. The derived classes can override these methods and can define new ones.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

iter:

number of iterations used to compute the estimates

crit:

value of the criterion function

wt:

weights

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "Cov", directly.

Methods

isClassic

signature(obj = "CovRobust"): Will return FALSE, since this is a 'Robust' object

getMeth

signature(obj = "CovRobust"): Return the name of the particular robust method used (as a character string)

show

signature(object = "CovRobust"): display the object

plot

signature(x = "CovRobust"): plot the object

getRaw

signature(obj = "CovRobust"): Return the object with the reweighted estimates replaced by the raw ones (only relevant for CovMcd, CovMve and CovOgk)

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Cov-class, CovMcd-class, CovMest-class, CovOgk-class

Examples

     data(hbk)
     hbk.x <- data.matrix(hbk[, 1:3])
     cv <- CovMest(hbk.x)               # it is not possible to create an object of
                                        # class CovRobust, since it is a VIRTUAL class
     cv
     summary(cv)                        # summary method for class CovRobust
     plot(cv)                           # plot method for class CovRobust

Stahel-Donoho Estimates of Multivariate Location and Scatter

Description

Compute a robust estimate of location and scale using the Stahel-Donoho projection based estimator

Usage

CovSde(x, nsamp, maxres, tune = 0.95, eps = 0.5, prob = 0.99, 
seed = NULL, trace = FALSE, control)

Arguments

Details

The projection based Stahel-Donoho estimator posses very good statistical properties, but it can be very slow if the number of variables is too large. It is recommended to use this estimator if n <= 1000 and p<=10 or n <= 5000 and p<=5. The number of subsamples required is calculated to provide a breakdown point of eps with probability prob and can reach values larger than the larger integer value - in such case it is limited to .Machine$integer.max. Of course you could provide nsamp in the call, i.e. nsamp=1000 but this will not guarantee the required breakdown point of th eestimator. For larger data sets it is better to use CovMcd or CovOgk. If you use CovRobust, the estimator will be selected automatically according on the size of the data set.

Value

An S4 object of class CovSde-class which is a subclass of the virtual class CovRobust-class.

Note

The Fortran code for the Stahel-Donoho method was taken almost with no changes from package robust which in turn has it from the Insightful Robust Library (thanks to by Kjell Konis).

Author(s)

Valentin Todorov valentin.todorov@chello.at and Kjell Konis kjell.konis@epfl.ch

References

R. A. Maronna and V.J. Yohai (1995) The Behavior of the Stahel-Donoho Robust Multivariate Estimator. Journal of the American Statistical Association 90 (429), 330–341.

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

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovSde(hbk.x)

## the following four statements are equivalent
c0 <- CovSde(hbk.x)
c1 <- CovSde(hbk.x, nsamp=2000)
c2 <- CovSde(hbk.x, control = CovControlSde(nsamp=2000))
c3 <- CovSde(hbk.x, control = new("CovControlSde", nsamp=2000))

## direct specification overrides control one:
c4 <- CovSde(hbk.x, nsamp=100,
             control = CovControlSde(nsamp=2000))
c1
summary(c1)
plot(c1)

## Use the function CovRobust() - if no estimation method is
##  specified, for small data sets CovSde() will be called
cr <- CovRobust(hbk.x)
cr

Stahel-Donoho Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates Stahel-Donoho estimates of multivariate location and scatter.

Objects from the Class

Objects can be created by calls of the form new("CovSde", ...), but the usual way of creating CovSde objects is a call to the function CovSde which serves as a constructor.

Slots

iter, crit, wt:

from the "CovRobust" class.

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovSde" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovSde, Cov-class, CovRobust-class

Examples

showClass("CovSde")

S Estimates of Multivariate Location and Scatter

Description

Computes S-Estimates of multivariate location and scatter based on Tukey's biweight function using a fast algorithm similar to the one proposed by Salibian-Barrera and Yohai (2006) for the case of regression. Alternativley, the Ruppert's SURREAL algorithm, bisquare or Rocke type estimation can be used.

Usage

    CovSest(x, bdp = 0.5, arp = 0.1, eps = 1e-5, maxiter = 120,
        nsamp = 500, seed = NULL, trace = FALSE, tolSolve = 1e-14,
        scalefn, maxisteps=200, 
        initHsets = NULL, save.hsets = missing(initHsets) || is.null(initHsets),
        method = c("sfast", "surreal", "bisquare", "rocke", "suser", "sdet"), 
        control, t0, S0, initcontrol)

Arguments

Details

Computes multivariate S-estimator of location and scatter. The computation will be performed by one of the following algorithms:

FAST-S

An algorithm similar to the one proposed by Salibian-Barrera and Yohai (2006) for the case of regression

SURREAL

Ruppert's SURREAL algorithm when method is set to 'surreal'

BISQUARE

Bisquare S-Estimate with method set to 'bisquare'

ROCKE

Rocke type S-Estimate with method set to 'rocke'

Except for the last algorithm, ROCKE, all other use Tukey biweight loss function. The tuning parameters used in the loss function (as determined by bdp) are returned in the slots cc and kp of the result object. They can be computed by the internal function .csolve.bw.S(bdp, p).

Value

An S4 object of class CovSest-class which is a subclass of the virtual class CovRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at, Matias Salibian-Barrera matias@stat.ubc.ca and Victor Yohai vyohai@dm.uba.ar. See also the code from Kristel Joossens, K.U. Leuven, Belgium and Ella Roelant, Ghent University, Belgium.

References

M. Hubert, P. Rousseeuw and T. Verdonck (2012) A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics 21(3), 618–637.

M. Hubert, P. Rousseeuw, D. Vanpaemel and T. Verdonck (2015) The DetS and DetMM estimators for multivariate location and scatter. Computational Statistics and Data Analysis 81, 64–75.

H.P. Lopuhaä (1989) On the Relation between S-estimators and M-estimators of Multivariate Location and Covariance. Annals of Statistics 17 1662–1683.

D. Ruppert (1992) Computing S Estimators for Regression and Multivariate Location/Dispersion. Journal of Computational and Graphical Statistics 1 253–270.

M. Salibian-Barrera and V. Yohai (2006) A fast algorithm for S-regression estimates, Journal of Computational and Graphical Statistics, 15, 414–427.

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

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

library(rrcov)
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
cc <- CovSest(hbk.x)
cc

## summry and different types of plots
summary(cc)                         
plot(cc)                            
plot(cc, which="dd")
plot(cc, which="pairs")
plot(cc, which="xydist")

## the following four statements are equivalent
c0 <- CovSest(hbk.x)
c1 <- CovSest(hbk.x, bdp = 0.25)
c2 <- CovSest(hbk.x, control = CovControlSest(bdp = 0.25))
c3 <- CovSest(hbk.x, control = new("CovControlSest", bdp = 0.25))

## direct specification overrides control one:
c4 <- CovSest(hbk.x, bdp = 0.40,
             control = CovControlSest(bdp = 0.25))
c1
summary(c1)
plot(c1)

## Use the SURREAL algorithm of Ruppert
cr <- CovSest(hbk.x, method="surreal")
cr

## Use Bisquare estimation
cr <- CovSest(hbk.x, method="bisquare")
cr

## Use Rocke type estimation
cr <- CovSest(hbk.x, method="rocke")
cr

## Use Deterministic estimation
cr <- CovSest(hbk.x, method="sdet")
cr

S Estimates of Multivariate Location and Scatter

Description

This class, derived from the virtual class "CovRobust" accomodates S Estimates of multivariate location and scatter computed by the ‘Fast S’ or ‘SURREAL’ algorithm.

Objects from the Class

Objects can be created by calls of the form new("CovSest", ...), but the usual way of creating CovSest objects is a call to the function CovSest which serves as a constructor.

Slots

iter, crit, wt:

from the "CovRobust" class.

iBest, nsteps, initHsets:

parameters for deterministic S-estimator (the best initial subset, number of concentration steps to convergence for each of the initial subsets, and the computed initial subsets, respectively).

cc, kp

tuning parameters used in Tukey biweight loss function, as determined by bdp. Can be computed by the internal function .csolve.bw.S(bdp, p).

call, cov, center, n.obs, mah, method, singularity, X:

from the "Cov" class.

Extends

Class "CovRobust", directly. Class "Cov", by class "CovRobust".

Methods

No methods defined with class "CovSest" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovSest, Cov-class, CovRobust-class

Examples

showClass("CovSest")

Reaven and Miller diabetes data

Description

The data set contains five measurements made on 145 non-obese adult patients classified into three groups.

The three primary variables are glucose intolerance (area under the straight line connecting glucose levels), insulin response to oral glucose (area under the straight line connecting insulin levels) and insulin resistance (measured by the steady state plasma glucose (SSPG) determined after chemical suppression of endogenous insulin secretion). Two additional variables, the relative weight and fasting plasma glucose, are also included.

Reaven and Miller, following Friedman and Rubin (1967), applied cluster analysis to the three primary variables and identified three clusters: "normal", "chemical diabetic", and "overt diabetic" subjects. The column group contains the classifications of the subjects into these three groups, obtained by current medical criteria.

Usage

data(diabetes)

Format

A data frame with the following variables:

rw

relative weight, expressed as the ratio of actual weight to expected weight, given the person's height.

fpg

fasting plasma glucose level.

glucose

area under plasma glucose curve after a three hour oral glucose tolerance test (OGTT).

insulin

area under plasma insulin curve after a three hour oral glucose tolerance test (OGTT).

sspg

Steady state plasma glucose, a measure of insulin resistance.

group

the type of diabetes: a factor with levels normal, chemical and overt.

Source

Reaven, G. M. and Miller, R. G. (1979). An attempt to define the nature of chemical diabetes using a multidimensional analysis. Diabetologia 16, 17–24. Andrews, D. F. and Herzberg, A. M. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker, Springer-Verlag, Ch. 36.

References

Reaven, G. M. and Miller, R. G. (1979). An attempt to define the nature of chemical diabetes using a multidimensional analysis. Diabetologia 16, 17–24.

Friedman, H. P. and Rubin, J. (1967). On some invariant criteria for grouping data. Journal of the American Statistical Association 62, 1159–1178.

Hawkins, D. M. and McLachlan, G. J., 1997. High-breakdown linear discriminant analysis. Journal of the American Statistical Association 92 (437), 136–143.

Examples

data(diabetes)
(cc <- Linda(group~insulin+glucose+sspg, data=diabetes))
(pr <- predict(cc))

Fish Catch Data Set

Description

The Fish Catch data set contains measurements on 159 fish caught in the lake Laengelmavesi, Finland.

Usage

data(fish)

Format

A data frame with 159 observations on the following 7 variables.

