Well-Separated Clusters and Optimal Fuzzy Partitions

Abstract

Two separation indices are considered for partitions P = {X₁, …, X_k} of a finite data set X in a general inner product space. Both indices increase as the pairwise distances between the subsets X_i become large compared to the diameters of X_i Maximally separated partitions p' are defined and it is shown that as the indices of p' increase without bound, the characteristic functions of X_i' in P' are approximated more and more closely by the membership functions in fuzzy partitions which minimize certain fuzzy extensions of the k-means squared error criterion function.     

A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters

Abstract

Two fuzzy versions of the k-means optimal, least squared error partitioning problem are formulated for finite subsets X of a general inner product space. In both cases, the extremizing solutions are shown to be fixed points of a certain operator T on the class of fuzzy, k-partitions of X, and simple iteration of T provides an algorithm which has the descent property relative to the least squared error criterion function. In the first case, the range of T consists largely of ordinary (i.e. non-fuzzy) partitions of X and the associated iteration scheme is essentially the well known ISODATA process of Ball and Hall. However, in the second case, the range of T consists mainly of fuzzy partitions and the associated algorithm is new; when X consists of k compact well separated (CWS) clusters, X_i , this algorithm generates a limiting partition with membership functions which closely approximate the characteristic functions of the clusters X_i . However, when X is not the union of k CWS clusters, the limiting partition is truly fuzzy in the sense that the values of its component membership functions differ substantially from 0 or 1 over certain regions of X. Thus, unlike ISODATA, the “fuzzy” algorithm signals the presence or absence of CWS clusters in X. Furthermore, the fuzzy algorithm seems significantly less prone to the “cluster-splitting” tendency of ISODATA and may also be less easily diverted to uninteresting locally optimal partitions. Finally, for data sets X consisting of dense CWS clusters embedded in a diffuse background of strays, the structure of X is accurately reflected in the limiting partition generated by the fuzzy algorithm. Mathematical arguments and numerical results are offered in support of the foregoing assertions.     

FUZZY GAMES IIM THE ANALYSIS OF OPTIONS

Fuzzy games model conflict situations with imprecise information, which may be in the strategy sets of the players, or in their preferences or both. This imprecision is due to the ill-defined nature of the context in contra distinction to imperfect and incomplete information in games, where Bayesian methods have been considered.

In fuzzy games, each player sets a threshold level for preference comparison. This leads to the concept of λ_irationality for the i^th player and a corresponding (λ₁ …λ_n)equilibrium for the game. Metagames are particular mathematical structures which attempt to model a player's problem-solving cognitive model in conjunction with the mathematical game. In applications we consider a technique termed the analysis of options technique. We shall present a modification of the underlying algorithm to incorporate fuzzy preferences and λ_irationality.     

A consensus‐ and harmony‐based feedback mechanism for multiple attribute group decision making with correlated intuitionistic fuzzy sets

In this study, an interactive consensus model is proposed for correlated multiple attribute group decision making (MAGDM) problems with intuitionistic triangular fuzzy numbers (ITFNs). The harmony degree (HD) is investigated to determine the degree of maintaining experts' original information while the consensus level is defined as the proximity degree (PD) between an expert and other experts on three levels: evaluation elements of alternatives, alternatives, and decision matrices. Combining HD and PD, a three‐dimensional feedback mechanism is proposed to identify discordant experts, alternatives, and corresponding preference values that contribute less to consensus, and provides advice to reach a higher consensus level. Additionally, visual representation of experts' consensus position within the group is provided. Furthermore, a graphical simulation of future consensus and harmony status, if the recommended values were to be implemented, is also provided. Therefore, our proposed feedback mechanism guarantees that it increases the consensus level of the set of experts while maintaining, as much as possible, experts' original information. Then, the PD‐induced intuitionistic triangular fuzzy correlated averaging (PD‐IITFCA) operator is investigated to aggregate the interactive individual opinions between experts. Finally, the intuitionistic triangular fuzzy correlated averaging (ITFCA) operator is developed to aggregate the evaluation elements of alternatives under correlative attributes. Based on the score and accurate functions of ITFNs, an order relation is proposed to obtain the final solution of alternatives.     

