AbstractÐThis paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data. The adjective ªunsupervisedº is justified by two properties of the algorithm: 1) it is capable of selecting the number of components and 2) unlike the standard expectation-maximization (EM) algorithm, it does not require careful initialization. The proposed method also avoids another drawback of EM for mixture fitting: the possibility of convergence toward a singular estimate at the boundary of the parameter space. The novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models; instead, we seamlessly integrate estimation and model selection in a single algorithm. Our technique can be applied to any type of parametric mixture model for which it is possible to write an EM algorithm; in this paper, we illustrate it with experiments involving Gaussian mixtures. These experiments testify for the good performance of our approach. Index TermsÐFinite mixtures, unsupervised learning, model selection, minimum message length criterion, Bayesian methods, expectation-maximization algorithm, clustering. 1

We present an algorithm that infers the model structure of a mixture of factor analysers using an ecient and deterministic variational approximation to full Bayesian integration over model parameters. This procedure can automatically determine the optimal number of components and the local dimensionality of each component (i.e. the number of factors in each factor analyser). Alternatively it can be used to infer posterior distributions over number of components and dimensionalities. Since all parameters are integrated out the method is not prone to over tting. Using a stochastic procedure for adding components it is possible to perform the variational optimisation incrementally and to avoid local maxima. Results show that the method works very well in practice and correctly infers the number and dimensionality of nontrivial synthetic examples. By importance sampling from the variational approximation we show how to obtain unbiased estimates of the true evidence, the exa...

Abstract. Given an arbitrary data set, to which no particular parametrical, statistical or geometrical structure can be assumed, different clustering algorithms will in general produce different data partitions. In fact, several partitions can also be obtained by using a single clustering algorithm due to dependencies on initialization or the selection of the value of some design parameter. This paper addresses the problem of finding consistent clusters in data partitions, proposing the analysis of the most common associations performed in a majority voting scheme. Combination of clustering results are performed by transforming data partitions into a co-association sample matrix, which maps coherent associations. This matrix is then used to extract the underlying consistent clusters. The proposed methodology is evaluated in the context of k-means clustering, a new clustering algorithm- voting-k-means, being presented. Examples, using both simulated and real data, show how this majority voting combination scheme simultaneously handles the problems of selecting the number of clusters, and dependency on initialization. Furthermore, resulting clusters are not constrained to be hyper-spherically shaped. 1

I consider the problem of finding all the modes of a mixture of multivariate Gaussian distributions, which has applications in clustering and regression. I derive exact formulas for the gradient and Hessian and give a partial proof that the number of modes cannot be more than the number of components, and are contained in the convex hull of the component centroids. Then, I develop two exhaustive mode search algorithms: one based on combined quadratic maximisation and gradient ascent and the other one based on a fixed-point iterative scheme. Appropriate values for the search control parameters are derived by taking into account theoretical results regarding the bounds for the gradient and Hessian of the mixture. The significance of the modes is quantified locally (for each mode) by error bars, or confidence intervals (estimated using the values of the Hessian at each mode); and globally by the sparseness of the mixture, measured by its differential entropy (estimated through bounds). I conclude with some reflections about bump-finding.

One of the most difficult problems in cluster analysis is the identification of the number of groups in a data set. Most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations. In this paper we develop a simple yet powerful non-parametric method for choosing the number of clusters based on distortion, a quantity that measures the average distance, per dimension, between each observation and its closest cluster center. Our technique is computationally efficient and straightforward to implement. We demonstrate empirically its effectiveness, not only for choosing the number of clusters but also for identifying underlying structure, on a wide range of simulated and real world data sets. In addition, we give a rigorous theoretical justification for the method based on information theoretic ideas. Specifically, results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the distortion in both the presence and absence of clustering. Finally, we note that these ideas potentially can be extended to a wide range of other statistical model selection problems. 1