Weight

Weight of the fish (in grams)

Length1

Length from the nose to the beginning of the tail (in cm)

Length2

Length from the nose to the notch of the tail (in cm)

Length3

Length from the nose to the end of the tail (in cm)

Height

Maximal height as % of Length3

Width

Maximal width as % of Length3

Species

Species

Details

The Fish Catch data set contains measurements on 159 fish caught in the lake Laengelmavesi, Finland. For the 159 fishes of 7 species the weight, length, height, and width were measured. Three different length measurements are recorded: from the nose of the fish to the beginning of its tail, from the nose to the notch of its tail and from the nose to the end of its tail. The height and width are calculated as percentages of the third length variable. This results in 6 observed variables, Weight, Length1, Length2, Length3, Height, Width. Observation 14 has a missing value in variable Weight, therefore this observation is usually excluded from the analysis. The last variable, Species, represents the grouping structure: the 7 species are 1=Bream, 2=Whitewish, 3=Roach, 4=Parkki, 5=Smelt, 6=Pike, 7=Perch. This data set was also analyzed in the context of robust Linear Discriminant Analysis by Todorov (2007), Todorov and Pires (2007).

Source

Journal of Statistical Education, Fish Catch Data Set, [https://jse.amstat.org/datasets/fishcatch.dat.txt] accessed November, 2023.

References

Todorov, V. (2007 Robust selection of variables in linear discriminant analysis, Statistical Methods and Applications, 15, 395–407, doi:10.1007/s10260-006-0032-6.

Todorov, V. and Pires, A.M. (2007) Comparative performance of several robust linear discriminant analysis methods, REVSTAT Statistical Journal, 5, 63–83.

Examples

    data(fish)

    # remove observation #14 containing missing value
    fish <- fish[-14,]

    # The height and width are calculated as percentages 
    #   of the third length variable
    fish[,5] <- fish[,5]*fish[,4]/100
    fish[,6] <- fish[,6]*fish[,4]/100
 
    # plot a matrix of scatterplots
    pairs(fish[1:6],
          main="Fish Catch Data",
          pch=21,
          bg=c("red", "green3", "blue", "yellow", "magenta", "violet", 
          "turquoise")[unclass(fish$Species)])

Fruit data set

Description

A data set that contains the spectra of six different cultivars of the same fruit (cantaloupe - Cucumis melo L. Cantaloupensis group) obtained from Colin Greensill (Faculty of Engineering and Physical Systems, Central Queensland University, Rockhampton, Australia). The total data set contained 2818 spectra measured in 256 wavelengths. For illustrative purposes are considered only three cultivars out of it, named D, M and HA with sizes 490, 106 and 500, respectively. Thus the data set thus contains 1096 observations. For more details about this data set see the references below.

Usage

data(fruit)

Format

A data frame with 1096 rows and 257 variables (one grouping variable – cultivar – and 256 measurement variables).

Source

Colin Greensill (Faculty of Engineering and Physical Systems, Central Queensland University, Rockhampton, Australia).

References

Hubert, M. and Van Driessen, K., (2004). Fast and robust discriminant analysis. Computational Statistics and Data Analysis, 45(2):301–320. doi:10.1016/S0167-9473(02)00299-2.

Vanden Branden, K and Hubert, M, (2005). Robust classification in high dimensions based on the SIMCA Method. Chemometrics and Intelligent Laboratory Systems, 79(1-2), pp. 10–21. doi:10.1016/j.chemolab.2005.03.002.

Hubert, M, Rousseeuw, PJ and Verdonck, T, (2012). A Deterministic Algorithm for Robust Location and Scatter. Journal of Computational and Graphical Statistics, 21(3), pp 618–637. doi:10.1080/10618600.2012.672100.

Examples

 data(fruit)
 table(fruit$cultivar)

Accessor methods to the essential slots of Cov and its subclasses

Description

Accessor methods to the slots of objects of classCov and its subclasses

Usage

getCenter(obj)
getCov(obj)
getCorr(obj)
getData(obj)
getDistance(obj)
getEvals(obj)
getDet(obj)
getShape(obj)
getFlag(obj, prob=0.975)
getMeth(obj)
isClassic(obj)
getRaw(obj)

Arguments

Methods

obj = "Cov"

generic functions - see getCenter, getCov, getCorr, getData, getDistance, getEvals, getDet, getShape, getFlag, isClassic

obj = "CovRobust"

generic functions - see getCenter, getCov, getCorr, getData, getDistance, getEvals, getDet, getShape, getFlag, getMeth, isClassic

Calculates the points for drawing a confidence ellipsoid

Description

A simple function to calculate the points of a confidence ellipsoid, by default dist=qchisq(0.975, 2)

Usage

getEllipse(loc = c(0, 0), cov = matrix(c(1, 0, 0, 1), ncol = 2), crit = 0.975)

Arguments

Value

A matrix with two columns containing the calculated points.

Author(s)

Valentin Todorov, valentin.todorov@chello.at

Examples

data(hbk)
cc <- cov.wt(hbk)
e1 <- getEllipse(loc=cc$center[1:2], cov=cc$cov[1:2,1:2])
e2 <- getEllipse(loc=cc$center[1:2], cov=cc$cov[1:2,1:2], crit=0.99)
plot(X2~X1, data=hbk,
    xlim=c(min(X1, e1[,1], e2[,1]), max(X1,e1[,1], e2[,1])),
    ylim=c(min(X2, e1[,2], e2[,2]), max(X2,e1[,2], e2[,2])))
lines(e1, type="l", lty=1, col="red")
lines(e2, type="l", lty=2, col="blue")
legend("topleft", legend=c(0.975, 0.99), lty=1:2, col=c("red", "blue"))

Accessor methods to the essential slots of Pca and its subclasses

Description

Accessor methods to the slots of objects of class Pca and its subclasses

Arguments

Methods

obj = "Pca"

Accessors for object of class Pca

obj = "PcaRobust"

Accessors for object of class PcaRobust

obj = "PcaClassic"

Accessors for object of class PcaClassic

Hemophilia Data

Description

The hemophilia data set contains two measured variables on 75 women, belonging to two groups: n1=30 of them are non-carriers (normal group) and n2=45 are known hemophilia A carriers (obligatory carriers).

Usage

data(hemophilia)

Format

A data frame with 75 observations on the following 3 variables.

AHFactivity

AHF activity

AHFantigen

AHF antigen

gr

group - normal or obligatory carrier

Details

Originally analized in the context of discriminant analysis by Habemma and Hermans (1974). The objective is to find a procedure for detecting potential hemophilia A carriers on the basis of two measured variables: X1=log10(AHV activity) and X2=log10(AHV-like antigen). The first group of n1=30 women consists of known non-carriers (normal group) and the second group of n2=45 women is selected from known hemophilia A carriers (obligatory carriers). This data set was also analyzed by Johnson and Wichern (1998) as well as, in the context of robust Linear Discriminant Analysis by Hawkins and McLachlan (1997) and Hubert and Van Driessen (2004).

Source

Habemma, J.D.F, Hermans, J. and van den Broek, K. (1974) Stepwise Discriminant Analysis Program Using Density Estimation in Proceedings in Computational statistics, COMPSTAT'1974 (Physica Verlag, Heidelberg, 1974, pp 101–110).

References

Johnson, R.A. and Wichern, D. W. Applied Multivariate Statistical Analysis (Prentice Hall, International Editions, 2002, fifth edition)

Hawkins, D. M. and McLachlan, G.J. (1997) High-Breakdown Linear Discriminant Analysis J. Amer. Statist. Assoc. 92 136–143.

Hubert, M., Van Driessen, K. (2004) Fast and robust discriminant analysis, Computational Statistics and Data Analysis, 45 301–320.

Examples

data(hemophilia)
plot(AHFantigen~AHFactivity, data=hemophilia, col=as.numeric(as.factor(gr))+1)
##
## Compute robust location and covariance matrix and 
## plot the tolerance ellipses
(mcd <- CovMcd(hemophilia[,1:2]))
col <- ifelse(hemophilia$gr == "carrier", 2, 3) ## define clours for the groups
plot(mcd, which="tolEllipsePlot", class=TRUE, col=col)

Johns Hopkins University Ionosphere database.

Description

”This radar data was collected by a system in Goose Bay, Labrador. This system consists of a phased array of 16 high-frequency antennas with a total transmitted power on the order of 6.4 kilowatts. The targets were free electrons in the ionosphere. "good" radar returns are those showing evidence of some type of structure in the ionosphere. "bad" returns are those that do not; their signals pass through the ionosphere. Received signals were processed using an autocorrelation function whose arguments are the time of a pulse and the pulse number. There were 17 described by 2 attributes per pulse number, corresponding to the complex values returned by the function resulting from the complex electromagnetic signal.” [UCI archive]

Usage

data(ionosphere)

Format

A data frame with 351 rows and 33 variables: 32 measurements and one (the last, Class) grouping variable: 225 'good' and 126 'bad'.

The original dataset at UCI contains 351 rows and 35 columns. The first 34 columns are features, the last column contains the classification label of 'g' and 'b'. The first feature is binary and the second one is only 0s, one grouping variable - factor with labels 'good' and 'bad'.

Source

Source: Space Physics Group; Applied Physics Laboratory; Johns Hopkins University; Johns Hopkins Road; Laurel; MD 20723

Donor: Vince Sigillito (vgs@aplcen.apl.jhu.edu)

The data have been taken from the UCI Repository Of Machine Learning Databases at https://archive.ics.uci.edu/ml/datasets/ionosphere

This data set, with the original 34 features is available in the package mlbench and a different data set (refering to the same UCI repository) is available in the package dprep (archived on CRAN).

References

Sigillito, V. G., Wing, S. P., Hutton, L. V., and Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.

Examples

 data(ionosphere)
 ionosphere[, 1:6] |> pairs()

Check if a covariance matrix (object of class 'Cov') is singular

Description

Returns TRUE if the covariance matrix contained in a Cov-class object (or derived from) is singular.

Usage

    ## S4 method for signature 'Cov'
isSingular(obj)

Arguments

Methods

isSingular

signature(x = Cov): Check if a covariance matrix (object of class Cov-class) is singular.

See Also

Cov-class, CovClassic, CovRobust-class.

Examples

data(hbk)
cc <- CovClassic(hbk)
isSingular(cc)

Class "Lda" - virtual base class for all classic and robust LDA classes

Description

The class Lda serves as a base class for deriving all other classes representing the results of classical and robust Linear Discriminant Analisys methods

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

call:

the (matched) function call.

prior:

prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the common covariance matrix

ldf:

a matrix containing the linear discriminant functions

ldfconst:

a vector containing the constants of each linear discriminant function

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

covobj:

object of class "Cov" containing the estimate of the common covariance matrix of the centered data. It is not NULL only in case of method "B".

control:

object of class "CovControl" specifying which estimate and with what estimation options to use for the group means and common covariance (or NULL for classical linear discriminant analysis)

Methods

predict

signature(object = "Lda"): calculates prediction using the results in object. An optional data frame or matrix in which to look for variables with which to predict. If omitted, the training data set is used. If the original fit used a formula or a data frame or a matrix with column names, newdata must contain columns with the same names. Otherwise it must contain the same number of columns, to be used in the same order.

show

signature(object = "Lda"): prints the results

summary

signature(object = "Lda"): prints summary information

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

LdaClassic, LdaClassic-class, LdaRobust-class

Examples

showClass("Lda")

Linear Discriminant Analysis

Description

Performs a linear discriminant analysis and returns the results as an object of class LdaClassic (aka constructor).

