The output of results

If the value pairs were imported from a CSV file, the file name is output in the header.

Data from:  "hopfenwachstum.csv"

Saturation limit:  6
     Dark figure:  1

f(t) = 4,0189/(0,66981 + 5,3302*e^(-0,3562*t))

Inflection point W(5,8226/3)

Maximum growth rate f'(xw) = 0,53433

8 Values 
Coeff.of determin. = 0,99383916
Correlation coeff. = 0,99691482
Standard deviation = 0,16172584

The program determines the logistic function f(t)  in the form: 

The parameters are    a1 = f (0)·S ,  a2 = f(0) ,  a3 = S - f(0) , and  a4 = -k·S .

S  is the saturation limit, that is, the value that the function approaches asymptotically.
f(0)  is the function value at the point t = 0 , which does not have to match the first measured value.

In addition, the inflection point of the function is determined, that is, the point from which the slope decreases again.
The function value at the inflection point is always equal to half the saturation limit so  f (tw) = ½·S .
The derivative f '(tw)  at the inflection point provides the maximum growth rate,

The parameters of the logistic function are determined as follows:

  1. Step: Form the reciprocal function of f(t)   to get the sum from the denominator to the numerator.
  2. Step: Log both sides to get the exponent t .
  3. Step: Bring the equation to the form h(t) = m·t + b .
  4. Step: Perform a linear regression for the value pairs  ( t  | h(t) ) .
  5. Step: Undo the transformation for  m  and  b .

Linear regression also provides the determination coefficient, the correlation coefficient and the standard deviation.

The output form of the results is a simple text editor with the usual functions such as cut  , copy   and paste  .
For example, you can add comments to the results, copy them to the clipboard and paste them into your word processor.