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

This chapter discusses the indefinite integrals of elementary functions. To integrate an arbitrary rational function, the integral part must be separated out, if there is an integral part, and then the integral part and the remainder must be integrated separately. Integration of the remainder, which is then a proper rational function, is based on the decomposition of the fraction into elementary fractions, which are the so-called partial fractions. If some of the roots of the equation are imaginary, the fractions that represent conjugate roots of the equation are then grouped together. The chapter further presents equation forms containing certain binomials.

3.-4 – definite integrals of elementary functions

RKHS Approach to Detection and Estimation Problems-Part IV: Non-Gaussian Detection

This chapter presents a list of equation forms containing elliptic integrals and functions, integration of elliptic integrals with respect to the modulus, complete elliptic integrals, and other definite integral functions. The chapter also focuses on Meijer's and MacRobert's functions, including combinations of the functions G and E and Bessel functions, combinations of the functions G and E, and other special functions.

Indefinite integrals of special functions

This chapter discusses the matrices and related results. The identity matrix is a diagonal matrix I in which all entries in the leading diagonal is unity. The null matrix is all zeros. An m × n matrix A is equivalent to an m × n matrix B if, and only if, B = PAQ for suitable nonsingular m × m and n × n matrices P and Q, respectively. A quadratic form involving the n real variables that are associated with the real n × n matrix A = [a ij ] is the scalar expression.

I.M. Ryzhik

Papers

3.-4 – definite integrals of elementary functions

RKHS Approach to Detection and Estimation Problems-Part IV: Non-Gaussian Detection

Indefinite integrals of special functions

Matrices and related results

Table of Integrals, Series and Products: CD-ROM Version 1.0.