Independent Component Analysis is becoming a popular exploratory method for analysing complex data such as that from FMRI experiments. The application of such 'model-free' methods, however, has been somewhat restricted both by the view that results can be uninterpretable and by the lack of ability to quantify statistical significance. We present an integrated approach to Probabilistic ICA for FMRI data that allows for non-square mixing in the presence of Gaussian noise. We employ an objective estimation of the amount of Gaussian noise through Bayesian analysis of the true dimensionality of the data, i.e. the number of activation and non-Gaussian noise sources. Reduction of the data to this 'true' subspace before the ICA decomposition automatically results in an estimate of the noise, leading to the ability to assign significance to voxels in ICA spatial maps. Estimation of the number of intrinsic sources not only enables us to carry out probabilistic modelling, but also achieves an asymptotically unique decomposition of the data. This reduces problems of interpretation, as each final independent component is now much more likely to be due to only one physical or physiological process. We also describe other improvements to standard ICA, such as temporal pre-whitening and variance normafisation of timeseries, the latter being particularly useful in the context of dimensionality reduction when weak activation is present. We discuss the use of prior information about the spatiotemporal nature of the source processes, and an alternative-hypothesis testing approach for inference, using Gaussian mixture models. The performance of our approach is illustrated and evaluated on real and complex artificial FMRI data, and compared to the spatio-temporal accuracy of restfits obtaine...

In this work, we present a unified bottom-up and top-down automatic model selection based approach for modelling complex activities of multiple objects in cluttered scenes. An activity of multiple objects is represented based on discrete scene events and their behaviours are modelled by reasoning about the temporal and causal correlations among different events. This is significantly different from the ma-jority of the existing techniques that are centred on object tracking followed by trajectory matching. In our approach, object-independent events are detected and classified by unsupervised clustering us-ing Expectation-Maximisation (EM) and classified using automatic model selection based on Schwarz’s Bayesian Information Criterion (BIC). Dynamic Probabilistic Networks (DPNs) are formulated for mod-elling the temporal and causal correlations among discrete events for robust and holistic scene-level be-haviour interpretation. In particular, we developed a Dynamically Multi-Linked Hidden Markov Model (DML-HMM) based on the discovery of salient dynamic interlinks among multiple temporal processes corresponding to multiple event classes. A DML-HMM is built using BIC based factorisation result-ing in its topology being intrinsically determined by the underlying causality and temporal order among events. Extensive experiments are conducted on modelling activities captured in different indoor and

We explore the idea of evidence accumulation (EAC) for combining the results of multiple clusterings. First, a clustering ensemble- a set of object partitions, is produced. Given a data set (n objects or patterns in d dimensions), different ways of producing data partitions are: (1)- applying different clustering algorithms, and (2)- applying the same clustering algorithm with different values of parameters or initializations. Further, combinations of different data representations (feature spaces) and clustering algorithms can also provide a multitude of significantly different data partitionings. We propose a simple framework for extracting a consistent clustering, given the various partitions in a clustering ensemble. According to the EAC concept, each partition is viewed as an independent evidence of data organization, individual data partitions being combined, based on a voting mechanism, to generate a new n × n similarity matrix between the n patterns. The final data partition of the n patterns is obtained by applying a hierarchical agglomerative clustering algorithm on this matrix. We have developed a theoretical framework for the analysis of the proposed clustering combination strategy and its evaluation, based on the concept of mutual information between data partitions. Stability of the results is evaluated using bootstrapping techniques. A detailed discussion of an evidence accumulation-based clustering algorithm, using a split and merge strategy based on the K-means clustering algorithm, is presented. Experimental results of the proposed method on several synthetic and real data sets are compared with other combination strategies, and with individual clustering results produced by well known clustering algorithms.

Consider the problem of fitting a finite Gaussian mixture, with an unknown number of components, to observed data. This paper proposes a new minimum description length (MDL) type criterion, termed MMDL (for mixture MDL), to select the number of components of the model. MMDL is based on the identification of an "equivalent sample size", for each component, which does not coincide with the full sample size. We also introduce an algorithm based on the standard expectationmaximization (EM) approach together with a new agglomerative step, called agglomerative EM (AEM). The experiments here reported have shown that MMDL outperforms existing criteria of comparable computational cost. The good behavior of AEM, namely its good robustness with respect to initialization, is also illustrated experimentally.

This paper investigates Hidden Markov Models (HMMs) in which the observations are generated from an autoregressive (AR) model. The overall model performs nonstationary spectral analysis and automatically segments a time series into discrete dynamic regimes. Because learning in HMMs is sensitive to initial conditions, we initialise the HMM model with parameters derived from a cluster analysis of Kalman filter coefficients. An important aspect of the Kalman filter implementation is that the state noise is estimated on-line. This allows for an initial estimation of AR parameters for each of the different dynamic regimes. These estimates are then fine-tuned with the HMM model. The method is demonstrated on a number of synthetic problems and on electroencephalogram (EEG) data. 1 Introduction Autoregressive (AR) models are a well-known parametric technique for the spectral estimation of stationary signals [1]. The standard AR model can also be used for the spectral estimation of nonstationa...