Antecedentes de interpretación Urbana de la Suite No 1 BWV 1007 para Cello Solo de Johann Sebastian Bach.

3.3

Density-Connected Subspace Clustering

3.3.1 Clusters as Density-Connected Sets

Our approach SUBCLU is based on the formal definitions of density-connected clusters underlying the algorithm DBSCAN. The original formal definition of the clustering notion for the entire feature space were presented and dis- cussed in Section 2.2. In the following, we adopt these definitions for the problem of subspace clustering.

Definition 3.1 (ε-neighborhood in a subspace)

Let ε∈ IR+₀, S ⊆ A and o∈ DB. The ε-neighborhood of o in S, denoted by NS

ε(o), is defined by

N_εS(o) ={x∈DB |dist(πS(o), πS(x))≤ε}.

Definition 3.2 (core point in a subspace)

Let ε∈IR+₀, k∈IN, andS ⊆ A. A point o∈DB is called core point in S, denoted by CoreSε,k(o) if its ε-neighborhood in S contains at least k points,

formally:

CoreSε,k(o)⇔ | NεS(o)| ≥k.

Definition 3.3 (direct density-reachability in a subspace)

Let ε ∈ IR, k ∈ IN, and S ⊆ A. A point p ∈ DB is directly density- reachablefrom q∈DB in S if q is a core point in S andp is an element of NS

ε (q), formally:

DirReachSε,k(q, p)⇔CoreSε,k(q) ∧ p∈ NεS(q).

Definition 3.4 (density-reachability in a subspace)

Let ε∈IR+₀,k∈IN, andS ⊆ A. A pointp∈DB isdensity-reachable from q ∈ DB in S if there is a chain of points p₁, . . . ,_pn, p1 = q, pn = p such

thatpi+1 is directly density-reachable from pi, formally:

ReachSε,k(q, p)⇔

∃p₁, . . . ,_pn∈DB : p1 =q ∧ pn=p ∧

Definition 3.5 (density-connectivity in a subspace)

Let ε ∈ IR+₀, k ∈ IN, and S ⊆ A. A point p ∈ DB is density-connected

to a point q ∈ DB in S if there is a point o such that both p and q are density-reachable from o, formally:

ConnectSε,k(q, p)⇔

∃o∈DB : ReachSε,k(o, q) ∧ ReachSε,k(o, p).

Definition 3.6 (density-connected set in a subspace)

Let ε ∈ IR+₀, k ∈ IN, and S ⊆ A. A non-empty subset C ⊆ DB is called a density-connected set in S if all points in C are density-connected in S, formally:

ConSetSε,k(C)⇔ ∀o, q ∈C : ConnectSε,k(o, q).

Finally, a density-connected cluster is defined as a set of density-connected points which is maximal w.r.t. density-reachability [EKSX96]. This defini- tion can easily be adopted to clusters in a particular subspace.

3.3.2 Monotonicity of Density-Connected Sets

A straightforward approach would be to run DBSCAN in all possible sub- spaces to detect all density-connected clusters. The problem is that the number of subspaces is 2d. A more effective strategy would be to use the clustering information of previous subspaces in the process of generating all clusters and drop all subspaces that cannot contain any density-connected clusters.

Unfortunately, density-connected clusters are not monotonic, i.e. ifC ⊆ DB is a density-connected cluster in subspace S ⊆ A, it need not be a density-connected cluster in anyT ⊆S. The reason for this is that inT the density-connected clusterCmay not be maximal w.r.t. density-reachability. There may be additional points which are not inCbut are density-reachable inT from a point inC.

3.3 Density-Connected Subspace Clustering 39 However, density-connected sets are monotonic. In fact, if C⊆DB is a density-connected set in a subspaceS⊆ AthenCis also a density-connected set in any subspace T ⊆S.

Lemma 3.1 (monotonicity)

Let ε∈IR+₀, k∈IN, o, q ∈DB, C ⊆DB, where C 6=∅ and S⊆ A. Then the following monotonicity properties hold:

∀T ⊆S: (1) CoreSε,k(o)⇒CoreTε,k(o) (2) DirReachSε,k(o, q)⇒DirReachTε,k(o, q) (3) ReachSε,k(o, q)⇒ReachTε,k(o, q) (4) ConnectSε,k(o, q)⇒ConnectTε,k(o, q) (5) ConSetSε,k(o, q)⇒ConSetTε,k(o, q) Proof. (1) CoreSε,k(o) ⇔ | NεS(o)| ≥k ⇔ |{x|dist(πS(o), πS(x))≤ε}| ≥k ⇔ |{x| p s X ai∈S (πai(o)−πai(x))p ≤ε}| ≥k (T⊆S) ⇒ |{x| p s X ai∈T (πai(o)−πai(x))p ≤ε}| ≥k ⇔ |{x|dist(πT(o), πT(x))≤ε}| ≥k ⇔ | N_εT(o)| ≥k ⇔_CoreT_ε,k(o) (2) DirReachSε,k(o, q) ⇔CoreSε,k(o)∧q∈ NεS(o) ⇔_CoreS_ε,k(o)∧dist(πS(o), πS(q))≤ε ⇔_CoreS_ε,k(o)∧ p s X ai∈S (πai(o)−πai(q))p ≤ε (T⊆S) (1) =⇒ _CoreT ε,k(o)∧ p s X ai∈T (πai(o)−πai(q))p ≤ε ⇔_CoreT_ε,k(o)∧dist(πT(o), πT(q))≤ε ⇔_CoreT_ε,k(o)∧q∈ N_εT(o)

⇔_DirReachT_ε,k(o, q) (3) ReachSε,k(o, q) ⇔ ∃p1, . . . ,pn∈DB : p1 =o∧pn=q ∧ ∀i∈ {1. . . n−1} : DirReachSε,k(pi, pi+1) (T⊆S) (2) =⇒ ∃p₁, . . . ,_pn∈DB : p1 =o∧pn=q ∧ ∀i∈ {1. . . n−1} : DirReachTε,k(pi, pi+1) ⇔ReachTε,k(o, q)

(4) ConnectSε,k(o, q) ⇔ ∃x∈DB : ReachSε,k(x, o)∧ReachSε,k(x, q)

(T⊆S) (3) =⇒ ∃x∈DB : ReachTε,k(x, o)∧ReachTε,k(x, q) ⇔ConnectTε,k(o, q) (5) ConSetSε,k(C) ⇔ ∀o, q ∈C : ConnectSε,k(o, q) (T⊆S) (4) =⇒ ∀o, q∈C : ConnectTε,k(o, q) ⇔_ConSetT_ε,k(C)

The monotonicity of density-connectivity is illustrated in Figure 3.3. In Figure 3.3(a),p and q are density-connected viaoin the subspace spanned by attributes A and B. Thus, p and q are also density-connected via o in each subspaceAand B ofAB. The inverse conclusion is depicted in Figure 3.3(b): p and q are not density-connected in subspace B. Thus, they are also not density-connected in the superspaceAB although they are density- connected in subspaceA via o.

The inversion of Lemma 3.1(5) is the key idea for an efficient bottom- up algorithm to detect the density-connected sets in all subspaces of high- dimensional data. We do not have to examine any subspace S if at least oneTi ⊂S contains no cluster, i.e. no density-connected set. On the other

hand, we have to test each subspaceS if allTi⊂Scontain clusters whether

3.4 The Algorithm SUBCLU 41