1.5. MATHEMATICAL MODELING 41
Modeling
A good description of the model will begin with an organized and complete
description of important factors and observations. The description will often
use data gathered from observations of the problem. It will also include the
statement of scientific laws and relations that apply to the important factors.
From there, the model must summarize and condense the observations into
a small set of hypotheses that capture the essence of the observations. The
small set of hypotheses is a restatement of the problem, changing the problem
from a descriptive, even colloquial, question into a precise formulation that
moves the statement from the general to the specific. This sets the stage for
the modeler to demonstrate a clear link between the listed assumptions and
the building of the model.
The hypotheses translate into a mathematical structure that becomes the
heart of the mathematical model. Many mathematical models, particularly
those from physics and engineering, become a single equation but mathe-
matical models need not be a single concise equation. Mathematical models
may be a regression relation, either a linear regression, an exponential re-
gression or a polynomial regression. The choice of regression model should
explicitly follow from the hypotheses since the growth rate is an important
consequence of the observations. The mathematical model may be a linear or
nonlinear optimization model, consisting of an objective function and a set
of constraints. Again the choice of linear or nonlinear functions for the ob-
jective and constraints should explicitly follow from the nature of the factors
and the observations. For dynamic situations, the observations often involve
some quantity and its rates of change. The hypotheses express some con-
nection between these quantities and the mathematical model then becomes
a differential equation, either linear or nonlinear depending on the explicit
details of the scientific laws relating the factors considered. For discretely
sampled data instead of continuous time expressions the model may become
a difference equation. If an important feature of the observations and factors
is noise or randomness, then the model may be a probability distribution or
a stochastic process. The classical models from science and engineering usu-
ally take one of these classical equation-like forms but not all mathematical
models need to follow this format. Models may be a connectivity graph, or
a group of transformations.
If the number of variables is more than a few, or the relations are compli-
cated to write in a concise mathematical expression then the model can be a