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Topic: Gaussian process regression

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Thomas Bayes

	Gaussian process - Wikipedia, the free encyclopedia
		The Ornstein-Uhlenbeck process is a stationary Gaussian process.
		The Brownian bridge is a Gaussian process whose increments are not independent.
		Inference of continuous values with a Gaussian process prior is known as Gaussian process regression, or Kriging.
	en.wikipedia.org /wiki/Gaussian_process (283 words)

	Kriging - Wikipedia, the free encyclopedia
		Kriging is a regression technique used in geostatistics to approximate or interpolate data.
		This prior takes the form of a Gaussian process: N samples from a function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location of two points.
		The resulting posterior distribution is also a Gaussian, with a mean and covariance that can be simply computed from the observed values, their variance, and the kernel matrix derived from the prior.
	en.wikipedia.org /wiki/Kriging (1050 words)

	The Gaussian Processes Web Site
		The simplest uses of Gaussian process models are for (the conjugate case of) regression with Gaussian noise.
		Gaussian processes are in my view the simplest and most obvious way of defining flexible Bayesian regression and classification models, but despite some past usage, they appear to have been rather neglected as a general-purpose technique.
		Regression is done with a Gaussian process and classification is achieved by regressing on the labels ("least-squares-classification" or "label-regression") using a Gaussian process.
	www.gaussianprocess.org (10721 words)

	Machine Learning Lunch Talk (Site not responding. Last check: 2007-10-31)
		I will first provide an introduction to Gaussian process regression models, including an overview of how Gaussian processes have been used in the spatial statistics literature under the name of kriging.
		The success of the Gaussian process approach hinges on the covariance function, used to describe how the response function is related at different points in the space of the predictor variables, thereby imposing smoothness constraints on the regression function.
		Preliminary results suggest that the nonstationary Gaussian process model is competitive with successful spline-based methods when using a small number of predictor variables.
	www.cs.cmu.edu /People/reinf/OldFiles/web/talks-2002/02-12-16.paciorek.html (280 words)

	Conference Schedule
		Gaussian processes, on the other hand, are ideal for representing nonlinear relationships.
		The benefit of using a Gaussian process for this is the meaningful probabilistic representation of the function.
		We employ a variational framework, where we seek a Gaussian process approximation to the posterior distribution of the state of a system whose dynamics are governed by a stochastic differential equation.
	www.dcs.shef.ac.uk /ml/gpip/schedule.html (1536 words)

	Alexander G. Gray's N-Body Page
		Gaussian process regression is a Bayesian regression method, related to the method of kriging, whose principal required computation is the inversion of an N×N matrix.
		Gaussian process regression is an example of a more general situation within statistics/machine learning of a kernel matrix-vector multiplication.
		A preliminary demonstration of this for a 1-million-point dataset is in (Gray,
	www.cs.cmu.edu /~agray/nbody.html (990 words)

	Documentation for GPML Matlab Code (Site not responding. Last check: 2007-10-31)
		Gaussian process classification (GPC) demonstrates implementations of Laplace and EP approximation methods for binary GP classification.
		Approximation methods for GPR demonstrates the methods of subset of datapoints (SD), subset of regressors (SR) and projected process (PP) approximations.
		A table of other sources of useful Gaussian process software, unrelated to the book, may be found
	www.gaussianprocess.org /gpml/code/matlab/doc (272 words)

	TKK / LCE Models and Methods: A Gaussian Process, 'demo_2ingp'
		The training data has an unknown Gaussian noise and can be seen in the figure 1.
		We are using the same data that was used in the regression model demonstration demo_2input.
		The effectiveness of Gaussian Process can be seen from the amount of samples needed.
	www.lce.hut.fi /research/mm/mcmcstuff/demo_2ingp.shtml (179 words)

	pairwise relations
		Gaussian process is partially justified as its maximum entropy property when the covariance between observations are known.
		Gaussian processes is a theory combining macro and micro explanations.
		Where the macro explanations are actually a natural assembly of micro explanations (due to property of Gaussian distribution) and amazingly, this process is not computational intractable.
	www.cse.ogi.edu /~zhengdon/pairwise_relations.htm (315 words)

	Approximate Inference for Robust Gaussian Process Regression
		Gaussian process (GP) priors have been successfully used in non-parametric Bayesian regression and classification models.
		This article provides a general summary of how expectation-propagation can be used for approximate inference in Gaussian process models.
		Furthermore we present a case study describing its implementation for a new robust variant of Gaussian process regression.
	www.kyb.tuebingen.mpg.de /publication.html?publ=3265 (149 words)

