Logistic Curves: An Interactive Demonstration
This applet is designed to provide hands-on understanding of
logistic curves. There are
instructions below.
How to use this applet:
On the bottom panel, you will see the current values for
the following parameters:
or K = Saturation, or the
level at which the curve levels off. This represented by the blue line on the graph.
or tm = Midpoint,
or "turning point" of the curve, or the time at which growth begins to
decrease. In analytical terms, tm is the inflection point of the
curve. This is represented by the purple circle
on the graph.
or Dt
= Growth Time, or the time during which the curve grows from 10% to
90% of its saturation level. When Dt is negative,
the curve decreases over time. This is represented by the
green hash marks on the graph.
Changing parameters: To change a parameter, click on the
corresponding button on the right panel. The button will turn red, as
will the parameter's marker on the graph, which means you can change
the value of that parameter.
Fitting curves: Finally, you can do a least-squares regression
to find the best-fitting curve by dragging the 
onto the graph. This regression method is
taken from Numerical Recipes by Press, et. al. Note that the
algorithm is iterative, so if the curve doesn't look right, you may
need to do this a few times before the parameters converge to their
best values.
Locking parameters: Because sometimes a parameter will not
converge, it may be useful to lock some of the parameters to a constant
value while fitting the others. To do this, drag the 
onto the 
next to the parameter you wish to lock.
When you try to fit a curve, the algorithm will hold constant the
parameters that you have locked. Naturally, if you wish to unlock a
parameter, drag the onto
the 
again.
Browsing through data sets: There are four sets of data to explore;
to look at the other sets, select
one from the selection bar at the top of the applet.
What is a logistic curve?
Logistic curves are used to model growth and diffusion, like
the height of a sunflower, the aggregate length of railroads, or the
cumulative works of an author. Such curves are S-shaped, following the
equation
y(t) = K / exp( (-ln(81)/Dt) * (t-tm)) )
Logistic curves can be fit to time-series data using
Levenberg-Marquardt least-squares regression. In simple terms, this
method works by taking initial values for the three parameters we
described above, and adjusts them according to their derivatives.
After a number of iterations, the parameters converge to some final
values. (Because of arbitrary limits on the number of iterations,
sometimes the fitting algorithm must be run more than once; i.e., you may
need to drag the "F" more than once before the values converge.)
For a detailed description of the mathematics of logistic curves,
see our Loglet web page.
For more applications of logistic curves, see our paper on
modelling populations.
For a more formal treatment of the Levenberg-Marquardt method, read
the relevant sections of Chapter 15 in Numerical
Recipes, from which the code originates. (I went through the small
trouble writing Java wrapper class for it) You will need either a
PostScript viewer (like Ghostview) or Acrobat Reader to browse the
book on-line; these viewers are freely available.
Written by Jason Yung, December 1997
