4. Process Modeling 4.1. Introduction to Process Modeling 4.1.3. What are process models used for? 4.1.3.1. Prediction
More on Prediction	As mentioned on the preceding page, the primary goals of prediction are to determine the value of the regression function or future value(s) of the response variable that are associated with a specific combination of predictor variable values. For each of type of prediction, the predicted values are computed by plugging the value(s) of the predictor variable(s) into the regression equation after estimating the unknown parameters from the data. The differences between the different types of predictions arise in the computation of the uncertainties. These differences are illustrated conceptually below using the Pressure/Temperature example from a few pages earlier.
Example	Suppose in this case the predictor variable value of interest is a temperature of 47 degrees. Computing the predicted value using the equation yields a predicted pressure of 192.4655.
	Of course, if the pressure/temperature experiment was repeated, the parameters of the regression function estimated from the data would differ slightly each time because of the randomness in the data and the need to sample a limited amount of data. Different parameter estimates would yield different predicted values in turn. The plot below illustrates the type of slight variation that would occur in a repeated experiment.
Prediction Result from Repeated Experiment
Prediction Uncertainty	A critical part of prediction is an assessment of how much a predicted value will fluctuate due to the noise in the data. Without that information there is no basis to compare a predicted value to a target value or to another prediction. Any method used for prediction should include an assessment of the uncertainty in the predicted value(s). Fortunately it is often the case that the data used to fit the model to a process can also be used to compute the uncertainty of predictions from the model. In the pressure/temperature example a confidence interval for the value of the regresion function at 47 degrees can be computed from the data used to fit the model. The plot below shows a 99% confidence interval produced using the original data. This interval gives the range of plausible pressure values for a temperature of 47 degrees based on the parameter estimates and the noise in the data.
99% Confidence Interval for Pressure at T=47
Confidence Intervals vs. Prediction Intervals	Because the confidence interval above is an interval for the value of the regression function, the uncertainty only includes the noise that is inherent in the estimates of the regression parameters. The noise in the this type of prediction is generally less than the noise in a single measurement because the data is essentially averaged (in a way that depends on the statistical method being used) to determine each parameter estimate. On the other hand, to determine the uncertainty in the predicted value of a new measurement, both the uncertainty in the estimated parameters and the uncertainty of the new measurement must be accounted for. This means that the interval for a new measurement will be wider than the interval for the value of the regression function. Intervals for a new measurement are typically called prediction intervals rather than confidence intervals to remind you that they include the extra variability for a new measurement. 192.4655 +/- 11.8070 is a 99% prediction interval for a new pressure measurement for T=47 degrees.
Tolerance Intervals	Like a prediction interval, a tolerance interval brackets the plausible values of new measurements from the process being modeled. However, instead of bracketing the value of a single measurement or a fixed number of measurements, a tolerance interval brackets a specified percentage of all future measurements for a given set of predictor variable values. For example, to monitor future pressure measurements at 47 degrees for extreme values, either low or high, a tolerance interval that brackets 98% of all future measurements with high confidence could be used. If a future value then fell outside of the interval, the system could be would then be checked to ensure that everything was working correctly. A 99% tolerance interval that captures 98% of all future pressure measurements at a temperature of 47 degrees is 192.4655 +/- 14.5810. This interval is wider than the prediction interval for a single measurement because it is designed to capture a larger proportion of all future measurements. The explanation of tolerance intervals is confusing because there are two percentages for used in the description of the interval. One, in this case 99%, describes how confident we are that the interval will capture the quantity that we want it to capture. The other, 98%, describes what the target quantity is, which in this case that is 98% of all future measurements at T=47 degrees.
More Info	For more information on the interpretation and computation confidence, prediction and tolerance intervals see Section 5.1