A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

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Gaussian Processes for Machine Learning

This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

Conditional heteroskedasticity in asset returns: a new approach

This paper focuses on the class of speech enhancement systems which capitalize on the major importance of the short-time spectral amplitude (STSA) of the speech signal in its perception. A system which utilizes a minimum mean-square error (MMSE) STSA estimator is proposed and then compared with other widely used systems which are based on Wiener filtering and the "spectral subtraction" algorithm. In this paper we derive the MMSE STSA estimator, based on modeling speech and noise spectral components as statistically independent Gaussian random variables. We analyze the performance of the proposed STSA estimator and compare it with a STSA estimator derived from the Wiener estimator. We also examine the MMSE STSA estimator under uncertainty of signal presence in the noisy observations. In constructing the enhanced signal, the MMSE STSA estimator is combined with the complex exponential of the noisy phase. It is shown here that the latter is the MMSE estimator of the complex exponential of the original phase, which does not affect the STSA estimation. The proposed approach results in a significant reduction of the noise, and provides enhanced speech with colorless residual noise. The complexity of the proposed algorithm is approximately that of other systems in the discussed class.

Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator

The partition of unity finite element method: Basic theory and applications

Since the work of Bishop and Heberlein, a number of contingent valuation experiments have appeared involving discrete responses which are analyzed by logit or similar techniques. This paper addresses the issues of how the logit models should be formulated to be consistent with the hypothesis of utility maximization and how measures of compensating and equivalent surplus should be derived from the fitted models. Two distinct types of welfare measures are introduced and then estimated from Bishop and Heberlein's data.

Welfare Evaluations in Contingent Valuation Experiments with Discrete Responses

Publisher Summary
This chapter discusses elementary functions. The trigonometric and hyperbolic sines and cosines are related by the following identities: ch x = cos ix and cos x = ch ix. Because of this duality, every relation involving trigonometric functions has its formal counterpart involving the corresponding hyperbolic functions and vice versa. In many cases, both pairs of relationships are meaningful. The chapter further presents equations representing sums of trigonometric and hyperbolic functions and sums of powers of trigonometric functions of multiple angles. It also describes the expansion of trigonometric and hyperbolic functions in power series.

1 – elementary functions

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Table of Integrals, Series, and Products

Publisher Summary
This chapter discusses integral inequalities. It presents the properties of integrals and integral inequalities and two mean value theorems and explains the differentiation of definite integral containing a parameter with the use of certain equations in cases where limits are finite and infinite. It further describes the Cauchy–Schwarz–Buniakowsky inequality for integrals, Holder's inequality for integrals, Minkowski's inequality for integrals, Chebyshev's inequality for integrals, Young's inequality for integrals, Steffensen's inequality for integrals, and Ostrowski's inequality for integrals.

12 – integral inequalities

Publisher Summary
This chapter discusses the special functions. A single-valued function f(z) of a complex variable, which is not a constant, is said to be elliptic if it has two periods 2ω1 and 2ω2. The ratio of the periods of an analytic function cannot be a real number. For an elliptic function f(z), the z-plane can be partitioned into parallelograms. A nonconstant elliptic function has a finite number of poles in a period parallelogram; it can have no more than two simple and one second-order pole in such a parallelogram. The sum of the residues of an elliptic function, with respect to all the poles belonging to a period parallelogram, is equal to zero. The difference between the sum of all the zeros and the sum of all the poles of an elliptic function that are located in a period parallelogram is equal to one of its periods.

8-9 – special functions

This chapter discusses Fourier and Laplace transforms. The inversion of the Laplace transform is accomplished for analytic functions f (p) of order O (p -k ) with k > 1 by means of the inversion integral. The inversion of the Fourier transform is accomplished by means of the inversion integral. The Fourier sine and cosine transforms of the function f(x), denoted by F s (ξ) and F c (ξ), respectively, are defined by the integrals. The functions f(x) and F s ( ξ ) are called a Fourier sine transform pair, and the functions f(x) and F c ( ξ ) a Fourier cosine transform pair, and knowledge of either F s ( ξ ) or F c ( ξ ) enables f(x) to be recovered. The inversions of the Fourier sine and Fourier cosine transform are accomplished by means of the inversion integral.

I.M. Ryzhik

Papers

1 – elementary functions

Table of Integrals, Series, and Products

12 – integral inequalities

8-9 – special functions

Fourier and laplace transforms