Usage

LdaClassic(x, ...)

## Default S3 method:
LdaClassic(x, grouping, prior = proportions, tol = 1.0e-4, ...)

Arguments

Value

Returns an S4 object of class LdaClassic

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Lda-class, LdaClassic-class,

Examples

## Example anorexia
library(MASS)
data(anorexia)

## rrcov: LdaClassic()
lda <- LdaClassic(Treat~., data=anorexia)
predict(lda)@classification

## MASS: lda()
lda.MASS <- lda(Treat~., data=anorexia)
predict(lda.MASS)$class

## Compare the prediction results of MASS:::lda() and LdaClassic()
all.equal(predict(lda)@classification, predict(lda.MASS)$class)

Class "LdaClassic" - Linear Discriminant Analysis

Description

Contains the results of a classical Linear Discriminant Analysis

Objects from the Class

Objects can be created by calls of the form new("LdaClassic", ...) but the usual way of creating LdaClassic objects is a call to the function LdaClassic which serves as a constructor.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the common covariance matrix

ldf:

a matrix containing the linear discriminant functions

ldfconst:

a vector containing the constants of each linear discriminant function

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

Extends

Class "Lda", directly.

Methods

No methods defined with class "LdaClassic" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

LdaRobust-class, Lda-class, LdaClassic

Examples

showClass("LdaClassic")

Robust Linear Discriminant Analysis by Projection Pursuit

Description

Performs robust linear discriminant analysis by the projection-pursuit approach - proposed by Pires and Branco (2010) - and returns the results as an object of class LdaPP (aka constructor).

Usage

LdaPP(x, ...)
## S3 method for class 'formula'
LdaPP(formula, data, subset, na.action, ...)
## Default S3 method:
LdaPP(x, grouping, prior = proportions, tol = 1.0e-4,
                 method = c("huber", "mad", "sest", "class"),
                 optim = FALSE,
                 trace=FALSE, ...)

Arguments

Details

Currently the algorithm is implemented only for binary classification and in the following will be assumed that only two groups are present.

The PP algorithm searches for low-dimensional projections of higher-dimensional data where a projection index is maximized. Similar to the original Fisher's proposal the squared standardized distance between the observations in the two groups is maximized. Instead of the sample univariate mean and standard deviation (T,S) robust alternatives are used. These are selected through the argument method and can be one of

huber

the pair (T,S) are the robust M-estimates of location and scale

mad

(T,S) are the Median and the Median Absolute Deviation

sest

the pair (T,S) are the robust S-estimates of location and scale

class

(T,S) are the mean and the standard deviation.

The first approximation A1 to the solution is obtained by investigating a finite number of candidate directions, the unit vectors defined by all pairs of points such that one belongs to the first group and the other to the second group. The found solution is stored in the slots raw.ldf and raw.ldfconst.

The second approximation A2 (optional) is performed by a numerical optimization algorithm using A1 as initial solution. The Nelder and Mead method implemented in the function optim is applied. Whether this refinement will be used is controlled by the argument optim. If optim=TRUE the result of the optimization is stored into the slots ldf and ldfconst. Otherwise these slots are set equal to raw.ldf and raw.ldfconst.

Value

Returns an S4 object of class LdaPP-class

Warning

Still an experimental version! Only binary classification is supported.

Author(s)

Valentin Todorov valentin.todorov@chello.at and Ana Pires apires@math.ist.utl.pt

References

Pires, A. M. and A. Branco, J. (2010) Projection-pursuit approach to robust linear discriminant analysis Journal Multivariate Analysis, Academic Press, Inc., 101, 2464–2485.

See Also

Linda, LdaClassic

Examples

##
## Function to plot a LDA separation line
##
lda.line <- function(lda, ...)
{
    ab <- lda@ldf[1,] - lda@ldf[2,]
    cc <- lda@ldfconst[1] - lda@ldfconst[2]
    abline(a=-cc/ab[2], b=-ab[1]/ab[2],...)
}

data(pottery)
x <- pottery[,c("MG", "CA")]
grp <- pottery$origin
col <- c(3,4)
gcol <- ifelse(grp == "Attic", col[1], col[2])
gpch <- ifelse(grp == "Attic", 16, 1)

##
## Reproduce Fig. 2. from Pires and branco (2010)
##
plot(CA~MG, data=pottery, col=gcol, pch=gpch)

## Not run: 

ppc <- LdaPP(x, grp, method="class", optim=TRUE)
lda.line(ppc, col=1, lwd=2, lty=1)

pph <- LdaPP(x, grp, method="huber",optim=TRUE)
lda.line(pph, col=3, lty=3)

pps <- LdaPP(x, grp, method="sest", optim=TRUE)
lda.line(pps, col=4, lty=4)

ppm <- LdaPP(x, grp, method="mad", optim=TRUE)
lda.line(ppm, col=5, lty=5)

rlda <- Linda(x, grp, method="mcd")
lda.line(rlda, col=6, lty=1)

fsa <- Linda(x, grp, method="fsa")
lda.line(fsa, col=8, lty=6)

## Use the formula interface:
##
LdaPP(origin~MG+CA, data=pottery)       ## use the same two predictors
LdaPP(origin~., data=pottery)           ## use all predictor variables

##
## Predict method
data(pottery)
fit <- LdaPP(origin~., data = pottery)
predict(fit)

## End(Not run)

Class "LdaPP" - Robust method for Linear Discriminant Analysis by Projection-pursuit

Description

The class LdaPP represents an algorithm for robust linear discriminant analysis by projection-pursuit approach. The objects of class LdaPP contain the results of the robust linear discriminant analysis by projection-pursuit approach.

Objects from the Class

Objects can be created by calls of the form new("LdaPP", ...) but the usual way of creating LdaPP objects is a call to the function LdaPP which serves as a constructor.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the common covariance matrix

raw.ldf:

a matrix containing the raw linear discriminant functions - see Details in LdaPP

raw.ldfconst:

a vector containing the raw constants of each raw linear discriminant function - see Details in LdaPP

ldf:

a matrix containing the linear discriminant functions

ldfconst:

a vector containing the constants of each linear discriminant function

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

Extends

Class "LdaRobust", directly. Class "Lda", by class "LdaRobust", distance 2.

Methods

predict

signature(object = "LdaPP"): calculates prediction using the results in object. An optional data frame or matrix in which to look for variables with which to predict. If omitted, the training data set is used. If the original fit used a formula or a data frame or a matrix with column names, newdata must contain columns with the same names. Otherwise it must contain the same number of columns, to be used in the same order. If the argument raw=TRUE is set the raw (obtained by the first approximation algorithm) linear discriminant function and constant will be used.

Author(s)

Valentin Todorov valentin.todorov@chello.at and Ana Pires apires@math.ist.utl.pt

References

Pires, A. M. and A. Branco, J. (2010) Projection-pursuit approach to robust linear discriminant analysis Journal Multivariate Analysis, Academic Press, Inc., 101, 2464–2485.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

LdaRobust-class, Lda-class, LdaClassic, LdaClassic-class, Linda, Linda-class

Examples

showClass("LdaPP")

Class "LdaRobust" is a virtual base class for all robust LDA classes

Description

The class LdaRobust searves as a base class for deriving all other classes representing the results of the robust Linear Discriminant Analysis methods

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the common covariance matrix

ldf:

a matrix containing the linear discriminant functions

ldfconst:

a vector containing the constants of each linear discriminant function

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

Extends

Class "Lda", directly.

Methods

No methods defined with class "LdaRobust" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Lda-class, LdaClassic-class,

Examples

showClass("LdaRobust")

Robust Linear Discriminant Analysis

Description

Robust linear discriminant analysis based on MCD and returns the results as an object of class Linda (aka constructor).

Usage

Linda(x, ...)

## Default S3 method:
Linda(x, grouping, prior = proportions, tol = 1.0e-4,
                 method = c("mcd", "mcdA", "mcdB", "mcdC", "fsa", "mrcd", "ogk"),
                 alpha=0.5, l1med=FALSE, cov.control, trace=FALSE, ...)

Arguments

.

Details

details

Value

Returns an S4 object of class Linda

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Hawkins, D.M. and McLachlan, G.J. (1997) High-Breakdown Linear Discriminant Analysis, Journal of the American Statistical Association, 92, 136–143.

Todorov V. (2007) Robust selection of variables in linear discriminant analysis, Statistical Methods and Applications, 15, 395–407, doi:10.1007/s10260-006-0032-6.

Todorov, V. and Pires, A.M. (2007) Comparative Performance of Several Robust Linear Discriminant Analysis Methods. REVSTAT Statistical Journal, 5, p 63–83.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMcd, CovMrcd

Examples

## Example anorexia
library(MASS)
data(anorexia)

## start with the classical estimates
lda <- LdaClassic(Treat~., data=anorexia)
predict(lda)@classification

## try now the robust LDA with the default method (MCD with pooled whitin cov matrix)
rlda <- Linda(Treat~., data= anorexia)
predict(rlda)@classification

## try the other methods
Linda(Treat~., data= anorexia, method="mcdA")
Linda(Treat~., data= anorexia, method="mcdB")
Linda(Treat~., data= anorexia, method="mcdC")

## try the Hawkins&McLachlan method
## use the default method
grp <- anorexia[,1]
grp <- as.factor(grp)
x <- anorexia[,2:3]
Linda(x, grp, method="fsa")

## Do DA with Linda and method mcdB or mcdC, when some classes
## have very few observations. Use L1 median instead of MCD
##  to compute the group means (l1med=TRUE).

data(fish)

# remove observation #14 containing missing value
fish <- fish[-14,]

# The height and width are calculated as percentages 
#   of the third length variable
fish[,5] <- fish[,5]*fish[,4]/100
fish[,6] <- fish[,6]*fish[,4]/100

table(fish$Species) 
Linda(Species~., data=fish, l1med=TRUE)
Linda(Species~., data=fish, method="mcdC", l1med=TRUE)

Class "Linda" - Robust method for LINear Discriminant Analysis

Description

Robust linear discriminant analysis is performed by replacing the classical group means and withing group covariance matrix by robust equivalents based on MCD.

