Choosing a Good Model Function
We would like to know the "best" answer to the question
"What is the net change in CO2 concentration over a 24 hour
time period?" Before we can we can get very far with this, we need to
identify some of the relevant issues.
Use each of the five models to approximately determine how much
CO2 accumulated in the San Marcos River during a 24-hour period
using "Data Set 1." This data set gives rates of carbon-dioxide during a time
interval [0,24]. To do this, select a Model Function (such as
"Piecewise Linear" and "Cubic Spline") and press the button labelled
"Integrate the model function." The integral of the model over the
selected interval of integration will be display under the graph of the model function.
Record your answers in
the table you were given.
One way of determining whether a particular model
is valid for a given set of data is to compare the properties of the
model functions with the properties of the system being modeled.
Argue whether you think CO2 production by plants and animals in the
is a continuous function of time. Is it differentiable? Which
models have these properties?
Another important issue is whether the way the data was collected will
react in some strange way with a given kind of model function.
We know from the definition of the integral as a limit of
approximating sums that if we collect enough data, at least the
piecewise constant models will get closer and closer to the true value
of the integral (provided the function we are modeling is continuous).
However, because of the time and expense of collecting and processing
data, researchers usually prefer to use as few data points as possible
to get good results.
Integrate each of the five models a second time using data set 2.
Data set 2 consists of the CO2 rate measurements taken every
other hour during the day. Again record your answers in
the table you were given. This data
will be used in the next question.
Based on your observations:
- Which model functions seem to behave the worst with fewer data
points? Explain your observations.
- Which model functions would be the most sensible to use if data
were taken every few seconds? What practical considerations start
becoming important with very, very large data sets?
- What kinds of things would you have to know about the underlying
phenomenon in order to have much faith in a model function based on
relatively few data points?
Previous:Introduction to Models of
The Geometry Center Calculus Development Team
A portion of this lab is based on a problem appearing in
the Harvard Consortium Calculus book, Hughes-Hallet, et al,
1994, p. 174
Last modified: Mon Jan 8 13:04:47 1996