Seite - 358 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 Themean-shift isdefinedby m(x)= k ∑ i=1 1 rn+2g ( d(x,xi)2 r2 ) ∑ki=1 1 rn+2g ( d(x,xi)2 r2 )logx(xi)∝∇fKrfgr , wherem(x) is in the tangentspaceatx. Thealgorithmmoves fromx to expx(m(x))until convergence toa localmaximum.Thepointsof thespacearesegmentedaccordingto the localmaximatowhich theyconverge. In order to assess the quality of unsupervised classification,weuse the notion of Silhouette, see [36],whichcomputes foreachpointaproximitycriterionwithrespect tootherpointsof thesame clusterandotherpointsofdifferentclusters (seeFigure4). Letxbe in theclusterA.Werespectively define a(x)=miny∈Ad(x,y)andb(x)=miny =Ad(x,y), theminimumdistance topointsof thesame (resp. other) class(es). TheSilhouetteofx is a(x)−b(x) max{a(x),b(x)}, which takesvaluesbetween−1and1, respectively,whenthedatapoint is considered“badly”and “well” clustered. The average of all the silhouettes provides an indication of the relevance of the classification.Onecanrepresentgraphically thesilhouetteprofilebyplotting foreachclasshorizontal segmentsof the lengthof thesilhouettevalue (seeFigure5). Figure4. Intraandinterclusterdistances. Figure5.Exampleofsilhouette. 358