L1 and L2 cross-validation for density estimation with special reference to orthogonal expansions

L₁ and L₂ cross-validation criteria are studied for a wide class of kernel estimators, estimators that have expansions as linear combinations of prescribed functions. In particular, the L₁ and L₂ cross-validation objective functions to be minimized are described for such estimators, and specialized for estimators based on orthogonal series. An alternative L₂ cross-validation criterion due to Scott and Terrell is adapted for these estimators. A new family of such estimators, “decreasing policy” estimators, is introduced.Results are given for both continuous and discrete densities, but the main emphasis is on the discrete case. In the multinomial case, new predictors are compared with some of the estimators proposed by Good, Fienberg and Holland, and Stone, and studied by Stone (1974), and consistency is discussed.     

The equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets

In this paper, with the help of the convergence of sequences of fuzzy numbers with respect to the Lebesgue measure, we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the L_p metric. We prove that these convergences are equivalent under proper conditions. In addition, by applying our result, we give a new characterization of compact sets in fuzzy number space with the sendograph metric.     

A linear programming algorithm for the simple model for discrete chebychev curve fitting

The L_x norm has been widely studied as a criterion for curve-fitting problems. This procedure is well suited to problems in numerical analysis [7], where errors due to round-off are assumed to have an underlying uniform distribution. TheL_x norm, or Chebychev problem, allows for the “worst case”, in the sense of requiring the largest absolute error to be a minimum. Stiefel [9] developed a method called the “exchange method” for finding Chebychev estimates. The method presented here differs from the exchange method in several important respects: We use a “reduced basis” although determinants are used instead of finding an explicit basis inverse; and we develop a procedure for multiple pivots, or skipping extreme point solutions. The algorithm is specialized to solve the problem with an intercept term and a single independent variable. Results of the computational experience with a computer code version of the algorithm are presented.     

Fuzzy measures for a correlation coefficient of fuzzy numbers under TW(the weakest t-norm)-based fuzzy arithmetic operations

Recently, the sup-min convolution based on Zadeh’s extension principle has been used by Liu and Kao [Fuzzy measures for correlation coefficient of fuzzy numbers, Fuzzy Sets and Systems 128 (2002) 267-275], to calculate a fuzzy correlation coefficient. They used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient. It is well known that T_W (the weakest t-norm)-based addition and multiplication preserve the shape of L-R fuzzy numbers. In this paper, we consider the computational aspect of the T_W-based extension principle when the principle is applied to a correlation coefficient of L-R fuzzy numbers. We give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.     

L 1 shortest paths among polygonal obstacles in the plane

We present an algorithm for computingL ₁ shortest paths among polygonal obstacles in the plane. Our algorithm employs the “continuous Dijkstra” technique of propagating a “wavefront” and runs in timeO(E logn) and spaceO(E), wheren is the number of vertices of the obstacles andE is the number of “events.” By using bounds on the density of certain sparse binary matrices, we show thatE =O(n logn), implying that our algorithm is nearly optimal. We conjecture thatE =O(n), which would imply our algorithm to be optimal. Previous bounds for our problem were quadratic in time and space. Our algorithm generalizes to the case of fixed orientation metrics, yielding anO(n?^?1/2 log² n) time andO(n?^?1/2) space approximation algorithm for finding Euclidean shortest paths among obstacles. The algorithm further generalizes to the case of many sources, allowing us to compute anL ₁ Voronoi diagram for source points that lie among a collection of polygonal obstacles.     

The construction of fuzzy least squares estimators in fuzzy linear regression models

A new concept and method of imposing imprecise (fuzzy) input and output data upon the conventional linear regression model is proposed. Under the considerations of fuzzy parameters and fuzzy arithmetic operations (fuzzy addition and multiplication), we propose a fuzzy linear regression model which has the similar form as that of conventional one. We conduct the h-level (conventional) linear regression models of fuzzy linear regression model for the sake of invoking the statistical techniques in (conventional) linear regression analysis for real-valued data. In order to determine the sign (nonnegativity or nonpositivity) of fuzzy parameters, we perform the statistical testing hypotheses and evaluate the confidence intervals. Using the least squares estimators obtained from the h-level linear regression models, we can construct the membership functions of fuzzy least squares estimators via the form of “Resolution Identity” which is well-known in fuzzy sets theory. In order to obtain the membership degree of any given estimate taken from the fuzzy least squares estimator, optimization problems have to be solved. We also provide two computational procedures to deal with those optimization problems.     

