Fuzzy Clustering Outline Similaritybased Fuzzy ClusteringSFC Similaritybased Fuzzy
Fuzzy Clustering
Outline � Similarity-based Fuzzy Clustering(SFC) � Similarity-based Fuzzy Classification � TSK-Type Fuzzy Neural Network � Principal Component Analysis Similarity-based Fuzzy Clustering(PC-SFC)
SFC-Diagram
SFC � Input data: � First layer: mj, i is the i-th feature in the center of j-th cluster, σj, i represents the i-th feature in the variance of j-th cluster � Second � Third layer � Fourth layer ρ is the user-defined threshold
SFC � O(4)=0 ◦ Increase a new cluster CJ+1 σ0 is the user-defined value and SJ+1 represent the number of the data in the CJ+1 � O(4)=1 ◦ Update the cluster Ca
SFC-Example There does not have any cluster, so we create a new cluster CJ+1(C 1) for the first input data.
SFC-Example � Calculate the Gaussian � Calculate the similarity of each cluster � Find the maximum similarity a=1, Compare with the threshold ρ O(3)≤ρ, so we create a new cluster CJ+1(C 2) �
SFC-Example � Calculate the Gaussian � Find the maximum similarity a=2, Compare with the threshold ρ O(3)>ρ, so we update the cluster C 2 �
SFC-Example
SFC-Result
SFC-Discuss � Input order 0. 25 0. 33 0. 09 0. 21 � The number of feature is large
SFC-Merge � Input: A={C 1, C 2, …CJ} � Calculate the similarity between the cluster i, j represent the cluster Ci and Cj, k represents the feature
SFC-Merge � Merge |qj| cluster
SFC-Merge example τ=0. 36 Θ=0. 1
SFC-Merge example � G 1, 3>τ, so Q={ {2}, {1, 3} }. � We merge C 1 and C 3 to get a new cluster Ca’
SFC-Merge result
Similarity-based fuzzy classification
Similarity-based fuzzy classification � Input: (xp, y(p)), y(p) is the category vector with k components � SFC+Least square ◦ SFC P=1, 2, …m, j=1, 2, …, J, Cj represent the j-th cluster ◦ Least square
Similarity-based fuzzy classification � Least square
TSK-Type Fuzzy Neural Network � Fuzzy if-then rules
TSK-Type Fuzzy Neural Network
TSK-Type Fuzzy Neural Network � Layer 1: � Layer 2: � Layer 3: normalized the degree
TSK-Type Fuzzy Neural Network � Layer 4: � Layer 5:
TSK-Type Fuzzy Neural Network � Suppose that tv=(pv, qv) be the v-th training pattern, where pv=[p 1 v, …, pnv] is the input vector and qv is the desired output. � Let v. o(5) and v. oj(3) denote the actual output of layer 5 and the actual output of node j in layer 3.
TSK-Type Fuzzy Neural Network � We compute the linear system AX=B q represents the desired output and b is the weight
PC-SFC � SFC-Result
PCA � Find out the eigenvalue {λ 1, λ 2, …, λn} of the covariance matrix, where λ 1≥ λ 2≥…≥ λn, and calculate their corresponding eigenvector{w 1, w 2, …wn}. � The eigenvalue is the variance in its corresponding eigenvector.
PC-SFC � First: calculate the distance between the input and the center in the new basis Wj, i. T is the eigenvector � Second: � Third: � Fourth: ‣ Five:
PC-SFC � O(5)=0 ◦ Increase a new cluster CJ+1 � O(5)=1 W is the new basis matrix which is composed of the eigenvector, and I is the identity matrix ◦ Update the cluster Ca Find out the eigenvalue {λ 1, λ 2, …, λn} of ∑’a, where λ 1≥ λ 2≥…≥ λn, and calculate their corresponding eigenvector{wa, 1, wa, 2, …wa, n}
PC-SFC-Example There does not have any cluster, so we create a new cluster CJ+1(C 1) for the first input data.
PC-SFC-Example � Calculate the distance � Computer the Gaussian � Calculate the similarity of each cluster � Find the maximum similarity a=1, Compare with the threshold ρ O(4)>ρ, so we update the cluster C 1 �
PC-SFC-Example � Compare with the threshold ρ O(3)>ρ, so we update the cluster C 1
PC-SFC-Example � Calculate the distance � Computer the Gaussian � Calculate the similarity of each cluster � Find the maximum similarity a=1, Compare with the threshold ρ O(4)≤ρ, so we create a new cluster CJ+1(C 2) �
PC-SFC-Example � Compare with the threshold ρ O(4)≤ρ, so we create a new cluster CJ+1(C 2)
PC-SFC-Result PC-SFC Result
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