Inference inferential models

A conceptually different and relatively new example of an inferential model, motivated by human performance problems specifically, is nonlinear causal resource analysis (NCRA) [Kondraske, 1988 Vasta and Kondraske, 1994]. Quantitative task demands, in terms of performance variables that characterize the involved subsystems, are inferred from a population data set that includes measures of subsystem performance resource availabilities (e.g., speed, accuracy, etc.) and overall performance on the task in question. This method is based on the following simple concept Consider a sample of 100 people, each with a known amount of cash (e.g., a fairly even distribution from 0 to 10,000). Each person is asked to try to purchase a specific computer, the cost of which is unknown. In the subgroup that was able to make the purchase (some would not have enough cash), the individual who had the least amount of cash provides the key clue. That amount of cash availability provides an estimate of the computer s cost (i.e., the unknown value). Thus, in human performance, demand is inferred from resource availabdities. 

Two very different approaches to inferential statistics exist the classical or fre-quentist approach and the Bayesian approach. Each approach is used to draw conclusions (or inferences) regarding the magnitude of some unknown quantity, such as the intercept and slope of a dose-response model. The key difference between classical... 

One solution to this problem is to employ inferential control, where process measurements that can be obtained more rapidly are used with a mathematical model to infer the value of the controlled variable, as illustrated in Figure 12. For example, if the overhead product stream in a distillation column cannot be analysed on-line, measurement of a selected tray temperature may be used to infer the actual composition. If necessary, the parameters in the model may be updated, if composition measurement become available, as illustrated by the second measuring device in Figure 12 (dashed lines). 

The concentration of the product B, CB, is not measured on-line and a measurement is only available hourly from a lab. The control of the concentration is therefore based on inferential control in loop 4 using the reactor temperature T. The inferential controller will then, from a model of the process, infer what the concentration CB is and use this inferred measurement as the signal to the controller CC3 (where the first C refers to Concentration). 

Available measurements. For controlled variables that are not directly measurable, measurements have to be inferred by measurements of secondary variables and/or laboratory analysis of samples. Good inference relies on reliable models. In addition, the results of laboratory analysis, usually produced much less frequently than inferential estimates, have to be fused with the inferential estimates produced by secondary measurements. 

D. B. Rubin first suggested PPC in 1984 (31) as a tool for constructing inferential procedures in modem statistical data analysis. In this approach a model is estimated directly from the index data, and then a new set of data is generated through the simulation of the resulting model. The simulated data set is compared with the index data to see if the model s deficiencies have a noticeable effect on the substantive inferences (9). The basic approach for PPC within the context of PPK modeling is as follows ... 

If data are collected from a random population (X, Y) from a bivariate normal distribution and predictions about Y given X are desired, then from the previous paragraphs it may be apparent that the linear model assuming fixed x is applicable because the observations are independent, normally distributed, and have constant variance with mean 0o + 0iX. Similar arguments can be made if inferences are to be made on X given Y. Thus, if X and Y are random, all calculations and inferential methods remain the same as if X were fixed. 

With linear models, exact inferential procedures are available for any sample size. The reason is that as a result of the linearity of the model parameters, the parameter estimates are unbiased with minimum variance when the assumption of independent, normally distributed residuals with constant variance holds. The same is not true with nonlinear models because even if the residuals assumption is true, the parameter estimates do not necessarily have minimum variance or are unbiased. Thus, inferences about the model parameter estimates are usually based on large sample sizes because the properties of these estimators are asymptotic, i.e., are true as n —> oo. Thus, when n is large and the residuals assumption is true, only then will nonlinear regression parameter estimates have estimates that are normally distributed and almost unbiased with minimum variance. As n increases, the degree of unbiasedness and estimation variability will increase. 

Inferential analysis [20, 21] is not a spectroscopy but could have a bearing upon the use of all process analysis techniques. It is a term being used to describe measurements that are not made but are inferred from other properties of the process under scrutiny. These methods rely upon process models being available for the process concerned. The value of this approach, quite apart from the fact that no expensive equipment is needed, is that it can give an indication of a measurement when it is impossible to extract a sample without it undergoing change or where inserting a probe is impractical. Inferential methods can also be useful to provide values between the frequency of the installed measurement devices or indeed when the measurement devices are off-line for maintenance purposes. The quality of an intermediate or a product, can in some instances be inferred from the values of temperature, pressure and flow rates in the area of the process under consideration. 

Inferential control [11,17,18] is an early model-based approach for process control. This control strategy is useful when the main dryer controlled variable (i.e., product moisture content) cannot be measured directly due to some technical difficulties or due to insufficient economic justification for its measurement. Inferential control uses the values of measured outputs (e.g., product or gas temperature and humidity) together with the process model to infer the value of the unmeasured control variable. These estimates are used to adjust the values of the manipulated variables in order to keep the moisture content at the desired levels (Figure 57.5). This control policy can also be used to counteract the disturbances as it is less expensive to infer these disturbances from other available process measurements rather than by measuring them directly. 

A CSTR is used to produce a specialty chemical. The reaction is exothermic and exhibits first-order kinetics. Laboratory analyses for the product quality are time-consuming, requiring several hours to complete. No on-line composition measurement has been found satisfactory. It has been suggested that composition can be inferred from the exit temperature of the CSTR. Using the linearized CSTR model in Example 4.8, determine whether this inferential control approach would be feasible. Assume that measurements of feed fiow rate, feed temperature, and coolant temperature are available. 

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