4. Process
Modeling
4.1. Introduction
to Process Modeling
4.1.3. What
are process models used for?

4.1.3.2.

Prediction

More on Prediction

As mentioned earlier,
the goal of prediction is to determine future value(s) of the response
variable that are associated with a specific combination of predictor
variable values. As in estimation,
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 difference between estimation and prediction arises
only in the computation of the uncertainties. These differences are
illustrated below using the Pressure/Temperature
example in parallel with the example
illustrating estimation.

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 were
repeated, the estimates of the parameters of the regression function
obtained 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, in turn, yield different predicted
values. The plot below illustrates the type of slight variation that could
occur in a repeated experiment.

Predicted Value from a 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 for comparing a predicted value to a
target value or to another prediction. As a result, 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
prediction 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% prediction interval produced using the original data. This interval
gives the range of plausible values for a single future pressure
measurement observed at a temperature of 47 degrees based on the parameter
estimates and the noise in the data.

99% Prediction Interval for Pressure at
T=47

Length of Prediction Intervals

Because the prediction interval is an interval for the
value of a single new measurement from the process, the uncertainty
includes the noise that is inherent in the estimates of the regression
parameters and the uncertainty of the new measurement. This means that the
interval for a new measurement will be wider than the confidence interval
for the value of the regression function. These intervals are called
prediction intervals rather than confidence intervals because the latter
are for parameters, and a new measurement is a random variable, not a
parameter.

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
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 potentially
confusing because there are two percentages 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
of prediction and tolerance intervals, see Section
5.1.