Objects from the Class

Objects can be created by calls of the form new("Linda", ...) but the usual way of creating Linda objects is a call to the function Linda which serves as a constructor.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the common covariance matrix

ldf:

a matrix containing the linear discriminant functions

ldfconst:

a vector containing the constants of each linear discriminant function

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

l1med:

wheather L1 median was used to compute group means.

Extends

Class "LdaRobust", directly. Class "Lda", by class "LdaRobust", distance 2.

Methods

No methods defined with class "Linda" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

LdaRobust-class, Lda-class, LdaClassic, LdaClassic-class

Examples

showClass("Linda")

Hosking and Wallis Data Set, Table 3.2

Description

The data on annual maximum streamflow at 18 sites with smallest drainage area basin in southeastern USA contains the sample L-moments ratios (L-CV, L-skewness and L-kurtosis) as used by Hosking and Wallis (1997) to illustrate the discordancy measure in regional freqency analysis (RFA).

Usage

data(lmom32)

Format

A data frame with 18 observations on the following 3 variables.

L-CV

L-coefficient of variation

L-skewness

L-coefficient of skewness

L-kurtosis

L-coefficient of kurtosis

Details

The sample L-moment ratios (L-CV, L-skewness and L-kurtosis) of a site are regarded as a point in three dimensional space.

Source

Hosking, J. R. M. and J. R. Wallis (1997), Regional Frequency Analysis: An Approach Based on L-moments. Cambridge University Press, p.49, Table 3.2

References

Neykov, N.M., Neytchev, P.N., Van Gelder, P.H.A.J.M. and Todorov V. (2007), Robust detection of discordant sites in regional frequency analysis, Water Resources Research, 43, W06417, doi:10.1029/2006WR005322

Examples

    data(lmom32)

    # plot a matrix of scatterplots
    pairs(lmom32,
          main="Hosking and Wallis Data Set, Table 3.3",
          pch=21,
          bg=c("red", "green3", "blue"))

    mcd<-CovMcd(lmom32)
    mcd
    plot(mcd, which="dist", class=TRUE)
    plot(mcd, which="dd", class=TRUE)

    ##  identify the discordant sites using robust distances and compare 
    ##  to the classical ones
    mcd <- CovMcd(lmom32)
    rd <- sqrt(getDistance(mcd))
    ccov <- CovClassic(lmom32)
    cd <- sqrt(getDistance(ccov))
    r.out <- which(rd > sqrt(qchisq(0.975,3)))
    c.out <- which(cd > sqrt(qchisq(0.975,3)))
    cat("Robust: ", length(r.out), " outliers: ", r.out,"\n")
    cat("Classical: ", length(c.out), " outliers: ", c.out,"\n")

Hosking and Wallis Data Set, Table 3.3

Description

The data on annual maximum streamflow at 17 sites with largest drainage area basins in southeastern USA contains the sample L-moments ratios (L-CV, L-skewness and L-kurtosis) as used by Hosking and Wallis (1997) to illustrate the discordancy measure in regional freqency analysis (RFA).

Usage

data(lmom33)

Format

A data frame with 17 observations on the following 3 variables.

L-CV

L-coefficient of variation

L-skewness

L-coefficient of skewness

L-kurtosis

L-coefficient of kurtosis

Details

The sample L-moment ratios (L-CV, L-skewness and L-kurtosis) of a site are regarded as a point in three dimensional space.

Source

Hosking, J. R. M. and J. R. Wallis (1997), Regional Frequency Analysis: An Approach Based on L-moments. Cambridge University Press, p.51, Table 3.3

References

Neykov, N.M., Neytchev, P.N., Van Gelder, P.H.A.J.M. and Todorov V. (2007), Robust detection of discordant sites in regional frequency analysis, Water Resources Research, 43, W06417, doi:10.1029/2006WR005322

Examples

    data(lmom33)

    # plot a matrix of scatterplots
    pairs(lmom33,
          main="Hosking and Wallis Data Set, Table 3.3",
          pch=21,
          bg=c("red", "green3", "blue"))

    mcd<-CovMcd(lmom33)
    mcd
    plot(mcd, which="dist", class=TRUE)
    plot(mcd, which="dd", class=TRUE)

    ##  identify the discordant sites using robust distances and compare 
    ##  to the classical ones
    mcd <- CovMcd(lmom33)
    rd <- sqrt(getDistance(mcd))
    ccov <- CovClassic(lmom33)
    cd <- sqrt(getDistance(ccov))
    r.out <- which(rd > sqrt(qchisq(0.975,3)))
    c.out <- which(cd > sqrt(qchisq(0.975,3)))
    cat("Robust: ", length(r.out), " outliers: ", r.out,"\n")
    cat("Classical: ", length(c.out), " outliers: ", c.out,"\n")

Computer Hardware

Description

A data set containing relative CPU performance data of 209 machines on 8 variables. are predictive, one (PRP) is the goal field and one (ERP) is the linear regression's guess. The estimated relative performance values were estimated by the authors using a linear regression method. See their article (Ein-Dor and Feldmesser, CACM 4/87, pp 308-317) for more details on how the relative performance values were set.

Usage

data(machines)

Format

A data frame with 209 rows and 8 variables The variables are as follows:

• MMIN: minimum main memory in kilobytes (integer)

• MMAX: maximum main memory in kilobytes (integer)

• CACH: cache memory in kilobytes (integer)

• CHMIN: minimum channels in units (integer)

• CHMAX: maximum channels in units (integer)

• PRP: published relative performance (integer)

• ERP: estimated relative performance from the original article (integer)

Source

UCI Archive

References

Phillip Ein-Dor and Jacob Feldmesser (1987), Attributes of the performance of central processing units: A relative performance prediction model, Communications of the ACM, 30, 4, pp 308-317.

M. Hubert, P. J. Rousseeuw and T. Verdonck (2009), Robust PCA for skewed data and its outlier map, Computational Statistics & Data Analysis, 53, 2264–2274.

Examples

 data(machines)

 ## Compute the medcouple of each variable of the Computer hardware data
     data.frame(MC=round(apply(machines, 2, mc),2))

 ## Plot a pairwise scaterplot matrix
     pairs(machines[,1:6])

     mcd <- CovMcd(machines[,1:6])
     plot(mcd, which="pairs")

 ##  Remove the rownames (too long)
     rownames(machines) <- NULL

 ## Start with robust PCA based on MCD (P << n)
     (pca1 <- PcaHubert(machines, k=3))
     plot(pca1, main="ROBPCA-MCD", off=0.03)

 ## PCA with the projection algoritm of Hubert
     (pca2 <- PcaHubert(machines, k=3, mcd=FALSE))
     plot(pca2, main="ROBPCA-SD", off=0.03)

 ## PCA with the adjusted for skewness algorithm of Hubert et al (2009)
     (pca3 <- PcaHubert(machines, k=3, mcd=FALSE, skew=TRUE))
     plot(pca3, main="ROBPCA-AO", off=0.03)

Marona and Yohai Artificial Data

Description

Simple artificial data set generated according the example by Marona and Yohai (1998). The data set consists of 20 bivariate normal observations generated with zero means, unit variances and correlation 0.8. The sample correlation is 0.81. Two outliers are introduced (i.e. these are 10% of the data) in the following way: two points are modified by interchanging the largest (observation 19) and smallest (observation 9) value of the first coordinate. The sample correlation becomes 0.05. This example provides a good example of the fact that a multivariate outlier need not be an outlier in any of its coordinate variables.

Usage

data(maryo)

Format

A data frame with 20 observations on 2 variables. To introduce the outliers x[9,1] with x[19,1] are interchanged.

Source

R. A. Marona and V. J. Yohai (1998) Robust estimation of multivariate location and scatter. In Encyclopedia of Statistical Sciences, Updated Volume 2 (Eds. S.Kotz, C.Read and D.Banks). Wiley, New York p. 590

Examples

data(maryo)
getCorr(CovClassic(maryo))          ## the sample correlation is 0.81

## Modify 10%% of the data in the following way:
##  modify two points (out of 20) by interchanging the 
##  largest and smallest value of the first coordinate
imin <- which(maryo[,1]==min(maryo[,1]))        # imin = 9
imax <- which(maryo[,1]==max(maryo[,1]))        # imax = 19
maryo1 <- maryo
maryo1[imin,1] <- maryo[imax,1]
maryo1[imax,1] <- maryo[imin,1]

##  The sample correlation becomes 0.05
plot(maryo1)
getCorr(CovClassic(maryo1))         ## the sample correlation becomes 0.05
getCorr(CovMcd(maryo1))      ## the (reweighted) MCD correlation is 0.79

Octane data

Description

The octane data contains near infrared absorbance spectra (NIR) of n=39 gasoline samples over p=226 wavelengths ranging from 1102 nm to 1552 nm with measurements every two nanometers. For each of the 39 production gasoline samples the octane number y was measured. Six of the samples (25, 26, and 36-39) contain added alcohol.

Usage

data(octane)

Format

A data frame with 39 observations and 227 columns, the wavelengts are by column and the first variable is the dependent one (y).

Source

K.H. Esbensen, S. Schoenkopf and T. Midtgaard Multivariate Analysis in Practice, Trondheim, Norway: Camo, 1994.

References

M. Hubert, P. J. Rousseeuw, K. Vanden Branden (2005), ROBPCA: a new approach to robust principal components analysis, Technometrics, 47, 64–79.

P. J. Rousseeuw, M. Debruyne, S. Engelen and M. Hubert (2006), Robustness and Outlier Detection in Chemometrics, Critical Reviews in Analytical Chemistry, 36(3–4), 221–242.

Examples

data(octane)

octane <- octane[, -1]    # remove the dependent variable y

pca=PcaHubert(octane, k=10)
screeplot(pca, type="lines")

pca2 <- PcaHubert(octane, k=2)
plot(pca2, id.n.sd=6)

pca7 <- PcaHubert(octane, k=7)
plot(pca7, id.n.sd=6)

Olive Oil Data

Description

This dataset consists of 120 olive oil samples on measurements on 25 chemical compositions (fatty acids, sterols, triterpenic alcohols) of olive oils from Tuscany, Italy (Armanino et al. 1989). There are 4 classes corresponding to different production areas. Class 1, Class 2, Class 3, and Class 4 contain 50, 25, 34, and 11 observations, respectively.

Usage

data(olitos)

Format

A data frame with 120 observations on the following 26 variables.

X1

Free fatty acids

X2

Refractive index

X3

K268

X4

delta K

X5

Palmitic acid

X6

Palmitoleic acid

X7

a numeric vector

X8

a numeric vector

X9

a numeric vector

X10

a numeric vector

X11

a numeric vector

X12

a numeric vector

X13

a numeric vector

X14

a numeric vector

X15

a numeric vector

X16

a numeric vector

X17

a numeric vector

X18

a numeric vector

X19

a numeric vector

X20

a numeric vector

X21

a numeric vector

X22

a numeric vector

X23

a numeric vector

X24

a numeric vector

X25

a numeric vector

grp

a factor with levels 1 2 3 4

Source

Prof. Roberto Todeschini, Milano Chemometrics and QSAR Research Group https://michem.unimib.it

References

C. Armanino, R. Leardi, S. Lanteri and G. Modi, 1989. Chemometric analysis of Tuscan olive oils. Cbemometrics and Intelligent Laboratoty Sysiem, 5: 343–354.

