Approximate Prediction Intervals Research Articles

Statistical Prediction of Future Sports Records Based on Record Values

Point prediction of future record values based on sequences of previous lower or upper records is considered by means of the method of maximum product of spacings, where the underlying distribution is assumed to be a power function distribution and a Pareto distribution, respectively. Moreover, exact and approximate prediction intervals are discussed and compared with regard to their expected lengths and their percentages of coverage. The focus is on deriving explicit expressions in the point and interval prediction procedures. Predictions and forecasts are of interest, e.g., in sports analytics, which is gaining more and more attention in several sports disciplines. Previous works on forecasting athletic records have mainly been based on extreme value theory. The presented statistical prediction methods are exemplarily applied to data from various disciplines of athletics as well as to data from American football based on fantasy football points according to the points per reception scoring scheme. The results are discussed along with basic assumptions and the choice of underlying distributions.

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

The prediction intervals of remaining useful life based on constant stress accelerated life test data

In practice, the prediction of the remaining useful life (RUL) of the product is a very important issue. This paper considers the prediction intervals of the RUL of the product at the normal operating stress level based on the exponential or Weibull constant stress accelerated life test (CSALT) data. For the exponential CSALT with type II censoring, we derive the exact and approximate prediction intervals for the RUL of the product. Using the asymptotic normality of the maximum likelihood estimation and the bootstrap method, we provide the two prediction intervals for the RUL of the product based on the exponential CSALT type I censored data. Under the Weibull CSALT with type II censoring, we use the generalized inferential procedure to obtain the generalized prediction interval of the RUL of the product. We also get the bootstrap prediction interval of the RUL of the product based on the Weibull CSALT type I censored data. The performance of the proposed prediction intervals is assessed by Monte Carlo simulation. The simulation results show that the coverage probabilities of the proposed prediction intervals are very close to the nominal confidence level. Finally, two examples are used to illustrate the proposed methods.

Statistical prediction of failure times under generalized progressive hybrid censoring in a simple step-stress accelerated competing risks model

Based on a simple step-stress accelerated competing risks model from generalized exponential distribution, the statistical prediction problem of unobserved failure times under generalized progressive hybrid censoring is investigated. Using the cumulative exposure model, maximum likelihood predictors, conditional median predictors, approximate prediction intervals and conditional prediction intervals of model parameters and unobserved competing failure times are obtained. To support and illustrate the prediction methods, an available dataset is analyzed and a simulation study is carried out. The numerical results of predictors of model parameters and competing failure times show that the prediction methods have a good performance.

Inference about Weibull Distribution Using Upper Record Values

This paper studies the frequentist inference about the shape and scale parameters of the two-parameter Weibull distribution using upper record values. The exact sampling distribution of the MLE of the shape parameter is derived. The asymptotic normality of the MLEs of both parameters are obtained. Based on these results this paper proposes various confidence intervals of the two parameters. Assuming one parameter is known certain testing procedures are proposed. Furthermore, approximate prediction interval for the immediately consequent record value is derived too. Conclusions are made based on intensive simulations.

A Prediction Interval Estimator for the Original Response When Using Box–Cox Transformations

Motivated by electron microscopy experiments, we develop an approximate prediction interval on the response variable Y, where it is assumed that a normal-theory linear model is fit using a transformed version of Y, and the transformation type is contained in the Box–Cox family. We derive a closed-form approximation to the kth moment of the original response variable Y, which is then used to estimate the mean and variance of Y, given parameter estimates obtained from fitting the model in the transformed domain. Chebychev's inequality is then used to construct a 100(1 – α)% prediction-interval estimator on Y. Using Monte Carlo simulation, we assess the width performance of our proposed Chebychev prediction interval, relative to that obtained by employing a more common interval-construction approach. General results suggest that, for a given level of expected coverage, the proposed interval estimator will achieve a smaller mean and variance of the interval width estimates, especially as the number of degrees of freedom beyond that required to estimate model terms is small. We apply our method to two experimental-data sets, one involving a standard 2k design and the other involving a 2k design with a split-plot error structure.

