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Definitions and notation

Consider the two-dimensional space in which our data set D exists. If there are m data points, then we define D as

\begin{displaymath}D = \{(t_1,d_1),\,\ldots ,(t_i,d_i),\,\ldots,(t_m,d_m)\}
\end{displaymath}

where ti usually represents time, while di represents the growing variable (e.g., number of organisms, percent of saturation).

Suppose we want to fit a logistic curve of n components to the model. Then we will require 3n parameters, represented as a $n \times 3$ matrix $\bold P$, where the ith row describes the ith component:

   \begin{displaymath}
\bold P = \left[ \begin{array}{ccc}
\Delta t_1 & \kappa_1 ...
...ts & \\
\Delta t_n & \kappa_n & t_{mn}
\end{array} \right]
\end{displaymath}

Thus a loglet can be alternatively specified by

   \begin{displaymath}
\bold{N}(t, \bold P) = \sum_{i=1}^{n} \frac{\bold P_{i2}}{1...
...t[ {-\frac{\ln(81)}{\bold P_{i1}}}(t - \bold P_{i3}) \right] }
\end{displaymath}

Figure 5A shows a hypothetical data set (the circles) and a fitted loglet with n=2 and

\begin{eqnarray*}\bold P = \left[ \begin{array}{ccc}
20 & 50 & 30 \\
25 & 60 & 60 \end{array} \right].
\end{eqnarray*}


Gaussian noise was added to the data to dataset to show the residuals.


  
Figure 5: A hypothetical bi-logistic data set(A), the residuals of the fitted curve(B), and the decompositions in raw form (C) and with the Fisher-Pry transform applied (D).
\resizebox{4in}{!}{\includegraphics{ex_bilog.eps}}


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Next: Decomposition Up: The Mathematics of Loglet Analysis Previous: The Mathematics of Loglet Analysis
Perrin S Meyer
1998-07-14