Abstract
We develop a cluster process which is invariant with respect to unknown affine transformations of the feature space without knowing the number of clusters in advance. Specifically, our proposed method can identify clusters invariant under (I) orthogonal transformations, (II) scaling-coordinate orthogonal transformations, and (III) arbitrary nonsingular linear transformations corresponding to models I, II, and III, respectively and represent clusters with the proposed heatmap of the similarity matrix. The proposed Metropolis-Hasting algorithm leads to an irreducible and aperiodic Markov chain, which is also efficient at identifying clusters reasonably well for various applications. Both the synthetic and real data examples show that the proposed method could be widely applied in many fields, especially for finding the number of clusters and identifying clusters of samples of interest in aerial photography and genomic data.
Highlights
Clustering of objects invariant with respect to affine transformations of feature vectors is an important research topic since objects may be recorded via different angles and positions so that their coordinates may vary and their nearest neighbors may belong to other clusters
The affine transformations consist of three types: (1) index permutations, rotation, onescaling on all variables, and location-translation transformations that are under the first type of covariance structures and named model I whose transformation and covariance structure σ 2Id were adopted by Vogt et al (2010); (2) each variable may have different scaling transformations that are under the second type of covariance structures and named model II; (3) the variables are transformed by a nonsingular matrix that is named model III, where the observed variables may be linear combinations of some latent variables in model I
6 Concluding discussion The proposed clustering method is invariant under different groups of affine transformations and computationally efficient
Summary
Clustering of objects invariant with respect to affine transformations of feature vectors is an important research topic since objects may be recorded via different angles and positions so that their coordinates may vary and their nearest neighbors may belong to other clusters. The affine transformations consist of three types: (1) index permutations, rotation, onescaling on all variables, and location-translation transformations that are under the first type of covariance structures and named model I whose transformation and covariance structure σ 2Id were adopted by Vogt et al (2010); (2) each variable may have different scaling transformations that are under the second type of covariance structures and named model II; (3) the variables are transformed by a nonsingular matrix that is named model III, where the observed variables may be linear combinations of some latent variables in model I These models cover fairly general situations of clustering in nature. We assume that the random partition of objects follows Ewens distribution (Ewens 1972), and we propose a likelihood of the responses which is invariant respect to affine transformations. We adopt the CRP prior for partition B which implies k = ∞ in the population with the proposed Gaussian likelihood to get the affine-invariant clusters.
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