HDDC is a model-based clustering method. It is based on the Gaussian Mixture Model and on the idea that the data lives in subspaces with a lower dimension than the dimension of the original space. It uses the Expectation - Maximisation algorithm to estimate the parameters of the model.
A matrix or a data frame of observations, assuming the rows are the observations and the columns the variables. Note that NAs are not allowed.
K
A vector of integers specifying the number of clusters for which the BIC and the parameters are to be calculated; the function keeps the parameters which maximises the BIC. Note that the length of the vector K can't be larger than 20. Default is 1:10.
model
A character string vector, or an integer vector indicating the models to be used. The available models are: "AkjBkQkDk" (default), "AkBkQkDk", "ABkQkDk", "AkjBQkDk", "AkBQkDk", "ABQkDk", "AkjBkQkD", "AkBkQkD", "ABkQkD", "AkjBQkD", "AkBQkD", "ABQkD", "AjBQD", "ABQD". It is not case sensitive and integers can be used instead of names, see details for more information. Several models can be used, if it is, only the results of the one which maximizes the BIC criterion is kept. To run all models, use model="ALL".
threshold
A float stricly within 0 and 1. It is the threshold used in the Cattell's Scree-Test.
com_dim
It is used only for common dimensions models. The user can give the common dimension he wants. If used, it must be an integer. Its default is set to NULL.
itermax
The maximum number of iterations allowed. The default is 60.
eps
A positive double. It is the stopping criterion: the algorithm stops when the difference between two successive Log Likelihoods is lower than ‘eps’.
graph
It is for comparison sake only, when several estimations are run at the same time (either when using several models, or when using cross-validation to select the best dimension/threshold). If graph = TRUE, the plot of the results of all estimations is displayed. Default is FALSE.
algo
A character string indicating the algorithm to be used. The available algorithms are the Expectation-Maximisation ("EM"), the Classification E-M ("CEM") and the Stochastic E-M ("SEM"). The default algorithm is the "EM".
d
Either “Cattell” (default) or “BIC”. See details for more information.
init
A character string or a vector of clusters. It is the way to initialize the E-M algorithm. There are five ways of initialization: “kmeans” (default), “param”, “random” or “mini-em”. See details for more information. It can also be directly initialized with a vector containing the prior classes of the observations.
show
Use show = FALSE to settle off the informations that may be printed.
mini.nb
A vector of integers of length two. This parameter is used in the “mini-em” initialization. The first integer sets how many times the algorithm is repeated; the second sets the maximum number of iterations the algorithm will do each time. For example, if init=“mini-em” and mini.nb=c(5,10), the algorithm wil be lauched 5 times, doing each time 10 iterations; finally the algorithm will begin with the initialization that maximizes the log-likelihood.
scaling
Logical: whether to scale the dataset (mean=0 and standard-error=1 for each variable) or not. By default the data is not scaled.
min.individuals
This parameter is used to control for the minimum population of a class. If the population of a class becomes stricly inferior to 'min.individuals' then the algorithm stops and gives the message: 'pop<min.indiv.'. Here the meaning of "population of a class" is the sum of its posterior probabilities. The value of 'min.individuals' cannot be lower than 2.
noise.ctrl
This parameter avoids to have a too low value of the 'noise' parameter b. It garantees that the dimension selection process do not select too many dimensions (which leads to a potential too low value of the noise parameter b). When selecting the intrinsic dimensions using Cattell's scree-test or BIC, the function doesn't use the eigenvalues inferior to noise.ctrl, so that the intrinsic dimensions selected can't be higher or equal to the order of these eigenvalues.
...
Any arguments that can be used by the function "kmeans" when this initialization is selected (namely: algorithm; nstart; iter.max).
Details
Some information on the signification of the model names:
Akj are the parameters of the classes subspaces:
if Akj: each class has its parameters and there is one parameter for each dimension
if Ak: the classes have different parameters but there is only one per class
if Aj: all the classes have the same parameters for each dimension (it's a particular case with a common orientation matrix)
if A: all classes have the same one parameter
Bk are the noises of the classes subspaces:
If Bk: each class has its proper noise
if B: all classes have the same noise
Qk is the orientation matrix of each class:
if Qk: all classes have its proper orientation matrix
if Q: all classes have the same orientation matrix
Dk is the intrinsic dimension of each class:
if Dk: the dimensions are free and proper to each class
if D: the dimension is common to all classes
The model “ALL” will compute all the models, give their BIC and keep the model with the highest BIC value.
