From: <¥Ñ Microsoft Internet Explorer 5 Àx¦s> Subject: 4.1.3.2. Prediction Date: Mon, 9 May 2005 22:38:44 +0800 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C554E7.DB791010" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C554E7.DB791010 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd132.htm 4.1.3.2. Prediction = =
4. Process=20 Modeling
4.1. In= troduction=20 to Process Modeling
4.1.3. W= hat=20 are process models used for?

## Prediction

More on Prediction As mentioned e= arlier,=20 the goal of prediction is to determine future value(s) of the = response=20 variable that are associated with a specific combination of = predictor=20 variable values. As in = estimation,=20 the predicted values are computed by plugging the value(s) of the=20 predictor variable(s) into the regression=20 equation, after estimating the unknown parameters=20 from the data. The difference between estimation and prediction = arises=20 only in the computation of the uncertainties. These differences = are=20 illustrated below using the Pressure/Temperature=20 example in parallel with the example=20 illustrating estimation.
Example Suppose in this case the predictor variable value = of=20 interest is a temperature of 47 degrees. Computing the predicted = value=20 using the equation

yields=20 a predicted pressure of 192.4655.
Of course, if the pressure/temperature experiment = were=20 repeated, the estimates of the parameters of the regression = function=20 obtained from the data would differ slightly each time because of = the=20 randomness in the data and the need to sample a limited amount of = data.=20 Different parameter estimates would, in turn, yield different = predicted=20 values. The plot below illustrates the type of slight variation = that could=20 occur in a repeated experiment.
Predicted Value from a Repeated=20 Experiment
Prediction Uncertainty A critical part of prediction is an assessment of = how much=20 a predicted value will fluctuate due to the noise in the data. = Without=20 that information there is no basis for comparing a predicted value = to a=20 target value or to another prediction. As a result, any method = used for=20 prediction should include an assessment of the uncertainty in the=20 predicted value(s). Fortunately it is often the case that the data = used to=20 fit the model to a process can also be used to compute the = uncertainty of=20 predictions from the model. In the pressure/temperature example a=20 prediction interval for the value of the regresion function at 47 = degrees=20 can be computed from the data used to fit the model. The plot = below shows=20 a 99% prediction interval produced using the original data. This = interval=20 gives the range of plausible values for a single future pressure=20 measurement observed at a temperature of 47 degrees based on the = parameter=20 estimates and the noise in the data.
99% Prediction Interval for = Pressure at=20 T=3D47
Length of Prediction Intervals = Because the prediction interval is an interval for = the=20 value of a single new measurement from the process, the = uncertainty=20 includes the noise that is inherent in the estimates of the = regression=20 parameters and the uncertainty of the new measurement. This means = that the=20 interval for a new measurement will be wider than the confidence = interval=20 for the value of the regression function. These intervals are = called=20 prediction intervals rather than confidence intervals because the = latter=20 are for parameters, and a new measurement is a random variable, = not a=20 parameter.
Tolerance Intervals Like a prediction interval, a tolerance interval = brackets=20 the plausible values of new measurements from the process being = modeled.=20 However, instead of bracketing the value of a single measurement = or a=20 fixed number of measurements, a tolerance interval brackets a = specified=20 percentage of all future measurements for a given set of predictor = variable values. For example, to monitor future pressure = measurements at=20 47 degrees for extreme values, either low or high, a tolerance = interval=20 that brackets 98% of all future measurements with high confidence = could be=20 used. If a future value then fell outside of the interval, the = system=20 would then be checked to ensure that everything was working = correctly. A=20 99% tolerance interval that captures 98% of all future pressure=20 measurements at a temperature of 47 degrees is 192.4655 +/- = 14.5810. This=20 interval is wider than the prediction interval for a single = measurement=20 because it is designed to capture a larger proportion of all = future=20 measurements. The explanation of tolerance intervals is = potentially=20 confusing because there are two percentages used in the = description of the=20 interval. One, in this case 99%, describes how confident we are = that the=20 interval will capture the quantity that we want it to capture. The = other,=20 98%, describes what the target quantity is, which in this case = that is 98%=20 of all future measurements at T=3D47 degrees.