Robust Improper Maximum Likelihood Clustering
Description
rimle
searches for G
approximately Gaussianshaped
clusters with/without noise/outliers. The method's tuning controlling
the noise
level is fixed and is to be provided by the user or will be guessed by
the function in
a rather quick and dirty way (otrimle
performs a more
sophisticated datadriven choice).
Usage
1 2 3 
Arguments
data 
A numeric vector, matrix, or data frame of observations. Rows correspond
to observations and columns correspond to variables. Categorical
variables and 
G 
An integer specifying the number of clusters. 
initial 
An integer vector specifying the initial cluster
assignment with 
logicd 
A number 
npr.max 
A number in 
erc 
A number 
det.min 
Lower bound for the minimum determinant of covariance matrices. This
is only active if 
cmstep 
A logical value. When set to 
em.iter.max 
An integer value specifying the maximum number of iterations allowed in the underlying EMalgorithm. 
em.tol 
Stopping criterion for the the underlying EMalgorithm. An EM iteration
stops if two successive improper loglikelihood values are within

monitor 
Set the verbosity level of tracing messages. Possible values
are 
Details
The rimle
function allows to approximate the RIMLE solution
with two different versions of the underlying EMtype algorithm.
 ECMalgorithm:
cmstep=TRUE

The RIMLE solution is obtained based on the ECMalgorithm proposed in Coretto and Hennig (2016). In this case both the eigenratio constraint and the noise proportion constraint are enforced in each conditional Mstep of the algorithm.
 Approximate EMalgorithm:
cmstep=FALSE

This corresponds to the algorithm proposed in Coretto and Hennig (2015). In this case covariance matrices are regularized in each step based on
det.min
, and the eigenratio constraint is applied only at the end of the EM iteration.
The ECMalgorithm is the default choice. The Approximate
EMalgorithm is often slower than the ECMalgorithm by a
factor of two. Furthermore the Approximate EMalgorithm is
more prone to lead to problems indicated by code=0
(see
Valuesection below) because of numerical degeneracies
connected to a low value of min.det
.
There may be datasets for which the function does not provide a
solution based on default arguments. This corresponds to
code=0
and flag=1
or 2 or 3 in the output (see
Valuesection below). This usually happens when some (or all) of the
following circumstances occur: (i) log(icd)
is too
large; (ii) erc
is too large; (iii) npr.max is too large;
(iv) choice of the initial partition. In these cases it is suggested
to find a suitable interval of icd
values by using the
otrimle
function. The Details section of
otrimle
suggests several actions to take
whenever a code=0
nonsolution occurs.
Value
An S3 object of class 'rimle'
. Output components are as follows:
code 
An integer indicator for the convergence.

flag 
A character string containing one or more flags related to
the EM iteration at the optimal icd.

iter 
Number of iterations performed in the underlying EMalgorithm. 
logicd 
Value of the 
iloglik 
Value of the improper likelihood. 
criterion 
Value of the OTRIMLE criterion. 
npr 
Estimated expected noise proportion. 
cpr 
Vector of estimated expected cluster proportions (notice that 
mean 
A matrix of dimension 
cov 
An array of size 
tau 
A matrix of dimension 
smd 
A matrix of dimension 
cluster 
A vector of integers denoting cluster assignments for each
observation. It's 
size 
A vector of integers with sizes (counts) of each cluster. 
References
Coretto, P. and C. Hennig (2015). Robust improper maximum likelihood: tuning, computation, and a comparison with other methods for robust Gaussian clustering. To appear on the Journal of the American Statistical Association. arXiv preprint at arXiv:1406.0808 with (supplement).
Coretto, P. and C. Hennig (2016). Consistency, breakdown robustness, and algorithms for robust improper maximum likelihood clustering. arXiv preprint at arXiv:1309.6895.
See Also
plot.rimle
,
InitClust
,
otrimle
,
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47  ## Load Swiss banknotes data
data(banknote)
x < banknote[,1]
## 
## EXAMPLE 1:
## Perform RIMLE with default inputs
## 
set.seed(1)
a < rimle(data=x, G=2)
print(a)
## Plot clustering
plot(a, data=x, what="clustering")
## PP plot of the clusterwise empirical weighted squared Mahalanobis
## distances against the target distribution pchisq(, df=ncol(data))
plot(a, what="fit")
plot(a, what="fit", cluster=1)
## 
## EXAMPLE 2:
## Compare solutions for different choices of logicd
## 
set.seed(1)
## Case 1: noiseless solution, that is fit a pure Gaussian Mixture Model
b1 < rimle(data=x, G=2, logicd=Inf)
plot(b1, data=x, what="clustering")
plot(b1, what="fit")
## Case 2: low noise level
b2 < rimle(data=x, G=2, logicd=100)
plot(b2, data=x, what="clustering")
plot(b2, what="fit")
## Case 3: medium noise level
b3 < rimle(data=x, G=2, logicd=10)
plot(b3, data=x, what="clustering")
plot(b3, what="fit")
## Case 3: large noise level
b3 < rimle(data=x, G=2, logicd=5)
plot(b3, data=x, what="clustering")
plot(b3, what="fit")
