MatheAss 9.0Stochastics

Growth models

Regression is about determining the unknown parameters of a growth model or a given function for a series of measurement data in such a way that the final model adapts to the data as best as possible.
Frequently considered models are:

Linear Growth
With linear growth, the rate of change, i.e. the derivation of the growth function, is constant.
The corresponding diagram is a straight line.
Exponential Growth
With exponential growth, the rate of change is proportional to the population:   ƒ'(t) ∼ ƒ(t)
Limited Growth
Whith limited growth, the rate of change is proportional to the saturation deficit, that is the difference between the saturation limit S  and the population:   ƒ'(t) ∼ (S − ƒ(t))
Logistic Growth
With logistic growth, it is assumed that the population grows essentially exponentially at the beginning, but that growth is slowed down more and more as the saturation limit is approached. It is therefore assumed that the rate of change is proportional to both the population and the saturation deficit. This results in the differential equation:   ƒ'(t) = k · ƒ(t) · (S − ƒ(t))
Which has the solution:

For a given saturation limit S , the program determines the initial value ƒ(0)  and the proportionality factor k  for adapting the function ƒ(t)  to the given value pairs.

Method

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

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

S  is the saturation limit, that is, the value that the function approaches asymptotically.
ƒ(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  ƒ(tw) = ½·S .
The derivative ƒ'(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 ƒ(t)   to get the sum from the denominator to the numerator.
  2. Step: Take the logarithm of 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.

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