Prediction intervals, types

Another type of prediction interval that might be of interest is a one-sided upper prediction interval to contain the standard deviation of a future sample of m observations, again with a specified degree of confidence, 100(1 - a)%. This is called the standard deviation prediction interval (SDPI). The form of this equation is... 

Some would argue that if there is sufficient information upon which to quantify ranges, then there is also likely information upon which to base a judgement regarding the type of distribution that could be used to describe uncertainty in the input. Interval methods, however, may be useful for problems in which there may be complex dependencies between inputs but for which the dependencies are not known. A disadvantage of interval methods is that the predicted intervals for a model output can be quite wide, since they are informed only by the end-points of the ranges for each input. Interval methods can be useful as a quality... 

Three kinds of reference intervals have been suggested tolerance interval, prediction interval, and interpercentile interval. The choice from among these types of intervals may be important for certain well-defined statistical problems, but their numerical differences are negligible when based on at least 100 reference values. 

There are two types of prediction intervals which can be constructed in the regression problem, prediction intervals for a population mean (the mean response y for a given x ) and prediction intervals for individual observations (i.e. the prediction interval for a particular patient). Conceptually, the difference between the two intervals is subtle. In the first case, one is interested in an interval for the population mean at a given value of X. In the second case, an individual observation from the population is of interest. In practice the difference between the two can be substantial since the prediction interval for the population mean is more narrow than that for an individual. To illustrate this difference, consider... 

Comparisons between observed data and model predictions must be made on a consistent basis, i.e., apples with apples and oranges with oranges. Since models provide a continuous timeseries, any type of statistic can be produced such as daily maximums, minimums, averages, medians, etc. However, observed data are usually collected on infrequent intervals so only certain statistics can be reliably estimated. Validation of aquatic chemical fate and transport models is often performed by comparing both simulated and observed concentration values and total chemical loadings obtained from multiplying the flow and the concentration values. Whereas the model supplies flow and concentration values in each time step, the calculated observed loads are usually based on values interpolated between actual flow and sample measurements. The frequency of sample collection will affect the validity of the resulting calculated load. Thus, the model user needs to be aware of how observed chemical loads are calculated in order to assess the veracity of the values. 

As recently as 1965, Thoma and Stewart predicted that alterations in reaction rates [in the presence of the cycloamyloses] should be anticipated whose magnitude and sign will fluctuate with the reaction type, and added that at the present juncture, it is impossible to sort out confidently. . . which factors may contribute importantly to raising or lowering the activation energy of the reaction. In the short interval between 1965 and the present, a wide variety of cycloamylose-induced rate accelerations and decelerations have, indeed, been revealed. More importantly, rate alterations imposed by the cycloamyloses can now be explained with substantially more confidence. The reactions of derivatives of carboxylic acids and organo-phosphorus compounds with the cycloamyloses, for example, proceed to form covalent intermediates. Other types of reactions appear to be influenced by the dielectric properties of the microscopic cycloamylose cavity. Still other reactions may be affected by the geometrical requirements of the inclusion process. 

The basis of all performance criteria are prediction errors (residuals), yt - yh obtained from an independent test set, or by CV or bootstrap, or sometimes by less reliable methods. It is crucial to document from which data set and by which strategy the prediction errors have been obtained furthermore, a large number of prediction errors is desirable. Various measures can be derived from the residuals to characterize the prediction performance of a single model or a model type. If enough values are available, visualization of the error distribution gives a comprehensive picture. In many cases, the distribution is similar to a normal distribution and has a mean of approximately zero. Such distribution can well be described by a single parameter that measures the spread. Other distributions of the errors, for instance a bimodal distribution or a skewed distribution, may occur and can for instance be characterized by a tolerance interval. 

Cardiac conduction Flecainide slows cardiac conduction in most patients to produce dose-related increases in PR, QRS, and QT intervals. The degree of lengthening of PR and QRS intervals does not predict either efficacy or the development of cardiac adverse effects. Patients may develop new first-degree AV heart block. Use caution and consider dose reductions. The JT interval (QT minus QRS) only widens approximately 4% on the average. Rare cases of torsade de pointes-type arrhythmias have occurred. 

