Growth rates and the bell'' view
Definitions and notation
The Mathematics of Loglet Analysis

## Decomposition

Now consider the set of n logistic components , where

where xi is an arbitrary subspace of t. (The components ciare analogous to Ni(t) as discussed in equation (7).) A typical choice for xi is the interval over which ci grows from 10% to 90% of its saturation level, namely .

Restricting the domain of ci gives us a criterion for decomposing the associated data set D into subsets Dj ( ) corresponding to each component logistic ci. A subset Dj contains a point (ti,di) if :

where is the adjusted value of di. The adjustment is subtracting out the effects'' from other components, leaving us with the (approximate) contribution of component ci to this data point. In other words,

In Figure 5C, the hypothetical data set plotted in Figure 5A is decomposed into subsets D1 (circles) and D2 (crosses); similarly, the fitted curve is also decomposed into its component logistics. Note that the subsets Dj are not necessarily mutually exclusive in the t domain; in this example, D1 and D2 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 Dj are transformed in a similar manner:

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

Next: Growth rates and the bell'' view Up: The Mathematics of Loglet Analysis Previous: Definitions and notation
Perrin S Meyer
1998-07-14