Predictable and Unpredictable Changes in Party Support: A Method for Long-Range Daily Election Forecasting from Opinion Polls

This paper outlines several methods for forecasting the next British general election on a daily basis from 20 months prior using opinion polls. It discusses their performance for previous electoral cycles and shows that the two models with the best historical record lead to substantially different predictions for 2015, but they can be averaged. The historical relationship between the polls and the vote suggests that government support rises substantially in the run up to elections, that Conservatives outperform but Labour underperform relative to the polls, and parties generally recover from low points or decline from high ones. Approximate prediction intervals and probabilities for key events are also generated. Despite the Conservatives trailing in the polls by 7 points in early October 2013, the models suggest a substantial Conservative lead at the 2015 election with a 64% chance of being the largest party and 42% chance of an overall majority. The estimated probability of a hung parliament (40%) is instructive for understanding the operation of the electoral system.

Computed tomography (CT) angiography for confirmation of the clinical diagnosis of brain death

The diagnosis of death using neurological criteria (brain death) has profound social, legal and ethical implications. The diagnosis can be made using standard clinical tests examining for brain function, but in some patient populations and in some countries additional tests may be required. Computed tomography (CT) angiography, which is currently in wide clinical use, has been identified as one such test. To assess from the current literature the sensitivity of CT cerebral angiography as an additional confirmatory test for diagnosing death using neurological criteria, following satisfaction of clinical neurological criteria for brain death. We performed comprehensive literature searches to identify studies that would assess the diagnostic accuracy of CT angiography (the index test) in cohorts of adult patients, using the diagnosis of brain death according to neurological criteria as the target condition. We searched the Cochrane Central Register of Controlled Trials (CENTRAL) (The Cochrane Library 2012, Issue 5) and the following databases from January 1992 to August 2012: MEDLINE; EMBASE; BNI; CINAHL; ISI Web of Science; BioMed Central. We also conducted searches in regional electronic bibliographic databases and subject-specific databases (MEDION; IndMed; African Index Medicus). A search was also conducted in Google Scholar where we reviewed the first 100 results only. We handsearched reference lists and conference proceedings to identify primary studies and review articles. Abstracts were identified by two authors. Methodological assessment of studies using the QUADAS-2 tool and further data extraction for re-analysis were performed by three authors. We included in this review all large case series and cohort studies that compared the results of CT angiography with the diagnosis of brain death according to neurological criteria. Uniquely, the reference standard was the same as the target condition in this review. We reviewed all included studies for methodological quality according to the QUADAS-2 criteria.We encountered significant heterogeneity in methods used to interpret CT angiography studies and therefore, where possible, we re-analysed the published data to conform to a standard radiological interpretation model. The majority of studies (with one exception) were not designed to include patients who were not brain dead, and therefore overall specificity was not estimable as part of a meta-analysis. Sensitivity, confidence and prediction intervals were calculated for both as-published data and as re-analysed to a standardized interpretation model. Ten studies were found including 366 patients in total. We included eight studies in the as-published data analysis, comprising 337 patients .The methodological quality of the studies was overall satisfactory, however there was potential for introduction of significant bias in several specific areas relating to performance of the index test and to the timing of index versus reference tests.Results demonstrated a sensitivity estimate of 0.84 (95% confidence interval (CI) 0.69 to 0.93). The 95% approximate prediction interval was very wide (0.34 to 0.98). Data in three studies were available as a four-vessel interpretation model and the data could be re-analysed to a four-vessel interpretation model in a further five studies, comprising 314 patient events. Results demonstrated a similar sensitivity estimate of 0.85 (95% CI 0.77 to 0.91) but with an improved 95% approximate prediction interval (0.56 to 0.96). The available evidence cannot support the use of CT angiography as a mandatory test, or as a complete replacement for neurological testing, in the management pathway of patients who are suspected to be clinically brain dead.CT angiography may be useful as a confirmatory or add-on test following a clinical diagnosis of death, assuming that clinicians are aware of the relatively low overall sensitivity. Consensus on a standard radiological interpretation protocol for future published studies would facilitate further meta-analysis.

