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 non-spatial applications are also mentioned. Some key words: Agricultural experiments; Bayesian image analysis; Conditional autoregressions; Dempster's algorithm; Geographical epidemiology; Geostatistics; Intrinsic autoregressions; Multi-way tables; Prior distributions; Spatial statistics; Surface reconstruction; Texture analysis. 1 Introduction

Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to in nite (countably or continuous) index sets. GPs have been applied in a large number of elds 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.

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.

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 high-quality random functions, including those with non-fractal 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.

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Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socio-economic 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 long-memory dependence. Either phenomenon has been modeled and explained by self-affine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-affinity 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, pre-specified fractal properties and power-law correlations is available. The new models suggest a test for self-affinity that assesses coupling and decoupling of local and global behavior.

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. Image-based 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 Dallas-Fort Worth International Airport. In the resulting accident there were many injuries-two of them serious-and 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 all-night 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

Abstract—We introduce an extended class of cardinal L L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L L-spline 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 L-spline. We show that this smoothing spline estimator has a stable representation in a B-spline-like 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 model-based 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.

Abstract: Since radial positive definite functions on R d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier-Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier-Bessel transform at the origin.

We present a family of spatio-temporal models which are geared to provide time-forward predictions in environmental applications where data is spatially sparse but temporally rich. That is