Environmental Modeling - Mathematics and Statistics for the Environment - page 78

model and possibly information about the data, an appropriate model-fitting method is used to esti-
mate the unknown parameters in the model. When the parameter estimates have been made, the
model is then carefully assessed to see if the underlying assumptions of the analysis appear plau-
sible. If the assumptions seem valid, the model can be used to answer the scientific or engineering
questions that prompted the modeling effort. If the model validation identifies problems with the
current model, however, then the modeling process is repeated using information from the model
validation step to select or fit an improved model.
The three basic steps of process modeling described in the paragraph above assume that the data
have already been collected and that the same dataset can be used to fit all of the candidate models.
Although this is often the case in mode-building situations, one variation on the basic model-build-
ing sequence comes up when additional data are needed to fit a newly hypothesized model based on
a model fit to the initial data. In this case, two additional steps, experimental design and data collec-
tion, can be added to the basic sequence between model selection and mode-fitting.

3.3 WHAT ARE MODELS USED FOR?
Models are used for four main purposes:

1. Estimation

2. Prediction

3. Calibration

4. Optimization

A brief explanation of the different uses of models is provided below (NIST, 2012):

• Estimation —The goal of estimation is to determine the value of the regression function
(i.e., the average value of the response variable) for a particular combination of the values
of the predictor variables. Regression function values can be estimated for any combina-
tion of predictor variable values, including values for which no data have been measured or
observed. Function values estimated for points within the observed space of predictor vari-
able values are sometimes called interpolations . Estimation of regression function values
for points outside the observed space of predictor variable values, called extrapolations ,
are sometimes necessary but require caution.
• Prediction —The goal of prediction is to determine either

1. The value of a new observation of the response variable, or

2. The values of a specified proportion of all future observations of the response variable
for a particular combination of the values of the predictor variables. Predictions can be
made for any combination of predictor variable values, including values for which no
data have been measured or observed. As in the case of estimation, predictions made
outside the observed space of predictor variable values are sometimes necessary but
require caution.
• Calibration —The goal of calibration is to quantitatively relate measurements made using
one measurement system to those of another measurement system. This is done so that
measurements can be compared in common units or to tie results from a relative measure-
ment method to absolute units.
• Optimization —Optimization is performed to determine the values of process inputs that
should be used to obtain the desired process output. Typical optimization goals might be
to maximize the yield of a process, to minimize the processing time required to fabricate
a product, or to hit a target product specification with minimum variation in order to main-
tain specified tolerances.