28,212 research outputs found
A New Functional Clustering Method with Combined Dissimilarity Sources and Graphical Interpretation
Clustering is an essential task in functional data analysis. In this study, we propose a framework for a clustering procedure based on functional rankings or depth. Our methods naturally combine various types of between-cluster variation equally, which caters to various discriminative sources of functional data; for example, they combine raw data with transformed data or various components of multivariate functional data with their covariance. Our methods also enhance the clustering results with a visualization tool that allows intrinsic graphical interpretation. Finally, our methods are model-free and nonparametric and hence are robust to heavy-tailed distribution or potential outliers. The implementation and performance of the proposed methods are illustrated with a simulation study and applied to three real-world applications
A Bayesian alternative to mutual information for the hierarchical clustering of dependent random variables
The use of mutual information as a similarity measure in agglomerative
hierarchical clustering (AHC) raises an important issue: some correction needs
to be applied for the dimensionality of variables. In this work, we formulate
the decision of merging dependent multivariate normal variables in an AHC
procedure as a Bayesian model comparison. We found that the Bayesian
formulation naturally shrinks the empirical covariance matrix towards a matrix
set a priori (e.g., the identity), provides an automated stopping rule, and
corrects for dimensionality using a term that scales up the measure as a
function of the dimensionality of the variables. Also, the resulting log Bayes
factor is asymptotically proportional to the plug-in estimate of mutual
information, with an additive correction for dimensionality in agreement with
the Bayesian information criterion. We investigated the behavior of these
Bayesian alternatives (in exact and asymptotic forms) to mutual information on
simulated and real data. An encouraging result was first derived on
simulations: the hierarchical clustering based on the log Bayes factor
outperformed off-the-shelf clustering techniques as well as raw and normalized
mutual information in terms of classification accuracy. On a toy example, we
found that the Bayesian approaches led to results that were similar to those of
mutual information clustering techniques, with the advantage of an automated
thresholding. On real functional magnetic resonance imaging (fMRI) datasets
measuring brain activity, it identified clusters consistent with the
established outcome of standard procedures. On this application, normalized
mutual information had a highly atypical behavior, in the sense that it
systematically favored very large clusters. These initial experiments suggest
that the proposed Bayesian alternatives to mutual information are a useful new
tool for hierarchical clustering
Robust EM algorithm for model-based curve clustering
Model-based clustering approaches concern the paradigm of exploratory data
analysis relying on the finite mixture model to automatically find a latent
structure governing observed data. They are one of the most popular and
successful approaches in cluster analysis. The mixture density estimation is
generally performed by maximizing the observed-data log-likelihood by using the
expectation-maximization (EM) algorithm. However, it is well-known that the EM
algorithm initialization is crucial. In addition, the standard EM algorithm
requires the number of clusters to be known a priori. Some solutions have been
provided in [31, 12] for model-based clustering with Gaussian mixture models
for multivariate data. In this paper we focus on model-based curve clustering
approaches, when the data are curves rather than vectorial data, based on
regression mixtures. We propose a new robust EM algorithm for clustering
curves. We extend the model-based clustering approach presented in [31] for
Gaussian mixture models, to the case of curve clustering by regression
mixtures, including polynomial regression mixtures as well as spline or
B-spline regressions mixtures. Our approach both handles the problem of
initialization and the one of choosing the optimal number of clusters as the EM
learning proceeds, rather than in a two-fold scheme. This is achieved by
optimizing a penalized log-likelihood criterion. A simulation study confirms
the potential benefit of the proposed algorithm in terms of robustness
regarding initialization and funding the actual number of clusters.Comment: In Proceedings of the 2013 International Joint Conference on Neural
Networks (IJCNN), 2013, Dallas, TX, US
Fuzzy clustering of univariate and multivariate time series by genetic multiobjective optimization
Given a set of time series, it is of interest to discover subsets that share similar properties. For instance, this may be useful for identifying and estimating a single model that may fit conveniently several time series, instead of performing the usual identification and estimation steps for each one. On the other hand time series in the same cluster are related with respect to the measures assumed for cluster analysis and are suitable for building multivariate time series models. Though many approaches to clustering time series exist, in this view the most effective method seems to have to rely on choosing some features relevant for the problem at hand and seeking for clusters according to their measurements, for instance the autoregressive coe±cients, spectral measures or the eigenvectors of the covariance matrix. Some new indexes based on goodnessof-fit criteria will be proposed in this paper for fuzzy clustering of multivariate time series. A general purpose fuzzy clustering algorithm may be used to estimate the proper cluster structure according to some internal criteria of cluster validity. Such indexes are known to measure actually definite often conflicting cluster properties, compactness or connectedness, for instance, or distribution, orientation, size and shape. It is argued that the multiobjective optimization supported by genetic algorithms is a most effective choice in such a di±cult context. In this paper we use the Xie-Beni index and the C-means functional as objective functions to evaluate the cluster validity in a multiobjective optimization framework. The concept of Pareto optimality in multiobjective genetic algorithms is used to evolve a set of potential solutions towards a set of optimal non-dominated solutions. Genetic algorithms are well suited for implementing di±cult optimization problems where objective functions do not usually have good mathematical properties such as continuity, differentiability or convexity. In addition the genetic algorithms, as population based methods, may yield a complete Pareto front at each step of the iterative evolutionary procedure. The method is illustrated by means of a set of real data and an artificial multivariate time series data set.Fuzzy clustering, Internal criteria of cluster validity, Genetic algorithms, Multiobjective optimization, Time series, Pareto optimality
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