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The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Comparison of Radial Basis Function Interpolants
 In Multivariate Approximation. From CAGD to Wavelets
, 1995
"... This paper compares radial basis function interpolants on different spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm of the error functional. The results pose some new questions for further research. x1. Introduction ..."
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Cited by 25 (7 self)
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This paper compares radial basis function interpolants on different spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm of the error functional. The results pose some new questions for further research. x1. Introduction We consider interpolation of realvalued functions f defined on a set \Omega ` IR d ; d 1. These functions are evaluated on a set X := fx 1 ; : : : ; xNX g of NX 1 pairwise distinct points x 1 ; : : : ; xNX in \Omega\Gamma If N 2; d 2 and\Omega ` IR d are given with\Omega containing at least an interior point, it is well known that there is no Ndimensional space of continuous functions on\Omega that contains a unique interpolant for every f and every set X = fx 1 ; : : : ; xNX g ae\Omega ` IR d consisting of N = NX data points. Thus the family of interpolants must necessarily depend on X. This can easily be achieved by using translates \Phi(x \Gamma x j ) of a single continu...
The Reliability of Curvature Estimates from Linear Elastic Tactile Sensors
 Proceedings of the 1995 IEEE International Conference on Robotics and Automation
, 1995
"... This papers analyzes the reliability of radius of curvature estimates from tactile sensor data. A linear elastic model is used to fit the indenter parameters, load, location, and curvature, to the sensor output. It was found that both contact models and calibration techniques could dramatically effe ..."
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Cited by 8 (1 self)
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This papers analyzes the reliability of radius of curvature estimates from tactile sensor data. A linear elastic model is used to fit the indenter parameters, load, location, and curvature, to the sensor output. It was found that both contact models and calibration techniques could dramatically effect the bias and variance of the estimated indenter parameters. The Fourier series is found to be an appropriate basis in which to analyze both the calibration of tactile sensors and the problem of bandlimited shape interpretation. 1 Introduction Recently it has been shown that the human tactile sensory system is capable of fine shape discrimination from static touch [6]. Robotic tactile sensors have also been shown to have the capability of providing curvature information [5], however results using finiteelement models of tactile sensors indicate that reliable shape classification is hard [3]. It is well known that the rubber layer on the finger acts as a spatial low pass filter and that t...
Tactile Sensing and Control of a Planar Manipulator
 Electrical Engineering and Computer Science, University of California at Berkeley
, 1994
"... Tactile Sensing and Control of a Planar Manipulator by Edward John Nicolson Doctor of Philosophy in Engineering  Electrical Engineering and Computer Sciences University of California at Berkeley Professor Ronald S. Fearing, Chair This dissertation explores the shape sensing capabilities of cylindri ..."
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Cited by 6 (1 self)
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Tactile Sensing and Control of a Planar Manipulator by Edward John Nicolson Doctor of Philosophy in Engineering  Electrical Engineering and Computer Sciences University of California at Berkeley Professor Ronald S. Fearing, Chair This dissertation explores the shape sensing capabilities of cylindrical tactile sensing fingers. Starting with an elastostatic model for the deformation of rubber fingers, sensor spacing and depth requirements are determined to allow reconstruction of subsurface strain fields with insignificant aliasing. Given this bandlimited version of the strain field, theoretical limits are found to classification and scaling of the perceived indentation. These theoretical results lead to the design of a silicone rubber tactile sensor which is characterized and calibrated to the model. The reliability of curvature estimates from the sensor is then determined. Finally, use of the sensor during manipulation is demonstrated. A spatial frequency domain model for the deformat...
On Separability, Positivity and Minimum Uncertainty in TimeFrequency Energy Distributions
 IEEE Trans. Signal Proc
, 1998
"... Gaussian signals play a very special role in classical timefrequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty or positivity and separability of WignerVille distributions. We investigate here some of the logical connections which exist between ..."
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Cited by 5 (0 self)
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Gaussian signals play a very special role in classical timefrequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty or positivity and separability of WignerVille distributions. We investigate here some of the logical connections which exist between these different features, and we discuss some examples and counterexamples of their extension to more general joint distributions within Cohen's class and the affine class. 1 Introduction Let us consider Gaussian signals of the form g(t) = C e \Gammafft 2 ; (1) with C 2 C and ff 2 R + . Besides the fact that their Fourier transform is also Gaussian, namely that G(f) j Z +1 \Gamma1 g(t) e \Gammai2ßf t dt = C r ß ff e \Gamma ß 2 f 2 ff ; such signals happen to play a very special role in classical timefrequency analysis, and this is so for at least three different reasons: 1. Minimum uncertainty. They are the only minimizers for the timefrequency uncertainty relation [...
Stationary space–time Gaussian fields and their time autoregressive representation
, 2002
"... We compare two different modelling strategies for continuous space discrete time data. The first strategy is in the spirit of Gaussian kriging. The model is a general stationary space–time Gaussian field where the key point is the choice of a parametric form for the covariance function. In the main, ..."
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Cited by 4 (0 self)
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We compare two different modelling strategies for continuous space discrete time data. The first strategy is in the spirit of Gaussian kriging. The model is a general stationary space–time Gaussian field where the key point is the choice of a parametric form for the covariance function. In the main, covariance functions that are used are separable in space and time. Nonseparable covariance functions are useful in many applications, but construction of these is not easy. The second strategy is to model the time evolution of the process more directly. We consider models of the autoregressive type where the process at time t is obtained by convolving the process at time t ¡ 1 and adding spatially correlated noise. Under speci�c conditions, the two strategies describe two different formulations of the same stochastic process. We show how the two representations look in different cases. Furthermore, by transforming timedynamic convolution models to Gaussian fields we can obtain new covariance functions and by writing a Gaussian field as a timedynamic convolution model, interesting properties are discovered. The computational aspects of the two strategies are discussed through experiments on a dataset of daily UK temperatures. Although algorithms for performing estimation, simulation, and so on are easy to do for
Characterization of a Class of Sigmoid Functions With Applications to Neural Networks
 Neural Networks
, 1996
"... this paper undertakes a study of two classes of sigmoids: the ..."
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Cited by 3 (0 self)
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this paper undertakes a study of two classes of sigmoids: the
unknown title
, 2002
"... MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries ..."
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MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries