a.

• PQR and XYZ are clustered
• distance is low for PQR and distance is high for XYZ and vice versa

b.

They are all somewhat similar to each other, but the range of attribute one for PQR attributes 1 and 2 is  ≥ 15, but the range is <= 4 for attributes 3,4,5,6. So PQR is more similar for attributes 3,4,5,6 than for attributes 1,2

For XYZ, the ranges are ordered by attributes smallest to greatest (most similar to least similar) \(a_3 < a_1 = a_6 = a_5 < a_4\)

c.

PQRXYZ (no good sub clusters). There are no obvious clusterings.

d.

a₄, and a₅

e.

b is the one we want to do subspace clustering on, because there are no good clusters in the current dimensionality

f.

• PQR (defining attributes a₂ and a₃ )
• XYZ (defining attributes a₄ and a₅ )

g.

They are axis aligned, meaning that axis aligned algorithms are most effective (clique, random-projections)

h.

Clique algorithm is vunerable to binning, meaning that some points will be mis-clustered

i.

• Q and R are the randomly selected points
• PQR is the subspace cluster (a₂ and a₃ are defining attributes)

2.

a.

linear combinations of existing dimensions, but not axis aligned

b.

The data is not axis aligned, meaning random projection cannot "chunk" data effectively.

c.

Projective k-means, as it can handle linear cominations of existing dimensions

d.

• choosing the number of clusters is hard
• PCA is expensive and must be applied many times

3.

a.

There are usually > 5 points in each cell (as far as I can tell, the graph is hard to read)

b.

• estimate the number of cells that are filled change 0.1 increment dimension to 0.2 increments
• 12 lines
• Each line is 5 cells long
• 60 cells would need to be evalutated