# Dimensionality Reduction using Maximum Variance Unfolding (MVU)

# Original correlation matrix with values
1.000000	-0.985771	0.978874
-0.985771	1.000000	-0.954350
0.978874	-0.954350	1.000000

# Original correlation matrix with only signs
1  	1 -	1
2 -	2  	2 -
3  	3 -	3

# Kernel matrix
6.545272	-8.342967	1.797695
-8.342967	10.672854	-2.329886
1.797695	-2.329886	0.532191

# Sorted Normalised Eigenvalues (ei) of Kernel matrix
e1 0.997598
e2 0.002402
e3 0.000000

# Number of Principal Components (Nv) that account for (theta)
Nv=1
theta=0.997000

# Sorted Eigenvectors (Vj) of Kernel matrix
V1(PCA1)	V2(PCA2)	V3(PCA3)
-0.607381	0.545669	-0.577350
0.776254	0.253173	-0.577350
-0.168873	-0.798842	-0.577350

# Objectives picked by each sorted PCA denoted by 1 else by 0
V1(PCA1)	V2(PCA2)	V3(PCA3)
f1 1		f1 1		f1 1
f2 1		f2 1		f2 0
f3 1		f3 1		f3 1

# Objectives set after reduction by Eigenvalue Analysis
Fe = { f1 f2 f3 }

# Matrix of objectives correlated based just on their signs (Equation 3.1)
  1	  2	  3
1 0	1 0	1 1
2 0	2 0	2 0
3 1	3 0	3 0

# Computation of Tcor (Equation 4)
Tcor = 1.0-e1(1.0-M2sigma/M) = 1.0-0.997598(1.0-1/3) = 0.334934

# Without Threshold: Identically Correlated set of objectives in Fe
S1={ f1 f3 }
S2={ f2 }
S3={ f3 f1 }

# Threshold Based: Identically Correlated set of objectives in Fe
S1={ f1 f3 }
S2={ f2 }
S3={ f3 f1 }

# Objective selection score (ci=sum(ei*|fij|))
f1 0.605923
f2 0.774390
f3 0.168467

# Variance accounted by each objective over all Principal Components (ciM=sum(ei*fij^2))
f1 0.368741
f2 0.601277
f3 0.029982

# Sorted variance accounted by each objective over all Principal Components (ciM=sum(ei*fij^2))
f2 0.601277
f1 0.368741
f3 0.029982

# Final Set
(Fs) = 1 2
Size = 2