Permutation-based finite implicative fuzzy associative memories

Implicative fuzzy associative memories (IFAMs) are single layer feedforward fuzzy neural networks whose synaptic weights and threshold values are given by implicative fuzzy learning. Despite an excellent tolerance with respect to either pasitive or negative noise, IFAMs are not suited for patterns corrupted by mixed noise. This paper presents a solution to this problem. Precisely, we first introduce the class of finite IFAMs by replacing the unit interval by a finite chain L. Then, we generalize both finite IFAMs and their dual versions by means of a permutation on L. The resulting models are referred to as permutation-based finite IFAMs (π-IFAMs). We show that a π-IFAM can be viewed as a finite IFAM, but defined on an alternative lattice structure (L,?). Thus, π-IFAMs also exhibit optimal absolute storage capacity and one step convergence in the autoassociative case. Furthermore, computational experiments revealed that a certain π-IFAM, called Lukasiewicz π_μ-IFAM, outperformed several other associative memory models for the reconstruction of gray-scale patterns corrupted by salt and pepper noise.     

Self-similar scalar cosmologies

We study several scalar cosmological models under the self-similar approach. We deduce, by stating and proving general theorems, which is the exact form that must follow the scalar field and the potential in the frameworks of a single scalar field and non-interacting (with a perfect fluid) models. The proofs are carried out by two methods: the matter collineation approach and the Lie group method. The results obtained are absolutely general and valid for all Bianchi models and the flat FRW one. In order to study how the “constant” G may vary we propose, in a phenomenological way, how to incorporate a variable G in the framework of scalar models by modifying the Klein-Gordon equation. This approach is more general than the usual one in the context of the FRW symmetries. We deduce the exact form to be followed by each quantity in these new models. Therefore, to study how the “constants” G and Λ may vary, we propose three scenarios where such constants are considered as time functions: modified general relativity with a perfect fluid, the scalar cosmological models (“quintessence”) in the non-interacting case and a scalar-tensor model with a dynamical Λ. As an example, we study the case of Bianchi VI₀ geometry.     

The semiring of matrices over a finite chain

Let L_m denote the chain {0,1,2,…,m-1} with the usual ordering and M_n(L_m) the matrix semiring of all n×n matrices with elements in L_m. We firstly introduce some order-preserving semiring homomorphisms from M_n(L_m) to M(L_k). By using these homomorphisms, we show that a matrix over the finite chain L_m can be decomposed into the sum of some matrices over the finite chain L_k, where k<m. As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1-25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring M_n(L_m) to the direct product of the h copies of the semiring M_n(L_k) and discuss Green’s relations on the multiplicative semigroup of the semiring M_n(L_m). We think that some results obtained in this paper is useful for the study of fuzzy matrices.     

On convergence of kernel estimators of density with variable window width by dependent observations

In [1, 2] was studied a new type of nonparametric kernel estimators of probability density, whose window width varies depending on the sample, i.e., are data-based. These estimators were called adaptive. New estimators of density are superior in the rate of convergence to classical Rosenblatt-Parzen estimators. However, these valuable properties of estimators were obtained assuming that observations are independent. In this paper, we study properties of these adaptive estimators but assuming that the sample is realization of the stationary in the narrow sense random sequence. The simulation examples for the adaptive estimator constructed by dependent observations which is generated by autoregressive models are represented. The results of the investigation prove the advantage of the adaptive estimator over the classical Rosenblatt-Parzen estimator in the sense of the mean-square error. The rate of mean-square convergence of the limiting estimator (the so-called “ideal” estimator) to the true unknown density according to the dependent sample is found. The consistency of the adaptive estimator constructed by stationary dependent observations is proved.     

