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Ellipsoidal Mixture Models for Clustering: an Application to Web-based Education Data

Saujanna Jafreen
Department of Natural Sciences, Daffodil International University, Dhaka-1207, Bangladesh

Taslim S. Mallick
Department of Natural Sciences, Daffodil International University, Dhaka-1207, Bangladesh

Jafar A. Khan
Department of Statistics, Biostatistics and Informatics, University of Dhaka, Dhaka-1000, Bangladesh

Abstract/ Executive Summary:

This study considers the Gaussian mixture models for clustering. Since spherical and diagonal models occur very rarely in practice and analysis can be simplified when these models are implemented, we focus on the ellipsoidal models. EM algorithm is used to fit these models to a real data set related to an adaptive educational electronic course. Misclassification rates and Bayesian Information Criteria (BIC) values are used for comparison.

Keywords: Model-based clustering, Gaussian mixture models, EM algorithm

Location: Department of Natural Sciences, Daffodil International University, Dhaka-1207, Bangladesh

Start Date:

End Date:

Program Area: Knowledge Management

Non-commodity: Database Management

Objectives:

The objective of this study is to compare the performance of several Gaussians mixture models. EM algorithm is used to fit these models to a real data set related to an Adaptive Educational Electronic Course obtained from UCI machine learning repository. Misclassification rates and Bayesian Information Criteria (BIC) values are used for comparison.

Methodology:

Mixture models for clustering assume that the data are generated by a finite mixture of underlying probability distributions. Suppose the data set × consists of n independent multivariate observations xi to be divided into K clusters C₁, C₂, ..., C_K^.The Expectation-Maximization (EM) algorithm is usually employed to fit a model when data contain missing values. The algorithm first replaces the missing values with some ‘initial values’ and fits the model to the ‘complete data’. Then the algorithm alternates between the E-step (where the missing values are replaced by values expected under the current model) and the M-step (where the model has fitted again by maximizing the likelihood). The algorithm stops when the estimates of two successive iterations are very close. In the clustering problem, we do not know which cluster an observation belongs to. In that sense, clustering data contain missing values, where all the values of the clustering variable are ‘missing’. Therefore, the EM algorithm can be used to obtain the estimates of these missing values. In order to fit a particular Gaussian Mixture Model, the desired number of clusters K is specified. Then the model parameters π_k, µ_k, Σ_k, k = 1, 2, ..., K, are estimated by the EM algorithm. To implement the EM algorithm, we require the initial estimates of the model parameters which are obtained either arbitrarily or by implementing a hierarchical clustering algorithm. Then the Expectation step (E-step) and the Maximization step (M-step) alternate until convergence. The E-step estimates the conditional probability of each observation belonging to each cluster, given the current parameter estimates. The M-step estimates the model parameters given the current conditional probabilities of the class occurrence. In order to select the number of clusters K and the covariance structure, the Bayesian Information Criteria (BIC) (Schwarz 1978) is used. BIC is the value of the maximized log-likelihood with a penalty for the number of parameters in the model. The value of K is varied from 1 to 9 and the model with the largest BIC score is selected. In the next section, we fit different mixture models to domain-dependent data of students’ educational activities and classify their knowledge level using the EM algorithm. One of the main goals in machine learning is to develop an algorithm that can best classify a user into one among different possible categories. User-models are commonly used in interactive systems, where the system adapts its behavior according to users' specific needs. Based on the acquired information from the user, the user-model classifies the user as one of the K possible categories according to which an interactive system adapts its behavior. In web-based adaptive courses, a user-model collects students’ data and uses them to predict (classify) their knowledge level about the course. In this paper, we analyze students' data (n = 258) related to an Adaptive Educational Electronic Course (AEEC) obtained from the machine learning repository of the University of California Irvine (http://archive.ics.uci.edu/ml). Kahraman et al. (2013) proposed a user-model where K NN classifier is used combined with weights of the predictors to classify users knowledge status of the course. The objective of this paper is to use a classifier that fits different mixture models using EM algorithm and compare the predicted knowledge levels of students enrolled in AEEC with that of Kahraman et al. (2013). Similar to Kahraman et al. (2013), we use five predictors, namely degree of study time for AEEC (STG), degree of repetition number (SCG), performance in exams (PEG), degree of study time in prerequisite objects (STR) and learning status of prerequisite objects (LPR), where STG, SCG and PEG are the features about learning objects and others are the features about the prerequisite objects. The domain-dependent data on these five features are obtained by real-valued functions over the range of 0 to 1. The response of the user-model is the student’s current knowledge level which can be any one of 4 levels, high, medium, low and very low.

Source of Information: Bangladesh J. Sci. Res. 28(1): 1-7, 2015 (June)

Article Link (Online Journal):

Funding Source:

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Total Budget (Tk.) :

Achievement/Findings:

This paper investigates the performance of several Gaussian Mixture Models. The spherical and diagonal models are not interesting from the practical point of view. Therefore, we focus on the ellipsoidal models EEE, EEV, VEV and VVV. The models are applied to a real data set related to an Adaptive Educational Electronic Course obtained from UCI machine learning repository. Only EEE identifies correctly that the data have 4 clusters, and the BIC value for this model is the largest. On the other hand, EEV is the best model with respect to the misclassification rate. This indicates that model selection based on BIC scores does not guarantee the best model with respect to other criteria.

Knowledge Management: Journal