K-Means Clustering Problem

Ahmad Sabiq

Febri Maspiyanti

Indah Kuntum Khairina

Wiwin Farhania

Yonatan

What is k-means?

• To partition n objects into k clusters, based on

attributes.

– Objects of the same cluster are close � their

attributes are related to each other.

– Objects of different clusters are far apart � their

attributes are very dissimilar.

Algorithm

• Input: n objects, k (integer k ≤ n)

• Output: k clusters

• Steps: 1. Select k initial centroids.

2. Calculate the distance between each object and each centroid.

3. Assign each object to the cluster with the nearest centroid.

4. Recalculate each centroid.

5. If the centroids don’t change, stop (convergence).

Otherwise, back to step 2.

• Complexity: O(k.n.d.total_iteration)

Initialization

• Why is it important? What does it affect?

– Clustering result � local optimum!

– Total iteration / complexity

Good Initialization

3 clusters with 2 iterations…

Bad Initialization

3 clusters with 4 iterations…

Initialization Methods

1. Random

2. Forgy

3. Macqueen

4. Kaufman

Random

• Algorithm:

1. Assigns each object to a random cluster.

2. Computes the initial centroid of each cluster.

Random

Random

Random

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35

Forgy

• Algorithm:

1. Chooses k objects at random and uses them as the initial

centroids.

Forgy

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35

MacQueen

• Algorithm:

1. Chooses k objects at random and uses them as the initial

centroids.

2. Assign each object to the cluster with the nearest

centroid.

3. After each assignment, recalculate the centroid.

MacQueen

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35

MacQueen

MacQueen

MacQueen

MacQueen

MacQueen

MacQueen

MacQueen

MacQueen

MacQueen

Kaufman

Kaufman

Kaufman

Kaufman

Kaufman

Kaufman

Kaufman

Kaufman

Kaufman

d = 24,33

D = 15,52

C = 0

Kaufman

C = 0

C = 0

C = 0

C = 0

C = 0

Kaufman

C = 0

C = 0

C = 0

C = 0

C = 0

∑C1 = 2,74

Kaufman

∑C1 = 2,74

∑C2 = 12,,21

∑C3 = 12,36

∑C3 = 8,38

∑C5 = 52,55

∑C6 = 55,88

∑C7 = 53,77

∑C8 = 51,16

∑C9 = 42,69

Kaufman

∑C1 = 2,74

∑C2 = 12,,21

∑C3 = 12,36

∑C3 = 8,38

∑C5 = 52,55

∑C6 = 55,88

∑C7 = 53,77

∑C8 = 51,16

∑C9 = 42,69

Reference

1. J.M. Peña, J.A. Lozano, and P. Larrañaga. An Empirical Comparison of Four Initialization Methods for the K-Means Algorithm. Pattern Recognition Letters, vol. 20, pp. 1027–1040. 1999.

2. J.R. Cano, O. Cordón, F. Herrera, and L. Sánchez. A Greedy Randomized Adaptive Search Procedure Applied to the Clustering Problem as an Initialization Process Using K-Means as a Local Search Procedure. Journal of Intelligent and Fuzzy Systems, vol. 12, pp. 235 – 242. 2002.

3. L. Kaufman and P.J. Rousseeuw. Finding Groups in Data: An Introduction to Cluster Analysis. Wiley. 1990.

Questions

1. Kenapa inisialisasi penting pada k-means?

2. Metode inisialisasi apa yang memiliki greedy

choice property?

3. Jelaskan kompleksitas O(nkd) pada metode

Random.