Improved closed-form prediction intervals for binomial and Poisson distributions

Abstract The problems of constructing prediction intervals for the binomial and Poisson distributions are considered. Available approximate, exact and conditional methods for both distributions are reviewed and compared. Simple approximate prediction intervals based on the joint distribution of the past samples and the future sample are proposed. Exact coverage studies and expected widths of prediction intervals show that the new prediction intervals are comparable to or better than the available ones in most cases. The methods are illustrated using two practical examples.

Approximate Confidence and Prediction Intervals for Least Squares Support Vector Regression

Bias-corrected approximate 100(1-α)% pointwise and simultaneous confidence and prediction intervals for least squares support vector machines are proposed. A simple way of determining the bias without estimating higher order derivatives is formulated. A variance estimator is developed that works well in the homoscedastic and heteroscedastic case. In order to produce simultaneous confidence intervals, a simple Šidák correction and a more involved correction (based on upcrossing theory) are used. The obtained confidence intervals are compared to a state-of-the-art bootstrap-based method. Simulations show that the proposed method obtains similar intervals compared to the bootstrap at a lower computational cost.

EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA

SummaryThe most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non‐negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non‐negative data. We explore exponential smoothing state space models for non‐negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models that do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non‐Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short‐term forecasting performance of the various models using data on the weekly sales of over 300 items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one‐step‐ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed.

Evaluation of threshold limit methods for sensory data

Current approaches to sensory thresholds, such as geometric means and logistic regression, ignore any formal consideration of uncertainty and variability. Various alternative methods based on approximate confidence and prediction intervals about the logistic regression were examined. All methods tended to provide the same ranking among different analyte/media combinations evaluated. Formal statistical conclusions could be made for thresholds based on interval analyses, but not for geometric mean or logistic regression. Methods based on prediction intervals consistently estimated the highest thresholds. Interval-based methods varied with the level of confidence required, as well as the number of panelists and concentrations tested. The geometric mean method yielded the most consistent estimates across a range of panel sizes.

One-Sided Approximate Prediction Intervals for at Least p of m Observations From a Gamma Population at Each of r Locations

We develop simultaneous approximate statistical prediction limits for a gamma-distributed random variable. Specifically, we develop an upper prediction limit (UPL) for p of m future samples at each of r locations, based on a previous sample of n measurements. A typical example is the environmental monitoring problem in which the distribution of an analyte of concern is typically non-Gaussian, simultaneous determinations are required for multiple locations (e.g., ground-water monitoring wells), and, in the event of an initial exceedance of the prediction limit, one or more verification samples are obtained to confirm evidence of an impact on the environment. For example, consider a ground-water monitoring program with r wells and the requirement that at least p = 1 of the m = 2 samples in each of the r wells be below the UPL. We provide derivation of simultaneous approximate gamma UPLs, illustration of the relevance of the gamma distribution to environmental data, a limited simulation study of type I and II error rates achieved using the method and comparison with normal and nonparametric alternatives, tables that aid computation, and an example using ground-water monitoring data.

Prediction Intervals of an Ordered Observation from One-Parameter Exponential Distribution Based on Multiple Type II Censored Samples

This paper provides a suitable pivotal quantity to present the prediction interval of the j(superscript th) future ordered-observation in a sample of size n from one-parameter exponential distribution in case of a multiple type II censored sample. In addition, we also discuss the approximate prediction interval, the best linear unbiased estimate and approximate maximum likelihood estimate of X(subscript j) based on the censored sample. As in application, the total duration time in a life test and the failure time of a j-out-of-n system may be predicated. Finally, a simulated study and three illustrative examples are included.

PREDICTION INTERVALS OF AN ORDERED OBSERVATION FROM ONE-PARAMETER EXPONENTIAL DISTRIBUTION BASED ON MULTIPLE TYPE II CENSORED SAMPLES

ABSTRACT This paper provides a suitable pivotal quantity to present the prediction interval of the j th future ordered-observation in a sample of size n from one-parameter exponential distribution in case of a multiple type II censored sample. In addition, we also discuss the approximate prediction interval, the best linear unbiased estimate and approximate maximum likelihood estimate of X j based on the censored sample. As in applications, the total duration time in a life test and the failure time of a j-out-of-n system may be predicted. Finally, a simulated study and three illustrative examples are included.

