Computes Mahalanobis 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 normal population. FPCs may overlap, are not necessarily exhausting and do not need a specification of the number of clusters.
Note that while fixmahal 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.
fixmahal(dat, n = nrow(as.matrix(dat)), p = ncol(as.matrix(dat)),
method = "fuzzy", cgen = "fixed",
ca = NA, ca2 = NA,
calpha = ifelse(method=="fuzzy",0.95,0.99),
calpha2 = 0.995,
pointit = TRUE, subset = n,
nc1 = 100+20*p,
startn = 18+p, mnc = floor(startn/2),
mer = ifelse(pointit,0.1,0),
distcut = 0.85, maxit = 5*n, iter = n*1e-5,
init.group = list(),
ind.storage = TRUE, countmode = 100,
plot = "none")
# S3 method for mfpc
summary(object, ...)
# S3 method for summary.mfpc
print(x, maxnc=30, ...)
# S3 method for mfpc
plot(x, dat, no, bw=FALSE, main=c("Representative FPC No. ",no),
xlab=NULL, ylab=NULL,
pch=NULL, col=NULL, ...)
# S3 method for mfpc
fpclusters(object, dat=NA, ca=object$ca, p=object$p, ...)
fpmi(dat, n = nrow(as.matrix(dat)), p = ncol(as.matrix(dat)),
gv, ca, ca2, method = "ml", plot,
maxit = 5*n, iter = n*1e-6)
fixmahal returns an object of class mfpc. This is a list
containing the components nc, g, means, covs, nfound, er, tsc,
ncoll, skc, grto, imatrix, smatrix, stn, stfound, ser, sfpc, ssig,
sto, struc, n, p, method, cgen, ca, ca2, cvec, calpha, pointit,
subset, mnc, startn, mer, distcut.
summary.mfpc returns an object of class summary.mfpc.
This is a list containing the components means, covs, stn,
stfound, sn, ser, tskip, skc, tsc, sim, ca, ca2, calpha, mer, method,
cgen, pointit.
fpclusters.mfpc returns a list of indicator vectors for the
representative FPCs of stable groups.
fpmi returns a list with the components mg, covg, md,
gv, coll, method, ca.
integer. Number of FPCs.
list of logical vectors. Indicator vectors of FPCs. FALSE
if ind.storage=FALSE.
list of numerical vectors. Means of FPCs. In
summary.mfpc, only for representative
FPCs of stable groups and sorted according to
ser.
list of numerical matrices. Covariance matrices of FPCs. In
summary.mfpc, only for representative
FPCs of stable groups and sorted according to
ser.
vector of integers. Number of findings for the FPCs.
numerical vector. Ratio of number of findings of FPCs to their
size. Under pointit=TRUE,
this can be taken as a measure of stability of FPCs.
integer. Number of algorithm runs leading to too small or too seldom found FPCs.
integer. Number of algorithm runs where collinear covariance matrices occurred.
integer. Number of skipped clusters.
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.mfpc, sorted according to ser.
vector of integers. Number of findings of members of
all groups of FPCs. In summary.mfpc, sorted according to
ser.
numerical vector. Ratio of number of findings of groups of
FPCs to their size. Under pointit=TRUE,
this can be taken as a measure of stability of FPCs. In
summary.mfpc, sorted from largest to smallest.
vector of integers. Numbers of representative FPCs of all groups.
vector of integers of length stn.
Numbers of representative FPCs of the stable groups.
vector of integers. Numbers of groups ordered
according to largest ser.
vector of integers. Number of group an FPC belongs to.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments, if cgen has been "fixed". Else
numerical vector of length nc (see below), giving the
final values of ca for all FPC. In fpmi, tuning
constant for the iterated FPC.
see arguments.
numerical vector of length n for
cgen="auto". The values for the
tuning constant ca corresponding to the cluster sizes from
1 to n.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments.
see arguments.
vector of integers. Number of points of representative FPCs.
integer. Number of algorithm runs leading to skipped FPCs.
vector of integers. Size of intersections between
representative FPCs of stable groups. See sseg.
mean vector.
covariance matrix.
