Results 1  10
of
165
A tutorial on support vector regression
, 2004
"... In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing ..."
Abstract

Cited by 828 (3 self)
 Add to MetaCart
In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.
Approximate Statistical Tests for Comparing Supervised Classification Learning Algorithms
, 1998
"... This article reviews five approximate statistical tests for determining whether one learning algorithm outperforms another on a particular learning task. These tests are compared experimentally to determine their probability of incorrectly detecting a difference when no difference exists (type I err ..."
Abstract

Cited by 718 (9 self)
 Add to MetaCart
This article reviews five approximate statistical tests for determining whether one learning algorithm outperforms another on a particular learning task. These tests are compared experimentally to determine their probability of incorrectly detecting a difference when no difference exists (type I error). Two widely used statistical tests are shown to have high probability of type I error in certain situations and should never be used: a test for the difference of two proportions and a paireddifferences t test based on taking several random traintest splits. A third test, a paireddifferences t test based on 10fold crossvalidation, exhibits somewhat elevated probability of type I error. A fourth test, McNemar’s test, is shown to have low type I error. The fifth test is a new test, 5 × 2 cv, based on five iterations of twofold crossvalidation. Experiments show that this test also has acceptable type I error. The article also measures the power (ability to detect algorithm differences when they do exist) of these tests. The crossvalidated t test is the most powerful. The 5×2 cv test is shown to be slightly more powerful than McNemar’s test. The choice of the best test is determined by the computational cost of running the learning algorithm. For algorithms that can be executed only once, McNemar’s test is the only test with acceptable type I error. For algorithms that can be executed 10 times, the 5×2 cv test is recommended, because it is slightly more powerful and because it directly measures variation due to the choice of training set.
Prediction With Gaussian Processes: From Linear Regression To Linear Prediction And Beyond
 Learning and Inference in Graphical Models
, 1997
"... The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. Th ..."
Abstract

Cited by 231 (4 self)
 Add to MetaCart
(Show Context)
The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. This leads in to a more general discussion of Gaussian processes in section 4. Section 5 deals with further issues, including hierarchical modelling and the setting of the parameters that control the Gaussian process, the covariance functions for neural network models and the use of Gaussian processes in classification problems. PREDICTION WITH GAUSSIAN PROCESSES: FROM LINEAR REGRESSION TO LINEAR PREDICTION AND BEYOND 2 1 Introduction In the last decade neural networks have been used to tackle regression and classification problems, with some notable successes. It has also been widely recognized that they form a part of a wide variety of nonlinear statistical techniques that can be used for...
Sparse Gaussian processes using pseudoinputs
 Advances in Neural Information Processing Systems 18
, 2006
"... We present a new Gaussian process (GP) regression model whose covariance is parameterized by the the locations of M pseudoinput points, which we learn by a gradient based optimization. We take M ≪ N, where N is the number of real data points, and hence obtain a sparse regression method which has O( ..."
Abstract

Cited by 218 (13 self)
 Add to MetaCart
(Show Context)
We present a new Gaussian process (GP) regression model whose covariance is parameterized by the the locations of M pseudoinput points, which we learn by a gradient based optimization. We take M ≪ N, where N is the number of real data points, and hence obtain a sparse regression method which has O(M 2 N) training cost and O(M 2) prediction cost per test case. We also find hyperparameters of the covariance function in the same joint optimization. The method can be viewed as a Bayesian regression model with particular input dependent noise. The method turns out to be closely related to several other sparse GP approaches, and we discuss the relation in detail. We finally demonstrate its performance on some large data sets, and make a direct comparison to other sparse GP methods. We show that our method can match full GP performance with small M, i.e. very sparse solutions, and it significantly outperforms other approaches in this regime. 1
Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification
, 1997
"... Abstract. Gaussian processes are a natural way of defining prior distributions over functions of one or more input variables. In a simple nonparametric regression problem, where such a function gives the mean of a Gaussian distribution for an observed response, a Gaussian process model can easily be ..."
Abstract

Cited by 153 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Gaussian processes are a natural way of defining prior distributions over functions of one or more input variables. In a simple nonparametric regression problem, where such a function gives the mean of a Gaussian distribution for an observed response, a Gaussian process model can easily be implemented using matrix computations that are feasible for datasets of up to about a thousand cases. Hyperparameters that define the covariance function of the Gaussian process can be sampled using Markov chain methods. Regression models where the noise has a t distribution and logistic or probit models for classification applications can be implemented by sampling as well for latent values underlying the observations. Software is now available that implements these methods using covariance functions with hierarchical parameterizations. Models defined in this way can discover highlevel properties of the data, such as which inputs are relevant to predicting the response. 1
Fast Forward Selection to Speed Up Sparse Gaussian Process Regression
 IN WORKSHOP ON AI AND STATISTICS 9
, 2003
"... We present a method for the sparse greedy approximation of Bayesian Gaussian process regression, featuring a novel heuristic for very fast forward selection. Our method is essentially as fast as an equivalent one which selects the "support" patterns at random, yet it can outperform random ..."
Abstract

Cited by 105 (7 self)
 Add to MetaCart
We present a method for the sparse greedy approximation of Bayesian Gaussian process regression, featuring a novel heuristic for very fast forward selection. Our method is essentially as fast as an equivalent one which selects the "support" patterns at random, yet it can outperform random selection on hard curve fitting tasks. More importantly, it leads to a sufficiently stable approximation of the log marginal likelihood of the training data, which can be optimised to adjust a large number of hyperparameters automatically.
A Bayesian Committee Machine
 NEURAL COMPUTATION
, 2000
"... The Bayesian committee machine (BCM) is a novel approach to combining estimators which were trained on different data sets. Although the BCM can be applied to the combination of any kind of estimators the main foci are Gaussian process regression and related systems such as regularization networks a ..."
Abstract

Cited by 94 (7 self)
 Add to MetaCart
The Bayesian committee machine (BCM) is a novel approach to combining estimators which were trained on different data sets. Although the BCM can be applied to the combination of any kind of estimators the main foci are Gaussian process regression and related systems such as regularization networks and smoothing splines for which the degrees of freedom increase with the number of training data. Somewhat surprisingly, we nd that the performance of the BCM improves if several test points are queried at the same time and is optimal if the number of test points is at least as large as the degrees of freedom of the estimator. The BCM also provides a new solution for online learning with potential applications to data mining. We apply the BCM to systems with fixed basis functions and discuss its relationship to Gaussian process regression. Finally, we also show how the ideas behind the BCM can be applied in a nonBayesian setting to extend the input dependent combination of estimators.
An empirical bayesian strategy for solving the simultaneous sparse approximation problem
 IEEE Trans. Sig. Proc
, 2007
"... Abstract—Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent 1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were ..."
Abstract

Cited by 87 (14 self)
 Add to MetaCart
Abstract—Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent 1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weightspace necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys priorbased Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches. Index Terms—Automatic relevance determination, empirical Bayes, multiple response models, simultaneous sparse approximation, sparse Bayesian learning, variable selection. I.
Gaussian Process Priors with Uncertain Inputs  Application to MultipleStep Ahead Time Series Forecasting
, 2003
"... We consider the problem of multistep ahead prediction in time series analysis using the nonparametric Gaussian process model. kstep ahead forecasting of a discretetime nonlinear dynamic system can be performed by doing repeated onestep ahead predictions. For a statespace model of the form ..."
Abstract

Cited by 75 (15 self)
 Add to MetaCart
(Show Context)
We consider the problem of multistep ahead prediction in time series analysis using the nonparametric Gaussian process model. kstep ahead forecasting of a discretetime nonlinear dynamic system can be performed by doing repeated onestep ahead predictions. For a statespace model of the form y t # f#y t## ;:::;y t#L #, the prediction of y at time t # k is based on the point estimates of the previous outputs. In this paper, we show how, using an analytical Gaussian approximation, we can formally incorporate the uncertainty about intermediate regressor values, thus updating the uncertainty on the current prediction.
Gaussian Processes  A Replacement for Supervised Neural Networks?
"... These lecture notes are based on the work of Neal (1996), Williams and ..."
Abstract

Cited by 59 (0 self)
 Add to MetaCart
These lecture notes are based on the work of Neal (1996), Williams and