next Growth rates and the ``bell'' view
previous Definitions and notation
up The Mathematics of Loglet Analysis

Decomposition

Now consider the set of n logistic components $C = \{c_1,\,\ldots,c_n\}$, where

\begin{displaymath}c_i = \bold{N_i}(x_i,\bold P_i)
\end{displaymath}

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 $(t_{mi} -\Delta t_i, t_{mi} + \Delta t_i)$.

Restricting the domain of ci gives us a criterion for decomposing the associated data set D into subsets Dj ( $1 \leq j
\leq n$) corresponding to each component logistic ci. A subset Dj contains a point (ti,di) if $t_{mi} -\Delta t_i \leq t_i \leq t_{mi} + \Delta t_i$:

\begin{displaymath}D_j = \{(t_i,d^\ast_i)\;\vert\; t_i \in x_i\}.
\end{displaymath}

where $d^\ast_i$ 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,

\begin{displaymath}d^\ast_i = d_i - \sum_{j \neq i} \frac{\bold P_{j2}}{1 +
\t...
...t[ {-\frac{\ln(81)}{\bold P_{j1}}}(t - \bold P_{j3}) \right] }
\end{displaymath}

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 $\bold c_i$ is

\begin{displaymath}FP(c_i) = \frac{\displaystyle \frac {\bold c_i}{\kappa_i}}{1 - \displaystyle\frac{\bold c_i}{\kappa_i}}
\end{displaymath}

where plotting $\bold x_i$ vs. $\log(FP(c_i)$ produces a straight line. The component data subsets Dj are transformed in a similar manner:

\begin{displaymath}FP(d^\ast_i) = \frac{\displaystyle\frac{d^\ast_i}{\kappa_i}}
{1 - \displaystyle\frac{d^ast_i}{\kappa_i}}
\end{displaymath}

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


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