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Clustering student skill set profiles in a unit hypercube using mixtures of multivariate betas. (English) Zbl 1416.62334

Summary: This paper presents a finite mixture of multivariate betas as a new model-based clustering method tailored to applications where the feature space is constrained to the unit hypercube. The mixture component densities are taken to be conditionally independent, univariate unimodal beta densities (from the subclass of reparameterized beta densities given by L. Bagnato and A. Punzo [Comput. Stat. 28, No. 4, 1571–1597 (2013; Zbl 1306.65024)]. The EM algorithm used to fit this mixture is discussed in detail, and results from both this beta mixture model and the more standard Gaussian model-based clustering are presented for simulated skill mastery data from a common cognitive diagnosis model and for real data from the Assistment System online mathematics tutor [M. Feng et al., “Addressing the assessment challenge with an online system that tutors as it assesses”, User Model. User-Adapted Interact. 19, No. 3, 243–266 (2009; doi:10.1007/s11257-009-9063-7)]. The multivariate beta mixture appears to outperform the standard Gaussian model-based clustering approach, as would be expected on the constrained space. Fewer components are selected (by BIC-ICL) in the beta mixture than in the Gaussian mixture, and the resulting clusters seem more reasonable and interpretable.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)

Citations:

Zbl 1306.65024

Software:

R
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References:

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