Weibull Prediction Intervals for a Future Number of Failures

This article evaluates exact coverage probabilities of approximate prediction intervals for the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the first in-service inspection of the sample. The failure time of such units is modeled with a Weibull distribution having a given shape parameter value. We illustrate the use of the procedures by using data from a nuclear power plant heat exchanger. The results suggest that the likelihood-based prediction intervals perform better than the alternatives.

A simple non-linear model in incidence prediction.

A simple model is proposed for incidence prediction. The model is non-linear in parameters but linear in time, following models in environmental cancer epidemiology. Assuming a Poisson distribution for the age and period specific numbers of incident cases approximate confidence and prediction intervals are calculated. The major advantage of this model over current models is that age-specific predictions can be made with greater accuracy. The model also preserves in the period of prediction the age pattern of incidence rates existing in the data. It may be fitted with any package which includes an iteratively reweighted least squares algorithm, for example GLIM. Cancer incidence predictions for the Stockholm-Gotland Oncological Region in Sweden are presented as an example.

Validation of a decision support system for use in drug development: pharmacokinetic data.

Single dose pharmacokinetic data from several individuals can be used to predict the fraction of the population that is expected to be within a therapeutic range. Without having some measure of its reliability, however, that prediction is only likely to marginally influence critical drug development decision making. The system (Forecaster) described generates an approximate prediction interval that contains the original prediction and where, for example, the probability is approximately 85% that a similar prediction from a new set of data will also be within the range. The goal is to validate that the system functions as designed. The strategy requires having a Surrogate Population (SP), which is a large number (> or = 1500) of hypothetical individuals each represented by set of model parameter values having unique attributes. The SP is generated so that a sample taken from it will give data that is statistically indistinguishable from the available experimental data. The automated method for building the SP is described. Validation studies using 300 independent samples document that for this example the SP can be used to make useful predictions, and that the approximate prediction interval functions as designed. For the boundary conditions and assumptions specified, the Forecaster can make valid predictions of pharmacokinetic-based population targets that without a SP would not be possible. Finally, the approximate prediction interval does provide a useful measure of prediction reliability.

Some results on bootstrap prediction intervals

AbstractWe investigate the construction of a BCa‐type bootstrap procedure for setting approximate prediction intervals for an efficient estimator θm of a scalar parameter θ, based on a future sample of size m. The results are also extended to nonparametric situations, which can be used to form bootstrap prediction intervals for a large class of statistics. These intervals are transformation‐respecting and range‐preserving. The asymptotic performance of our procedure is assessed by allowing both the past and future sample sizes to tend to infinity. The resulting intervals are then shown to be second‐order correct and second‐order accurate. These second‐order properties are established in terms of min(m, n), and not the past sample size n alone.

Review: likelihood method for fitting Weibull log-linear models to accelerated life-test data : Alan J. Watkins. IEEE Transactions on Reliability, 43, 3, 361 (September 1994)

In practice, the prediction of the remaining useful life (RUL) of the product is a very important issue. This paper considers the prediction intervals of the RUL of the product at the normal operating stress level based on the exponential or Weibull constant stress accelerated life test (CSALT) data. For the exponential CSALT with type II censoring, we derive the exact and approximate prediction intervals for the RUL of the product. Using the asymptotic normality of the maximum likelihood estimation and the bootstrap method, we provide the two prediction intervals for the RUL of the product based on the exponential CSALT type I censored data. Under the Weibull CSALT with type II censoring, we use the generalized inferential procedure to obtain the generalized prediction interval of the RUL of the product. We also get the bootstrap prediction interval of the RUL of the product based on the Weibull CSALT type I censored data. The performance of the proposed prediction intervals is assessed by Monte Carlo simulation. The simulation results show that the coverage probabilities of the proposed prediction intervals are very close to the nominal confidence level. Finally, two examples are used to illustrate the proposed methods.