Theoretical analysis of picture fuzzy clustering

Journal of Computer Science and Cybernetics, V.34, N.1 (2018), 17–31 DOI 10.15625/1813-9663/34/1/12725 THEORETICAL ANALYSIS OF PICTURE FUZZY CLUSTERING PHAM THI MINH PHUONG1, PHAM HUY THONG, LE HOANG SON Vietnam National University, Ha Noi, Viet Nam 1phamthiminhphuong t60@hus.edu.vn
Abstract. Recently, picture fuzzy clustering (FC-PFS) has been introduced as a new computatio- nal intelligence tool for various problems in knowledge discovery and pattern recognition. However, an important question that was lacked in the related researches is examination of mathematical pro- perties behind the picture fuzzy clustering algorithm such as the convergence, the boundary or the convergence rate, etc. In this paper, we will prove that FC-PFS converges to at least one local minimum. Analysis on the loss function is also considered. Keywords. Convergence analysis, picture fuzzy sets, picture fuzzy clustering. 1. INTRODUCTION One of the most efficient tools in pattern recogntion and knowledge discovery is fuzzy clustering in which the uncertainty and vagueness of data can be handled sucessfully. Fuzzy clustering, as its reminiscent names recalled, uses a membership function to assign for each data elements in the original dataset. The decision of an appropriate cluster depends on the membership values, that is to say, a greater one implies the inclusion. Fuzzy clustering sucessfully handle the problem of crisp clustering in which a data element can belong to many clusters at the same time [1, 2]. However, it was deployed on the traditional fuzzy set, which shows some limitations in dealing with practical scenarios like voting [3]. A new extension of the fuzzy set called the Picture Fuzzy Set (PFS) was presented by Cuong in [3, 4] to handle such the problem. A PFS is characterized by three membership degrees: positive, neutral, and negative degrees. In the real case of voting applications, ‘positive’ refers to the support for a candidate, ‘negative’ in reverse shows the opposition, and ‘neutral’ reflects the hesistant group who do not agree and disagree. There are many other cases to demonstrate the usage and practical necessity of the PFS [5]. Picture Fuzzy Set has been applied to decision making problems as in the works of Wei [6, 7, 8, 9, 10]. In these researches, the authors have applied picture aggregation opeartors and picture fuzzy entropy in multi-attribute decision problems. Some new operators based on the cosine function and their weighted variants have been utilized for recommendation of products [8]. In [9], picture Bonferroni mean operators have been given in the view of software suppliers. The 2-tuple linguistic picture operators were also examined in [10, 11]. Yang et al. [12] extended the notion of picture fuzzy soft set. Other decision making pro- cedures in the picture fuzzy set can be retrieved in [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Wei [23] summarized some similarity measures in the picture fuzzy set. Indeed, Singh [24] proposed correlation coefficients for picture fuzzy sets. Zhang [25] designed Picture Fuzzy c© 2018 Vietnam Academy of Science & Technology 18 PM PHUONG, PH THONG, LH SON Filters. Other researches regarding picture operators, picture fuzzy rules and database can be found in [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Picture fuzzy clustering (FC-PFS) is a generalization of the traditional fuzzy clustering algorithm [38]. By adding a new membership to the fuzzy set to denote the vagueness of prototype parameters, the FC-PFS has already covered situations that require human opinions as in above. It gives precise results for clustering which has been proven through numerous researches recently [39, 40, 41, 42, 43, 44, 45]. FC-PFS showed significant roles in weather nowcasting from satellite image sequences [42], brain tumor segmentation [5], recommender systems [40], and stock prediction [39]. However, to create a solid constructed basis for the algorithm, it is necessary to per- form the theoretical analysis. Proving the convergence of picture fuzzy clustering is of an important role in understanding the algorithm and how it is evolved. In this paper, the con- vergence of the FC-PFS algorithm is proven and some properties of its such as the boundary of the loss function are expanded. The similarities and differences between this algorithm and other clustering methods are compared. Analysis on the loss function is also considered. Section 2 recalls the general definition of the picture fuzzy set. The convergence ac- companied with some propositions is proven followed by the Zangwill theorem in Section 3. Section 4 validates the way to calculate the boundary of the loss function and describe the changing of the loss function until convergence. The final section draws the conclusion and delineates the future research directions. 2. PRELIMINARY Definition 1. [3] A picture fuzzy set (PFS) E on the universe Y is E = (y, µE(y), ηE(y), γE(y)‖y ∈ Y , (1) where µE(y) ∈ [0, 1], ηE(y) ∈ [0, 1], and γE ∈ [0, 1] are the positive, neutral, and negative memberships of y in Y satisfying µE(y) + ηE(y) + γE(y) ≤ 1,∀y ∈ Y. (2) Definition 2. [38] Assume Y is a dataset ofN points inR dimensions and µkj = µkj(y), ηkj = ηkj(y), ξkj = ξkj(y), 1 ≤ j ≤ C, 1 ≤ k ≤ N, C is a number of clusters, Vj is the cluster center j, 1 ≤ j ≤ C, m is fuzzifier, α is exponent coefficient. The picture fuzzy clustering model is Jm = N∑ k=1 C∑ j=1 (µkj(2− ξkj))m‖Yk − Vj‖2 + N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj)→ min, (3) where ξkj = 1− (µkj + ηkj + γkj), (4) with constraints µkj + ηkj + ξkj ≤ 1, (5) µkj ∈ [0, 1], ηkj ∈ [0, 1], ξkj ∈ [0, 1], (6) THEORETICAL ANALYSIS OF PICTURE FUZZY CLUSTERING 19 C∑ j=1 (µkj(2− ξkj)) = 1, (7) C∑ j=1 (ηkj + ξkj C ) = 1. (8) Let us denote, Uc $ {U = [µkj ] ∈ Rc×n : µkj satisfies (1) ∀i, k}, Nc $ {N = [ηkj ] ∈ Rc×n : ηkj satisfies (1) ∀i, k}, Zc $ {Z = [ξkj ] ∈ Rc×n : ξkj satisfies (1) ∀i, k}. It was shown in [38] that (U∗, V ∗, N∗, Z∗) might be a local minimum of Jm if and only if for any m > 1 µ∗kj = 1 C∑ i=1 (2− ξ∗kj) (‖Yk − V ∗j ‖ ‖Yk − V ∗i ‖ ) 2 m−1 , (1 ≤ j ≤ C, 1 ≤ k ≤ N), (9) η∗kj = e−ξ ∗ kj C∑ i=1 e−ξ ∗ ki ( 1− 1 C C∑ i=1 ξ∗kj ) , (1 ≤ j ≤ C, 1 ≤ k ≤ N), (10) ξ∗kj = 1− (µ∗kj + η∗kj)− (1− (µ∗kj + η∗kj)α) 1 α , (1 ≤ j ≤ C, 1 ≤ k ≤ N), (11) V ∗j = N∑ k=1 (µ∗kj(2− ξ∗kj))mYk N∑ k=1 (µ∗kj(2− ξ∗kj))m , (1 ≤ j ≤ C, 1 ≤ k ≤ N). (12) The following describes the FC-PFS algorithm [38]. Picture Fuzzy Clustering algorithm 1. Input: Data Y with N elements; C is number of clusters, threshold ; fuzzifier m; exponent α; maxstep ≥ 0. 2. Initialize µ0kj ← random, η0kj ← random, ξ0kj ← random, (1 ≤ j ≤ C), (1 ≤ k ≤ N) satisfying constraints (5-8). 3. For each iteration t, update µtkj , η t kj , ξ t kj , V t j following equations (9-12) respectively. 4. Until ‖µt − µt−1‖+ ‖ηt − ηt−1‖+ ‖ξt − ξt−1‖ maxstep, stop. 5. Output: matrices µ, η, ξ and centers V . 20 PM PHUONG, PH THONG, LH SON 3. CONVERGENCE OF PICTURE FUZZY CLUSTERING In this section, we explore some propositions which ensure the convergence of FC-PFS. Proposition 1. Let φ : Uc → R, φ(U) := Jm(U, V,N,Z), where V,N,Z are fixed. Then, U∗ ∈Mfc is a strict local minimum of φ if and only if U∗ is calculated as in eq. (9). Proof. Since µkj has two constrains in eqs. (4), (6), we consider the relaxed minimization of φ(U) via Lagrange multipliers. Let λ = (λ1, λ2, ..., λn) be the multipliers, and L(U, λ) be the Lagrangian L(U, λ) = N∑ k=1 C∑ j=1 (µkj(2− ξkj))m‖Yk − Vj‖2 + N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj) − λk  C∑ j=1 (µkj(2− ξkj))− 1  . Then, ∂L(U, λ) ∂µkj = mum−1kj (2− ξkj)m‖Yk − Vj‖2 − λk(2− ξkj) = 0 at U∗, and calculate the second-order derivative of L(U, λ) ∂ ∂µst ( ∂L(U, λ) ∂µkj ) = { (m− 1)mµm−2kj (2− ξkj)m‖Yk − Vj‖2 if s = k, t = j 0 otherwise. Now, substitute the updated formula of µkj into the second-order derivation of L(U, λ) to calculate the Hessian matrix H(U∗). It follows that H(U) is a nonzero entries matrix, whose diagonal elements are αkj,kj = (m− 1)m(µ∗kj)m−2(2− ξkj)m‖Yk − Vj‖2 = (m− 1)m(2− ξkj)m‖Yk − Vj‖2 1( C∑ i=1 (2− ξkj) (‖Yk − Vj‖2 ‖Yk − Vi‖2 ) 2 m−1 )m−2 > 0 and αst,kj = 0 for s 6= k and t 6= j. The Hessian of φ at U∗ has all positive eigenvalues which are αkj,kj , 1 ≤ k ≤ N, 1 ≤ j ≤ C. It is sufficient to show that U∗ is a strict local minimum of φ.  Next, we fix U ∈ Mc, N ∈ Nc, Z ∈ Zc and consider the minimization of Jm in variables V = {Vi}. THEORETICAL ANALYSIS OF PICTURE FUZZY CLUSTERING 21 Proposition 2. Let ψ : RCS → R, ψ(V ) $ Jm(U, V,N,Z), where U,N,Z are fixed. Then, V ∗ is a strict local minimum of ψ if and only if V ∗i , 1 ≤ i ≤ C is calculated via the updated formula in eq. (12). Proof. Since there is no constrains for V , in order to minimize ψ over RCS , it is necessary to require 5Viψ(V ∗) to vanish for every i, ∂ψ(V ) ∂Vj = N∑ k=1 (ukj(2− ξkj))m(−2Yk + 2Vj) = 0 at V ∗, then take the second-order derivative, ∂ ∂Vs ( ∂ψ(V ) ∂Vj ) =  N∑ k=1 (ukj(2− ξkj))m > 0 if s = j 0 otherwise. The Hessian matrix of ψ(V ) at V ∗ has all positive eigenvalues. Therefore, V ∗ is sufficient to be minimum point of ψ(V ).  A similar way to prove that η∗ is sufficient to minimize Jm when U, V, Z are fixed in their spaces. Proposition 3. Let f : N → R $ Jm(U, V,N,Z) where U, V, Z are fixed. Then, N∗ is a strict local minimum of f if and only if ηkj , 1 ≤ k ≤ n, 1 ≤ j ≤ C is calculated via eq. (10). Proof. Since each ηkj has it own constrains in eq.(7), we consider the minimization of f(N) via Lagrange multipliers obtained constrains. Let β = (β1, β2, ..., βn) be the multipliers, and L(N, β) be the Lagrangian L(N, β) = N∑ k=1 C∑ j=1 (µkj(2− ξkj))m‖Yk − Vj‖2 + N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj) − βk  C∑ j=1 ( ηkj + ξkj C ) − 1  . Since N∗ is the root of equation system, ∂L(N, β) ∂ηkj = ln ηkj + 1− βk + ξkj = 0, and ∂ ∂ηst ( ∂L(N, β) ∂ηkj ) =  1 ηkj > 0 if s = k; t = j 0 otherwise. The Hessian matrix of L(N∗, β) at N∗ has all positive eigenvalues and N∗ also minimizes f(N).  22 PM PHUONG, PH THONG, LH SON The updated formula for the neutral degree is h(ζkj) = 1−(µkj+ηkj)−(1−(µkj+ηkj)α) 1α , where h : Z → R which is based on the Yager’s operator. When the neutral and refusal de- grees of each elements increase, the entropy decreases. When the update of centroids change in minor variation, Jm may slowly increase to the convergence point. Now, to show that the algorithm makes Jm converge, we use the Zangwill theorem below. Proposition 4. [1] Let f : Df ∈ Rm → R and S = {y∗ ∈ Df : f(y∗) < f(y) ∀ y ∈ Bo(y∗, r)}. Let A : Df → Df be an iterative algorithm and yk+1 = E(yk); k = 0, 1, ... If E is continuous on Df \S, g is a descent function for A,S and the iterative sequences E(Yk) : k = 0, 1, 2, ...; y0 ∈ Df ⊂ K are contained in a compact set K ⊆ Df for arbitrary y0 ∈ Df , then for each iterative sequence Yk generated by E, we have either Yk terminates at a solution y∗ ∈ S or there exists a subsequence ykj ⊂ yk so that ykj → y∗ ∈ S. To apply the Zangwill theorem, we need to show that Jm is a descent function and the algorithm is continuous on [0, 1]4 \ S. Then, we only need to show that Jm is a descent function. Now, let Pm be the algorithm to update the parameters in eqs. (9-12). Proposition 5. Let S = {(U∗, V ∗, N∗, Z∗) : Jm(U∗, V ∗, N∗, Z∗) < Jm(U, V,N,Z),∀(U, V,N,Z) ∈ Bo((U∗, V ∗, N∗, Z∗), r)}. Then Jm is descent function for Pm, S. Proof. Since the norm function and the exponent function are continuous, we call the sum of products of such functions as Jm. Obviously, Jm is also continuous on Mfc ×RCS . Suppose (U, V,N,Z) /∈ S then Jm(Pm(U, V,N,Z)) = Jm(P1 ◦ P2 ◦ P3 ◦ P4(U, V,N,Z)) = Jm(P1 ◦ P2 ◦ P3(U, V,N, h(Z))) < Jm(P1 ◦ P2 ◦ P3(U, V,N,Z)) = Jm(P1 ◦ P2(U, V, f(N), Z)) < Jm(P1 ◦ P2(U, V,N,Z)) = Jm(P1(U,ψ(V ), N, Z)) < Jm(P1(U, V,N,Z)) = Jm(φ(U), V,N, Z) < Jm(U, V,N,Z). Hence Jm is a descent function.  