Kp were calculated by averaging the partitioning data collected after 2 hr in experiments conducted in the absence of OH. Equation (6.130) may be employed to model partitioning kinetics over short time intervals, but applying the particles used in these experiments as equilibrium models may not adequately predict partitioning over extended time intervals for many types of particles. Partitioning data and reaction rate constants calculated for 2,2, 5-trichlorobiphenyl and 2,2, 4,4, 5-PeCIBp by Sedlak and Andren (1994) are shown in Table 6.6. 

When the EPA considered exposures to insecticide residues in the home they identified at least six possible sources and routes these are given in Table 2.6. Their original approach apportioned the acceptable daily intake (ADI) between the various routes but it soon became clear that this was unrealistic because an individual was unlikely to be exposed via all routes on any one day. The EPA s present strategy is to develop an approach called micro-exposure event modelling. Micro-exposure event modelling is based on statistical data on the frequencies and levels of contamination of food, water, etc. and on behavioural information about the frequency of use of lawn/pet/timber treatments, etc. The combined data are assembled in a probabilistic model called LIFELINE which is able to predict the frequency and level of exposure to a group of hypothetical individuals over their lifetime.12 The model is also able to take account of the relative proportions of different types of accommodation, the incidence of pet ownership or any other data that will affect real levels of exposure. The output from the LIFELINE model allows the exposures of individuals in a population to be modelled over any interval from a single occasion to a lifetime. 

For the types of comparisons reported here it has generally been convenient to use steady state assumptions, but these clearly do not apply to conditions after forest spraying. Monitoring studies typically report rapid penetration of pesticides to forest streams followed by rapid dissipation of residues by a number of processes. Most published bioconcentration equations do not contain a time term and so they cannot readily be applied to short intervals when only a small fraction of the time to reach equilibrium would apply. The rate constants and other descriptive equations offer the possibility of predicting bioconcentration under non-equilibrium conditions. 

With the TSA developed in (115,130,132,165) it was possible to almost reach simulation intervals of 1 milisecond. This makes in principle possible to predict D as small as a few 10"12 cm2/s. Therefore the TSA seems to be a good choice to predict D for the type of diffusion processes encountered in packaging applications. But for this, the actual TSA algorithms must be developed to take also into account strong interactions between the penetrant and the host atoms, and the deformation of the polymer structure at the passage of complex penetrant molecules. 

The distribution of hydrolyzed V02+ as a function of pH at a total vanadium concentration of 10 xM is shown in Fig. 3. The curves in the distribution diagram also depend on the total vanadium concentration because of the dimer formation and the precipitation reactions. While distribution diagrams of this type for V02+ are incomplete, they nevertheless illustrate the interrelationship between some of the species present and are of predictive value below pH 6 and above pH 11, and possibly in the pH 6 to 11 interval provided one starts with a solution below pH 6 and slowly adds base. The unidentified soluble hydroxide species are less likely to form under those conditions. Species distribution diagrams for a number of V02+ complexes with several common ligands are given by Kraglen34. ... 

The classical mathematical theories by which certain types of uncertainty can be expressed are classical set theory and probability theory. In terms of set theory, uncertainty is expressed by any given set of possible alternatives in situations where only one of the alternatives may actually happen. For example, when an interval of values of a variable is predicted by a given mathematical model, the set of values in the interval represents a predictive uncertainty, when an unsettled historical question allows a set of possible answers rather than a unique one, the set represents a retrodictive uncertainty when medical diagnosis of a patient results in a set of possible diseases rather than a single disease, the set represents a diagnostic uncertainty when design requirements are specified in terms of sets of alternatives, the sets represent a prescriptive uncertainty. 

The most common method of identifying and monitoring corrosion is visual inspection. Evidence of leakage, staining, or a change in surface appearance can be an indication that some type of corrosion is taking place. Experience with certain types of equipment and processes may help dictate inspection intervals and areas on which to focus the inspection. Records of vessel operation, maintenance, and repair can be helpful in establishing a pattern of performance that will improve predictability and minimize down time. 

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