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276 CHAPTER 11 KIDNEY-INSPIRED ALGORITHM AND FUZZY CLUSTERING

Here, “C i ” indicates the ith candidate solution in “P.” F(C i ) indicates the fitness of “C i .” “n” and “m”
represent the number of instances of a particular dataset and number of cluster centers. “o r ” represents
the rth instance of the dataset, “c i,j ” is the jth cluster center of ith (C i ) solution in the population. Here
“k” is a constant and “d” is a small valued constant. The term “o r c i,j ” calculates the Euclidean dis-
tance, where “o r ” is the data object and “c i,j ” is the cluster center. The value of “d” is set to 0.1. Using
itr
F(C i ), the intra cluster distance is minimized. Based on the fitness of each C i , the best and worst clus-
itr itr itr
ter center are selected as C best and C worst respectively. In the solute movement phase, C best has
been used for improvising the solution in the population P by using Eq. (11.1) as follows:
0
C ¼ C i + rand C best C i Þ (11.7)
ð
i
0
In Eq. (11.7), C i , C i , and C best are the ith cluster center vector after the movement, the previous ith
cluster center vector, and the best cluster center vector (discovered so far) in the population P, respec-
tively. After the solute movement process, the resultant solutes (cluster centers) may be visualized as
0
itr
0
itr
0
M ¼{C 1 , C 2 , …C n }, M holds all cluster center vectors on iteration “itr.” Filtration rate (fr) is com-
puted on P using Eq. (11.2) as follows:
0 n 1
X
ðÞ
B FC i C
i¼1,C i 2P
B C
fr ¼ α B C (11.8)
B n C
@ A
In Eq. (11.8), objective function F(C i ) is calculated for each C i using Eq. (11.6). If new cluster center
0
vectors C i satisfied fr, i.e., if (rand(1) fr) then it is considered for FB. If not, it is added to W. Before
0
adding into FB, it is tested with the worst solution C worst , i.e., if F(C i ) is greater than or equal to F-
0 0
(C worst ), then C i is added into FB. If not, it is added into W. After that, all the C i that are added into
0
W go through the excretion process and all the C i that are added into FB go through the secretion pro-
0
cess. All the C i go through the movement of solute process using Eq. (11.1) as in Eq. (11.7), and are
checked with fr. If they satisfy the fr and are better than C worst , then they are added into FB. If not, they
0 0 0
are added into W after random modification on C i (i.e., C i ¼C i rand(1)). After completion of the
selection process of FB, the population P is updated using FB. This complete process is iterated until
itr
the stopping criteria are met or maximum iteration is reached. Finally, the best solution C best from the
itr
population P is selected, where C best is a vector of cluster centers. Consequently, this obtained cluster
centers is used with FCM for cluster analysis.
11.5.2.2 Cluster analysis using optimal cluster centers
Unlike the original working mechanism of FCM, which starts with randomly initialized cluster centers,
the obtained cluster centers from Section 11.5.2.1 are assigned as initial cluster centers of FCM while
performing clustering. With these cluster centers, the fuzzy partition matrix was evaluated using
Eq. (11.4) and the fuzzy membership objective value was observed. Then, the fuzzy cluster centers
are updated (Eq. 11.5) according to the fuzzy partition matrix. If a change in fuzzy objective values
in successive iterations is found to be insignificant, then the FCM iteration is stopped and the resultant
partition matrix is used for cluster analysis on the considered dataset.