Interpolation of Lipschitz functions

This paper describes a new computational approach to multivariate scattered data interpolation. It is assumed that the data is generated by a Lipschitz continuous function f. The proposed approach uses the central interpolation scheme, which produces an optimal interpolant in the worst case scenario. It provides best uniform error bounds on f, and thus translates into reliable learning of f. This paper develops a computationally efficient algorithm for evaluating the interpolant in the multivariate case. We compare the proposed method with the radial basis functions and natural neighbor interpolation, provide the details of the algorithm and illustrate it on numerical experiments. The efficiency of this method surpasses alternative interpolation methods for scattered data.

Construction of aggregation operators for automated decision making via optimal interpolation and global optimization

This paper examines methods of point wise construction of aggregation operators via optimal interpolation. It is shown that several types of application-specific requirements lead to interpolatory type constraints on the aggregation function. These constraints are translated into global optimization problems, which are the focus of this paper. We present several methods of reduction of the number of variables, and formulate suitable numerical algorithms based on Lipschitz optimization.

Can the choice of interpolation method explain the difference between swap prices and futures prices?

The standard model linking the swap rate to the rates in a contemporaneous strip of futures interest rate contracts typically produces biased estimates of the swap rate. Institutional differences usually require some form of interpolation to be employed and may in principle explain this empirical result. Using Australian data, we find evidence consistent with this explanation and show that model performance is greatly improved if an alternative interpolation method is used. In doing so, we also provide the first published Australian evidence on the accuracy of the futures-based approach to pricing interest rate swaps.

On the use of fuzzy rule interpolation techniques for monotonic multi-input fuzzy rule base models

Constructing a monotonicity relating function is important, as many engineering problems revolve around a monotonicity relationship between input(s) and output(s). In this paper, we investigate the use of fuzzy rule interpolation techniques for monotonicity relating fuzzy inference system (FIS). A mathematical derivation on the conditions of an FIS to be monotone is provided. From the derivation, two conditions are necessary. The derivation suggests that the mapped consequence fuzzy set of an FIS to be of a monotonicity order. We further evaluate the use of fuzzy rule interpolation techniques in predicting a consequent associated with an observation according to the monotonicity order. There are several findings in this article. We point out the importance of an ordering criterion in rule selection for a multi-input FIS before the interpolation process; and hence, the practice of choosing the nearest rules may not be true in this case. To fulfill the monotonicity order, we argue with an example that conventional fuzzy rule interpolation techniques that predict each consequence separately is not suitable in this case. We further suggest another class of interpolation techniques that predicts the consequence of a set of observations simultaneously, instead of separately. This can be accomplished with the use of a search algorithm, such as the brute force, genetic algorithm or etc.

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.

A new interpolation error expansion based reversible watermarking algorithm considering the human visual system

Reversible watermarking has merged over the past few years as a promising solution for copyright protection, especially for applications like remote sensing, medical imaging and military applications which require lossless recovery of the host media. In this paper, we aim to extend the additive interpolation error expansion technique in [16]. We will consider the human visual system (HVS) to improve the embedding rate while maintaining the image visual quality. To this end, the just noticeable difference (JND) is used to embed more watermark bits. The experimental results show that the proposed algorithm can improve the embedding rate while preserving the image visual quality. © 2014 IEEE.

Comparison of interpolation and central activation ratios as measures of muscle inactivation

The objective of this study was to investigate different methods of estimating muscle inactivation, derived from single and multiple voluntary contractions. Ten subjects performed maximal and submaximal leg extensor contractions to determine an interpolation (IT) or central activation ratio (CAR). A superimposed evoked force was compared with the force output of either a voluntary (CAR) or resting evoked contraction (IT ratio), or the ratios were inserted into regression equations (linear, polynomial, exponential). Linear-regression estimates of CAR using doublets and tetanus provided physiologically inaccurate values. Whereas IT ratios using doublets (IT-doublet) and tetanus (IT-tetanus) had a significant difference in only one interaction, IT-tetanus and CAR using a tetanus (CAR-tetanus) estimates provided the most extensive correlation within and between measures. Thus, tetanic stimulation superimposed upon single maximal or multiple contractions seems to provide the most valid measure of muscle inactivation when using the interpolated-twitch technique.

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.

`I`m less at risk than most guys`: gay men`s unrealistic optimism about becoming infected with HIV

Among the self-justifications that gay men use when engaging in high-risk sex is the thought that they are less at risk than most gay men. Two explanatory models of such 'unrealistic optimism' (UO) have been proposed: while the motivational account holds that UO arises because it serves the function of bringing comfort, the cognitive account holds that UO serves no particular function, being simply a by-product of normal cognitive strategies. This study investigated predictions derived from the motivational account. Gay men uninfected with HIV (n = 88) answered two test questions, requiring them to estimate, respectively, their own risk of becoming infected and that of the average gay man. The questions were presented in the two possible orders, and were either separated or not separated by unrelated filler material. The great majority of the men (89%) exhibited UO. Neither question order nor the interpolation of filler material affected responses to either test question. The results were inconsistent with the motivational account, but explicable in terms of the cognitive account. It seems that the cognitive account provides the better explanation of at least that form of UO measured in this study. Implications for AIDS educators are discussed.

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.

Self adjusted security architecture for mobile ad hoc networks (MANETs)

In this work we present a novel security architecture for MANETs that merges the clustering and the threshold key management techniques. The proposed distributed authentication architecture reacts with the frequently changing topology of the network and enhances the process of assigning the node's public key. In the proposed architecture, the overall network is divided into clusters where the clusterheads (CH) are connected by virtual networks and share the private key of the Central Authority (CA) using Lagrange interpolation. Experimental results show that the proposed architecture reaches to almost 95.5% of all nodes within an ad-hoc network that are able to communicate securely, 9 times faster than other architectures, to attain the same results. Moreover, the solution is fully decentralized to operate in a large-scale mobile network.

Constructing artificial images of facial expressions

This paper presents a method for construction of artificial images of facial expressions. The proposed fractal-based synthesis procedure called pixel-based correspondence works on 2D images and does not require any depth information. This method can generate artificial images of an object when only a single image is given. Using the proposed method, effective example-based facial analysis systems can be trained and utilised in various applications.

Review of applications of the Cutting Angle methods

The theory of abstract convexity provides us with the necessary tools for building accurate one-sided approximations of functions. Cutting angle methods have recently emerged as a tool for global optimization of families of abstract convex functions. Their applicability have been subsequently extended to other problems, such as scattered data interpolation. This paper reviews three different applications of cutting angle methods, namely global optimization, generation of nonuniform random variates and multivatiate interpolation.