However, in some cases, Jm will slowly increase because of updating of η. Because the difference between period centroids and the next centroids changes very small, it still guarantees that Jm converges. THEORETICAL ANALYSIS OF PICTURE FUZZY CLUSTERING 23 4. SOME PROPERTIES 4.1. Property of the loss function We consider the loss function Jm = N∑ k=1 C∑ j=1 (µkj(2− ξkj))m‖Yk − Vj‖2 + N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj). We also know that the first part J1 = N∑ k=1 C∑ j=1 (µkj(2 − ξkj))m‖Yk − Vj‖2 converges to a value called M . Now, we find the upper bound and lower bound of the second part. Let J2 = N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj). We see that ηkj(ln ηkj + ξkj) ≤ ηkj(ln ηkj + 1− ηkj) ≤ 0. Indeed, consider f(y) = y(ln y + 1− y) with y ∈ [0, 1]. Therefore, f ′(y) = 2 + ln y − 2y. f ′(y) = 0↔ y = 1 or y = −1 2 W ( − 2 e2 ) . From f(1) = 0 and f ( −1 2 W ( − 2 e2 )) < 0 we get f(y) ≤ 0, ∀y ∈ [0, 1]. Therefore, ηkj(ln ηkj + ξkj) ≤ 0 and it leads to J2 = N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj) ≤ 0. On the other hand, ηkjξkj ≥ 0. Let us consider g(y) = y. ln y where y ∈ [0, 1], we have g′(y) = 1 + ln y, g′(y) = 0↔ y = 1 e , and f ( 1 e ) = −1 e is the minimal value of this function. We have J2 = N∑ k=1 C∑ j=1 ηkj(ln ηkj + ξkj) ≥ N∑ k=1 C∑ j=1 ( −1 e + 0 ) = −1 e ×N × C. 24 PM PHUONG, PH THONG, LH SON Figure 1. Loss function of Haberman dataset [46] Figure 2. Loss function of Wdbc dataset [46] THEORETICAL ANALYSIS OF PICTURE FUZZY CLUSTERING 25 Figure 3. Loss function of Iris dataset [46] Figure 4. Loss function of Glass dataset [46] 26 PM PHUONG, PH THONG, LH SON Therefore, the upper bound of Jm is M , the lower bound is M − 1 e ×N × C. In Fig.1, the loss function of Habamen data takes 6 iterations to converge to the local minimum. Because of the update of η, it increases gradually from the second iteration but still reaches the convergence. The loss function of the Wdbc dataset in Fig.2 decreases slowly in each iteration and converges at the 16th iteration. Because the initialization elements are random, the value of the first iteration is also random. However, from the second step the value of J decre- ases significantly and from the 8th step, the stability appears and the loss function slowly converges. From the 2nd iteration in Figs.3 and 4, the loss functions of Iris and Glass datasets decrease but sometimes they increase slightly and converge to a stable point. 4.2. Property of centroid We can see the updating of centroids in each iteration similar to the update of parameters in Gradient Descent. From eq. (11), we have V t+1j = N∑ k=1 (µtkj(2− ξtkj))mYk N∑ k=1 (µtkj(2− ξtkj))m = V tj − 1 N∑ k=1 (µtkj(2− ξtkj))m . ( N∑ k=1 (µtkj(2− ξtkj))m(−2Yk + 2V tj ) ) = V tj − αtj .∇VjJ tm(V tj ), where αtj = 1 N∑ k=1 (µtkj(2− ξtkj))m . 5. CONCLUSIONS This paper presented some theoretical properties of FC-PFS and proved the convergence of this algorithm. We have pointed out that this algorithm converges to at least local minimum which guaranties to archive acceptable solutions. Specifically, Propositions 1 to 5 stated that the membership matrices and cluster centers converge if and only if their values are computed by updated equations. Moreover, the objective function is descent in the domain. Some properties of PF-PFS were also considered such as the boundary of the loss function. 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Smarandache, “Interval complex neutrosophic set: for- mulation and applications in decision-making”, International Journal of Fuzzy Systems, vol. 20, no. 3, pp. 986–999, 2018. Received on June 07, 2018 Revised on July 23, 2018