An entropy criterion for assessing the number of clusters in a mixture model

In this paper, we consider an entropy criterion to estimate the number of clusters arising from a mixture model. This criterion is derived from a relation linking the likelihood and the classification likelihood of a mixture. Its performance is investigated through Monte Carlo experiments, and it shows favorable results compared to other classical criteria.

---

Résumé Nous proposons un critère d'entropie pour évaluer le nombre de classes d'une partition en nous fondant sur un modèle de mélange de lois de probabilité. Ce critère se déduit d'une relation liant la vraisemblance et la vraisemblance classifiante d'un mélange. Des simulations de Monte Carlo illustrent ses qualités par rapport à des critères plus classiques.

     

1998-2002年《自然辩证法研究》的统计分析

受《自然辩证法研究》编辑部委托，本文对该杂志1998—2002年间的论文类型、作者群的年龄构成、职业、职称和地域分布进行了统计分析，并与1998年进行的1985—1997年间的统计结果进行了比较。尔后，我们在统计分析的基础上，简要地分析了该杂志的主要特色，适当地提出了一些可供参考的建议。     

Distributional Equivalence and Subcompositional Coherence in the Analysis of Compositional Data,Contingency Tables and Ratio-Scale Measurements

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyze the ratios of the data values. A common approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property and can be applied to a wider class of methods. This weighted log-ratio analysis is theoretically equivalent to “spectral mapping”, a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modeling. The weighted log-ratio methodology is used here to visualize frequency data in linguistics and chemical compositional data in archeology. The first author acknowledges research support from the Fundación BBVA in Madrid as well as partial support by the Spanish Ministry of Education and Science, grant MEC-SEJ2006-14098. The constructive comments of the referees, who also brought additional relevant literature to our attention, significantly improved our article.