	Anton Schwaighofer \| Publications
		Our solution works by building Gaussian process models for the distribution of signal strengths, as obtained in a series of calibration measurements.
		Gaussian process regression allows a simple analytical treatment of exact Bayesian inference and has been found to provide good performance, yet scales badly with the number of training data.
		In form of the support vector machine and Gaussian processes, kernel-based systems are currently very popular approaches to supervised learning.
	ida.first.fraunhofer.de /~anton/publications.html (1775 words)

	Publications Neural Information Processing Group, TU-Berlin
		We consider active data selection and test point rejection strategies for Gaussian process regression based on the variance of the posterior over target values.
		Gaussian process regression is viewed as transductive regression that provides target distributions for given points rather than selecting an explicit regression function.
		Test point rejection is performed using the estimated posterior variance as a confidence measure.
	ni.cs.tu-berlin.de /publications/abstract-seo00.html (126 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		Gaussian processes are very simple to use, but the extrapolation properties of `standard' Gaussian processes are different from those of neural networks.
		Chris Williams [1] recently showed how to implement a Gaussian process whose predictions are identical to those of an infinite neural network.
		Gaussian processes show promise for nonlinear data-modelling, but they are slow to use if the straightforward implementation is used and the data set size exceeds 1000.
	www.inference.phy.cam.ac.uk /teaching/projects/m.html (251 words)

	Support Vector Machine Regression
		One of the most important ideas in Support Vector Classification and Regression cases is that presenting the solution by means of small subset of training points gives enormous computational advantages.
		The problem of optimal parameter selection is further complicated by the fact that SVM model complexity (and hence its generalization performance) depends on all three parameters.
		One of the advantages of Support Vector Machine, and Support Vector Regression as the part of it, is that it can be used to avoid difficulties of using linear functions in the high dimensional feature space and optimization problem is transformed into dual convex quadratic programmes.
	members.tripod.com /kernelsvm (1058 words)

	Gaussian Processes Site
		gpros.ps.gz) gives a full introduction to the use of Gaussian processes and a discussion of their application to regression problems.
		This discusses both regression and classification using Gaussian processes, details efficient implementations and tackles some interesting real-world problems.
		David MacKay - has written a review paper on Gaussian processes by stealing figures out of my thesis and has a GP page.
	www.inference.phy.cam.ac.uk /mng10/GP (553 words)

	Computing Papers on Gaussian (Site not responding. Last check: 2007-10-31)
		In this paper, we investigate the average collusion attack and several basic nonlinear collusions on independent Gaussian fingerprints, and study their effectiveness and the impact on the perceptual quality.
		With unbounded Gaussian fingerprints, perceivable distortion may exist in the fingerprinted copies as well as the copies after the collusion attacks.
		To reduce the computation complexity in Gaussian component probability densities, the concept of quasi-Gaussian probability density is used to compute the simplified probabilities.
	computing.breinestorm.net /Gaussian (2512 words)

	Extensions of the Informative Vector Machine
		The informative vector machine (IVM) is a practical method for Gaussian process regression and classification.
		The IVM produces a sparse approximation to a Gaussian process by combining assumed density filtering with a heuristic for choosing points based on minimizing posterior entropy.
		First, we propose a novel noise model that allows the IVM to be applied to a mixture of labeled and unlabeled data.
	research.microsoft.com /research/pubs/view.aspx?type=Publication&id=1442 (122 words)

	Fast Gaussian Process Regression using KD-Trees
		The computation required for Gaussian process regression with $n$ training examples is about $O(n^3)$ during training and $O(n)$ for each prediction.
		This makes Gaussian process regression too slow for large data sets.
		In this paper, we present a fast approximation method, based on kd-trees, that significantly reduces both the prediction and the training times of Gaussian process regression.
	www.kyb.tuebingen.mpg.de /publication.html?publ=3606 (71 words)

	Testing monotonicity of regression, Subhashis Ghosal, Arusharka Sen, Aad W. van der Vaart
		We consider the problem of testing monotonicity of the regression function in a nonparametric regression model.
		Hall, P. and Heckman, N. Testing for monotonicity of a regression mean by calibrating for linear functions.
		Wright, F. The asymptotic behavior of monotone percentile regression estimates.
	projecteuclid.org /getRecord?id=euclid.aos/1015956707 (481 words)

