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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

$\displaystyle D = \{(t_1,d_1), \ldots ,(t_i,d_i), \ldots,(t_m,d_m)\} $

where $ t_i$ usually represents time, while $ d_i$ 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 $ i$th row describes the $ i$th component:

$\displaystyle \bold P = \left[ \begin{array}{ccc} \Delta t_1 & \kappa_1 & t_{m1} \\ & \vdots & \\ \Delta t_n & \kappa_n & t_{mn} \end{array} \right] $

Thus a loglet can be alternatively specified by

$\displaystyle \bold{N}(t, \bold P) = \sum_{i=1}^{n} \frac{\bold P_{i2}}{1 + \text{exp} \left[ {-\frac{\ln(81)}{\bold P_{i1}}}(t - \bold P_{i3}) \right] } $

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

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

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}}

next up previous contents
Next: Decomposition Up: The Mathematics of Loglet Previous: The Mathematics of Loglet   Contents
Jason Yung 2004-01-28