R. Todeschini, V. Consonni, A. Mauri, M. Pavan (2004) Software for the calculation of molecular descriptors. Pavan M. Talete slr, Milan, Italy, http://www.talete.mi.it

Examples

data(olitos)
cc <- Linda(grp~., data=olitos, method="mcdC", l1med=TRUE)
cc
pr <- predict(cc)
tt <- mtxconfusion(cc@grp, pr@classification, printit=TRUE)

Oslo Transect Data

Description

The oslo Transect data set contains 360 samples of different plant species collected along a 120 km transect running through the city of Oslo, Norway.

Usage

data(OsloTransect)

Format

A data frame with 360 observations on the following 38 variables.

X.ID

a numeric vector, unique ID of the sample

X.MAT

a factor with levels BBA BIL BWO FER MOS ROG SNE STW TWI

XCOO

a numeric vector, X coordinate

YCOO

a numeric vector, Y coordinate

XCOO_km

a numeric vector

YCOO_km

a numeric vector

X.FOREST

a factor with levels BIRSPR MIXDEC PINE SPRBIR SPRPIN SPRUCE

DAY

a numeric vector

X.WEATHER

a factor with levels CLOUD MOIST NICE RAIN

ALT

a numeric vector

X.ASP

a factor with levels E FLAT N NE NW S SE SW W

X.GRVEG

a factor with levels BLGR BLLY BLMOLI BLUE BLUGRA GRAS GRBLU GRFE GRMO LYLI MIX MOGR MOSS

X.FLITHO

a factor with levels CAMSED GNEID_O GNEIS_O GNEIS_R MAGM MICSH

Ag_ppb

a numeric vector

As_ash

a numeric vector

B

a numeric vector

Ba

a numeric vector

Ca

a numeric vector

Cd

a numeric vector

Co

a numeric vector

Cr

a numeric vector

Cu

a numeric vector

Fe

a numeric vector

Hg_ppb

a numeric vector

K

a numeric vector

La

a numeric vector

LOI

a numeric vector

Mg

a numeric vector

Mn

a numeric vector

Mo

a numeric vector

Ni

a numeric vector

P

a numeric vector

Pb

a numeric vector

S

a numeric vector

Sb

a numeric vector

Sr

a numeric vector

Ti

a numeric vector

Zn

a numeric vector

Details

Samples of different plant species were collected along a 120 km transect running through the city of Oslo, Norway (forty samples each of leaves, needles,roots or barks of several plant species), and the concentrations of 25 chemical elements for the sample materials are reported. The factors that influenced the observed element concentrations in the sample materials were investigated. This data set was used in Todorov and Filzmoser (2007) for illustration of the robust statistics for one-way MANOVA implemented in the function Wilks.test.

Source

REIMANN,C., ARNOLDUSSEN,A., BOYD,R., FINNE,T.E., NORDGULEN,Oe., VOLDEN,T. and ENGLMAIER,P. (2006) The Influence of a city on element contents of a terrestrial moss (Hylocomium splendens), The Science of the Total Environment 369 419–432.

REIMANN,C., ARNOLDUSSEN,A., BOYD,R., FINNE,T.E., KOLLER,F., NORDGULEN,Oe., and ENGLMAIER,P. (2007) Element contents in leaves of four plant species (birch, mountain ash, fern and spruce) along anthropogenic and geogenic concentration gradients, The Science of the Total Environment 377 416–433.

REIMANN,C., ARNOLDUSSEN,A., FINNE,T.E., KOLLER,F., NORDGULEN,Oe., and ENGLMAIER,P., (2007) Element contents in birch leaves, bark and wood under different anthropogenic and geogenic conditions, Applied Geochemistry, 22 1549–1566.

References

Todorov V. and Filzmoser P. (2007) Robust statistic for the one-way MANOVA, submetted to the Journal of Environmetrics.

Examples

data(OsloTransect)
str(OsloTransect)

##
##  Log-transform the numerical part of the data, 
##  choose the desired groups and variables and 
##  perform the classical Wilks' Lambda test
##
OsloTransect[,14:38] <- log(OsloTransect[,14:38])
grp <- OsloTransect$X.FLITHO
ind <- which(grp =="CAMSED" | grp == "GNEIS_O" |
    grp == "GNEIS_R" | grp=="MAGM")
(cwl <- Wilks.test(X.FLITHO~K+P+Zn+Cu,data=OsloTransect[ind,]))

##
## Perform now the robust MCD based Wilks' Lambda test. 
##  Use the already computed multiplication factor 'xd' and 
##  degrees of freedom 'xq' for the approximate distribution.
##

xd <- -0.003708238
xq <- 11.79073
(mcdwl <- Wilks.test(X.FLITHO~K+P+Zn+Cu,data=OsloTransect[ind,], 
    method="mcd", xd=xd, xq=xq))

Class "Pca" - virtual base class for all classic and robust PCA classes

Description

The class Pca searves as a base class for deriving all other classes representing the results of the classical and robust Principal Component Analisys methods

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

call:

Object of class "language"

center:

Object of class "vector" the center of the data

scale:

Object of class "vector" the scaling applied to each variable of the data

rank:

Object of class "numeric" the rank of the data matrix

loadings:

Object of class "matrix" the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors)

eigenvalues:

Object of class "vector" the eigenvalues

scores:

Object of class "matrix" the scores - the value of the projected on the space of the principal components data (the centred (and scaled if requested) data multiplied by the loadings matrix) is returned. Hence, cov(scores) is the diagonal matrix diag(eigenvalues)

k:

Object of class "numeric" number of (choosen) principal components

sd:

Object of class "Uvector" Score distances within the robust PCA subspace

od:

Object of class "Uvector" Orthogonal distances to the robust PCA subspace

cutoff.sd:

Object of class "numeric" Cutoff value for the score distances

cutoff.od:

Object of class "numeric" Cutoff values for the orthogonal distances

flag:

Object of class "Uvector" The observations whose score distance is larger than cutoff.sd or whose orthogonal distance is larger than cutoff.od can be considered as outliers and receive a flag equal to zero. The regular observations receive a flag 1

crit.pca.distances

criterion to use for computing the cutoff values for the orthogonal and score distances. Default is 0.975.

n.obs:

Object of class "numeric" the number of observations

eig0:

Object of class "vector" all eigenvalues

totvar0:

Object of class "numeric" the total variance explained (=sum(eig0))

Methods

getCenter

signature(obj = "Pca"): center of the data

getScale

signature(obj = "Pca"): return the scaling applied to each variable

getEigenvalues

signature(obj = "Pca"): the eigenvalues of the covariance/correlation matrix, though the calculation is actually done with the singular values of the data matrix)

getLoadings

signature(obj = "Pca"): returns the matrix loadings (i.e., a matrix whose columns contain the eigenvectors). The function prcomp returns this matrix in the element rotation.

getPrcomp

signature(obj = "Pca"): returns an S3 object prcomp for compatibility with the functions prcomp() and princomp(). Thus the standard plots screeplot() and biplot() can be used

getScores

signature(obj = "Pca"): returns the rotated data (the centred (and scaled if requested) data multiplied by the loadings matrix).

getSdev

signature(obj = "Pca"): returns the standard deviations of the principal components (i.e., the square roots of the eigenvalues of the covariance/correlation matrix, though the calculation is actually done with the singular values of the data matrix)

plot

signature(x = "Pca"): produces a distance plot (if k=rank) or distance-distance plot (ifk<rank)

print

signature(x = "Pca"): prints the results. The difference to the show() method is that additional parametesr are possible.

show

signature(object = "Pca"): prints the results

predict

signature(object = "Pca"): calculates prediction using the results in object. An optional data frame or matrix in which to look for variables with which to predict. If omitted, the scores are used. If the original fit used a formula or a data frame or a matrix with column names, newdata must contain columns with the same names. Otherwise it must contain the same number of columns, to be used in the same order. See also predict.prcomp and predict.princomp

screeplot

signature(x = "Pca"): plots the variances against the number of the principal component. See also plot.prcomp and plot.princomp

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaClassic, PcaClassic-class, PcaRobust-class

Examples

showClass("Pca")

Compute score and orthogonal distances for Principal Components (objects of class 'Pca')

Description

Compute score and orthogonal distances for an object (derived from)Pca-class.

Usage

    pca.distances(obj, data, r, crit=0.975)

Arguments

Details

This function calculates the score and orthogonal distances and the appropriate cutoff values for identifying outlying observations. The computed values are used to create a vector a of flags, one for each observation, identifying the outliers.

Value

An S4 object of class derived from the virtual class Pca-class - the same object passed to the function, but with the score and orthogonal distances as well as their cutoff values and the corresponding flags appended to it.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

M. Hubert, P. J. Rousseeuw, K. Vanden Branden (2005), ROBPCA: a new approach to robust principal components analysis, Technometrics, 47, 64–79.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

## PCA of the Hawkins Bradu Kass's Artificial Data
##  using all 4 variables
data(hbk)
pca <- PcaHubert(hbk)
pca.distances(pca, hbk, rankMM(hbk))

Score plot for Principal Components (objects of class 'Pca')

Description

Produces a score plot from an object (derived from) Pca-class.

Usage

    pca.scoreplot(obj, i=1, j=2, main, id.n, ...)

Arguments

Author(s)

Valentin Todorov valentin.todorov@chello.at

See Also

Pca-class, PcaClassic, PcaRobust-class.

Examples

require(graphics)

## PCA of the Hawkins Bradu Kass's Artificial Data
##  using all 4 variables
data(hbk)
pca <- PcaHubert(hbk)
pca
pca.scoreplot(pca)

Principal Components Analysis

Description

Performs a principal components analysis and returns the results as an object of class PcaClassic (aka constructor).

Usage

PcaClassic(x, ...)
## Default S3 method:
PcaClassic(x, k = ncol(x), kmax = ncol(x), 
    scale=FALSE, signflip=TRUE, crit.pca.distances = 0.975, trace=FALSE, ...)
## S3 method for class 'formula'
PcaClassic(formula, data = NULL, subset, na.action, ...)

Arguments

Value

An S4 object of class PcaClassic-class which is a subclass of the virtual class Pca-class.

Note

This function can be seen as a wrapper arround prcomp() from stats which returns the results of the PCA in a class compatible with the object model for robust PCA.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Pca-class, PcaClassic-class,

Class "PcaClassic" - Principal Components Analysis

Description

Contains the results of a classical Principal Components Analysis

Objects from the Class

Objects can be created by calls of the form new("PcaClassic", ...) but the usual way of creating PcaClassic objects is a call to the function PcaClassic which serves as a constructor.

