Residuals
Decomposition
The Mathematics of Loglet Analysis
Growth rates and the ``bell'' view
Just as the differential equation (1) reveals the
mechanism propelling its integral equation (2),
the rates of change of the component logistics provide clues to the
mechanisms propelling the composite logistic. Analyzing the rates of
change is often useful when yearly or percent per year tabulations
are applicable, as in the case of economic data. Recall
that the analytic form of the threeparameter logistic is:
The instantaneous rate of growth of the
logistic function is given by its derivative with respect to time:

(8) 
Figure 6:
The rates of change of the component logistics
(``The Bell View'')

Plotting (8) produces a bellshaped curve
similar, but not identical, to the normal distribution
function. Naturally, the bellshaped curve peaks at the midpoint
t_{m}; analytically, this is the point of inflection, and thus it is
an extremum of N(t). Panel B of Figure 6 shows the
derivative of the component logistics of our test function (Panel A is
shown again for comparison purposes).
The rate of change of the data subsets D_{j} is computed discretely,
creating the sets
,
in which each point E_{i} is
where
.
In other words, E_{i} contains the
discrete derivative from
to
.
Loglet Lab can decompose a logistic curve into its discrete components;
each component can be transformed using the FisherPry transform,
or their rate of change can be plotted.
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Perrin S Meyer
19980714