On the possibility of using complex values in fuzzy logic for representing inconsistencies

In science and engineering, there are “paradoxical” cases in which we have some arguments in favor of some statement A (so the degree to which A is known to be true is positive (nonzero)), and we have some arguments in favor of its negation ¬A, and we do not have enough information to tell which of these two statements is correct. Traditional fuzzy logic, in which “truth values” are described by numbers from the interval [0, 1], easily describes such “paradoxical” situations: the degree a to which the statement A is true and the degree 1−a to which its negation ¬A is true can both be positive. In this case, if we use traditional fuzzy &-operations (min or product), the “truth value” a&(1−a) of the statement A&¬A is positive, indicating that there is some degree of inconsistency in the initial beliefs. When we try to use fuzzy logic to formalize expert reasoning in the humanities, we encounter the problem that is humanities, in addition to the above-described paradoxical situations caused by the incompleteness of our knowledge, there are also true paradoxes, i.e., statements that are perceived as true and false at the same time. For such statements, A&¬A=“true.” The corresponding equality a&(1−a)=1 is impossible in traditional fuzzy logic (where a&(1−a) is always≤0.5), so, to formalize such true paradoxes, we must extend the set of truth values from the interval [0, 1]. In this paper we show that such an extension can be achieved if we allow truth values to be complex numbers. © 1998 John Wiley & Sons, Inc.     

The canonical representation of multiplication operation on triangular fuzzy numbers

The representation of multiplication operation on fuzzy numbers is very useful and important in the fuzzy system such as the fuzzy decision making. In this paper, we propose a new arithmetical principle and a new arithmetical method for the arithmetical operations on fuzzy numbers. The new arithmetical principle is the L⁻¹-R⁻¹ inverse function arithmetic principle. Based on the L⁻¹-R⁻¹ inverse function arithmetic principle, it is easy to interpret the multiplication operation with the membership functions of fuzzy numbers. The new arithmetical method is the graded multiple integrals representation method. Based on the graded multiple integrals representation method, it is easy to compute the canonical representation of multiplication operation on fuzzy numbers. Finally, the canonical representation is applied to a numerical example of fuzzy decision.     

The copying power of one-state tree transducers

One-state deterministic top-down tree transducers (or, tree homomorphisms) cannot handle “prime copying,” i.e., their class of output (string) languages is not closed under the operation L → {$(w$)^f(n) ∥ w?L, f(n) ? 1}, where f is any integer function whose range contains numbers with arbitrarily large prime factors (such as a polynomial). The exact amount of nonclosure under these copying operations is established for several classes of input (tree) languages. These results are relevant to the extended definable (or, restricted parallel level) languages, to the syntax-directed translation of context-free languages, and to the tree transducer hierarchy.     

On the rate of convergence for two-term recursions

LetB be a compact interval in ?,M=B×B and φ:M→B a map inC ₃ (M). Suppose that ξ is a fixed point of φ. We study the behaviour of the iteratesx _n+2=φ(x _n+1,x _n ) (x ₀,x ₁∈B). Of particular interest is the situation where ? _x (ξ,ξ)=? _y (ξ,ξ)=0. In case of the wellknown “Regula falsi” we also have ? _xx (ξ,ξ)=? _yy (ξ,ξ)=0 and the order of convergence is \(\tfrac{1}{2}(1 + \sqrt 5 )\) . We consider the case where ? _yy (ξ,ξ)≠0. It turns out that there is a constant γ∈(1,2) such that successive iterates gain factors γ, 2/γ, γ, 2/γ, ... on the number of valid decimals. Depending on the initial iteratesx ₀,x ₁ the number λ may range over all of (1, 2) such that in the extreme cases an additional iterative step may have virtually no effect on the number of correct digits or nearly doubles them.     

A faster convergence and concise interpretability TSK fuzzy classifier deep-wide-based integrated learning

Hierarchical TSK fuzzy system was proposed to approach the exponential growth of IF-THEN rules which named “fuzzy rule explosion”. However, it could not get better performance in few layers for instability of TSK fuzzy system, such that hierarchical TSK fuzzy system suffers from bad interpretability and slow convergence along with too much layers. To get a better solution, this study employs a faster convergence and concise interpretability TSK fuzzy classifier deep-wide-based integrated learning (FCCI-TSK) which has a wide structure to adopt several ensemble units learning in a meantime, and the best performer will be picked up to transfer its learning knowledge to next layer with the help of stacked generalization principle. The ensemble units are integrated by negative correlation learning (NCL). FCCI-TSK adjusts the input of the next layer with a better guidance such that it can quicken the speed of convergence and reduce the number of layers. Besides, leading with guidance, it can achieve higher accuracy and better interpretability with more simple structure. The contributions of this study include: (1) To enhance the performance of fuzzy classifier, we mix NCL and stacked generalization principle together in FCCI-TSK; (2) To overcome the phenomenon of “fuzzy rule explosion”, we adopt deep-wide integrated learning and information discarding to accelerate convergence and obtain concise interpretability in the meantime. Comparing with other 11 algorithms, the results on twelve UCI datasets show that FCCI-TSK has the best performance overall and the convergence of FCCI-TSK is also examined.     