Slots

call:

Object of class "language"

center:

Object of class "vector" the center of the data

scale:

Object of class "vector" the scaling applied to each variable

rank:

Object of class "numeric" the rank of the data matrix

loadings:

Object of class "matrix" the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors)

eigenvalues:

Object of class "vector" the eigenvalues

scores:

Object of class "matrix" the scores - the value of the projected on the space of the principal components data (the centred (and scaled if requested) data multiplied by the loadings matrix) is returned. Hence, cov(scores) is the diagonal matrix diag(eigenvalues)

k:

Object of class "numeric" number of (choosen) principal components

sd:

Object of class "Uvector" Score distances within the robust PCA subspace

od:

Object of class "Uvector" Orthogonal distances to the robust PCA subspace

cutoff.sd:

Object of class "numeric" Cutoff value for the score distances

cutoff.od:

Object of class "numeric" Cutoff values for the orthogonal distances

flag:

Object of class "Uvector" The observations whose score distance is larger than cutoff.sd or whose orthogonal distance is larger than cutoff.od can be considered as outliers and receive a flag equal to zero. The regular observations receive a flag 1

n.obs:

Object of class "numeric" the number of observations

eig0:

Object of class "vector" all eigenvalues

totvar0:

Object of class "numeric" the total variance explained (=sum(eig0))

Extends

Class "Pca", directly.

Methods

getQuan

signature(obj = "PcaClassic"): returns the number of observations used in the computation, i.e. n.obs

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaRobust-class, Pca-class, PcaClassic

Examples

showClass("PcaClassic")

Robust PCA based on a robust covariance matrix

Description

Robust PCA are obtained by replacing the classical covariance matrix by a robust covariance estimator. This can be one of the available in rrcov estimators, i.e. MCD, OGK, M or S estimator.

Usage

PcaCov(x, ...)
## Default S3 method:
PcaCov(x, k = ncol(x), kmax = ncol(x), cov.control=CovControlMcd(), 
    scale = FALSE, signflip = TRUE, crit.pca.distances = 0.975, trace=FALSE, ...)
## S3 method for class 'formula'
PcaCov(formula, data = NULL, subset, na.action, ...)

Arguments

.

Details

PcaCov, serving as a constructor for objects of class PcaCov-class is a generic function with "formula" and "default" methods. For details see the relevant references.

Value

An S4 object of class PcaCov-class which is a subclass of the virtual class PcaRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

## PCA of the Hawkins Bradu Kass's Artificial Data
##  using all 4 variables
    data(hbk)
    pca <- PcaCov(hbk)
    pca

## Compare with the classical PCA
    prcomp(hbk)

## or  
    PcaClassic(hbk)
    
## If you want to print the scores too, use
    print(pca, print.x=TRUE)

## Using the formula interface
    PcaCov(~., data=hbk)

## To plot the results:

    plot(pca)                    # distance plot
    pca2 <- PcaCov(hbk, k=2)  
    plot(pca2)                   # PCA diagnostic plot (or outlier map)
    
## Use the standard plots available for for prcomp and princomp
    screeplot(pca)    
    biplot(pca)    

Class "PcaCov" - Robust PCA based on a robust covariance matrix

Description

Robust PCA are obtained by replacing the classical covariance matrix by a robust covariance estimator. This can be one of the available in rrcov estimators, i.e. MCD, OGK, M, S or Stahel-Donoho estimator.

Objects from the Class

Objects can be created by calls of the form new("PcaCov", ...) but the usual way of creating PcaCov objects is a call to the function PcaCov which serves as a constructor.

Slots

quan:

Object of class "numeric" The quantile h used throughout the algorithm

call, center, rank, loadings, eigenvalues, scores, k, sd, od, cutoff.sd, cutoff.od, flag, n.obs, eig0, totvar0:

from the "Pca" class.

Extends

Class "PcaRobust", directly. Class "Pca", by class "PcaRobust", distance 2.

Methods

getQuan

signature(obj = "PcaCov"): ...

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaRobust-class, Pca-class, PcaClassic, PcaClassic-class

Examples

showClass("PcaCov")

Robust Principal Components based on Projection Pursuit (PP): GRID search Algorithm

Description

Computes an approximation of the PP-estimators for PCA using the grid search algorithm in the plane.

Usage

    PcaGrid(x, ...)
    ## Default S3 method:
PcaGrid(x, k = 0, kmax = ncol(x), 
        scale=FALSE, na.action = na.fail, crit.pca.distances = 0.975, trace=FALSE, ...)
    ## S3 method for class 'formula'
PcaGrid(formula, data = NULL, subset, na.action, ...)

Arguments

Details

PcaGrid, serving as a constructor for objects of class PcaGrid-class is a generic function with "formula" and "default" methods. For details see PCAgrid and the relevant references.

Value

An S4 object of class PcaGrid-class which is a subclass of the virtual class PcaRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

C. Croux, P. Filzmoser, M. Oliveira, (2007). Algorithms for Projection-Pursuit Robust Principal Component Analysis, Chemometrics and Intelligent Laboratory Systems, 87, 225.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    # multivariate data with outliers
    library(mvtnorm)
    x <- rbind(rmvnorm(200, rep(0, 6), diag(c(5, rep(1,5)))),
                rmvnorm( 15, c(0, rep(20, 5)), diag(rep(1, 6))))
    # Here we calculate the principal components with PCAgrid
    pc <- PcaGrid(x, 6)
    # we could draw a biplot too:
    biplot(pc)
    
    # we could use another objective function, and 
    # maybe only calculate the first three principal components:
    pc <- PcaGrid(x, 3, method="qn")
    biplot(pc)
    
    # now we want to compare the results with the non-robust principal components
    pc <- PcaClassic(x, k=3)
    # again, a biplot for comparision:
    biplot(pc)

Class "PcaGrid" - Robust PCA using PP - GRID search Algorithm

Description

Holds the results of an approximation of the PP-estimators for PCA using the grid search algorithm in the plane.

Objects from the Class

Objects can be created by calls of the form new("PcaGrid", ...) but the usual way of creating PcaGrid objects is a call to the function PcaGrid() which serves as a constructor.

Slots

call, center, scale, rank, loadings, eigenvalues, scores, k, sd, od, cutoff.sd, cutoff.od, flag, n.obs:

from the "Pca" class.

Extends

Class "PcaRobust", directly. Class "Pca", by class "PcaRobust", distance 2.

Methods

getQuan

signature(obj = "PcaGrid"): ...

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaRobust-class, Pca-class, PcaClassic, PcaClassic-class

Examples

showClass("PcaGrid")

ROBPCA - ROBust method for Principal Components Analysis

Description

The ROBPCA algorithm was proposed by Hubert et al (2005) and stays for 'ROBust method for Principal Components Analysis'. It is resistant to outliers in the data. The robust loadings are computed using projection-pursuit techniques and the MCD method. Therefore ROBPCA can be applied to both low and high-dimensional data sets. In low dimensions, the MCD method is applied.

Usage

PcaHubert(x, ...)
## Default S3 method:
PcaHubert(x, k = 0, kmax = 10, alpha = 0.75, mcd = TRUE, skew=FALSE,
maxdir=250, scale = FALSE, signflip = TRUE, crit.pca.distances = 0.975, trace=FALSE, ...)
## S3 method for class 'formula'
PcaHubert(formula, data = NULL, subset, na.action, ...)

Arguments

Details

PcaHubert, serving as a constructor for objects of class PcaHubert-class is a generic function with "formula" and "default" methods. The calculation is done using the ROBPCA method of Hubert et al (2005) which can be described briefly as follows. For details see the relevant references.

Let n denote the number of observations, and p the number of original variables in the input data matrix X. The ROBPCA algorithm finds a robust center M (p x 1) of the data and a loading matrix P which is (p x k) dimensional. Its columns are orthogonal and define a new coordinate system. The scores T, an (n x k) matrix, are the coordinates of the centered observations with respect to the loadings:

T=(X-M)P

The ROBPCA algorithm also yields a robust covariance matrix (often singular) which can be computed as

S=PLP^t

where L is the diagonal matrix with the eigenvalues l_1, \dots, l_k.

This is done in the following three main steps:

Step 1: The data are preprocessed by reducing their data space to the subspace spanned by the n observations. This is done by singular value decomposition of the input data matrix. As a result the data are represented using at most n-1=rank(X) without loss of information.

Step 2: In this step for each data point a measure of outlyingness is computed. For this purpose the high-dimensional data points are projected on many univariate directions, each time the univariate MCD estimator of location and scale is computed and the standardized distance to the center is measured. The largest of these distances (over all considered directions) is the outlyingness measure of the data point. The h data points with smallest outlyingness measure are used to compute the covariance matrix \Sigma_h and to select the number k of principal components to retain. This is done by finding such k that l_k/l_1 >= 10.E-3 and \Sigma_{j=1}^k l_j/\Sigma_{j=1}^r l_j >= 0.8 Alternatively the number of principal components k can be specified by the user after inspecting the scree plot.

Step 3: The data points are projected on the k-dimensional subspace spanned by the k eigenvectors corresponding to the largest k eigenvalues of the matrix \Sigma_h. The location and scatter of the projected data are computed using the reweighted MCD estimator and the eigenvectors of this scatter matrix yield the robust principal components.

Value

An S4 object of class PcaHubert-class which is a subclass of the virtual class PcaRobust-class.

Note

The ROBPCA algorithm is implemented on the bases of the Matlab implementation, available as part of LIBRA, a Matlab Library for Robust Analysis to be found at www.wis.kuleuven.ac.be/stat/robust.html

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

M. Hubert, P. J. Rousseeuw, K. Vanden Branden (2005), ROBPCA: a new approach to robust principal components analysis, Technometrics, 47, 64–79.

M. Hubert, P. J. Rousseeuw and T. Verdonck (2009), Robust PCA for skewed data and its outlier map, Computational Statistics & Data Analysis, 53, 2264–2274.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

## PCA of the Hawkins Bradu Kass's Artificial Data
##  using all 4 variables
    data(hbk)
    pca <- PcaHubert(hbk)
    pca

## Compare with the classical PCA
    prcomp(hbk)

## or  
    PcaClassic(hbk)
    
## If you want to print the scores too, use
    print(pca, print.x=TRUE)

## Using the formula interface
    PcaHubert(~., data=hbk)

## To plot the results:

    plot(pca)                    # distance plot
    pca2 <- PcaHubert(hbk, k=2)  
    plot(pca2)                   # PCA diagnostic plot (or outlier map)
    
## Use the standard plots available for prcomp and princomp
    screeplot(pca)    
    biplot(pca)    
    
## Restore the covraiance matrix     
    py <- PcaHubert(hbk)
    cov.1 <- py@loadings %*% diag(py@eigenvalues) %*% t(py@loadings)
    cov.1    

Class "PcaHubert" - ROBust method for Principal Components Analysis

Description

The ROBPCA algorithm was proposed by Hubert et al (2005) and stays for 'ROBust method for Principal Components Analysis'. It is resistant to outliers in the data. The robust loadings are computed using projection-pursuit techniques and the MCD method. Therefore ROBPCA can be applied to both low and high-dimensional data sets. In low dimensions, the MCD method is applied.