SUBJECTIVE MENTAL WORKLOAD ASSESSMENT BASED ON GENERALIZED FUZZY NUMBERS

In this article, a new method for subjective mental workload assessment is presented based on generalized fuzzy numbers. First, the linguistic values are modeled as generalized fuzzy numbers, which is more efficient in dealing with uncertain information and representing experts' opinions than normal fuzzy numbers due to the fact that the degree of confidence of decision makers' opinions is taken into consideration. In addition, compared with existing methods, the result of the proposed method is more reasonable because it can correctly determine the degree of similarity between the assessment and the linguistic terms. A numerical example is used to show that the proposed method can provide a useful way to handle subjective mental workload assessments in a linguistic manner.     

Two alternative ways for solving the coordination problem in multilevel optimization

The paper describes two new techniques for formulating the coupling between levels in multilevel optimization by linear decomposition, proposed as improvements over the original formulation, now several years old, that relied on explicit equality constraints which were shown by application experience as occasionally causing numerical difficulties. The two new techniques account for the coupling without using explicit equality constraints, thus avoiding the above difficulties and also reducing the computational cost of the procedure. The old and new formulations are presented in detail, illustrated by an example of a structural optimization. A generic version of the improved algorithm is also developed for applications to multidisciplinary systems not limited to structures.The symbolsU andL withX, Y, Z, e.g.XU, XL, denote upper and lower bounds on these variables; other symbols are defined where first used.Notation (vectors indicated by bold-face, or by {}) A vector of cross-sectional areas,A _i - C _i cumulative constraint of thei-th beam - D _i vector defined by (22) - g vector of constraintsg _k for a beam, e.g. stress limits, and local buckling,k = 1 NGB, partitioned in subsets of lengthsNGB _i, each subset corresponding to thei-th beam - G vector of constraintsG _k for the assembled structure, e.g. displacement limits,k = 1 NGA - I vector of cross-sectional moments of inertia,I _i,i = 1 NGB _i - KS(f _i) the Kreisselmeier-Steinhauser function of the set of functionsf _i,i = 1NF, defined by (5) and (5a) - L _i length of thei-th beam - NE number of beams in a framework - N _i vector of the end forces for thei-th beam - NP _i length of vectorP _i - NN _i length of vectorN _i - NSS number of subsystems - NX length of vectorX - P _i vector of parameters in optimization of thei-th beam, comprising elementsP _q,q = 1 NP _i - P _qi q-th element of vectorP _i - SA system analysis - SI input vector of lengthNSI intoSA - SO output vector of lengthNSO fromSA - SSA _i i-th subsystem black box analysis - SSC vector of geometrical variables determining the structure overall shape - SSI _i input vector of lengthNSSI _i intoSSA _i - SSO _i output vector of lengthNSSO _i fromSSA _i - TOL tolerance parameter set by user - U vector of displacements - W weight, equivalent to volume in a homogeneous structure - X vector of design variables at the system level (assembled structure) in optimization by decomposition - Y _i vector of design variables in thei-th beam optimization problem - Z vector of design variables in optimization without decomposition - user-defined coefficient in the KS function, e.g. (5)     

A new polynomial algorithm for a paralle identical scheduling problem

A precedence order is defined based on the release dates of jobs' direct successors. Using the defined precedence order and Heap Sort, a new polynomial algorithm is provided which aims to solve the parallel scheduling problem P|p = 1, r ,outtree|∑C . The new algorithm is shown to be more compact and easier to implement.     

Efficient Pairing Computation on Huff Curves

Abstract

Pairings on elliptic curves are currently of great interest due to their applications in a number of cryptographic protocols such as identity-based encryption, group signatures, short signatures, and the tripartite Diffie-Hellman. Miller's algorithm is the most commonly used method of computing Tate pairing. Denominator elimination can improve Miller's algorithm when the embedding degree has the form 2ⁱ3^j. However, if the embedding degree does not have the above form, how can the speed of Miller's algorithm be increased? In this article, the authors modified Miller's algorithm over Huff curves. It is about 20.38% faster than the original algorithm.     

