Monotone approximation of aggregation operators using least squares splines

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

Identification of general and double aggregation operators using monotone smoothing

Aggregation operators model various operations on fuzzy sets, such as conjunction, disjunction and averaging. Recently double aggregation operators have been introduced; they model multistep aggregation process. The choice of aggregation operators depends on the particular problem, and can be done by fitting the operator to empirical data. We examine fitting general aggregation operators by using a new method of monotone Lipschitz smoothing. We study various boundary conditions and constraints which determine specific types of aggregation.

Iterations

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Non-monotone averaging aggregation

We advance the theory of aggregation operators and introduce non-monotone aggregation methods based on minimization of a penalty for inputs disagreements. The application in mind is processing data sets which may contain noisy values. Our aim is to filter out noise while at the same time preserve signs of unusual values. We review various methods of robust estimators of location, and then introduce a new estimator based on penalty minimisation.

A monotonicity index for the monotone fuzzy modeling problem

In this paper, the problem of maintaining the (global) monotonicity and local monotonicity properties between the input(s) and the output of an FIS model is addressed. This is known as the monotone fuzzy modeling problem. In our previous work, this problem has been tackled by developing some mathematical conditions for an FIS model to observe the monotonicity property. These mathematical conditions are used as a set of governing equations for undertaking FIS modeling problems, and have been extended to some advanced FIS modeling techniques. Here, we examine an alternative to the monotone fuzzy modeling problem by introducing a monotonicity index. The monotonicity index is employed as an approximate indicator to measure the fulfillment of an FIS model to the monotonicity property. It allows the FIS model to be constructed using an optimization method, or be tuned to achieve a better performance, without knowing the exact mathematical conditions of the FIS model to satisfy the monotonicity property. Besides, the monotonicity index can be extended to FIS modeling that involves the local monotonicity problem. We also analyze the relationship between the FIS model and its monotonicity property fulfillment, as well as derived mathematical conditions, using the Monte Carlo method.

A new framework with similarity reasoning and monotone fuzzy rule relabeling for fuzzy inference systems

A complete and monotonically-ordered fuzzy rule base is necessary to maintain the monotonicity property of a Fuzzy Inference System (FIS). In this paper, a new monotone fuzzy rule relabeling technique to relabel a non-monotone fuzzy rule base provided by domain experts is proposed. Even though the Genetic Algorithm (GA)-based monotone fuzzy rule relabeling technique has been investigated in our previous work [7], the optimality of the approach could not be guaranteed. The new fuzzy rule relabeling technique adopts a simple brute force search, and it can produce an optimal result. We also formulate a new two-stage framework that encompasses a GA-based rule selection scheme, the optimization based-Similarity Reasoning (SR) scheme, and the proposed monotone fuzzy rule relabeling technique for preserving the monotonicity property of the FIS model. Applicability of the two-stage framework to a real world problem, i.e., failure mode and effect analysis, is further demonstrated. The results clearly demonstrate the usefulness of the proposed framework.

Parallel monotone spline interpolation and approximation on GPUs

Monotonicity preserving interpolation and approximation have received substantial attention in the last thirty years because of their numerous applications in computer aided-design, statistics, and machine learning [9, 10, 19]. Constrained splines are particularly popular because of their flexibility in modeling different geometrical shapes, sound theoretical properties, and availability of numerically stable algorithms [9,10,26]. In this work we examine parallelization and adaptation for GPUs of a few algorithms of monotone spline interpolation and data smoothing, which arose in the context of estimating probability distributions.

Weakly monotone averaging functions

Averaging behaviour of aggregation functions depends on the fundamental property of monotonicity with respect to all arguments. Unfortunately this is a limiting property that ensures that many important averaging functions are excluded from the theoretical framework. We propose a definition for weakly monotone averaging functions to encompass the averaging aggregation functions in a framework with many commonly used non-monotonic means. Weakly monotonic averages are robust to outliers and noise, making them extremely important in practical applications. We show that several robust estimators of location are actually weakly monotone and we provide sufficient conditions for weak monotonicity of the Lehmer and Gini means and some mixture functions. In particular we show that mixture functions with Gaussian kernels, which arise frequently in image and signal processing applications, are actually weakly monotonic averages. Our concept of weak monotonicity provides a sound theoretical and practical basis for understanding both monotone and non-monotone averaging functions within the same framework. This allows us to effectively relate these previously disparate areas of research and gain a deeper understanding of averaging aggregation methods. © Springer International Publishing Switzerland 2014.