Objects from the Class

Objects can be created by calls of the form new("PcaHubert", ...) but the usual way of creating PcaHubert objects is a call to the function PcaHubert which serves as a constructor.

Slots

alpha:

Object of class "numeric" the fraction of outliers the algorithm should resist - this is the argument alpha

quan:

The quantile h used throughout the algorithm

skew:

Whether the adjusted outlyingness algorithm for skewed data was used

ao:

Object of class "Uvector" Adjusted outlyingness within the robust PCA subspace

call, center, scale, rank, loadings, eigenvalues, scores, k, sd, od, cutoff.sd, cutoff.od, flag, n.obs, eig0, totvar0:

from the "Pca" class.

Extends

Class "PcaRobust", directly. Class "Pca", by class "PcaRobust", distance 2.

Methods

getQuan

signature(obj = "PcaHubert"): Returns the quantile used throughout the algorithm

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaRobust-class, Pca-class, PcaClassic, PcaClassic-class

Examples

showClass("PcaHubert")

Spherical Principal Components

Description

The Spherical Principal Components procedure was proposed by Locantore et al., (1999) as a functional data analysis method. The idea is to perform classical PCA on the data, projected onto a unit sphere. The estimates of the eigenvectors are consistent and the procedure is extremely fast. The simulations of Maronna (2005) show that this method has very good performance.

Usage

PcaLocantore(x, ...)
## Default S3 method:
PcaLocantore(x, k = ncol(x), kmax = ncol(x), delta = 0.001, 
    na.action = na.fail, scale = FALSE, signflip = TRUE, 
    crit.pca.distances = 0.975, trace=FALSE, ...)
## S3 method for class 'formula'
PcaLocantore(formula, data = NULL, subset, na.action, ...)

Arguments

Details

PcaLocantore, serving as a constructor for objects of class PcaLocantore-class is a generic function with "formula" and "default" methods. For details see the relevant references.

Value

An S4 object of class PcaLocantore-class which is a subclass of the virtual class PcaRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at The SPC algorithm is implemented on the bases of the available from the web site of the book Maronna et al. (2006) code https://www.wiley.com/legacy/wileychi/robust_statistics/

References

N. Locantore, J. Marron, D. Simpson, N. Tripoli, J. Zhang and K. Cohen K. (1999), Robust principal components for functional data. Test, 8, 1-28.

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

R. Maronna (2005). Principal components and orthogonal regression based on robust scales. Technometrics, 47, 264-273.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

## PCA of the Hawkins Bradu Kass's Artificial Data
##  using all 4 variables
    data(hbk)
    pca <- PcaLocantore(hbk)
    pca

## Compare with the classical PCA
    prcomp(hbk)

## or  
    PcaClassic(hbk)
    
## If you want to print the scores too, use
    print(pca, print.x=TRUE)

## Using the formula interface
    PcaLocantore(~., data=hbk)

## To plot the results:

    plot(pca)                    # distance plot
    pca2 <- PcaLocantore(hbk, k=2)  
    plot(pca2)                   # PCA diagnostic plot (or outlier map)
    
## Use the standard plots available for for prcomp and princomp
    screeplot(pca)    
    biplot(pca)    

Class "PcaLocantore" Spherical Principal Components

Description

The Spherical Principal Components procedure was proposed by Locantore et al., (1999) as a functional data analysis method. The idea is to perform classical PCA on the the data, \ projected onto a unit sphere. The estimates of the eigenvectors are consistent and the procedure is extremly fast. The simulations of Maronna (2005) show that this method has very good performance.

Objects from the Class

Objects can be created by calls of the form new("PcaLocantore", ...) but the usual way of creating PcaLocantore objects is a call to the function PcaLocantore which serves as a constructor.

Slots

delta:

Accuracy parameter

quan:

Object of class "numeric" The quantile h used throughout the algorithm

call, center, scale, rank, loadings, eigenvalues, scores, k, sd, od, cutoff.sd, cutoff.od, flag, n.obs, eig0, totvar0:

from the "Pca" class.

Extends

Class "PcaRobust", directly. Class "Pca", by class "PcaRobust", distance 2.

Methods

getQuan

signature(obj = "PcaLocantore"): ...

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaRobust-class, Pca-class, PcaClassic, PcaClassic-class

Examples

showClass("PcaLocantore")

Robust Principal Components based on Projection Pursuit (PP): Croux and Ruiz-Gazen (2005) algorithm

Description

A fast and simple algorithm for approximating the PP-estimators for PCA: Croux and Ruiz-Gazen (2005)

Usage

    PcaProj(x, ...)
    ## Default S3 method:
PcaProj(x, k = 0, kmax = ncol(x), scale=FALSE, 
        na.action = na.fail, crit.pca.distances = 0.975, trace=FALSE, ...)
    ## S3 method for class 'formula'
PcaProj(formula, data = NULL, subset, na.action, ...)

Arguments

Details

PcaProj, serving as a constructor for objects of class PcaProj-class is a generic function with "formula" and "default" methods. For details see PCAproj and the relevant references.

Value

An S4 object of class PcaProj-class which is a subclass of the virtual class PcaRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

C. Croux, A. Ruiz-Gazen (2005). High breakdown estimators for principal components: The projection-pursuit approach revisited, Journal of Multivariate Analysis, 95, 206–226.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

Examples

    # multivariate data with outliers
    library(mvtnorm)
    x <- rbind(rmvnorm(200, rep(0, 6), diag(c(5, rep(1,5)))),
                rmvnorm( 15, c(0, rep(20, 5)), diag(rep(1, 6))))
    # Here we calculate the principal components with PcaProj
    pc <- PcaProj(x, 6)
    # we could draw a biplot too:
    biplot(pc)

    # we could use another calculation method and another objective function, and
    # maybe only calculate the first three principal components:
    pc <- PcaProj(x, k=3, method="qn", CalcMethod="sphere")
    biplot(pc)

    # now we want to compare the results with the non-robust principal components
    pc <- PcaClassic(x, k=3)
    # again, a biplot for comparision:
    biplot(pc)

Class "PcaProj" - Robust PCA using PP - Croux and Ruiz-Gazen (2005) algorithm

Description

Holds the results of an approximation of the PP-estimators for PCA by a fast and simple algorithm: Croux and Ruiz-Gazen (2005) algorithm.

Objects from the Class

Objects can be created by calls of the form new("PcaProj", ...) but the usual way of creating PcaProj objects is a call to the function PcaProj() which serves as a constructor.

Slots

call, center, scale, rank, loadings, eigenvalues, scores, k, sd, od, cutoff.sd, cutoff.od, flag, n.obs:

from the "Pca" class.

Extends

Class "PcaRobust", directly. Class "Pca", by class "PcaRobust", distance 2.

Methods

getQuan

signature(obj = "PcaProj"): ...

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

PcaRobust-class, Pca-class, PcaClassic, PcaClassic-class

Examples

showClass("PcaProj")

Class "PcaRobust" is a virtual base class for all robust PCA classes

Description

The class PcaRobust searves as a base class for deriving all other classes representing the results of the robust Principal Component Analisys methods

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

call:

Object of class "language"

center:

Object of class "vector" the center of the data

loadings:

Object of class "matrix" the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors)

eigenvalues:

Object of class "vector" the eigenvalues

scores:

Object of class "matrix" the scores - the value of the projected on the space of the principal components data (the centred (and scaled if requested) data multiplied by the loadings matrix) is returned. Hence, cov(scores) is the diagonal matrix diag(eigenvalues)

k:

Object of class "numeric" number of (choosen) principal components

sd:

Object of class "Uvector" Score distances within the robust PCA subspace

od:

Object of class "Uvector" Orthogonal distances to the robust PCA subspace

cutoff.sd:

Object of class "numeric" Cutoff value for the score distances

cutoff.od:

Object of class "numeric" Cutoff values for the orthogonal distances

flag:

Object of class "Uvector" The observations whose score distance is larger than cutoff.sd or whose orthogonal distance is larger than cutoff.od can be considered as outliers and receive a flag equal to zero. The regular observations receive a flag 1

n.obs:

Object of class "numeric" the number of observations

Extends

Class "Pca", directly.

Methods

No methods defined with class "PcaRobust" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Pca-class, PcaClassic-class,

Examples

showClass("PcaRobust")

Methods for Function 'plot' in Package 'rrcov'

Description

Shows the Mahalanobis distances based on robust and/or classical estimates of the location and the covariance matrix in different plots. The following plots are available:

- index plot of the robust and mahalanobis distances

- distance-distance plot

- Chisquare QQ-plot of the robust and mahalanobis distances

- plot of the tolerance ellipses (robust and classic)

- Scree plot - Eigenvalues comparison plot

Usage

## S4 method for signature 'CovClassic'
plot(x, which = c("all","distance","qqchi2","tolellipse","screeplot"), 
        ask=(which=="all" && dev.interactive()), 
        cutoff, id.n, tol=1e-7, ...)
## S4 method for signature 'CovRobust'
plot(x, which = c("all","dd","distance","qqchi2","tolellipse","screeplot"), 
        classic=FALSE, ask=(which=="all" && dev.interactive()), 
        cutoff, id.n, tol=1e-7, ...)

Arguments

.

.

Methods

x = "Cov", y = "missing"

Plot mahalanobis distances for x.

x = "CovRobust", y = "missing"

Plot robust and classical mahalanobis distances for x.

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
cv <- CovClassic(hbk.x)
plot(cv)
rcv <- CovMest(hbk.x)
plot(rcv)

Archaic Greek Pottery data

Description

The Archaic Greek Pottery data set contains data on fragments of Greek pottery which were classified into two groups according to their origin: Attic or Eritrean. Six chemical variables, metallic oxide constituents, were measured: Si, Al, Fe, Ca and Ti. The main data set consists of 13 Attic objects and 14 Eritrean ones. There is a separate data set with 13 observations which can be used as a test data set. It consists of 4 observations classified as "probably Attic" and the remaining 9 as "probably Eritrean".

Usage

data(pottery)

Format

Two data frames with 27 an 13 observations on the following 7 variables.

SI

Si content

AL

Al content

FE

Fe content

MG

Mg content

CA

Ca content

TI

Ti content

origin

Origin - factor with two levels: Attic and Eritrean

Details

The Archaic Greek Pottery data set was first published by Stern and Descoeudres (1977) and later reproduced in Cooper and Weeks (1983) for illustration of linear discriminant analisys. The data set was used by Pires and Branco (2010) for illustration of their projection pursuit approach to linear discriminant analysis.