Mahalanobis distances.
logical (numerical, respectively, if method="fuzzy")
indicator vector of iterated FPC.
logical. TRUE means that singular covariance
matrices occurred during the iterations.
something that can be coerced to a
numerical matrix or vector. Data matrix, rows are points, columns
are variables.
fpclusters.rfpc
does not need specification of dat if fixmahal
has been run with ind.storage=TRUE.
optional positive integer. Number of cases.
optional positive integer. Number of independent variables.
a string. method="classical" means 0-1 weighting
of observations by Mahalanobis distances and use of the classical
normal covariance estimator. method="ml" uses the
ML-covariance estimator (division by n instead of n-1)
This is used in Hennig and Christlieb (2002).
method can also be "mcd" or "mve",
to enforce the use of robust centers and covariance matrices, see
cov.rob. This is experimental, not recommended at the
moment, may be very slowly and requires library lqs.
The default is
method="fuzzy", where weighted means and covariance matrices
are used (Hennig, 2005).
The weights are computed by wfu, i.e., a
function that is constant 1 for arguments smaller than ca, 0 for
arguments larger than ca2 and continuously linear in between.
Convergence is only proven for method="ml" up to now.
optional string. "fixed" means that the same tuning
constant ca is used for all iterations. "auto" means
that ca is generated dependently on the size of the current data
subset in each iteration by cmahal. This is
experimental.
optional positive number. Tuning constant, specifying
required cluster
separation. By default determined as calpha-quantile of the
chisquared distribution with p degrees of freedom.
optional positive number. Second tuning constant needed if
method="fuzzy".
By default determined as calpha2-quantile of the
chisquared distribution with p degrees of freedom.
number between 0 and 1. See ca.
number between 0 and 1, larger than calpha.
See ca2.
optional logical. If TRUE, subset fixed point
algorithms are started from initial configurations, which are built
around single points of the dataset, cf. mahalconf.
Otherwise, initial configurations are only specified by
init.group.
optional positive integer smaller or equal than n.
Initial configurations for the fixed point algorithm
(cf. mahalconf) are built from
a subset of subset points from the data. No effect if
pointit=FALSE. Default: all points.
optional positive integer. Tuning constant needed by
cmahal to generate ca automatically. Only
needed for cgen="auto".
optional positive integer. Size of the initial configurations. The default value is chosen to prevent that small meaningless FPCs are found, but it should be decreased if clusters of size smaller than the default value are of interest.
optional positive integer. Minimum size of clusters to be reported.
optional nonnegative number. FPCs (groups of them,
respectively, see details)
are only reported as stable if the ratio
of the number of their
findings to their number of points exceeds mer. This holds
under pointit=TRUE and subset=n. For subset<n,
the ratio is adjusted, but for small subset, the results
may extremely vary and have to be taken with care.
optional value between 0 and 1. A similarity
measure between FPCs, given in Hennig (2002), 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 integer. Maximum number of iterations per algorithm run (usually an FPC is found much earlier).
positive number. Algorithm stops when difference between
subsequent weight vectors is smaller than iter. Only needed
for method="fuzzy".
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, as in Hennig and Christlieb (2002).
optional logical. If TRUE,
then all indicator
vectors of found FPCs are given in the value of fixmahal.
May need lots of memory, but is a bit faster.
optional positive integer. Every countmode
algorithm runs fixmahal shows a message.
optional string. If "start", you get a scatterplot
of the first two variables to highlight the initial configuration,
"iteration" generates such a plot at each iteration,
"both" does both (this may be very time consuming).
The default is "none".
object of class mfpc, output of fixmahal.
object of class mfpc, output of fixmahal.
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. If NULL, a default text is used.
label for y-axis. If NULL, a default text is used.
plotting symbol, see par.
If NULL, the default is used.
plotting color, see par.
If NULL, the default is used.
logical vector (or, with method="fuzzy",
vector of weights between 0 and 1) 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 (crisp) Mahalanobis FPC is a data subset
that reproduces itself under the following operation:
Compute mean and covariance matrix estimator for the data
subset, and compute all points of the dataset for which the squared
Mahalanobis distance is smaller than ca.
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.
The current default is to compute fuzzy Mahalanobis FPCs, where the
points in the subset have a membership weight between 0 and 1 and give
rise to weighted means and covariance matrices.
