Assign and Update

Starting Centroids Matter

The same points can be assigned differently when the starting centroids change. This lesson compares two chosen starts and one exact update from each.

highlighted = computed this step

Start A

Use the same one-dimensional points: 0, 2, 8, and 10. Start A uses centroids 0 and 10, so points 0 and 2 go to c1, while 8 and 10 go to c2.

Start A: c1=0,  c2=10⇒[c1,c1,c2,c2]\text{Start A: } c_1=0,\; c_2=10\Rightarrow [c_1,c_1,c_2,c_2]
Starting centroids matterSame points with two chosen starting centroid pairs.starting centroids mattersame points: 0, 2, 8, 10same points, different starting centroids, different one-step clusters and new centroids; one deterministic step fromchosen starts; NOT convergence; NOT randomness; NOT optimality; NOT future behaviorstartcentroidsassigned clustersnew centroidsd^2 rowsStart Ac1=0, c2=10c1: 0 2; c2: 8 10c1'=1, c2'=9(0,100); (4,64); (64,4); (100,0)Start Bc1=0, c2=2c1: 0; c2: 2 8 10c1'=0, c2'=20/3(0,4); (4,0); (64,36); (100,64)assignment rule: choose the smaller squared distance; ties use the lowest centroid index

Start B

Start B uses centroids 0 and 2 on the same points. Now point 0 goes to c1, while 2, 8, and 10 go to c2.

Start B: c1=0,  c2=2⇒[c1,c2,c2,c2]\text{Start B: } c_1=0,\; c_2=2\Rightarrow [c_1,c_2,c_2,c_2]
Starting centroids matterSame points with two chosen starting centroid pairs.starting centroids mattersame points: 0, 2, 8, 10same points, different starting centroids, different one-step clusters and new centroids; one deterministic step fromchosen starts; NOT convergence; NOT randomness; NOT optimality; NOT future behaviorstartcentroidsassigned clustersnew centroidsd^2 rowsStart Ac1=0, c2=10c1: 0 2; c2: 8 10c1'=1, c2'=9(0,100); (4,64); (64,4); (100,0)Start Bc1=0, c2=2c1: 0; c2: 2 8 10c1'=0, c2'=20/3(0,4); (4,0); (64,36); (100,64)assignment rule: choose the smaller squared distance; ties use the lowest centroid index

Compare one step

Start A updates to centroids 1 and 9. Start B updates to centroids 0 and 20/3. Same points, different starting centroids, different one-step clusters and new centroids. This is one deterministic step from chosen starts, not a guarantee about convergence or the best clustering.

Start A new=(1,9),Start B new=(0,20/3)\text{Start A new}=( 1,9 ),\quad \text{Start B new}=( 0,20/3 )
Starting centroids matterSame points with two chosen starting centroid pairs.starting centroids mattersame points: 0, 2, 8, 10same points, different starting centroids, different one-step clusters and new centroids; one deterministic step fromchosen starts; NOT convergence; NOT randomness; NOT optimality; NOT future behaviorstartcentroidsassigned clustersnew centroidsd^2 rowsStart Ac1=0, c2=10c1: 0 2; c2: 8 10c1'=1, c2'=9(0,100); (4,64); (64,4); (100,0)Start Bc1=0, c2=2c1: 0; c2: 2 8 10c1'=0, c2'=20/3(0,4); (4,0); (64,36); (100,64)assignment rule: choose the smaller squared distance; ties use the lowest centroid index