Prediction Of Future Values Research Articles

Bayesian analysis of a growth curve model with power transformation, random effects and AR(1) dependence

In this paper we devote ourselves to a general growth curve model with power transformation, random effects and AR(1) dependence via a Bayesian approach. Two priors are proposed and both parameter estimation and prediction of future values are considered. Some numerical results with a set of real data are also given.

Prediction Variance and Information Worth of Observations in Time Series

The problem of developing measures of worth of observations in time series has not received much attention in the literature. Any meaningful measure of worth should naturally depend on the position of the observation as well as the objectives of the analysis, namely parameter estimation or prediction of future values. We introduce a measure that quantifies worth of a set of observations for the purpose of prediction of outcomes of stationary processes. The worth is measured as the change in the information content of the entire past due to exclusion or inclusion of a set of observations. The information content is quantified by the mutual information, which is the information theoretic measure of dependency. For Gaussian processes, the measure of worth turns out to be the relative change in the prediction error variance due to exclusion or inclusion of a set of observations. We provide formulae for computing predictive worth of a set of observations for Gaussian autoregressive moving‐average processs. For non‐Gaussian processes, however, a simple function of its entropy provides a lower bound for the variance of prediction error in the same manner that Fisher information provides a lower bound for the variance of an unbiased estimator via the Cramer‐Rao inequality. Statistical estimation of this lower bound requires estimation of the entropy of a stationary time series.

On estimation and prediction for temporally correlated longitudinal data

Abstract In this paper, from a Bayesian point of view, we consider estimation of parameters and prediction of future values for the longitudinal model proposed by Diggle (1988. Biometrics 44, 959–971). This model, called the repeated measures linear model, incorporates group mean, variability among individuals, serial correlation within an individual, and measurement error. Two different priors are employed by the Bayesian approach, one is the noninformative prior and the other is composed of inverse gamma distributions. Given the noninformative prior, it is shown that the resulting approximate estimates of the regression coefficients are the same as those derived by the restricted maximum likelihood estimation. Markov chain Monte Carlo methods are also used to obtain more accurate Bayesian inference for parameters as well as prediction of future values. For parameter estimation and prediction of future values, the advantages of the Bayesian approach over the maximum likelihood method and the restricted maximum likelihood method are demonstrated by both real and simulated data.

Bayesian analysis of a general growth curve model with predictions using power transformations and AR(1) autoregressive dependence

In this paper, we consider a Bayesian analysis of the unbalanced (general) growth curve model with AR(1) autoregressive dependence, while applying the Box-Cox power transformations. We propose exact, simple and Markov chain Monte Carlo approximate parameter estimation and prediction of future values. Numerical results are illustrated with real and simulated data.

Prediction of shot noise

Prediction of future values of a shot noise process observed on a discrete lattice of points is considered. The shot magnitudes are assumed to be independent and identically distributed and to arrive via a Poisson process; the effect of each shot dissipates and/or accumulates according to a known shot function. Conditional mean and linear point predictors of future process values are developed. Distributional prediction, obtained through saddlepoint approximation of the conditional distributions of the process, is also explored.

Model Selection, Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.

Bayesian analysis of growth curves with AR(1) dependence

In this paper we consider Bayesian analysis of the generalized growth curve model when the covariance matrix Σ = σ 2 C where C = ( ϱ i − j ), σ 2 > 0 and −1 < ϱ < 1 are unknown. We consider both parameter estimation and prediction of future values. Results are illustrated with real and simulated data.

Analysis of longitudinal data in craniofacial research: some strategies.

Although it is generally acknowledged that longitudinal data provide the most information on growth and development and other time-dependent phenomena, such data are often analyzed by conventional (cross-sectional) statistical methods. This widespread practice ignores the distinctive characteristics (e.g., covariance structure) of longitudinal data and may yield misleading results. The purpose of this article is to present some strategies and make available computer programs for the appropriate analysis of longitudinal data. User-friendly PC programs for the estimation of average growth curves, computation of tracking indices, prediction of future values, diagnosis, classification, clustering, estimation of missing values, and testing hypotheses concerning individual and group differences are presented. Benefits of these methods over the usual techniques are illustrated with the example of maxillary growth in the rhesus monkey.

A decomposition of some serially-structured variance matrices

SUMMARY Serially-structured variance matrices which can be decomposed into lower-triangular band matrices are defined. These may be thought of as the variance matrices of finite realizations of generalized autoregressive-moving average processes. A large number of serially-generated stochastic processes are shown to be of this type. To estimate parameters in a model by maximum likelihood, or related techniques, with data in which the errors are correlated it is often necessary to handle a large error covariance matrix. The evaluation of elements in this matrix and repeated inversion in the course of an iterative estimation procedure can be prohibitively expensive in computer time. For this reason, ways of evaluating the likelihood have been proposed for particular types of covariance structure without explicit use of the error variance matrix. For example, Duncan & Jones (1966) used a Fourier transform to find an approximation to the likelihood of any stationary error process and Harvey & Phillips (1979) used the Kalman filter to evaluate the likelihood of a linear model in the presence of an autoregress- ive-moving average error process. Matis & Wehrly (1979) and Sandland & McGilchrist (1979), both in a special review issue of Biometrics, used the particular properties of Markov processes to estimate parameters respectively in simple cases of stochastic compartment models and stochastic differential equations. In ? 2 a decomposition into lower-triangular band matrices is found for a large class of serially-structured variance matrices. This enables a wider range of covariance struc- tures to be used than before. In particular, it is a useful tool when fitting higher-order models of the type proposed by Matis & Wehrly (1979) and Sandland & McGilchrist (1979) and when other sources of error have to be included. It also simplifies the prediction of future values for particular error processes and the simulation of serially-correlated observations. The form of decomposition shows the class to be a generalization of

Vitality, Community, Creativity: Prediction of Future Values, by Alfred P. Bernhart. Printed by University of Toronto Press and distributed by the Author, 23 Cheriton Avenue, Toronto, Ontario, Canada: [x +] 389 pp., illustr., 23.6 × 16.2 × 2.0 cm, [no price indicated], [1976

Vitality, Community, Creativity: Prediction of Future Values, by Alfred P. Bernhart. Printed by University of Toronto Press and distributed by the Author, 23 Cheriton Avenue, Toronto, Ontario, Canada: [x +] 389 pp., illustr., 23.6 × 16.2 × 2.0 cm, [no price indicated], [1976