How to fully represent expert information about imprecise properties in a computer system: random sets,fuzzy sets,and beyond: an overview

To help computers make better decisions, it is desirable to describe all our knowledge in computer-understandable terms. This is easy for knowledge described in terms on numerical values: we simply store the corresponding numbers in the computer. This is also easy for knowledge about precise (well-defined) properties which are either true or false for each object: we simply store the corresponding “true” and “false” values in the computer. The challenge is how to store information about imprecise properties. In this paper, we overview different ways to fully store the expert information about imprecise properties. We show that in the simplest case, when the only source of imprecision is disagreement between different experts, a natural way to store all the expert information is to use random sets; we also show how fuzzy sets naturally appear in such random set representation. We then show how the random set representation can be extended to the general (“fuzzy”) case when, in addition to disagreements, experts are also unsure whether some objects satisfy certain properties or not.     

Asymptotics for Lp-norms of kernel estimators of densities

Let X₁, X₂,… be independent identically distributed random variables with a smooth density function ⨍. We present central limit theorems for weighted L_p-distances of kernel estimators ⨍_n of ⨍ from ⨍. These results cover the important cases of L₁ and L₂ (mean square error) asymptotics, without assuming that ⨍ has finite support.     

Optimisation of a non linear fuzzy function

 We present a first study concerning the optimization of a non linear fuzzy function f depending both on a crisp variable and a fuzzy number: therefore the function value is a fuzzy number. More specifically, given a real fuzzy number ?∈F and the function f(a,x):R ²→R, we consider the fuzzy extension induced by f, f˜ : F × R → F, f˜(?,x) = Y˜. If K is a convex subset of R, the problem we consider is “maximizing”f˜(?,x), xˉ∈ K. The first problem is the meaning of the word “maximizing”: in fact it is well-known that ranking fuzzy numbers is a complex matter. Following a general method, we introduce a real function (evaluation function) on real fuzzy numbers, in order to get a crisp rating, induced by the order of the real line. In such a way, the optimization problem on fuzzy numbers can be written in terms of an optimization problem for the real-valued function obtained by composition of f with a suitable evaluation function. This approach allows us to state a necessary and sufficient condition in order that ∈K is the maximum for f˜ in K, when f(a,x) is convex-concave (Theorem 4.1).     

L2–L∞ fuzzy control for Markov jump systems with neutral time-delays

The L₂–L_∞ fuzzy control problem is considered for nonlinear stochastic Markov jump systems with neutral time-delays. By means of Takagi–Sugeno fuzzy models, the fuzzy controller systems and the overall closed-loop fuzzy dynamics are constructed. A sufficient condition is firstly established on the stochastic stability using stochastic Lyapunov–Krasovskii functional. Then in terms of linear matrix inequalities techniques, the sufficient conditions on the existence of mode-dependent state feedback L₂–L_∞ fuzzy controller are presented and proved respectively for constant and time varying case. Finally, the design problems are formulated as optimization algorithms. Simulation results are exploited to illustrate the effectiveness of the developed techniques.     

On the new solutions for a fully fuzzy linear system

In this paper, we propose a new method to present a fuzzy trapezoidal solution, namely “suitable solution”, for a fully fuzzy linear system (FFLS) based on solving two fully interval linear systems (FILSs) that are 1-cut and 0-cut of the related FILS. After some manipulations, two FILSs are transformed to 2n crisp linear equations and 4n crisp linear nonequations and n crisp nonlinear equations. Then, we propose a nonlinear programming problem (NLP) to computing simultaneous (synchronic) equations and nonequations. Moreover, we define two other new solutions namely, “fuzzy surrounding solution” and “fuzzy peripheral solution” for an FFLS. It is shown that the fuzzy surrounding solution is placed in a tolerable fuzzy solution set and the fuzzy peripheral solution is placed in a controllable fuzzy solution set. Finally, some numerical examples are given to illustrate the ability of the proposed methods.