	Software
		The goal of Instance Filtering is to reduce both the skewed class distribution and the data set size by eliminating negative instances, while preserving positive ones as much as possible.
		This process is performed on both the training and test set, with the effect of reducing the learning and classification time, while maintaining or improving the prediction accuracy.
		A library in MATLAB for classification, regression, clustering,....
	www.kernel-machines.org /software.html (505 words)

	Abstract for ``Regression and classification using Gaussian process priors'' (Site not responding. Last check: 2007-10-31)
		Abstract for ``Regression and classification using Gaussian process priors''
		Gaussian processes are a natural way of specifying prior distributions over functions of one or more input variables.
		Neal, R. (1997) ``Monte Carlo implementation of Gaussian process models for Bayesian regression and classification'', Technical Report No. 9702, Dept. of Statistics, University of Toronto, 24 pages: abstract, postscript, pdf, associated software.
	www.cs.toronto.edu /~radford/valencia.abstract.html (269 words)

	Conference Materials
		Learning curves for Gaussian process regression are well understood when the `student' model happens to match the `teacher' (true data generation process).
		In lower dimensions, plateaux also appear, and the learning curve remains dependent on the mismatch between student and teacher even in the asymptotic limit of a large number of training examples.
		Learning with excessively strong smoothness assumptions can be particularly dangerous: For example, a student with a standard radial basis function covariance function will learn a rougher teacher function only logarithmically slowly.
	cognet.mit.edu /library/conferences/paper?paper_id=55212 (84 words)

	Online GP Toolbox - Reference Documentation (Site not responding. Last check: 2007-10-31)
		The OGP software package implements general Bayesian inference using Gaussian Processes as latent variables.
		The package allows the usage of a large variety of likelihood functions ranging from the standard normal noise assumption for regression tasks to the use of local inverse models in data assimilation.
		Gaussian Process regression using a fixed set of basis vectors.
	www.ncrg.aston.ac.uk /Projects/SSGP/code/www/index.html (291 words)

	Neil Lawrence's Gaussian Process Software Available Online
		GP-LVM and Gaussian Process Regression using sparse approximations described at NIPS 2005 by Snelson and Ghahramani as well as extensions given by Quinonero-Candela and Rasmussen
		A Gaussian process interpretation of Kernel Fisher Discriminants.
		A set of Gaussian process demos, sampling from covariance functions etc..
	www.dcs.shef.ac.uk /~neil/gpsoftware.html (335 words)

	Hamilton Institute Systems Modelling Publications (Site not responding. Last check: 2007-10-31)
		Yunong Zhang, Leithead, W. Approximate implementation of the logarithm of the matrix determinant in Gaussian process regression, Jour.
		Leithead, W. E., Yunong Zhang, Leith, D. Approximation of the log-determinant & BFGS updating for efficient use in gaussian processes, Proc.
		Leithead, W. E., Neo, K. S., Leith, D. Gaussian regression based on models with two stochastic processes, Proc.
	www.hamilton.may.ie /systemsmodelling/pubs.php (465 words)

	Hamilton Institute (Site not responding. Last check: 2007-10-31)
		Exploiting Hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process, Applied Mathematics and Computation, in press.
		Approximate implementation of logarithm of matrix determinant in Gaussian processes, Journal of Statistical Computation and Simulation, accepted and in press.
		Inference of disjoint linear and nonlinear sub-domains of a nonlinear mapping.
	www.hamilton.may.ie /pubs_Leithead.htm (487 words)

	publication (Site not responding. Last check: 2007-10-31)
		Shi, J.Q., B. Wang, Murray-Smith, R. and Titterington, D.M. Gaussian process functional regression modeling for batch data.
		Shi, J.Q., Murray-Smith, R. and Titterington, D.M. Hierarchical Gaussian process mixtures for regression.
		Shi, J.Q., Murray-Smith, R. and Titterington, D. Bayesian regression and classification using mixtures of Gaussian process.
	www.staff.ncl.ac.uk /j.q.shi/publication.html (460 words)

	Machine Learning Reading Group Meetings at Columbia
		His research interests are probabilistic models with a particular focus on Gaussian processes.
		Each group is modeled using a Dirichlet process (DP) mixture, which provides a nonparametric prior for the number of components within each group.
		We describe a generic way of generalizing the sparse Bayesian Gaussian process Informative Vector Machine (IVM) to such multi process models, emphasizing the key techniques which are required for an efficient solution (exploiting matrix structure, numerical quadrature).
	www.cs.columbia.edu /learning/meetings.html (3502 words)

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