The new weights are then obtained by computing the weight function
wfu of the squared Mahalanobis distances, i.e.,
full weight for squared distances smaller than ca, zero
weight for squared distances larger than ca2 and
decreasing weights (linear function of squared distances)
in between.
A fixed point algorithm is started from the whole dataset,
algorithms are started from the subsets specified in
init.group, and further algorithms are started from further
initial configurations as explained under subset and in the
function mahalconf.
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: A similarity matrix is computed
between 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 often enough (cf. parameter mer) 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 number of points
of an FPC, adjusted suitably if subset<n.
Usually only the representative FPCs of stable groups
are of interest.
Default tuning constants are taken from Hennig (2005).
Generally, the default settings are recommended for
fixmahal. For large datasets, the use of
init.group together with pointit=FALSE
is useful. Occasionally, mnc and startn may be chosen
smaller than the default,
if smaller clusters are of interest, but this may lead to too many
clusters. 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.
Singular covariance matrices during the iterations are handled by
solvecov.
summary.mfpc gives a summary about the representative FPCs of
stable groups.
plot.mfpc is a plot method for the representative FPC of stable
group no. no. It produces a scatterplot, where
the points belonging to the FPC are highlighted, the mean is and
for p<3 also the region of the FPC is shown. For p>=3,
the optimal separating projection computed by batcoord
is shown.
fpclusters.mfpc produces a list of indicator vectors for the
representative FPCs of stable groups.
fpmi is called by fixmahal for a single fixed point
algorithm and will usually not be executed alone.
Hennig, C. (2002) Fixed point clusters for linear regression: computation and comparison, Journal of Classification 19, 249-276.
Hennig, C. (2005) Fuzzy and Crisp Mahalanobis Fixed Point Clusters, in Baier, D., Decker, R., and Schmidt-Thieme, L. (eds.): Data Analysis and Decision Support. Springer, Heidelberg, 47-56.
Hennig, C. and Christlieb, N. (2002) Validating visual clusters in large datasets: Fixed point clusters of spectral features, Computational Statistics and Data Analysis 40, 723-739.
fixreg for linear regression fixed point clusters.
mahalconf, wfu, cmahal
for computation of initial configurations, weights, tuning constants.
sseg for indexing the similarity/intersection vectors
computed by fixmahal.
batcoord, cov.rob, solvecov,
cov.wml, plotcluster
for computation of projections, (inverted)
covariance matrices, plotting.
rFace for generation of example data, see below.
options(digits=2)
set.seed(20000)
face <- rFace(400,dMoNo=2,dNoEy=0, p=3)
# The first example uses grouping information via init.group.
initg <- list()
grface <- as.integer(attr(face,"grouping"))
for (i in 1:5) initg[[i]] <- (grface==i)
ff0 <- fixmahal(face, pointit=FALSE, init.group=initg)
summary(ff0)
cff0 <- fpclusters(ff0)
plot(face, col=1+cff0[[1]])
plot(face, col=1+cff0[[4]]) # Why does this come out as a cluster?
plot(ff0, face, 4) # A bit clearer...
# Without grouping information, examples need more time:
# ff1 <- fixmahal(face)
# summary(ff1)
# cff1 <- fpclusters(ff1)
# plot(face, col=1+cff1[[1]])
# plot(face, col=1+cff1[[6]]) # Why does this come out as a cluster?
# plot(ff1, face, 6) # A bit clearer...
# ff2 <- fixmahal(face,method="ml")
# summary(ff2)
# ff3 <- fixmahal(face,method="ml",calpha=0.95,subset=50)
# summary(ff3)
## ...fast, but lots of clusters. mer=0.3 may be useful here.
# set.seed(3000)
# face2 <- rFace(400,dMoNo=2,dNoEy=0)
# ff5 <- fixmahal(face2)
# summary(ff5)
## misses right eye of face data; with p=6,
## initial configurations are too large for 40 point clusters
# ff6 <- fixmahal(face2, startn=30)
# summary(ff6)
# cff6 <- fpclusters(ff6)
# plot(face2, col=1+cff6[[3]])
# plot(ff6, face2, 3)
# x <- c(1,2,3,6,6,7,8,120)
# ff8 <- fixmahal(x)
# summary(ff8)
# ...dataset a bit too small for the defaults...
# ff9 <- fixmahal(x, mnc=3, startn=3)
# summary(ff9)
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