Computes linear regression fixed point clusters (FPCs), i.e., subsets of the data, which consist exactly of the non-outliers w.r.t. themselves, and may be interpreted as generated from a homogeneous linear regression relation between independent and dependent variable. FPCs may overlap, are not necessarily exhausting and do not need a specification of the number of clusters.
Note that while fixreg has lots of parameters, only one (or
few) of them have usually to be specified, cf. the examples. The
philosophy is to allow much flexibility, but to always provide
sensible defaults.
fixreg(indep=rep(1,n), dep, n=length(dep),
p=ncol(as.matrix(indep)),
ca=NA, mnc=NA, mtf=3, ir=NA, irnc=NA,
irprob=0.95, mncprob=0.5, maxir=20000, maxit=5*n,
distcut=0.85, init.group=list(),
ind.storage=FALSE, countmode=100,
plot=FALSE)# S3 method for rfpc
summary(object, ...)
# S3 method for summary.rfpc
print(x, maxnc=30, ...)
# S3 method for rfpc
plot(x, indep=rep(1,n), dep, no, bw=TRUE,
main=c("Representative FPC No. ",no),
xlab="Linear combination of independents",
ylab=deparse(substitute(indep)),
xlim=NULL, ylim=range(dep),
pch=NULL, col=NULL,...)
# S3 method for rfpc
fpclusters(object, indep=NA, dep=NA, ca=object$ca, ...)
rfpi(indep, dep, p, gv, ca, maxit, plot)
fixreg returns an object of class rfpc. This is a list
containing the components nc, g, coefs, vars, nfound, er, tsc,
ncoll, grto, imatrix, smatrix, stn, stfound, sfpc, ssig, sto, struc,
n, p, ca, ir, mnc, mtf, distcut.
summary.rfpc returns an object of class summary.rfpc.
This is a list containing the components coefs, vars, stfound,
stn, sn, ser, tsc, sim, ca, ir, mnc, mtf.
fpclusters.rfpc returns a list of indicator vectors for the
representative FPCs of stable groups.
rfpi returns a list with the components coef, var, g,
coll, ca.
integer. Number of FPCs.
list of logical vectors. Indicator vectors of FPCs. FALSE
if ind.storage=FALSE.
list of numerical vectors. Regression coefficients of
FPCs. In summary.rfpc, only for representative
FPCs of stable groups and sorted according to
stfound.
list of numbers. Error variances of FPCs. In
summary.rfpc, only for representative
FPCs of stable groups and sorted according to
stfound.
vector of integers. Number of findings for the FPCs.
numerical vector. Expectation ratios of FPCs. Can be taken as a stability measure.
integer. Number of algorithm runs leading to too small or too seldom found FPCs.
integer. Number of algorithm runs where collinear regressor matrices occurred.
vector of integers. Numbers of FPCs to which algorithm
runs led, which were started by init.group.
vector of integers. Size of intersection between
FPCs. See sseg.
numerical vector. Similarities between
FPCs. See sseg.
integer. Number of representative FPCs of stable groups. In
summary.rfpc sorted according to stfound.
vector of integers. Number of findings of members of
all groups of FPCs. In
summary.rfpc sorted according to stfound.
vector of integers. Numbers of representative FPCs.
vector of integers. As sfpc, but only for stable
groups.
vector of integers. Number of representative FPC of most, 2nd most, ..., often found group of FPCs.
vector of integers. Number of group an FPC belongs to.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments.
vector of integers. Number of points of representative FPCs.
numerical vector. Expectation ratio for stable groups.
vector of integers. Size of intersections between
representative FPCs of stable groups. See sseg.
vector of regression coefficients.
error variance.
logical indicator vector of iterated FPC.
logical. TRUE means that singular covariance
matrices occurred during the iterations.
numerical matrix or vector. Independent
variables.
Leave out for clustering one-dimensional data.
fpclusters.rfpc
does not need specification of indep if fixreg
was run with ind.storage=TRUE.
numerical vector. Dependent variable.
fpclusters.rfpc
does not need specification of dep if fixreg
was run with ind.storage=TRUE.
optional positive integer. Number of cases.
optional positive integer. Number of independent variables.
optional positive number. Tuning constant, specifying
required cluster
separation. By default determined automatically as a
function of n and p, see function can,
Hennig (2002a).
optional positive integer. Minimum size of clusters
to be reported.
By default determined automatically as a function of
mncprob. See Hennig (2002a).
optional positive integer. FPCs must be found at
least mtf times to be reported by summary.rfpc.
optional positive integer. Number of algorithm runs.
By default determined
automatically as a function of n, p, irnc,
irprob, mtf,
maxir. See function itnumber and Hennig (2002a).
optional positive integer. Size of the smallest
cluster to be found with
approximated probability irprob.
optional value between 0 and 1. Approximated
probability for a cluster of size irnc to be found.
optional value between 0 amd 1. Approximated
probability for a cluster of size mnc to be found.
optional integer. Maximum number of algorithm runs.
optional integer. Maximum number of iterations per algorithm run (usually an FPC is found much earlier).
optional value between 0 and 1. A similarity
measure between FPCs, given in Hennig (2002a), and the corresponding
Single Linkage groups of FPCs with similarity larger
than distcut are computed.
A single representative FPC is selected for each group.
optional list of logical vectors of length
n.
Every vector indicates a starting configuration for the fixed
point algorithm. This can be used for datasets with high
dimension, where the vectors of init.group indicate cluster
candidates found by graphical inspection or background
knowledge.
optional logical. If TRUE,
then all indicator
vectors of found FPCs are given in the value of fixreg.
May need lots of memory, but is a bit faster.
optional positive integer. Every countmode
algorithm runs fixreg shows a message.
optional logical. If TRUE, you get a scatterplot
of first independent vs. dependent variable at each iteration.
object of class rfpc, output of fixreg.
object of class rfpc, output of fixreg.
positive integer. Maximum number of FPCs to be reported.
positive integer. Number of the representative FPC to be plotted.
optional logical. If TRUE, plot is black/white,
FPC is
indicated by different symbol. Else FPC is indicated red.
plot title.
label for x-axis.
label for y-axis.
plotted range of x-axis. If NULL, the range of the
plotted linear combination of independent variables is used.
plotted range of y-axis.
plotting symbol, see par.
If NULL, the default is used.
plotting color, see par.
If NULL, the default is used.
logical vector of length n. Indicates the initial
configuration for the fixed point algorithm.
additional parameters to be passed to plot
(no effects elsewhere).
Christian Hennig christian.hennig@unibo.it https://www.unibo.it/sitoweb/christian.hennig/en/
A linear regression FPC is a data subset
that reproduces itself under the following operation:
Compute linear regression and error variance estimator for the data
subset, and compute all points of the dataset for which the squared
residual is smaller than ca times the error variance.
Fixed points of this operation can be considered as clusters,
because they contain only
non-outliers (as defined by the above mentioned procedure) and all other
points are outliers w.r.t. the subset.
fixreg performs ir fixed point algorithms started from
random subsets of size p+2 to look for
FPCs. Additionally an algorithm is started from the whole dataset,
and algorithms are started from the subsets specified in
init.group.
Usually some of the FPCs are unstable, and more than one FPC may
correspond to the same significant pattern in the data. Therefore the
number of FPCs is reduced: FPCs with less than mnc points are
ignored. Then a similarity matrix is computed between the remaining
FPCs. Similarity between sets is defined as the ratio between
2 times size of
intersection and the sum of sizes of both sets. The Single Linkage
clusters (groups)
of level distcut are computed, i.e. the connectivity
components of the graph where edges are drawn between FPCs with
similarity larger than distcut. Groups of FPCs whose members
are found mtf times or more are considered as stable enough.
A representative FPC is
chosen for every Single Linkage cluster of FPCs according to the
maximum expectation ratio ser. ser is the ratio between
the number of findings of an FPC and the estimated
expectation of the number of findings of an FPC of this size,
called expectation ratio and
computed by clusexpect.
Usually only the representative FPCs of stable groups
are of interest.
The choice of the involved tuning constants such as ca and
ir is discussed in detail in Hennig (2002a). Statistical theory
is presented in Hennig (2003).
Generally, the default settings are recommended for
fixreg. In cases where they lead to a too large number of
algorithm runs (e.g., always for p>4), the use of
init.group together with mtf=1 and ir=0
is useful. Occasionally, irnc may be chosen
smaller than the default,
if smaller clusters are of interest, but this may lead to too many
clusters and too many algorithm runs. Decrease of
ca will often lead to too many clusters, even for homogeneous
data. Increase of ca will produce only very strongly
separated clusters. Both may be of interest occasionally.
rfpi is called by fixreg for a single fixed point
algorithm and will usually not be executed alone.
summary.rfpc gives a summary about the representative FPCs of
stable groups.
plot.rfpc is a plot method for the representative FPC of stable
group
no. no. It produces a scatterplot of the linear combination of
independent variables determined by the regression coefficients of the
FPC vs. the dependent variable. The regression line and the region
of non-outliers determined by ca are plotted as well.
fpclusters.rfpc produces a list of indicator vectors for the
representative FPCs of stable groups.
Hennig, C. (2002) Fixed point clusters for linear regression: computation and comparison, Journal of Classification 19, 249-276.
Hennig, C. (2003) Clusters, outliers and regression: fixed point clusters, Journal of Multivariate Analysis 86, 183-212.
fixmahal for fixed point clusters in the usual setup
(non-regression).
regmix for clusterwiese linear regression by mixture
modeling ML.
can, itnumber for computation of the default
settings.
clusexpect for estimation of the expected number of
findings of an FPC of given size.
itnumber for the generation of the number of fixed point
algorithms.
minsize for the smallest FPC size to be found with a given
probability..
sseg for indexing the similarity/intersection vectors
computed by fixreg.
set.seed(190000)
options(digits=3)
data(tonedata)
attach(tonedata)
tonefix <- fixreg(stretchratio,tuned,mtf=1,ir=20)
summary(tonefix)
# This is designed to have a fast example; default setting would be better.
# If you want to see more (and you have a bit more time),
# try out the following:
if (FALSE) {
set.seed(1000)
tonefix <- fixreg(stretchratio,tuned)
# Default - good for these data
summary(tonefix)
plot(tonefix,stretchratio,tuned,1)
plot(tonefix,stretchratio,tuned,2)
plot(tonefix,stretchratio,tuned,3,bw=FALSE,pch=5)
toneclus <- fpclusters(tonefix,stretchratio,tuned)
plot(stretchratio,tuned,col=1+toneclus[[2]])
tonefix2 <- fixreg(stretchratio,tuned,distcut=1,mtf=1,countmode=50)
# Every found fixed point cluster is reported,
# no matter how instable it may be.
summary(tonefix2)
tonefix3 <- fixreg(stretchratio,tuned,ca=7)
# ca defaults to 10.07 for these data.
summary(tonefix3)
subset <- c(rep(FALSE,5),rep(TRUE,24),rep(FALSE,121))
tonefix4 <- fixreg(stretchratio,tuned,
mtf=1,ir=0,init.group=list(subset))
summary(tonefix4)
}
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