Fuzzy knowledge combination

A general answer is given to what one should conclude from disagreeing experts. the answer is generalized further to incorporate the experts' credibility weights. the answer rests on a wide range of intuitively based epistemic axioms, scientific and philosophical conjectures, and formal mathematical relationships. A recurring theme is the making of Bellman - Zadeh fuzzy decisions, wherein a decision is the intersection of fuzzy goal and fuzzy constraint subsets of some space of alternatives. Another result is that measures of central tendency, such as the arithmetic mean, make poor knowledge combination operators. Formally, fuzzy knowledge combination operators are sought. the function space of knowledge combination operators ø: K″ → K is shrunk by imposing successive axioms. the final shrunken set is said to consist of admissible knowledge combination operators. Some of its mathematical properties are explored and a simple admissible operator is finally chosen. Knowledge sources X_i: S → K are mappings from epistemic stimuli or questions into a knowledge response set K. the uncertainty of the underlying epistemic situations is captured by the cardinality of K and by the fuzziness of its partial ordering. Admissible knowledge combination operators Aggregate knowledge responses in some desirable way. the arithmetic mean is not admissible. Nor in general is a probabilistic framework even definable in the abstract poset setting employed by this theory. the fuzzy knowledge combination theory is extended by associating general credibility weights with the knowledge sources. A new set of weighting axioms is required to satisfy certain intuitions and to satisfy the admissibility axioms. General weighting functions are obtained and thereby weighted admissible operators are obtained. the weighted mean still proves inadmissible. Appendix I contains a technical glossary and summary of the proposed fuzzy knowledge combination theory. Appendix II contains proofs of the probabilistic uncertainty theorems required for the uncertainty testbed used in the theory.     

On aggregating uncertain information by type‐2 OWA operators for soft decision making

Yager's ordered weighted averaging (OWA) operator has been widely used in soft decision making to aggregate experts' individual opinions or preferences for achieving an overall decision. The traditional Yager's OWA operator focuses exclusively on the aggregation of crisp numbers. However, human experts usually tend to express their opinions or preferences in a very natural way via linguistic terms. Type‐2 fuzzy sets provide an efficient way of knowledge representation for modeling linguistic terms. In order to aggregate linguistic opinions via OWA mechanism, we propose a new type of OWA operator, termed type‐2 OWA operator, to aggregate the linguistic opinions or preferences in human decision making modeled by type‐2 fuzzy sets. A Direct Approach to aggregating interval type‐2 fuzzy sets by type‐2 OWA operator is suggested in this paper. Some examples are provided to delineate the proposed technique. © 2010 Wiley Periodicals, Inc.     

Observer-based fuzzy control of fuzzy time-delay systems with parametric uncertainties

In this paper, non-linear systems with time-varying delays and parametric uncertainties are represented by an equivalent Takagi–Sugeno-type fuzzy model. Based on the Convex combination property, Lyapunov criterion and Razumikhin theorem, some sufficient conditions are derived under which the parallel-distributed fuzzy control can stabilize the whole uncertain fuzzy time-delay systems asymptotically. On the other hand, if the states are not all available, the fuzzy state observers are proposed to estimate all states of the fuzzy systems for fuzzy control. By satisfying some criteria, the stabilization, estimation and robustness of the fuzzy time-delay systems are also guaranteed. Moreover, if all the time-delays τ _i ( t ) of the fuzzy systems are the same for all the rules (i.e. τ _i (t) = τ _j (t) = τ for all i ≠ j), this paper proposes the simpler and less conservative criteria. The practical example based on the continuous stirred tank reactor model and a numerical example are given to illustrate the control design and its effectiveness.     

Exploring fusion architecture for a common operational picture

Every commander's dream is to have a graphic picture of the unfolding battlespace to show the locations and movements of all entities along with extra prompter information. The DoD command concepts have evolved to yield the common operational picture (COP) and four-stage hierarchy of information fusion. We explore an architecture for refining the fusion for building a more accurate picture. It uses a central processing center to fuse tracks from multiple tracking centers with a cognitive approach that associates local tracks with central tracks and refines estimates via our new fuzzy clustering algorithm. A refinement of target identification at the central tracker is based on the local track IDs, which resolves conflicting identities in the local tracks of the same target. Situation assessment (SA) and force threat assessment (TA) are approached using our fuzzy classifier with built-in fuzzy clustering, but these are not fully developed here due to their complexity. We also propose a dual distributed-centralized tracker that establishes central tracks with both fuzzy clustering and an adaptive α–β filter and fuses the resulting tracks.     

