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52
Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions
 CONSTRUCTIVE APPROXIMATION
, 1986
"... Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke. ..."
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Cited by 275 (3 self)
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Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.
On Conditional and Intrinsic Autoregressions
, 1995
"... This paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using Dempster's (1972) algorithm or an appropriate modification. The approach partly represents a synthesis of standard geostatistical and Gaussian Markov random field formulation ..."
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Cited by 76 (6 self)
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This paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using Dempster's (1972) algorithm or an appropriate modification. The approach partly represents a synthesis of standard geostatistical and Gaussian Markov random field formulations. Some nonspatial applications are also mentioned. Some key words: Agricultural experiments; Bayesian image analysis; Conditional autoregressions; Dempster's algorithm; Geographical epidemiology; Geostatistics; Intrinsic autoregressions; Multiway tables; Prior distributions; Spatial statistics; Surface reconstruction; Texture analysis. 1 Introduction
Gaussian processes for machine learning
 International Journal of Neural Systems
, 2004
"... Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to infinite (countably or continuous) index sets. GPs have been applied in a large number of fields to a diverse range of ends, and very many deep theoretical analyses of various properties are available. ..."
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Cited by 67 (15 self)
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Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to infinite (countably or continuous) index sets. GPs have been applied in a large number of fields to a diverse range of ends, and very many deep theoretical analyses of various properties are available. This paper gives an introduction to Gaussian processes on a fairly elementary level with special emphasis on characteristics relevant in machine learning. It draws explicit connections to branches such as spline smoothing models and support vector machines in which similar ideas have been investigated. Gaussian process models are routinely used to solve hard machine learning problems. They are attractive because of their flexible nonparametric nature and computational simplicity. Treated within a Bayesian framework, very powerful statistical methods can be implemented which offer valid estimates of uncertainties in our predictions and generic model selection procedures cast as nonlinear optimization problems. Their main drawback of heavy computational scaling has recently been alleviated by the introduction of generic sparse approximations [13, 78, 31]. The mathematical literature on GPs is large and often uses deep
Classes of kernels for machine learning: a statistics perspective
 Journal of Machine Learning Research
, 2001
"... In this paper, we present classes of kernels for machine learning from a statistics perspective. Indeed, kernels are positive definite functions and thus also covariances. After discussing key properties of kernels, as well as a new formula to construct kernels, we present several important classes ..."
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Cited by 59 (2 self)
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In this paper, we present classes of kernels for machine learning from a statistics perspective. Indeed, kernels are positive definite functions and thus also covariances. After discussing key properties of kernels, as well as a new formula to construct kernels, we present several important classes of kernels: anisotropic stationary kernels, isotropic stationary kernels, compactly supportedkernels, locally stationary kernels, nonstationary kernels, andseparable nonstationary kernels. Compactly supportedkernels andseparable nonstationary kernels are of prime interest because they provide a computational reduction for kernelbased methods. We describe the spectral representation of the various classes of kernels and conclude with a discussion on the characterization of nonlinear maps that reduce nonstationary kernels to either stationarity or local stationarity.
Generalized Stochastic Subdivision
 ACM Transactions on Graphics
, 1987
"... This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functi ..."
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Cited by 35 (2 self)
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This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functions. The generalized construction is suitable for generating a variety of perceptually distinct highquality random functions, including those with nonfractal spectra and directional or oscillatory characteristics. It is argued that a spectral modeling approach provides a more powerful and somewhat more intuitive perceptual characterization of random processes than does the fractal model. Synthetic textures and terrains are presented as a means of visually evaluating the generalized subdivision technique. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three Dimensional Graphics and Realism <F11.
Generalized smoothing splines and the optimal discretization of the Wiener filter
 IEEE Trans. Signal Process
, 2005
"... Abstract—We introduce an extended class of cardinal L Lsplines, where L is a pseudodifferential operator satisfying some admissibility conditions. We show that the L Lspline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional L ..."
