Predicting with linear models

Introduction

We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models. For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in glam and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.

Best linear unbiased prediction(BLUP)

Best Linear Unbiased Prediction (BLUP) has become the most widely accepted method for genetic evaluation of domestic livestock. Since its introduction, the method has evolved and despite this there is no simple text on the application of linear models to the prediction of breeding values. This book has been written with a good balance of theory and application to fill this gap. Equations for partitioning breeding values into contributions from various sources of information are derived under the various models. Recent developments in the analysis of longitudinal data with random regression models and the inclusion of genetic marker information in the evaluation of animals have been incorporated. Overall the book has been thoroughly updated since the first edition was published in 1996.

Linear prediction in audio signals

While linear prediction (LP) has become immensely popular in speech modelling, it does not seem to provide a good approach for modelling audio signals. This is somewhat surprising, since a tonal signal consisting of a number of sinusoids can be perfectly predicted based on an (all-pole) LP model with a model order that is twice the number of sinusoids. We provide an explanation why this result cannot simply be extrapolated to LP of audio signals. If noise is taken into account in the tonal signal model, a low-order all-pole model appears to be only appropriate when the tonal components are uniformly distributed in the SyQuest interval. Based on this observation, different alternatives to the conventional LP model can be suggested. Either the model should be changed to a pole-zero, a high-order all-pole, or a pitch prediction model, or the conventional LP model should be preceded by an appropriate frequency transform, such as a frequency warping or down sampling. By comparing these alternative LP models to the conventional LP model in terms of frequency estimation accuracy, residual spectral flatness, and perceptual frequency resolution, we obtain several new and promising approaches to LP-based audio modelling.

Linear prediction (LP) is a widely used and well-understood technique for the analysis, modeling, and coding of speech signals. Its success can be attributed to its correspondence with the speech generation process. The vocal tract can be modeled as a slowly time-varying, low-order all-pole filter, while the glottal excitation can be represented either by a white noise sequence (for unvoiced sounds), or by an impulse train generated by periodic vibrations of the vocal chords (for voiced sounds). By using this so-called source-filter model, a speech segment can be whitened with a cascade of a formant predictor for removing short-term correlation, and a pitch predictor for removing long-term correlation.