Source

STERN, W. B. and DESCOEUDRES, J.-P. (1977) X-RAY FLUORESCENCE ANALYSIS OF ARCHAIC GREEK POTTERY Archaeometry, Blackwell Publishing Ltd, 19, 73–86.

References

Cooper, R.A. and Weekes, A.J.. 1983 Data, Models, and Statistical Analysis, (Lanham, MD: Rowman & Littlefield).

Pires, A. M. and A. Branco, J. (2010) Projection-pursuit approach to robust linear discriminant analysis Journal Multivariate Analysis, Academic Press, Inc., 101, 2464–2485.

Examples

data(pottery)
x <- pottery[,c("MG", "CA")]
grp <- pottery$origin

##
## Compute robust location and covariance matrix and
## plot the tolerance ellipses
library(rrcov)
(mcd <- CovMcd(x))
col <- c(3,4)
gcol <- ifelse(grp == "Attic", col[1], col[2])
gpch <- ifelse(grp == "Attic", 16, 1)
plot(mcd, which="tolEllipsePlot", class=TRUE, col=gcol, pch=gpch)

##
## Perform classical LDA and plot the data, 0.975 tolerance ellipses
##  and LDA separation line
##
x <- pottery[,c("MG", "CA")]
grp <- pottery$origin
lda <- LdaClassic(x, grp)
lda
e1 <- getEllipse(loc=lda@center[1,], cov=lda@cov)
e2 <- getEllipse(loc=lda@center[2,], cov=lda@cov)

plot(CA~MG, data=pottery, col=gcol, pch=gpch,
    xlim=c(min(MG,e1[,1], e2[,1]), max(MG,e1[,1], e2[,1])),
    ylim=c(min(CA,e1[,2], e2[,2]), max(CA,e1[,2], e2[,2])))

ab <- lda@ldf[1,] - lda@ldf[2,]
cc <- lda@ldfconst[1] - lda@ldfconst[2]
abline(a=-cc/ab[2], b=-ab[1]/ab[2], col=2, lwd=2)

lines(e1, type="l", col=col[1])
lines(e2, type="l", col=col[2])

##
## Perform robust (MCD) LDA and plot data, classical and
##  robust separation line
##
plot(CA~MG, data=pottery, col=gcol, pch=gpch)
lda <- LdaClassic(x, grp)
ab <- lda@ldf[1,] - lda@ldf[2,]
cc <- lda@ldfconst[1] - lda@ldfconst[2]
abline(a=-cc/ab[2], b=-ab[1]/ab[2], col=2, lwd=2)
abline(a=-cc/ab[2], b=-ab[1]/ab[2], col=4, lwd=2)

rlda <- Linda(x, grp, method="mcd")
rlda
ab <- rlda@ldf[1,] - rlda@ldf[2,]
cc <- rlda@ldfconst[1] - rlda@ldfconst[2]
abline(a=-cc/ab[2], b=-ab[1]/ab[2], col=2, lwd=2)

Class "PredictLda" - prediction of "Lda" objects

Description

The prediction of a "Lda" object

Objects from the Class

Objects can be created by calls of the form new("PredictLda", ...) but most often by invoking 'predict' on a "Lda" object. They contain values meant for printing by 'show'

Slots

classification:

a factor variable containing the classification of each object

posterior:

a matrix containing the posterior probabilities

x:

matrix with the discriminant scores

ct:

re-classification table of the training sample

Methods

show

signature(object = "PredictLda"): Prints the results

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Lda-class

Examples

showClass("PredictLda")

Class "PredictQda" - prediction of "Qda" objects

Description

The prediction of a "Qda" object

Objects from the Class

Objects can be created by calls of the form new("PredictQda", ...) but most often by invoking 'predict' on a "Qda" object. They contain values meant for printing by 'show'

Slots

classification:

a factor variable containing the classification of each object

posterior:

a matrix containing the posterior probabilities

x:

matrix with the discriminant scores

ct:

re-classification table of the training sample

Methods

show

signature(object = "PredictQda"): prints the results

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Qda-class

Examples

showClass("PredictQda")

Class "Qda" - virtual base class for all classic and robust QDA classes

Description

The class Qda serves as a base class for deriving all other classes representing the results of classical and robust Quadratic Discriminant Analisys methods

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

call:

the (matched) function call.

prior:

prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the group covariance matrices

covinv:

the inverse of the group covariance matrices

covdet:

the determinants of the group covariance matrices

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

control:

object of class "CovControl" specifying which estimate and with what estimation options to use for the group means and covariances (or NULL for classical discriminant analysis)

Methods

predict

signature(object = "Qda"): calculates prediction using the results in object. An optional data frame or matrix in which to look for variables with which to predict. If omitted, the scores are used. If the original fit used a formula or a data frame or a matrix with column names, newdata must contain columns with the same names. Otherwise it must contain the same number of columns, to be used in the same order.

show

signature(object = "Qda"): prints the results

summary

signature(object = "Qda"): prints summary information

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

QdaClassic, QdaClassic-class, QdaRobust-class

Examples

showClass("Qda")

Quadratic Discriminant Analysis

Description

Performs quadratic discriminant analysis and returns the results as an object of class QdaClassic (aka constructor).

Usage

QdaClassic(x, ...)

## Default S3 method:
QdaClassic(x, grouping, prior = proportions, tol = 1.0e-4, ...)

Arguments

Value

Returns an S4 object of class QdaClassic

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Qda-class, QdaClassic-class,

Class "QdaClassic" - Quadratic Discriminant Analysis

Description

Contains the results of classical Quadratic Discriminant Analysis

Objects from the Class

Objects can be created by calls of the form new("QdaClassic", ...) but the usual way of creating QdaClassic objects is a call to the function QdaClassic which serves as a constructor.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the group covariance matrices

covinv:

the inverse of the group covariance matrices

covdet:

the determinants of the group covariance matrices

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

control:

Object of class "CovControl" inherited from class Qda specifying which estimate and with what estimation options to use for the group means and covariances. It is always NULL for classical discriminant analysis.

Extends

Class "Qda", directly.

Methods

No methods defined with class "QdaClassic" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

QdaRobust-class, Qda-class, QdaClassic

Examples

showClass("QdaClassic")

Robust Quadratic Discriminant Analysis

Description

Performs robust quadratic discriminant analysis and returns the results as an object of class QdaCov (aka constructor).

Usage

QdaCov(x, ...)

## Default S3 method:
QdaCov(x, grouping, prior = proportions, tol = 1.0e-4,
                 method = CovControlMcd(), ...)

Arguments

Details

details

Value

Returns an S4 object of class QdaCov

Warning

Still an experimental version!

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

CovMcd

Examples

## Example anorexia
library(MASS)
data(anorexia)

## start with the classical estimates
qda <- QdaClassic(Treat~., data=anorexia)
predict(qda)@classification

## try now the robust LDA with the default method (MCD with pooled whitin cov matrix)
rqda <- QdaCov(Treat~., data= anorexia)
predict(rqda)@classification

## try the other methods
QdaCov(Treat~., data= anorexia, method="sde")
QdaCov(Treat~., data= anorexia, method="M")
QdaCov(Treat~., data= anorexia, method=CovControlOgk())

Class "QdaCov" - Robust methods for Quadratic Discriminant Analysis

Description

Robust quadratic discriminant analysis is performed by replacing the classical group means and withing group covariance matrices by their robust equivalents.

Objects from the Class

Objects can be created by calls of the form new("QdaCov", ...) but the usual way of creating QdaCov objects is a call to the function QdaCov which serves as a constructor.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the group covariance matrices

covinv:

the inverse of the group covariance matrices

covdet:

the determinants of the group covariance matrices

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

control:

Object of class "CovControl" specifying which estimate to use for the group means and covariances

Extends

Class "QdaRobust", directly. Class "Qda", by class "QdaRobust", distance 2.

Methods

No methods defined with class "QdaCov" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

QdaRobust-class, Qda-class, QdaClassic, QdaClassic-class

Examples

showClass("QdaCov")

Class "QdaRobust" is a virtual base class for all robust QDA classes

Description

The class QdaRobust searves as a base class for deriving all other classes representing the results of robust Quadratic Discriminant Analysis methods

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

call:

The (matched) function call.

prior:

Prior probabilities used, default to group proportions

counts:

number of observations in each class

center:

the group means

cov:

the group covariance matrices

covinv:

the inverse of the group covariance matrices

covdet:

the determinants of the group covariance matrices

method:

a character string giving the estimation method used

X:

the training data set (same as the input parameter x of the constructor function)

grp:

grouping variable: a factor specifying the class for each observation.

control:

Object of class "CovControl" specifying which estimate to use for the group means and covariances

Extends

Class "Qda", directly.

Methods

No methods defined with class "QdaRobust" in the signature.

Author(s)

Valentin Todorov valentin.todorov@chello.at

References

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. doi:10.18637/jss.v032.i03.

See Also

Qda-class, QdaClassic-class,

Examples

showClass("QdaRobust")

Methods for Function estimate in Package 'rrcov'

Description

Each concrete control class, like CovControlMest, CovControlOgk, etc., should implement an restimate method which will call the correponding (constructor)-function and will return the obtained S4 class, derived from CovRobust.

Usage

## S4 method for signature 'CovControlMest'
restimate(obj, x, ...)

Arguments

.

Methods

obj = "CovControlMcd"

Compute the MCD estimates of multivariate location and scatter by callingCovMcd

obj = "CovControlMest"

Compute the constrained M-estimates of multivariate location and scatter by callingCovMest

obj = "CovControlOgk"

Compute the Ortogonalized Gnanadesikan-Kettenring (OGK) estimates of multivariate location and scatter by callingCovOgk

Rice taste data

Description

The rice taste data consists of five inputs and a single output whose values are associated with subjective evaluations as follows: xl: flavor, x2: appearance, x3: taste, x4: stickiness, x5: toughness, y: overall evaluation. Sensory test data have been obtained by such subjective evaluations for 105 kinds of rice (e.g., Sasanishiki, Akita-Komachi, etc.). The data set was used by Nozaki et al. (1997) to demonstrate the high performance of a proposed for automatically generating fuzzy if-then rules from numerical data.

Usage

data(rice)

Format

A data frame with 105 observations on the following 6 variables:

Favor

compactness

Appearance

circularity

Taste

distance circularity

Stickiness

radius ratio

Toughness

principal axis aspect ratio

Overall_evaluation

maximum length aspect ratio

Source

Nozaki, K., Ishibuchi, H, and Tanaka, H. (1997) A simple but powerful heuristic method for generating fuzzy rules from numerical data Fuzzy Sets and Systems 86 3 p. 251–270.