Neural network regression with input uncertainty

It is generally assumed when using Bayesian inference methods for neural networks that the input data contains no noise or corruption. For real-world (errors in variable) problems this is clearly an unsafe assumption. This paper presents a Bayesian neural network framework which allows for input noise given that some model of the noise process exists. In the limit where this noise process is small and symmetric it is shown, using the Laplace approximation, that there is an additional term to the usual Bayesian error bar which depends on the variance of the input noise process. Further, by treating the true (noiseless) input as a hidden variable and sampling this jointly with the network's weights, using Markov Chain Monte Carlo methods, it is demonstrated that it is possible to infer the unbiassed regression over the noiseless input.

General bounds on Bayes errors for regression with Gaussian processes

Based on a simple convexity lemma, we develop bounds for different types of Bayesian prediction errors for regression with Gaussian processes. The basic bounds are formulated for a fixed training set. Simpler expressions are obtained for sampling from an input distribution which equals the weight function of the covariance kernel, yielding asymptotically tight results. The results are compared with numerical experiments.

The use of correlation and regression methods in optometry

Correlation and regression are two of the statistical procedures most widely used by optometrists. However, these tests are often misused or interpreted incorrectly, leading to erroneous conclusions from clinical experiments. This review examines the major statistical tests concerned with correlation and regression that are most likely to arise in clinical investigations in optometry. First, the use, interpretation and limitations of Pearson's product moment correlation coefficient are described. Second, the least squares method of fitting a linear regression to data and for testing how well a regression line fits the data are described. Third, the problems of using linear regression methods in observational studies, if there are errors associated in measuring the independent variable and for predicting a new value of Y for a given X, are discussed. Finally, methods for testing whether a non-linear relationship provides a better fit to the data and for comparing two or more regression lines are considered.

Probing three-way interactions in moderated multiple regression:Development and application of a slope difference test

Researchers often use 3-way interactions in moderated multiple regression analysis to test the joint effect of 3 independent variables on a dependent variable. However, further probing of significant interaction terms varies considerably and is sometimes error prone. The authors developed a significance test for slope differences in 3-way interactions and illustrate its importance for testing psychological hypotheses. Monte Carlo simulations revealed that sample size, magnitude of the slope difference, and data reliability affected test power. Application of the test to published data yielded detection of some slope differences that were undetected by alternative probing techniques and led to changes of results and conclusions. The authors conclude by discussing the test's applicability for psychological research. Copyright 2006 by the American Psychological Association.

Canonical solution groups of a homomorphism : manipulator kinematics, nonparametric regression and distributed object systems

The kinematic mapping of a rigid open-link manipulator is a homomorphism between Lie groups. The homomorphisrn has solution groups that act on an inverse kinematic solution element. A canonical representation of solution group operators that act on a solution element of three and seven degree-of-freedom (do!) dextrous manipulators is determined by geometric analysis. Seven canonical solution groups are determined for the seven do! Robotics Research K-1207 and Hollerbach arms. The solution element of a dextrous manipulator is a collection of trivial fibre bundles with solution fibres homotopic to the Torus. If fibre solutions are parameterised by a scalar, a direct inverse funct.ion that maps the scalar and Cartesian base space coordinates to solution element fibre coordinates may be defined. A direct inverse pararneterisation of a solution element may be approximated by a local linear map generated by an inverse augmented Jacobian correction of a linear interpolation. The action of canonical solution group operators on a local linear approximation of the solution element of inverse kinematics of dextrous manipulators generates cyclical solutions. The solution representation is proposed as a model of inverse kinematic transformations in primate nervous systems. Simultaneous calibration of a composition of stereo-camera and manipulator kinematic models is under-determined by equi-output parameter groups in the composition of stereo-camera and Denavit Hartenberg (DH) rnodels. An error measure for simultaneous calibration of a composition of models is derived and parameter subsets with no equi-output groups are determined by numerical experiments to simultaneously calibrate the composition of homogeneous or pan-tilt stereo-camera with DH models. For acceleration of exact Newton second-order re-calibration of DH parameters after a sequential calibration of stereo-camera and DH parameters, an optimal numerical evaluation of DH matrix first order and second order error derivatives with respect to a re-calibration error function is derived, implemented and tested. A distributed object environment for point and click image-based tele-command of manipulators and stereo-cameras is specified and implemented that supports rapid prototyping of numerical experiments in distributed system control. The environment is validated by a hierarchical k-fold cross validated calibration to Cartesian space of a radial basis function regression correction of an affine stereo model. Basic design and performance requirements are defined for scalable virtual micro-kernels that broker inter-Java-virtual-machine remote method invocations between components of secure manageable fault-tolerant open distributed agile Total Quality Managed ISO 9000+ conformant Just in Time manufacturing systems.

Analysis of spatial patterns in histological sections of brain tissue using a method based on regression

A method of determining the spatial pattern of any histological feature in sections of brain tissue which can be measured quantitatively is described and compared with a previously described method. A measurement of a histological feature such as density, area, amount or load is obtained for a series of contiguous sample fields. The regression coefficient (β) is calculated from the measurements taken in pairs, first in pairs of adjacent samples and then in pairs of samples taken at increasing degrees of separation between them, i.e. separated by 2, 3, 4,..., n units. A plot of β versus the degree of separation between the pairs of sample fields reveals whether the histological feature is distributed randomly, uniformly or in clusters. If the feature is clustered, the analysis determines whether the clusters are randomly or regularly distributed, the mean size of the clusters and the spacing of the clusters. The method is simple to apply and interpret and is illustrated using simulated data and studies of the spatial patterns of blood vessels in the cerebral cortex of normal brain, the degree of vacuolation of the cortex in patients with Creutzfeldt-Jacob disease (CJD) and the characteristic lesions present in Alzheimer's disease (AD). Copyright (C) 2000 Elsevier Science B.V.

Spatial pattern analysis of beta-amyloid (A beta) deposits in Alzheimer disease by linear regression

The spatial patterns of discrete beta-amyloid (Abeta) deposits in brain tissue from patients with Alzheimer disease (AD) were studied using a statistical method based on linear regression, the results being compared with the more conventional variance/mean (V/M) method. Both methods suggested that Abeta deposits occurred in clusters (400 to <12,800 mu m in diameter) in all but 1 of the 42 tissues examined. In many tissues, a regular periodicity of the Abeta deposit clusters parallel to the tissue boundary was observed. In 23 of 42 (55%) tissues, the two methods revealed essentially the same spatial patterns of Abeta deposits; in 15 of 42 (36%), the regression method indicated the presence of clusters at a scale not revealed by the V/M method; and in 4 of 42 (9%), there was no agreement between the two methods. Perceived advantages of the regression method are that there is a greater probability of detecting clustering at multiple scales, the dimension of larger Abeta clusters can be estimated more accurately, and the spacing between the clusters may be estimated. However, both methods may be useful, with the regression method providing greater resolution and the V/M method providing greater simplicity and ease of interpretation. Estimates of the distance between regularly spaced Abeta clusters were in the range 2,200-11,800 mu m, depending on tissue and cluster size. The regular periodicity of Abeta deposit clusters in many tissues would be consistent with their development in relation to clusters of neurons that give rise to specific neuronal projections.

Quantification of hydroxycinnamic acids and lignin in perennial forage and energy grasses by Fourier-transform infrared spectroscopy and partial least squares regression

Levels of lignin and hydroxycinnamic acid wall components in three genera of forage grasses (Lolium,Festuca and Dactylis) have been accurately predicted by Fourier-transform infrared spectroscopy using partial least squares models correlated to analytical measurements. Different models were derived that predicted the concentrations of acid detergent lignin, total hydroxycinnamic acids, total ferulate monomers plus dimers, p-coumarate and ferulate dimers in independent spectral test data from methanol extracted samples of perennial forage grass with accuracies of 92.8%, 86.5%, 86.1%, 59.7% and 84.7% respectively, and analysis of model projection scores showed that the models relied generally on spectral features that are known absorptions of these compounds. Acid detergent lignin was predicted in samples of two species of energy grass, (Phalaris arundinacea and Pancium virgatum) with an accuracy of 84.5%.

Statnote 17: using a regression line for prediction and calibration

Two types of prediction problem can be solved using a regression line viz., prediction of the ‘population’ regression line at the point ‘x’ and prediction of an ‘individual’ new member of the population ‘y1’ for which ‘x1’ has been measured. The second problem is probably the most commonly encountered and the most relevant to calibration studies. A regression line is likely to be most useful for calibration if the range of values of the X variable is large, if there is a good representation of the ‘x,y’ values across the range of X, and if several estimates of ‘y’ are made at each ‘x’. It is poor statistical practice to use a regression line for calibration or prediction beyond the limits of the data.

Statnote 16: fitting a regression line to data

Fitting a linear regression to data provides much more information about the relationship between two variables than a simple correlation test. A goodness of fit test of the line should always be carried out. Hence, ‘r squared’ estimates the strength of the relationship between Y and X, ANOVA whether a statistically significant line is present, and the ‘t’ test whether the slope of the line is significantly different from zero. In addition, it is important to check whether the data fit the assumptions for regression analysis and, if not, whether a transformation of the Y and/or X variables is necessary.

Statnote 18: comparison of regression lines

In many circumstances, it may be of interest to discover whether two or more regression lines are the same. Regression lines may differ in three properties, viz., in residual variance, in slope, and in elevation; all of which can be tested using analysis of covariance. If there are no significant differences between regression lines, an investigator may which to combine the data from different studies and fit a single regression line to the whole of the data.

Statnote 19: non-linear regression: fitting an exponential curve to data

Non-linear relationships are common in microbiological research and often necessitate the use of the statistical techniques of non-linear regression or curve fitting. In some circumstances, the investigator may wish to fit an exponential model to the data, i.e., to test the hypothesis that a quantity Y either increases or decays exponentially with increasing X. This type of model is straight forward to fit as taking logarithms of the Y variable linearises the relationship which can then be treated by the methods of linear regression.

Statnote 20: non-linear regression: fitting a general polynomial curve

In some circumstances, there may be no scientific model of the relationship between X and Y that can be specified in advance and indeed the objective of the investigation may be to provide a ‘curve of best fit’ for predictive purposes. In such an example, the fitting of successive polynomials may be the best approach. There are various strategies to decide on the polynomial of best fit depending on the objectives of the investigation.