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Cited by 28 (14 self)
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Abstract—We introduce an extended class of cardinal L Lsplines, where L is a pseudodifferential operator satisfying some admissibility conditions. We show that the L Lspline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional L P, subject to the interpolation constraint. Next, we consider the corresponding regularized least squares estimation problem, which is more appropriate for dealing with noisy data. The criterion to be minimized is the sum of a quadratic data term, which forces the solution to be close to the input samples, and a “smoothness” term that privileges solutions with small spline energies. Here, too, we find that the optimal solution, among all possible functions, is a cardinal L Lspline. We show that this smoothing spline estimator has a stable representation in a Bsplinelike basis and that its coefficients can be computed by digital filtering of the input signal. We describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational (which corresponds to the case of exponential splines). We justify these algorithms statistically by establishing an equivalence between L L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this modelbased formulation, the optimum operator L is the whitening filter of the process, and the regularization parameter is proportional to the noise variance. Thus, the proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm. It extends the standard Wiener solution by providing the optimal interpolation space. We also present a Bayesian interpretation of the algorithm. Index Terms—Nonparametric estimation, recursive filtering, smoothing splines, splines (polynomial and exponential), stationary processes, variational principle, Wiener filter. I.
Stochastic Models That Separate Fractal Dimension and Hurst Effect
 SIAM Review
, 2003
"... Fractal behavior and longrange dependence have been observed in an astonishing number of physical, biological, geological, and socioeconomic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a meas ..."
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Cited by 24 (5 self)
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Fractal behavior and longrange dependence have been observed in an astonishing number of physical, biological, geological, and socioeconomic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of longmemory dependence. Either phenomenon has been modeled and explained by selfaffine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical selfaffinity implies a linear relationship between fractal dimension and Hurst coe#cient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coefficient. Associated software for the synthesis of images with arbitrary, prespecified fractal properties and powerlaw correlations is available. The new models suggest a test for selfaffinity that assesses coupling and decoupling of local and global behavior.
Automated storm tracking for Terminal Air Traffic Control
 The Lincoln Laboratory Journal
, 1994
"... II Good estimates ofstorm motion are essential to improved air traffic control operations during times ofinclement weather. Automating such a service is a challenge, however, because meteorological phenomena exist as complex distributed systems that exhibit motion across a wide spectrum ofscales. Ev ..."
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Cited by 15 (1 self)
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II Good estimates ofstorm motion are essential to improved air traffic control operations during times ofinclement weather. Automating such a service is a challenge, however, because meteorological phenomena exist as complex distributed systems that exhibit motion across a wide spectrum ofscales. Even when viewed &om a fIXed perspective, these evolving dynamic systems can test the extent of our definition ofmotion, as well as any attempt at automated tracking ofthis motion. Imagebased motion detection and processing appear to provide the best route toward robust performance ofan automated tracking system. ON APRIL 14, 1993, AMERICAN AIRLINES FLIGHT 102 was unable to hold the runway while landing at DallasFort Worth International Airport. In the resulting accident there were many injuriestwo of them seriousand the plane (a DC 10) was irreparably damaged. Itwas raining at the airport that morning, and numerous thunderstorms were occurring throughout the area. The darkness of the early hour, the fatigue of the flight crew after an allnight flight, and the bad weather were all suspected causes of the accident. Although the National Transportation Safety Board officially concluded that the stormy weather was not a contributing factor to the crash (despite high cross winds from a severe storm passing over the airport, the aircraft was able to touch down on the runway [1]), the weather clearly did play an important role in the events ofthat day. The crew of Flight 102 had access to a variety of weather information that morning, including their Own radar. Their information sources included American
Selfsimilarity: Part II  Optimal estimation of fractallike processes
 IEEE SIGNAL PROCESSING MAGAZINE, THE IEEE TRANSACTIONS ON IMAGE PROCESSING (1992 TO 1995), AND THE IEEE SIGNAL PROCESSING LETTERS
, 2007
"... In a companion paper (see SelfSimilarity: Part I—Splines and Operators), we characterized the class of scaleinvariant convolution operators: the generalized fractional derivatives of order. We used these operators to specify regularization functionals for a series of Tikhonovlike leastsquares da ..."
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Cited by 10 (9 self)
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In a companion paper (see SelfSimilarity: Part I—Splines and Operators), we characterized the class of scaleinvariant convolution operators: the generalized fractional derivatives of order. We used these operators to specify regularization functionals for a series of Tikhonovlike leastsquares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically selfsimilar; that is, fractional Brownian motion (fBm) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of fBm, we adopt a distributional formulation due to Gel’fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) @ @ A A, where a @ AC ‘ “ are the noisy samples of the fBm @ A with Hurst exponent. We find that the conditional mean is a fractional spline of degree P, which proves that this class of functions is indeed optimal for the estimation of fractallike processes. The result also yields the optimal [minimum meansquare error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering.