We present a new method for obtaining local error bars for nonlinear regression, i.e., estimates of the confidence in predicted values that depend on the input. We approach this problem by applying a maximumlikelihood framework to an assumed distribution of errors. We demonstrate our method first on computer-generated data with locally varying, normally distributed target noise. We then apply it to laser data from the Santa Fe Time Series Competition where the underlying system noise is known quantization error and the error bars give local estimates of model misspecification. In both cases, the method also provides a weightedregression effect that improves generalization performance.

The TNM staging system has been used since the early 1960's to predict breast cancer patient outcome. In an attempt to increase prognosticaccuracy, many putative prognostic factors have been identified. Because the TNM stage model can not accommodate thesenew factors, the proliferation of factors in breast cancer has lead to clinical confusion. What is required is a new computerized prognostic system that can test putative prognostic factors and integrate the predictive factors with the TNM variables inorder to increase prognostic accuracy. Using the area under the curve of the receiver operating characteristic, we compare the accuracy of the following predictive models in terms of five year breast cancer-specific survival: pTNM staging system, principal componentanalysis, classification and regression trees, logistic regression, cascade correlation neural network, conjugate gradient descent neural, probabilistic neural network, and backpropagation neural network. Several statistical models are significantly more ac- 1064 HarryB.

We present a new method for obtaining local error bars for nonlinear regression, i.e., estimates of the confidence in predicted values that depend onthe input. We approach this problem by applying a maximumlikelihood frameworkto an assumed distribution of errors. We demonstrate our method first on computer-generated data with locally varying, normally distributed target noise. We then apply it to laser data from the Santa Fe Time Series Competition where the underlying system noise is known quantization error and the error bars give local estimates of model misspecification. In both cases, the method also provides a weightedregression effectthat improves generalization performance.

In this procedure one transforms variables to a basis in which the covariance isdiagonal and then projects out the low variance directions. While application of PCA to remove input variables is useful in some cases (Leen et al., 1990), there is no guarantee that low variance variables have little effect on error. We propose a saliency measure, based on PCA, that identifies those variables that have the least effect on error. Our proposed Principal Components Pruning algorithm applies this measure to obtain a simple and cheap pruning technique in the context of supervised learning. Fast Pruning Using Principal Components 37 Special Case: PCP in Linear Regression In unbiased linear models, one can bound the bias introduced from pruning the principal degrees of freedom in the model.

This paper is a multidisciplinary review of empirical, statistical learning from a graphical model perspective. Well-known examples of graphical models include Bayesian networks, directed graphs representing a Markov chain, and undirected networks representing a Markov field. These graphical models are extended to model data analysis and empirical learning using the notation of plates. Graphical operations for simplifying and manipulating a problem are provided including decomposition, differentiation, andthe manipulation of probability models from the exponential family. Two standard algorithm schemas for learning are reviewed in a graphical framework: Gibbs sampling and the expectation maximizationalgorithm. Using these operations and schemas, some popular algorithms can be synthesized from their graphical specification. This includes versions of linear regression, techniques for feed-forward networks, and learning Gaussian and discrete Bayesian networks from data. The paper concludes by sketching some implications for data analysis and summarizing how some popular algorithms fall within the framework presented. The main original contributions here are the decompositiontechniques and the demonstration that graphical models provide a framework for understanding and developing complex learning algorithms.

The bootstrap method offers an computation intensive alternative to estimate the predictive distribution for a neural network even if the analytic derivation is intractable. The available asymptotic results show that it is valid for a large number of linear, nonlinear and even nonparametric regression problems. It has the potential to model the distribution of estimators to a higher precision than the usual normal asymptotics. It even may be valid if the normal asymptotics fail. However, the theoretical properties of bootstrap procedures for neural networks - especially nonlinear models - have to be investigated more comprehensively.

The bootstrap method offers an computation intensive alternative to estimate the predictive distribution for a neural network even if the analytic derivation is intractable. The available asymptotic results show that it is valid for a large number of linear, nonlinear and even nonparametric regression problems. It has the potential to model the distribution of estimators to a higher precision than the usual normal asymptotics. It even may be valid if the normal asymptotics fail. However, the theoretical properties of bootstrap procedures for neural networks - especially nonlinear models - have to be investigated more comprehensively.

The bootstrap method offers an computation intensive alternative to estimate the predictive distribution for a neural network even if the analytic derivation is intractable. Theavailable asymptotic results show that it is valid for a large number of linear, nonlinear and even nonparametric regression problems. It has the potential tomodel the distribution of estimators to a higher precision than the usual normal asymptotics. It even may be valid if the normal asymptotics fail. However, the theoretical properties of bootstrap procedures for neural networks - especially nonlinear models - have to be investigated more comprehensively.

Two projection based feedforward network learning methods for modelfree regression problems are studied and compared in this paper: one is the popular back-propagation learning (BPL); the other is the projection pursuit learning (PPL).

G/SPLINES is an algorithm for building functional models of data. It uses genetic search to discover combinations of basis functions which are then used to build a least-squares regression model. Because it produces a population of models which evolve over time rather than a single model, it allows analysis not possible with other regression-based approaches. 1 INTRODUCTION G/SPLINES is a hybrid of Friedman's Multivariable Adaptive Regression Splines (MARS) algorithm (Friedman, 1990) with Holland's Genetic Algorithm (Holland, 1975). G/SPLINES has advantages over MARS in that it requires fewer least-squares computations, is easily extendable to non-spline basis functions, may discover models inaccessible to local-variable selection algorithms, and allows significantly larger problems to be considered. These issues are discussed in (Rogers, 1991). This paper begins with a discussion of linear regression models, followed by a description of the G/SPLINES algorithm, and finishes with a series of experiments illustrating its performance, robustness, and analysis capabilities.