Growth rates and the ``bell'' view

Definitions and notation

The Mathematics of Loglet Analysis

Now consider the set of *n* logistic components
,
where

where *x*_{i} is an arbitrary subspace of *t*. (The components *c*_{i}are analogous to *N*_{i}(*t*) as discussed in equation
(7).) A typical choice for *x*_{i} is the interval
over which *c*_{i} grows from 10% to 90% of its saturation level,
namely
.
Restricting the domain of *c*_{i} gives us a criterion for
decomposing the associated data set *D* into subsets *D*_{j} (
)
corresponding to each component logistic *c*_{i}. A subset
*D*_{j} contains a point (*t*_{i},*d*_{i}) if
:

where
is the *adjusted* value of *d*_{i}. The adjustment
is subtracting out the ``effects'' from other components,
leaving us with the (approximate) contribution of component *c*_{i} to this
data point. In other words,

In Figure 5C, the hypothetical data set plotted in Figure
5A is decomposed into subsets *D*_{1} (circles) and
*D*_{2} (crosses); similarly, the fitted curve is also decomposed into
its component logistics. Note that the subsets *D*_{j} are not
necessarily mutually exclusive in the *t* domain; in this example,
*D*_{1} and *D*_{2} share points with common *t*-values around *t* = 40.
At these times, we can see that there are two concurrent growth
processes; in addition, we can also quantify how much of the growth
can be attributed to each process.
As we saw in Figure 3, we can apply the Fisher-Pry
transform to each component and its corresponding data subset. This
is useful because it normalizes each component on a semi-logarithmic
scale, allowing for easy comparison when plotted on the same graph.
Moreover, it allows the fitting of logistics using linear least
squares. The Fisher-Pry transform of components
is

where plotting
vs.
produces
a straight line. The component data subsets *D*_{j} are transformed in
a similar manner:

Figure 5D shows the
Fisher-Pry decomposition of the hypothetical data.

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*Perrin S Meyer*

*1998-07-14*