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)

dat

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`

.

n

optional positive integer. Number of cases.

p

optional positive integer. Number of independent variables.

method

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.

cgen

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.

ca

optional positive number. Tuning constant, specifying
required cluster
separation. By default determined as `calpha`

-quantile of the
chisquared distribution with `p`

degrees of freedom.

ca2

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.

calpha

number between 0 and 1. See `ca`

.

calpha2

number between 0 and 1, larger than `calpha`

.
See `ca2`

.

pointit

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`

.

subset

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.

nc1

optional positive integer. Tuning constant needed by
`cmahal`

to generate `ca`

automatically. Only
needed for `cgen="auto"`

.

startn

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.

mnc

optional positive integer. Minimum size of clusters to be reported.

mer

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.

distcut

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.

maxit

optional integer. Maximum number of iterations per algorithm run (usually an FPC is found much earlier).

iter

positive number. Algorithm stops when difference between
subsequent weight vectors is smaller than `iter`

. Only needed
for `method="fuzzy"`

.

init.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, as in Hennig and Christlieb (2002).

ind.storage

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.

countmode

optional positive integer. Every `countmode`

algorithm runs `fixmahal`

shows a message.

plot

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

object of class `mfpc`

, output of `fixmahal`

.

x

object of class `mfpc`

, output of `fixmahal`

.

maxnc

positive integer. Maximum number of FPCs to be reported.

no

positive integer. Number of the representative FPC to be plotted.

bw

optional logical. If `TRUE`

, plot is black/white,
FPC is
indicated by different symbol. Else FPC is indicated red.

main

plot title.

xlab

label for x-axis. If `NULL`

, a default text is used.

ylab

label for y-axis. If `NULL`

, a default text is used.

pch

plotting symbol, see `par`

.
If `NULL`

, the default is used.

col

plotting color, see `par`

.
If `NULL`

, the default is used.

gv

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).

`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.

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.

# NOT RUN { 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) # }