On some significance tests in cluster analysis
type: Item
instance of: scholarly article
title: On some significance tests in cluster analysis
zbMATH Open document ID: 0587.62048
DOI: 10.1007/BF01908065
published in: Journal of Classification
publication date: 1985-01-01T00:00:00Z
review text: The author investigates the properties of several significance tests for distinguishing between the hypothesis H of a ''homogeneous'' population and an alternative A involving ''clustering'' or ''heterogeneity'', with emphasis on the case of multidimensional observations \(x_ 1,...,x_ n\in R^ p.\) Four types of test statistics are considered: the (s-th) largest gap between observations, their mean distance (or similarity), the minimum within-cluster sum of squares resulting from a k-means algorithm, and the resulting maximum F statistic. If, for a given significance level (error probability) a, such a test statistic exceeds the corresponding critical value \(c=c(a)\), the hypothesis H of homogeneity is rejected (e.g., in favor of a clustering structure A). The asymptotic distributions under H are given for \(n\to \infty\) and the asymptotic power of the tests is derived for neighboring alternatives \(A=A_ n\) approaching A. In particular, the asymptotic distribution of the maximum F statistic is obtained. Moreover, the asymptotic power of the gap test is characterized by a speed factor (log n)\({}^{-1}\) (for \(A_ n\) converging to H), and by a factor \(n^{-1/4}\) for tests based on the mean similarity.
Mathematics Subject Classification ID: 62F03
Mathematics Subject Classification ID: 62F05
Mathematics Subject Classification ID: 62H30
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3942733
zbMATH Keywords: cluster analysis
zbMATH Keywords: asymptotic normality
zbMATH Keywords: classification
zbMATH Keywords: significance tests
zbMATH Keywords: clustering
zbMATH Keywords: heterogeneity
zbMATH Keywords: mean distance
zbMATH Keywords: similarity
zbMATH Keywords: minimum within-cluster sum of squares
zbMATH Keywords: k-means algorithm
zbMATH Keywords: maximum F statistic
zbMATH Keywords: homogeneity
zbMATH Keywords: neighboring alternatives
zbMATH Keywords: asymptotic power
zbMATH Keywords: gap test
author: H.-H. Bock
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01908065
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description: scientific article; zbMATH DE number 3942733