Fuzzy Principal Component Analysis FPCA

For the data collected on p variables for n cases, PCA performs analyses in the -dimensional space defined by p variables and p-dimensional space defined by n cases, hi PCA, straight fines are sought that best fit the clouds of points in the vector spaces (of variables and cases), according to the least-squares criterion. This, in turn, yields the principal components (factors) that result in the maximum sums of squares for the orthogonal projections. Consequently, a lower-dimensional vector subspace is recovered that best represents the original vector space. Although the first factor is extracted so as to capture the variance to the maximum extent, it can seldom capture the variance in its entirety. What remains should, therefore, be recovered by another (second) factor, a third, and so on. However, the number of factors thus extracted will never exceed the number of original variables. 

Call DETERMiNEFuzzYMEMBERSHip(ot) with the value of a computed above, and determine the optimal value of the fuzzy membership degrees. 

Using the fuzzy membership degrees determined above, compute the fuzzy covariance matrix C, and compute its eigenvalues and eigenvectors these are the fuzzy principal components and the corresponding scatter values. 

Recall that a is the membership degree corresponding to the farthest outlier of the data set. In the algorithm above, the fuzzy covariance matrix is determined as 

---

Cundari, T.R., Sarbu, C. and Pop, H.F. (2002) Robust fuzzy principal component analysis (FPCA). A comparative study concerning interaction of carbon-hydrogen bonds with molybdenum—oxo bonds. /. Chem. Inf. Comp. Sci., 42, 1363. 

We compared fuzzy principal component analysis (FPCA) to classical PCA methods to characterize lanthanum, the 14 lanthanides, and the other... 

Fuzzy Principal Component Analysis (FPCA). A Comparative Study Concerning Interaction of Carbon-Hydrogen Bonds with Molybdenum-Oxo Bonds. 

---