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Prediction Variance (prediction + variance)
Selected AbstractsPrediction Variance and Information Worth of Observations in Time SeriesJOURNAL OF TIME SERIES ANALYSIS, Issue 4 2000Mohsen Pourahmadi 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. [source] Prediction variance and G-criterion location for split-plot designsQUALITY AND RELIABILITY ENGINEERING INTERNATIONAL, Issue 4 2009Wayne R. Wesley Abstract Prediction variance properties for completely randomized designs (CRD) are fairly well covered in the response surface literature for both spherical and cuboidal designs. This paper evaluates the impact of changes in the variance ratio on the prediction properties of second-order split-plot designs (SPD). It is shown that the variance ratio not only influences the value of the G-criterion but also its location, in contrast with the G-criterion tendencies in CRD. An analytical method, rather than a heuristic optimization algorithm, is used to compute the prediction variance properties, which include the maximum, minimum and integrated variances for second-order SPD. The analytical equations are functions of the design parameters, radius and variance ratio. As a result, the exact values for these quantities are reported along with the location of the maximum prediction variance used in the G-criterion. The two design spaces of the whole plot and the subplot are studied and as a result, relative efficiency values for these distinct spaces are suggested. Copyright © 2008 John Wiley & Sons, Ltd. [source] The price paid for the second-order advantage when using the generalized rank annihilation method (GRAM)JOURNAL OF CHEMOMETRICS, Issue 9 2001Nicolaas (Klaas) M. Faber Abstract In a ground-breaking paper, Linder and Sundberg developed a statistical framework for the calibration of bilinear data (Chemometrics Intell. Lab. Syst. 1998; 42: 159,178). Within this framework they formulated three different predictor construction methods (J. Chemometrics accepted), namely a so-called naive method, a least squares (LS) method and a refined version of the latter that takes account of the calibration uncertainty. They showed that the naive method is statistically less efficient than the others under the assumption of white noise. In the current work a close relationship is established between the generalized rank annihilation method (GRAM) and the naive method by comparing expressions for prediction variance. The main conclusion is that the relatively poor efficiency of GRAM is the price one pays for obtaining the second-order advantage with a single calibration sample. Copyright © 2001 John Wiley & Sons, Ltd. [source] Two-strata rotatability in split-plot central composite designsAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 4 2010Li Wang Abstract A rotatable design (Ann. Math. Stat. 1957; 28:195,241) for k factors is one such that the prediction variance is purely a function of distance from the design center. Of special interest in this paper is the rotatable central composite design (CCD), which most software packages use as the typical default choice for a second-order design. In many cases some factors are hard to change while others are easy to change, which creates a split-plot experiment. This paper establishes that the split-plot structure precludes the possibility of any second-order design being rotatable in the traditional sense. As an alternative this paper proposes the two-strata rotatable split-plot CCD, where the resulting prediction variance is a function of the whole plot (WP) distance and the subplot (SP) distance separately instead of the sum of them. The resulting design is rotatable in the WP space when the SP factors are held fixed, and vice versa. In the special case where the WP variance component is zero, the two-strata rotatable split-plot CCD becomes the standard rotatable CCD. Copyright © 2009 John Wiley & Sons, Ltd. [source] Assessment of uncertainty in computer experiments from Universal to Bayesian KrigingAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 2 2009C. Helbert Abstract Kriging was first introduced in the field of geostatistics. Nowadays, it is widely used to model computer experiments. Since the results of deterministic computer experiments have no experimental variability, Kriging is appropriate in that it interpolates observations at data points. Moreover, Kriging quantifies prediction uncertainty, which plays a major role in many applications. Among practitioners we can distinguish those who use Universal Kriging where the parameters of the model are estimated and those who use Bayesian Kriging where model parameters are random variables. The aim of this article is to show that the prediction uncertainty has a correct interpretation only in the case of Bayesian Kriging. Different cases of prior distributions have been studied and it is shown that in one specific case, Bayesian Kriging supplies an interpretation as a conditional variance for the prediction variance provided by Universal Kriging. Finally, a simple petroleum engineering case study presents the importance of prior information in the Bayesian approach. Copyright © 2009 John Wiley & Sons, Ltd. [source] Sensitivity analysis of prior model probabilities and the value of prior knowledge in the assessment of conceptual model uncertainty in groundwater modellingHYDROLOGICAL PROCESSES, Issue 8 2009Rodrigo Rojas Abstract A key point in the application of multi-model Bayesian averaging techniques to assess the predictive uncertainty in groundwater modelling applications is the definition of prior model probabilities, which reflect the prior perception about the plausibility of alternative models. In this work the influence of prior knowledge and prior model probabilities on posterior model probabilities, multi-model predictions, and conceptual model uncertainty estimations is analysed. The sensitivity to prior model probabilities is assessed using an extensive numerical analysis in which the prior probability space of a set of plausible conceptualizations is discretized to obtain a large ensemble of possible combinations of prior model probabilities. Additionally, the value of prior knowledge about alternative models in reducing conceptual model uncertainty is assessed by considering three example knowledge states, expressed as quantitative relations among the alternative models. A constrained maximum entropy approach is used to find the set of prior model probabilities that correspond to the different prior knowledge states. For illustrative purposes, a three-dimensional hypothetical setup approximated by seven alternative conceptual models is employed. Results show that posterior model probabilities, leading moments of the predictive distributions and estimations of conceptual model uncertainty are very sensitive to prior model probabilities, indicating the relevance of selecting proper prior probabilities. Additionally, including proper prior knowledge improves the predictive performance of the multi-model approach, expressed by reductions of the multi-model prediction variances by up to 60% compared with a non-informative case. However, the ratio between-model to total variance does not substantially decrease. This suggests that the contribution of conceptual model uncertainty to the total variance cannot be further reduced based only on prior knowledge about the plausibility of alternative models. These results advocate including proper prior knowledge about alternative conceptualizations in combination with extra conditioning data to further reduce conceptual model uncertainty in groundwater modelling predictions. Copyright © 2009 John Wiley & Sons, Ltd. [source] | |||||||||||||