An upper bound on the Bayesian error bars for generalized linear regression

In the Bayesian framework, predictions for a regression problem are expressed in terms of a distribution of output values. The mode of this distribution corresponds to the most probable output, while the uncertainty associated with the predictions can conveniently be expressed in terms of error bars. In this paper we consider the evaluation of error bars in the context of the class of generalized linear regression models. We provide insights into the dependence of the error bars on the location of the data points and we derive an upper bound on the true error bars in terms of the contributions from individual data points which are themselves easily evaluated.

A principled approach to interactive hierarchical non-linear visualization of high-dimensional data

Hierarchical visualization systems are desirable because a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex high-dimensional data sets. We extend an existing locally linear hierarchical visualization system PhiVis [1] in several directions: bf(1) we allow for em non-linear projection manifolds (the basic building block is the Generative Topographic Mapping -- GTM), bf(2) we introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree, bf(3) we describe folding patterns of low-dimensional projection manifold in high-dimensional data space by computing and visualizing the manifold's local directional curvatures. Quantities such as magnification factors [3] and directional curvatures are helpful for understanding the layout of the nonlinear projection manifold in the data space and for further refinement of the hierarchical visualization plot. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. We demonstrate the visualization system principle of the approach on a complex 12-dimensional data set and mention possible applications in the pharmaceutical industry.

Gaussian regression and optimal finite dimensional linear models

The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal regression is the posterior mean and its computation scales as O(n³), where n is the sample size. We show that the optimal m-dimensional linear model under a given prior is spanned by the first m eigenfunctions of a covariance operator, which is a trace-class operator. This is an infinite dimensional analogue of principal component analysis. The importance of Hilbert space methods to practical statistics is also discussed.

Semi-supervised learning of hierarchical latent trait models for data visualisation

An interactive hierarchical Generative Topographic Mapping (HGTM) ¸iteH_GTM has been developed to visualise complex data sets. In this paper, we build a more general visualisation system by extending the HGTM visualisation system in 3 directions: bf (1) We generalize HGTM to noise models from the exponential family of distributions. The basic building block is the Latent Trait Model (LTM) developed in ¸iteKaban_pami. bf (2) We give the user a choice of initializing the child plots of the current plot in either em interactive, or em automatic mode. In the interactive mode the user interactively selects ``regions of interest'' as in ¸iteH_GTM, whereas in the automatic mode an unsupervised minimum message length (MML)-driven construction of a mixture of LTMs is employed. bf (3) We derive general formulas for magnification factors in latent trait models. Magnification factors are a useful tool to improve our understanding of the visualisation plots, since they can highlight the boundaries between data clusters. The unsupervised construction is particularly useful when high-level plots are covered with dense clusters of highly overlapping data projections, making it difficult to use the interactive mode. Such a situation often arises when visualizing large data sets. We illustrate our approach on a toy example and apply our system to three more complex real data sets.

Semisupervised learning of hierarchical latent trait models for data visualization

Recently, we have developed the hierarchical Generative Topographic Mapping (HGTM), an interactive method for visualization of large high-dimensional real-valued data sets. In this paper, we propose a more general visualization system by extending HGTM in three ways, which allows the user to visualize a wider range of data sets and better support the model development process. 1) We integrate HGTM with noise models from the exponential family of distributions. The basic building block is the Latent Trait Model (LTM). This enables us to visualize data of inherently discrete nature, e.g., collections of documents, in a hierarchical manner. 2) We give the user a choice of initializing the child plots of the current plot in either interactive, or automatic mode. In the interactive mode, the user selects "regions of interest," whereas in the automatic mode, an unsupervised minimum message length (MML)-inspired construction of a mixture of LTMs is employed. The unsupervised construction is particularly useful when high-level plots are covered with dense clusters of highly overlapping data projections, making it difficult to use the interactive mode. Such a situation often arises when visualizing large data sets. 3) We derive general formulas for magnification factors in latent trait models. Magnification factors are a useful tool to improve our understanding of the visualization plots, since they can highlight the boundaries between data clusters. We illustrate our approach on a toy example and evaluate it on three more complex real data sets. © 2005 IEEE.

Probabilistic multiple model neural network based leak detection system:experimental study

This paper presents an effective decision making system for leak detection based on multiple generalized linear models and clustering techniques. The training data for the proposed decision system is obtained by setting up an experimental pipeline fully operational distribution system. The system is also equipped with data logging for three variables; namely, inlet pressure, outlet pressure, and outlet flow. The experimental setup is designed such that multi-operational conditions of the distribution system, including multi pressure and multi flow can be obtained. We then statistically tested and showed that pressure and flow variables can be used as signature of leak under the designed multi-operational conditions. It is then shown that the detection of leakages based on the training and testing of the proposed multi model decision system with pre data clustering, under multi operational conditions produces better recognition rates in comparison to the training based on the single model approach. This decision system is then equipped with the estimation of confidence limits and a method is proposed for using these confidence limits for obtaining more robust leakage recognition results.

