After the processing of fuzzy clustering, we get center matrix, fuzzy partition matrix and the values of the objective function during iterations.
After giving the dataset with the required number of clusters (c) to FCM as input, it generates U (membership matrix) and V (cluster center matrix).
The front of the water (swelling front) penetrated progressively toward the matrix center, while the matrix front (eroding front) apparently maintained the same position.
Intra- (bottom-left matrix triangle) and inter-hemisphere (top-right matrix triangle) correlations are shown for all pairwise comparisons between visual areas V1, V2, V3, hV4, VO1-2, and V3A-B for resting fixation (left matrix), resting eyes shut (center matrix), and movie viewing (right matrix) experiments.
As one of the main results, we showed that for every system ( S λ ) there is a naturally associated limiting Weyl disk D + consisting of complex 2 n × 2 n matrices, whose center P + and matrix radius R + can be explicitly calculated by the formulas in (2.12).
The empirical estimator of HSIC for a finite sample of points X and Y from x and y was shown in (Gretton et al., 2005) to be (3) where tr is the trace of the products of the matrices, H is a centering matrix (where δ i, j )=1 if i= j and δ i, j )=0 otherwise) and K and L are the kernel matrices of the two data sets of size n× n, n being the number of observations/individuals of the study.
The empirical estimator of HSIC for a finite sample of points X and Z from x and z with p x, z ) was shown in Gretton et al. (2005) to be (9) where tr is the trace of the products of the matrices, H is a centering matrix (where δ i, j )=1 if i= j and δ i, j )=0 otherwise), K and L are the kernel matrices on the two random variables of size m× m and m is the number of observations.
We denote G c as the centered kernel matrix as G c = PGP, where P is the centering matrix, G is the kernel matrix, I N is the N × N identity matrix, is a column vector of N ones.
The pseudo- F statistic is defined as where tr is the trace function of a matrix, H = X(X T X)−1 X T is the hat (projection) matrix of the design matrix X, G is Gower's centered matrix and n and m is the number of samples and the number of predictors, respectively.
With the M × M centering matrix H with elements we finally obtain the matrix B = HCH.
