Clustering Methods - A summary
Algorithms for clustering from feature vector
Called "Partitional clustering" in Jain and Dude as opposed to "hierarchical clustering"
Clustering feature vectors = finding separation between clusters but these are not known
<<Picture>>
We have a set of points S Want to find subsets $ S_1 , S_2 , \cdots , S_c $ such that $ S=S_1 \cup S_2 \cup \cdots \cup S_c $ and $ S_i \cap S_j = \phi $
need to define "clustering criteria" i.e., a measure of how natural the clustering is.
if c=2,
consider
$ J=tr(S_m ^{-1} S_w) $
where $ S_w $ is "within class scatter matrix"
$ S_w= \sum _{i=1, X_i \in S_1} ^d ||X_i - \mu _1||^2 + \sum _{i=1, X_i \in S_2} ^d ||X_i - \mu _2||^2 $ (2-1)
$ \mu _1 = \frac{1}{|S_1|} \sum _{X_i \in S_1} X_i $, $ \mu _2 = \frac{1}{|S_2|} \sum _{X_i \in S_2} X_i $ (2-2)
$ S_m $ is "mixture scatter matrix"
$ S_m= \sum _{i=1} ^d ||X_i - \mu||^2 $, $ \mu = \frac{1}{d} \sum _{i=1} ^{d} X_i $ (2-3)
Try to find $ S_1 $ and $ S_2 $ that minimize J.
Exhaustive search procedure
Examnple with 6 pattern $ X_1 , X_2 , \cdots , X_6 $
List all partition of 6 points into 2 sets.
<<Someting>>
Evaluate J for each partition
Cansider other J's
$ J=ln|S_m ^{-1} S_w| $ (2-4)
$ J=tr (S_m) - \mu (tr S_w -c) $ (2-5)
C is fixed constant, $ \mu $ : Largrange multiplier
$ J=\frac{tr(S_m)}{tr(S_w)} $ (2-6)
To speed up search "use iterative procedure"
Pick a partition at random
$ S_1={X_1, X_2, X_3}, S_2= {X_4, X_5, X_6} $ (2-7)
Compute J
Consider effect of moving
$ X_1 $ into $ S_2 \Rightarrow \Delta J_{12} $,
$ X_2 $ into $ S_2 \Rightarrow \Delta J_{22} $,
$ X_3 $ into $ S_3 \Rightarrow \Delta J_{33} $,
$ X_4 $ into $ S_1 \Rightarrow \Delta J_{41} $,
$ X_5 $ into $ S_1 \Rightarrow \Delta J_{51} $,
$ X_6 $ into $ S_1 \Rightarrow \Delta J_{61} $
Apply (Simultaneously) all the moves for which $ \Delta J $ is negative, repeat procedure
OR
Apply the move for which $ \Delta J $ is the most negative repeat procedure
Convergence?
If convergence, global minimum? No idea
If c>2, can use similar procedure
If c is unknown, try c=2,3,4, etc (hierarchical clustering)
Look at evolution of J for c increase (similarity scale)
<<Picture>>
An important J
"Square error criterion"
$ J=\sum _{j=1} ^{c} \sum _{X_i \in S_j}||X_i - \mu _ j||^2 $ (2-8)
where $ \mu _j = \frac{1}{|S_j|} \sum _{X_i \in S_j} X_i $ (2-9)
- Good when clusters are compact, well separated
- Sensitive to outliers
$ J=\sum _{j=1} ^{c} \sum _{X_i \in S_j} ||X_i - \mu_j||_{L1} $ is more robust to outliers
Can use other types of similarity measure
$ S(X_1, X_2)=\frac{X_1 \cdot X_2}{||X_1||||X_2||} $ (2-10)
to speed up optimization of J
Use "Nearest mean reclassification rule"
- choose initial partition $ S_1, S_2 , \cdots , S_c $ (2-11)
- calculate $ \mu _1 , \mu _2 , \cdots , \mu _c $
- reclsssify each $ X_i $ to the class of the nearest mean
- If cluster have changed, repeat
Note: $ X_i $ class of its nearest mean
is same as choosing the move for $ X_i $ that moves $ \Delta J $ as negative as possible because $ J=\sum _{j=1} ^{c} \sum _{X_i \in S_j} ||X_i - \mu _j||^2 $ (2-12)
If $ X_{io} \in S_{jo} \rightarrow X_{io} \in S_{\bar{j0}} $,
$ J \Rightarrow \sum _{j=1} ^{c} \sum _{X_i \in S_j} ||X_i - \mu_j||^2 - ||X_{io}-\mu _{jo}||^2 + ||X_{io} - \mu _{\bar {jo}}||^2 $ (2-13)
$ \Delta J $ is as negative as possible when $ ||X_{io} - \mu _{\bar {jo}}||^2 $ is $ \begin{smallmatrix} min\\ j \end{smallmatrix} ||X_i -\mu _j||^2 $ (2-14)
Can use FORGY, CLUSTER
Observation
$ J= \cdots = \sum _{j=1} ^c \frac{1}{|S_j|} \sum _{X_i \in S_j , X_k \in S_j} \frac{||X_i-X_k||^2}{2} $ (2-15)
This is a distance based clustering method.
No need for feature vector