Efficient and secure attribute-based signature for monotone predicates

Attribute-based signature (ABS) is a novel cryptographic primitive, which can make the signing party sign a message with fine-grained control over identifying information. ABS only reveals the fact that the verified message must be signed by a user with a set of attributes satisfying a predicate. Thus, ABS can hide any identifying information and make fine-grained control on signing. Presently, many attribute-based signature schemes have been proposed, but most of them are not very efficient. Maji et al. recently presented a complete definition and construction about ABS for monotone predicates and showed three instantiations under their framework for ABS. Although the most practical one of their instantiations is efficient, the instantiation is constructed in the generic group model and has been proved to be insecure. Then, Okamoto et al. proposed an attribute-based signature scheme in the standard model, which can support generalized non-monotone predicates over access structure. However, their scheme is not efficient in practice. In this paper, we present a framework for ABS and show a detailed security model for ABS. Under our framework, we present an attribute-based signature scheme for monotone predicates in the standard model, where we choose the Waters’ signature scheme as the prototype of our attribute-based signature scheme. Compared with the Maji’s scheme in the generic group model, the proposed scheme is constructed in the standard model. Furthermore, compared with the Okamoto’s scheme, the proposed scheme is more efficient by decreasing the computation cost.

Organizational factors in the diffusion of an industry standard: implementing an online documentation system for Australian exporters

At present, governments in many countries are actively engaged in the development of electronic trading and certification standards to enable the smooth operation of export markets. Standards and their usability underpin confidence in the operations of markets and their effective functioning. In institutional markets, an important role for government agencies lies in developing the initial specifications for standards for interoperable systems. Once these specifications are accepted, governments can then facilitate the eventual diffusion of a standard to the B2B marketplace. Acceptance of an industry standard can determine demand, which defines the viability of that market. In this paper, we describe an initiative by a government agency, the Australian Quarantine and Inspection Service (AQIS), in developing EXDOC, a standard for export documentation and the role that AQIS played in its implementation and diffusion. Our case study illustrates a successfully facilitated B2G implementation. It demonstrates how a standard can be supported and promulgated for the effective functioning of markets in the transition from manual to online export documentation. Once the overarching specifications for related industries have been established and diffused by government, opportunities arise for private sector markets to develop across these industries. Government agencies can promote the effective operation of standards for electronic markets. The EXDOC implementation and its iterations provide an exemplar of active engagement in the development of electronic trading and certification standards for an institutional market. Its successful diffusion provides a model of the implementation process for other export sectors and agencies.

Monotonicity preserving approximation of multivariate scattered data

This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.

Pointwise construction of Lipschitz aggregation operators

This paper establishes tight upper and lower bounds on Lipschitz aggregation operators considering their diagonal, opposite diagonal and marginal sections. Also we provide explicit formulae to determine the bounds. These are useful for construction of these type of aggregation operators, especially using interpolation schemata.

Construction of aggregation operators with noble reinforcement

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements.

Sequential technology adoption with asymmetric firms

We analyse the incentives and welfare implications of costly technology adoption in a two-period duopoly model where firms have different amounts of capital. We also extend our framework to an open economy set-up and examine the relationship between trade and technology adoption. Our findings are as follows. First, no monotone relationship exists between the threshold cost of adoption and capital shares. Second, an unequal distribution of capital, despite lessening competition, can increase total surplus. Third, trade generally encourages adoption of modern technology unless the share of capital for the adopters is too low.

Scope for further analytical solutions for constant flux infiltration into a semi-infinite soil profile or redistribution in a finite soil profile.

We attempt to generate new solutions for the moisture content form of the one-dimensional Richards' [1931] equation using the Lisle [1992] equivalence mapping. This mapping is used as no more general set of transformations exists for mapping the one-dimensional Richards' equation into itself. Starting from a given solution, the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions. We first seek new analytical solutions satisfying Richards' equation subject to a constant flux surface boundary condition for a semi-infinite dry soil, starting with the Burgers model. The first iteration produces an existing solution, while subsequent iterations are shown to endlessly reproduce this same solution. Next, we briefly consider the problem of redistribution in a finite-length soil. In this case, Lisle's equivalence mapping is generalized to account for arbitrary initial conditions. As was the case for infiltration, however, it is found that new analytical solutions are not generated using the equivalence mapping, although existing solutions are recovered.