Computing Prediction Intervals Research Articles

Bootstrap prediction intervals for power-transformed time series

In this paper, we propose a bootstrap procedure to construct prediction intervals for future values of a variable after a linear ARIMA model has been fitted to its power transformation. The procedure is easy to implement and provides a useful tool in empirical applications given that it is often the case that, for example, the log-transformation is modeled when the variable of interest for prediction is the original one. The advantages over existing methods for computing prediction intervals of power-transformed time series are that the proposed bootstrap intervals incorporate the variability due to parameter estimation and do not rely on distributional assumptions neither on the original variable nor on the transformed one. We derive the asymptotic distribution and show the good behavior of the bootstrap approach versus alternative procedures by means of Monte Carlo experiments. Finally, the procedure is illustrated by analyzing three real-time series data sets.

Time Series Based Errors and Empirical Errors in Fertility Forecasts in the Nordic Countries*

Summary We use ARCH time series models to derive model based prediction intervals for the Total Fertility Rate (TFR) in Norway, Sweden, Finland, and Denmark up to 2050. For the short term (5–10 yrs), expected TFR‐errors are compared with empirical forecast errors observed in historical population forecasts prepared by the statistical agencies in these countries since 1969. Medium‐term and long‐term (up to 50 years) errors are compared with error patterns based on so‐called naïve forecasts, i.e. forecasts that assume that recently observed TFR‐levels also apply for the future.

Why population forecasts should be probabilistic - illustrated by the case of Norway

Deterministic population forecasts do not give an appropriate indication of forecast uncertainty. Forecasts should be probabilistic, rather than deterministic, so that their expected accuracy can be assessed. We review three main methods to compute probabilistic forecasts, namely time series extrapolation, analysis of historical forecast errors, and expert judgement. We illustrate, by the case of Norway up to 2050, how elements of these three methods can be combined when computing prediction intervals for a population’s future size and age-sex composition. We show the relative importance for prediction intervals of various sources of variance, and compare our results with those of the official population forecast computed by Statistics Norway.

Managing uncertainty in call centres using Poisson mixtures

AbstractWe model a call centre as a queueing model with Poisson arrivals having an unknown varying arrival rate. We show how to compute prediction intervals for the arrival rate, and use the Erlang formula for the waiting time to compute the consequences for the occupancy level of the call centre. We compare it to the current practice of using a point estimate of the arrival rate (assumed constant) as forecast. Copyright © 2001 John Wiley & Sons, Ltd.

Prediction intervals for farima processes by bootstrap methods

In this paper we introduce a procedure to compute prediction intervals for FARIMA (p d q) processes, taking into account the variability due to model identification and parameter estimation. To this aim, a particular bootstrap technique is developed. The performance of the prediction intervals is then assessed and compared to that of stand­ard bootstrap percentile intervals. The methods are applied to the time series of Nile River annual minima.

Prediction Intervals for Neural Networks via Nonlinear Regression

Standard methods for computing prediction intervals in nonlinear regression can be effectively applied to neural networks when the number of training points is large. Simulations show, however, that these methods can generate unreliable prediction intervals on smaller datasets when the network is trained to convergence. Stopping the training algorithm prior to convergence, to avoid overfitting, reduces the effective number of parameters but can lead to prediction intervals that are too wide. We present an alternative approach to estimating prediction intervals using weight decay to fit the network and show via a simulation study that this method may be effective in overcoming some of the shortcomings of the other approaches.

Prediction Intervals for Neural Networks via Nonlinear Regression

Standard methods for computing prediction intervals in nonlinear regression can be effectively applied to neural networks when the number of training points is large. Simulations show, however, that these methods can generate unreliable prediction intervals on smaller datasets when the network is trained to convergence. Stopping the training algorithm prior to convergence, to avoid overfitting, reduces the effective number of parameters but can lead to prediction intervals that are too wide. We present an alternative approach to estimating prediction intervals using weight decay to fit the network and show via a simulation study that this method may be effective in overcoming some of the shortcomings of the other approaches.

Forecasting in the 1990s

This paper reviews recent developments in time series forecasting with particular emphasis on the use of multivariate and non-linear models, the results of recent forecasting competitions, the computation of prediction intervals and the effects of model uncertainty.

Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models

A class of nonlinear state-space models, characterized by a single source of randomness, is introduced. A special case, the model underpinning the multiplicative Holt-Winters method of forecasting, is identified. Maximum likelihood estimation based on exponential smoothing instead of a Kalman filter, and with the potential to be applied in contexts involving non-Gaussian disturbances, is considered. A method for computing prediction intervals is proposed and evaluated on both simulated and real data.

Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models

Abstract A class of nonlinear state-space models, characterized by a single source of randomness, is introduced. A special case, the model underpinning the multiplicative Holt-Winters method of forecasting, is identified. Maximum likelihood estimation based on exponential smoothing instead of a Kalman filter, and with the potential to be applied in contexts involving non-Gaussian disturbances, is considered. A method for computing prediction intervals is proposed and evaluated on both simulated and real data.

Computation of Time-Specified Tolerance Intervals for Hybrid Time Series with Nonequidistant Sampling, Illustrated for Plasma Growth Hormone

The ideal reference interval for a variable of clinical interest would be specific for all deterministic factors affecting that variable, including the time of sampling in relation to biological rhythms. In particular, growth hormone is characterized in children by circadian and ultradian variability, with high peaks of secretion occurring mainly during sleep. For clinical applications, the use of tolerance intervals has been recommended, and they should substitute, whenever possible, for prediction limits. In the case of hybrid data (time series of data collected from a group of subjects), such a tolerance interval could be very difficult to determine following a parametric approach similar to the procedure used for the computation of prediction intervals, especially when consideration of both within-subjects and among-subjects variances is wanted. Accordingly, we have developed a nonparametric method for the computation of such tolerance intervals. Because the method is based on bootstrap techniques, it does not require the assumption of normality or symmetry in the data and is also more appropriate when dealing with small samples. The method was used to establish time-qualified reference limits for a series of growth hormone sampled around the clock in groups of prepubertal children differentiated according to stature. The use of these tolerance intervals may eliminate many false-positive and false-negative diagnoses that might be obtained when relying on time-unspecified single samples. The provision of such tolerance limits introduces time-specification and time-structure evaluation into prevention, diagnosis, and treatment of growth disorders. (Chronobiology International, 14(4), 409–425, 1997)

A Simple Method of Computing Prediction Intervals for Time Series Forecasts

Theoretical approaches to computing prediction intervals require strong assumptions that do not appear to hold in practice. This paper presents an empirical approach to prediction intervals that assumes very little. During model-fitting, variances of the errors are computed at different forecast leadtimes. Using these variances, the Chebyshev inequality is applied to determine prediction intervals. Empirical evidence is presented to show that this approach gives reasonable results. For example, using the 111 series in the M-competition, 95% prediction intervals actually contain 95.8% of post-sample observations.