Instead of writing the model names, they can also be specified using an integer. 1 represents the most general model (“AkjBkQkDk”) while 14 is the most constrained (“ABQD”), the others number/name matching are given below. Note also that several models can be run at once, by using a vector of models (e.g. model = c("AKBKQKD","AKJBQKDK","AJBQD") is equivalent to model = c(8,4,13); to run the 6 first models, use model=1:6). If all the models are to be run, model="all" is faster than model=1:14.
AkjBkQkDk
1
AkjBkQkD
7
AkBkQkDk
2
AkBkQkD
8
ABkQkDk
3
ABkQkD
9
AkjBQkDk
4
AkjBQkD
10
AkBQkDk
5
AkBQkD
11
ABQkDk
6
ABQkD
12
AjBQD
13
ABQD
14
The parameter d, is used to select the intrinsic dimensions of the subclasses. Here are his definictions:
“Cattell”:
The Cattell's scree-test is used to gather the intrinsic dimension of each class. If the model is of common dimension (models 7 to 14), the scree-test is done on the covariance matrix of the whole dataset.
“BIC”:
The intrinsic dimensions are selected with the BIC criterion. See Bouveyron et al. (2010) for a discussion of this topic.
For common dimension models, the procedure is done on the covariance matrix of the whole dataset.
Note that "Cattell" (resp. "BIC") can be abreviated to "C" (resp. "B") and that this argument is not case sensitive.
The different initializations are:
“param”:
it is initialized with the parameters, the means being generated by a multivariate normal distribution and the covariance matrix being common to the whole sample
“mini-em”:
it is an initialization strategy, the classes are randomly initialized and the E-M algorithm makes several iterations, this action is repetead a few times (the default is 5 iterations and 10 times), at the end, the initialization choosen is the one which maximise the log-likelihood (see mini.nb for more information about its parametrization)
“random”:
the classes are randomly given using a multinomial distribution
“kmeans”:
the classes are initialized using the kmeans function (with: algorithm="Hartigan-Wong"; nstart=4; iter.max=50); note that the user can use his own arguments for kmeans using the dot-dot-dot argument
A prior class vector:
It can also be directly initialized with a vector containing the prior classes of the observations
The BIC criterion used in this function is to be maximized and is defined as 2*LL-k*log(n) where LL is the log-likelihood, k is the number of parameters and n is the number of observations.
Value
hddc returns an 'hdc' object; it's a list containing:
model
The name of the model.
K
The number of classes.
d
The dimensions of each class.
a
The parameters of each class subspace.
b
The noise of each class subspace.
mu
The mean of each variable for each class.
prop
The proportion of each class.
ev
The eigen values of the var/covar matrix.
Q
The orthogonal matrix of orientation of each class.
kname
The name of each class.
BIC
The BIC value of the model used.
scaling
The centers and the standard deviation of the original dataset.
likelihood
The Log Likelihood for each iteration.
class
The class vector obtained by the clustering.
posterior
The matrix of the probabilities to belong to a class for each observation and each class.
Author(s)
Laurent Berge, Charles Bouveyron and Stephane Girard
References
Bouveyron, C. Girard, S. and Schmid, C. (2007) “High-Dimensional Data Clustering”, Computational Statistics and Data Analysis, vol. 52 (1), pp. 502–519
Berge, L. Bouveyron, C. and Girard, S. (2012) “HDclassif: An R Package for Model-Based Clustering and Discriminant Analysis of High-Dimensional Data”, Journal of Statistical Software, 46(6), 1–29, url: http://www.jstatsoft.org/v46/i06/
See Also
hdda, predict.hdc, plot.hdc.
Examples
# Example 1:
data <- simuldata(1000, 1000, 50)
X <- data$X
clx <- data$clx
Y <- data$Y
cly <- data$cly
#clustering of the simulated dataset:
prms1 <- hddc(X, K=3, algo="CEM", init='param')
#class vector obtained by the clustering:
prms1$class
#We can look at the adjusted rand index to assess the goodness of fit
res1 <- predict(prms1, X, clx)
res2 <- predict(prms1, Y)
#the class predicted using hddc parameters on the test dataset:
res2$class
# Example 2:
data(Crabs)
#clustering of the Crabs dataset:
prms3 <- hddc(Crabs[,-1], K=4, algo="EM", init='mini-em')
res3 <- predict(prms3, Crabs[,-1], Crabs[,1])
#another example using the Crabs dataset
prms4 <- hddc(Crabs[,-1], K=1:8, graph=TRUE, model=c(1,2,7,9))
#model=c(1,2,7,9) is equivalent to model=c("AKJBKQKDK","AKBKQKDK","AKJBKQKD","ABKQKD")
res4 <- predict(prms4, Crabs[,-1], Crabs[,1])