Note: If you cannot run the applet, try
downloading this stand-alone app. Unzip in a
folder, and run in there as "java grexapp" (needs an installed Java Runtime
Environment, JRE). Here is the source.
Click or drag on the parameters area.
A family of growth curves with two shape parameters (a and b),
that includes most of the known univariate growth models as special cases.
It also describes probability distributions with an explicit form for
the cumulative, useful in computer simulation, including the Burr Type
III and Type XII, among others.
The growth curve equation is y = B-1[B-1(t,
b), a], where B is the negative Box-Cox
transformation B(x, c) = (1 - xc)
/ c if c ≠ 0, B(x,
0) = -ln x. Or B-1(x, c)
= (1 - c x)1/c if c ≠ 0,
B-1(x, 0) = e-x.
In general, t and y are subject to affine
transformations, with additional location and scale parameters.
Negative scale parameters reverse the t- and y-axis,
and are specified here by ticking the Reverse checkbox (both axes,
reversing only one would give decreasing curves). For details see Garcia,
O. "Unifying sigmoid univariate growth equations", Forest Biometry,
Modelling and Information Sciences 1, 63-68, 2005 (www.fbmis.info).
The graphs at the top show the y-over-t growth curve (or
distribution function) and the dy/dt-over-y
growth rate curve, corresponding to the shape parameters a
and b chosen with the mouse in the parameters diagram below.
The contours in the diagram (Figure 1 from Garcia 2005) indicate the
height of the inflection point, and common names of some growth-curve
instances are indicated (items in parenthesis correspond to reversed-axis
versions). Sigmoids have an inflection point and an upper asymptote.
Selecting the Auto checkbox chooses the appropriate axes
orientation for a sigmoid, when possible. Ticking the Density
checkbox displays the probability density function.
What's in a name? Attaching names to equations is
always risky. Very often assumptions about priority turn out to be
wrong. Naming the case a = 1/3, b = 0
after von Bertalanffy, although common, seems particularly unfair,
given that he used a general a ≥ 0 in the
1930's. These values contain the most useful parameter range of the
"Richards" function, as acknowledged in Richards' 1959 publication that
popularized it for modelling plant growth. In the forestry literature
the function is often referred to as "Chapman-Richards", although Chapman's
1961 application of it to growth in fishes follows a well-known 1957
fisheries monograph by Beverton and Holt. Recently, the Richards
function has been attributed to Mitscherlich, who only used instances
where 1/a is an integer.
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