A grouping of data objects such that the objects within a group are similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized Intra-cluster distances are minimized
Group observations into groups so that the observations belonging in the same group are similar, whereas observations in different groups are different
Basic questions:
What does “similar” mean
What is a good partition of the objects? I.e., how is the quality of a solution measured
The choice of initial points can have large influence
Clusters of different densities
Clusters of different sizes
Outliers can also cause a problem (Example?)
22.
Some alternatives to random initialization of the central points
Multiple runs
Helps, but probability is not on your side
Select original set of points by methods other than random . E.g., pick the most distant (from each other) points as cluster centers (kmeans++ algorithm)
Find the unlabelled point furthest from {1,2,…,i-1} and label it as i .
//Use d(x,S) = min y є S d(x,y) to identify the distance //of a point from a set
π (i) = argmin j<i d(i,j)
R i =d(i, π (i))
30.
The farthest-first traversal is a 2-approximation algorithm
Claim1 : R 1 ≥R 2 ≥… ≥R n
Proof:
R j =d(j, π (j)) = d(j,{1,2,…,j-1})
≤ d(j,{1,2,…,i-1}) //j > i
≤ d(i,{1,2,…,i-1}) = R i
31.
The farthest-first traversal is a 2-approximation algorithm
Claim 2: If C is the clustering reported by the farthest algorithm, then R(C)=R k+1
Proof:
For all i > k we have that
d(i, {1,2,…,k})≤ d(k+1,{1,2,…,k}) = R k+1
32.
The farthest-first traversal is a 2-approximation algorithm
Theorem: If C is the clustering reported by the farthest algorithm, and C * is the optimal clustering, then then R(C)≤2xR(C * )
Proof:
Let C* 1 , C* 2 ,…, C* k be the clusters of the optimal k-clustering.
If these clusters contain points {1,…,k} then R(C)≤ 2R(C * ) (triangle inequality)
Otherwise suppose that one of these clusters contains two or more of the points in {1,…,k} . These points are at distance at least R k from each other. Thus clusters must have radius
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