Fast Algorithm for Computing the Upper Endpoint of Sample Variance for Interval Data: Case of Sufficiently Accurate Measurements

When we have n results x₁,...,x_n of repeated measurement of the same quantity, the traditional statistical approach usually starts with computing their sample average E and their sample variance V. Often, due to the inevitable measurement uncertainty, we do not know the exact values of the quantities, we only know the intervals x_i of possible values of x₁ In such situations, for different possible values x_i∈ x_i, we get different values of the variance. We must therefore find the range V of possible values of V. It is known that in general, this problem is NP-hard. For the case when the measurements are sufficiently accurate (in some precise sense), it is known that we can compute the interval V in quadratic time O(n²). In this paper, we describe a new algorithm for computing V that requires time O(n log(n)) (which is much faster than O(n²)).     

A dynamically weight adjustment in the consensus reaching process for group decision-making with hesitant fuzzy preference relations

The consensus reaching process is a dynamic and iterative process for improving group's consensus level before making a final decision in group decision-making (GDM). As the experts will express their opinions under their own intellectual level from different aspects, it is natural that the experts’ weights should reflect their judgment information. This paper proposes a dynamic way to adjust weights of decision-makers (DMs) automatically when they are asked to give original judgment information for GDM problems, in which the DMs express their judgment information by hesitant fuzzy preference relations (HFPRs). Two indices, an individual consensus index of hesitant fuzzy preference relation (ICIHFPR) and a group consensus index of hesitant fuzzy preference relation (GCIHFPR), are introduced. Normalisation of HFPRs with different numbers of possible values is taken into consideration for better computation and comparison. An iterative consensus reaching algorithm is presented with DMs’ weighting vector changing in each consensus reaching process and the process terminates until both the ICIHFPR and GCIHFPR are controlled within predefined thresholds. Finally, an example is illustrated and comparative analyses demonstrate the effectiveness of the proposed methods.     

Graphic representation of the evolutionary processes of the compositions of multicomponent objects of any nature

A method for the graphic representation of evolutionary processes of compositions is described. Preparation of materials includes descending ranking of contents of components and standardization of the length of the obtained sequences while discarding excess contents. The following three parameters are calculated for the persistent distribution of contents: (1) information entropy H = −Σp _ilogp _i as a measure of complexity of the system’s composition, (2) anentropy A = −Σlogp _i as a measure of the composition purity, and (3) tolerance T = log[(Σ1/p _i)/n] as a measure of special purity. In order to represent the process of compositional change, paired diagrams with the axes En, An, and T are used. The obtained entropy diagrams describe separation and mixing processes that occur in nature, technology, and society in the most adequate manner.     

非下采样Contourlet域融合和参数化内核图割的SAR图像无监督水灾变化检测

目的 基于非下采样Contourlet变换（NSCT）融合策略可以有效地抑制背景信息增强变化区域的信息。但是融合后图像具有复杂的统计特征,传统的基于统计特征的变化检测难以实现。基于参数化内核图割的遥感图像分割不受统计特征的限制。为此提出了一种基于NSCT融合和参数化内核图割的SAR图像无监督水灾变化检测新算法。方法 将均值比差异图像和对数比差异图像采用基于NSCT的融合算法进行融合,将融合后的差异图像采用参数化内核图割算法进行前景/背景的分割,得到最终的变化检测结果。结果 融合后的差异图像利用前两种差异图像的互补信息提高了变化检测精度。算法不受统计模型限制,不需要先验知识,适用性强。结论 实验结果表明,本文算法的检测精度优于传统的变化检测方法。     

Hybrid curve fitting

We consider a parameterized family of closed planar curves and introduce an evolution process for identifying a member of the family that approximates a given unorganized point cloud {p _i } _{i =1,..., N} . The evolution is driven by the normal velocities at the closest (or foot) points (f _i ) to the data points, which are found by approximating the corresponding difference vectors p _i -f _i in the least-squares sense. In the particular case of parametrically defined curves, this process is shown to be equivalent to normal (or tangent) distance minimization, see [3], [19]. Moreover, it can be generalized to very general representations of curves. These include hybrid curves, which are a collection of parametrically and implicitly defined curve segments, pieced together with certain degrees of geometric continuity.