Stable Components and Layers

Abstract

Component graphs Γ0(F ) are defined for arrays of sets F . The path components of Γ0(F ) are the stable components of the array F . The stable components for the system of Lesnick complexes {L s,k (X)} for a finite data set X decompose into layers, which are themselves path components of a graph. Combinatorial scoring functions are defined for layers and stable components.

Stability for layer points

“…For a v-hierarchical clustering, H, i th -parameter branch points are defined as the branch points of the slice hierarchical clusterings, in the same way as i th -parameter layer points were defined as the layer points of the slice clusterings. These appear in [4] as vertical and horizontal branch points.…”

“…As in [4], we call the distances between elements of X the phase change numbers of X. Since X is finite, there are a finite number of phase change numbers,…”

“…Alternately, layers can be viewed as the path components of a graph whose vertices are the elements of Γ(H) with edges ( s, S) → ( t, S) for all s ≤ t with S ∈ H( s) ∩ H( t). This graph is the component graph defined in [4].…”

“…(See Section 3 for a brief introduction to these complexes.) This is the context in which layer points were originally introduced by Jardine in [4]. We consider the single parameter slices of this clustering where the density parameter is constant.…”

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In the first half this paper, we generalize the theory of layer points [4] to the more general context of v-hierarchical clusterings [5]. Layer points provide a compressed description of a hierarchical clustering by recording only the points where a cluster changes.

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