A comparison of linear forecasting models and neural networks:an application to Euro inflation and Euro Divisia

Linear models reach their limitations in applications with nonlinearities in the data. In this paper new empirical evidence is provided on the relative Euro inflation forecasting performance of linear and non-linear models. The well established and widely used univariate ARIMA and multivariate VAR models are used as linear forecasting models whereas neural networks (NN) are used as non-linear forecasting models. It is endeavoured to keep the level of subjectivity in the NN building process to a minimum in an attempt to exploit the full potentials of the NN. It is also investigated whether the historically poor performance of the theoretically superior measure of the monetary services flow, Divisia, relative to the traditional Simple Sum measure could be attributed to a certain extent to the evaluation of these indices within a linear framework. Results obtained suggest that non-linear models provide better within-sample and out-of-sample forecasts and linear models are simply a subset of them. The Divisia index also outperforms the Simple Sum index when evaluated in a non-linear framework. © 2005 Taylor & Francis Group Ltd.

On the relationship between Bayesian error bars and the input data density

We investigate the dependence of Bayesian error bars on the distribution of data in input space. For generalized linear regression models we derive an upper bound on the error bars which shows that, in the neighbourhood of the data points, the error bars are substantially reduced from their prior values. For regions of high data density we also show that the contribution to the output variance due to the uncertainty in the weights can exhibit an approximate inverse proportionality to the probability density. Empirical results support these conclusions.

DVMS 1.5: A user manual (the data visualization and modeling system)

The data available during the drug discovery process is vast in amount and diverse in nature. To gain useful information from such data, an effective visualisation tool is required. To provide better visualisation facilities to the domain experts (screening scientist, biologist, chemist, etc.),we developed a software which is based on recently developed principled visualisation algorithms such as Generative Topographic Mapping (GTM) and Hierarchical Generative Topographic Mapping (HGTM). The software also supports conventional visualisation techniques such as Principal Component Analysis, NeuroScale, PhiVis, and Locally Linear Embedding (LLE). The software also provides global and local regression facilities . It supports regression algorithms such as Multilayer Perceptron (MLP), Radial Basis Functions network (RBF), Generalised Linear Models (GLM), Mixture of Experts (MoE), and newly developed Guided Mixture of Experts (GME). This user manual gives an overview of the purpose of the software tool, highlights some of the issues to be taken care while creating a new model, and provides information about how to install & use the tool. The user manual does not require the readers to have familiarity with the algorithms it implements. Basic computing skills are enough to operate the software.

Contrast integration over area is extensive: a three-stage model of spatial summation

Classical studies of area summation measure contrast detection thresholds as a function of grating diameter. Unfortunately, (i) this approach is compromised by retinal inhomogeneity and (ii) it potentially confounds summation of signal with summation of internal noise. The Swiss cheese stimulus of T. S. Meese and R. J. Summers (2007) and the closely related Battenberg stimulus of T. S. Meese (2010) were designed to avoid these problems by keeping target diameter constant and modulating interdigitated checks of first-order carrier contrast within the stimulus region. This approach has revealed a contrast integration process with greater potency than the classical model of spatial probability summation. Here, we used Swiss cheese stimuli to investigate the spatial limits of contrast integration over a range of carrier frequencies (1–16 c/deg) and raised plaid modulator frequencies (0.25–32 cycles/check). Subthreshold summation for interdigitated carrier pairs remained strong (~4 to 6 dB) up to 4 to 8 cycles/check. Our computational analysis of these results implied linear signal combination (following square-law transduction) over either (i) 12 carrier cycles or more or (ii) 1.27 deg or more. Our model has three stages of summation: short-range summation within linear receptive fields, medium-range integration to compute contrast energy for multiple patches of the image, and long-range pooling of the contrast integrators by probability summation. Our analysis legitimizes the inclusion of widespread integration of signal (and noise) within hierarchical image processing models. It also confirms the individual differences in the spatial extent of integration that emerge from our approach.

Bayesian data assimilation

This thesis addresses data assimilation, which typically refers to the estimation of the state of a physical system given a model and observations, and its application to short-term precipitation forecasting. A general introduction to data assimilation is given, both from a deterministic and' stochastic point of view. Data assimilation algorithms are reviewed, in the static case (when no dynamics are involved), then in the dynamic case. A double experiment on two non-linear models, the Lorenz 63 and the Lorenz 96 models, is run and the comparative performance of the methods is discussed in terms of quality of the assimilation, robustness "in the non-linear regime and computational time. Following the general review and analysis, data assimilation is discussed in the particular context of very short-term rainfall forecasting (nowcasting) using radar images. An extended Bayesian precipitation nowcasting model is introduced. The model is stochastic in nature and relies on the spatial decomposition of the rainfall field into rain "cells". Radar observations are assimilated using a Variational Bayesian method in which the true posterior distribution of the parameters is approximated by a more tractable distribution. The motion of the cells is captured by a 20 Gaussian process. The model is tested on two precipitation events, the first dominated by convective showers, the second by precipitation fronts. Several deterministic and probabilistic validation methods are applied and the model is shown to retain reasonable prediction skill at up to 3 hours lead time. Extensions to the model are discussed.

Quantitative prediction of mouse class I MHC peptide binding affinity using support vector machine regression (SVR) models

Background - The binding between peptide epitopes and major histocompatibility complex proteins (MHCs) is an important event in the cellular immune response. Accurate prediction of the binding between short peptides and the MHC molecules has long been a principal challenge for immunoinformatics. Recently, the modeling of MHC-peptide binding has come to emphasize quantitative predictions: instead of categorizing peptides as "binders" or "non-binders" or as "strong binders" and "weak binders", recent methods seek to make predictions about precise binding affinities. Results - We developed a quantitative support vector machine regression (SVR) approach, called SVRMHC, to model peptide-MHC binding affinities. As a non-linear method, SVRMHC was able to generate models that out-performed existing linear models, such as the "additive method". By adopting a new "11-factor encoding" scheme, SVRMHC takes into account similarities in the physicochemical properties of the amino acids constituting the input peptides. When applied to MHC-peptide binding data for three mouse class I MHC alleles, the SVRMHC models produced more accurate predictions than those produced previously. Furthermore, comparisons based on Receiver Operating Characteristic (ROC) analysis indicated that SVRMHC was able to out-perform several prominent methods in identifying strongly binding peptides. Conclusion - As a method with demonstrated performance in the quantitative modeling of MHC-peptide binding and in identifying strong binders, SVRMHC is a promising immunoinformatics tool with not inconsiderable future potential.

Estimating conditional volatility with neural networks

It is well known that one of the obstacles to effective forecasting of exchange rates is heteroscedasticity (non-stationary conditional variance). The autoregressive conditional heteroscedastic (ARCH) model and its variants have been used to estimate a time dependent variance for many financial time series. However, such models are essentially linear in form and we can ask whether a non-linear model for variance can improve results just as non-linear models (such as neural networks) for the mean have done. In this paper we consider two neural network models for variance estimation. Mixture Density Networks (Bishop 1994, Nix and Weigend 1994) combine a Multi-Layer Perceptron (MLP) and a mixture model to estimate the conditional data density. They are trained using a maximum likelihood approach. However, it is known that maximum likelihood estimates are biased and lead to a systematic under-estimate of variance. More recently, a Bayesian approach to parameter estimation has been developed (Bishop and Qazaz 1996) that shows promise in removing the maximum likelihood bias. However, up to now, this model has not been used for time series prediction. Here we compare these algorithms with two other models to provide benchmark results: a linear model (from the ARIMA family), and a conventional neural network trained with a sum-of-squares error function (which estimates the conditional mean of the time series with a constant variance noise model). This comparison is carried out on daily exchange rate data for five currencies.

Developments of the generative topographic mapping

The generative topographic mapping (GTM) model was introduced by Bishop et al. (1998, Neural Comput. 10(1), 215-234) as a probabilistic re- formulation of the self-organizing map (SOM). It offers a number of advantages compared with the standard SOM, and has already been used in a variety of applications. In this paper we report on several extensions of the GTM, including an incremental version of the EM algorithm for estimating the model parameters, the use of local subspace models, extensions to mixed discrete and continuous data, semi-linear models which permit the use of high-dimensional manifolds whilst avoiding computational intractability, Bayesian inference applied to hyper-parameters, and an alternative framework for the GTM based on Gaussian processes. All of these developments directly exploit the probabilistic structure of the GTM, thereby allowing the underlying modelling assumptions to be made explicit. They also highlight the advantages of adopting a consistent probabilistic framework for the formulation of pattern recognition algorithms.

Efficient training of RBF networks for classification

Radial Basis Function networks with linear outputs are often used in regression problems because they can be substantially faster to train than Multi-layer Perceptrons. For classification problems, the use of linear outputs is less appropriate as the outputs are not guaranteed to represent probabilities. In this paper we show how RBFs with logistic and softmax outputs can be trained efficiently using algorithms derived from Generalised Linear Models. This approach is compared with standard non-linear optimisation algorithms on a number of datasets.