I doing a regression of an experimental curve, it best fits with a logistic law (best becauseI already compare it to some polynomial regressions with different degrees).
The logistic law is defined by
$$ f(z)= D+ \frac{A-D}{ 1+\Big( \frac{z}{C}\Big) ^B} $$
Usually the curve of a logistic law have a sigmoid shape defined as follows:
With $A$, $B$, $C$, and $D$ are the paramters of the logistic law.
While my experimental curve is fited by a logistic law, it does not have two dendings:
For example, for the first curve appearing in the second graph, the values of the logistic parameters that appear after fitting are: A=-11.39962, B= 0.988052231, C=0.01338638, and D=-1.493300048. hope this helps @ClaudeLeibovici . I plotted it here
My question is when a logistic law loses his first bending? I tried to derive some formulation for this interms of the variation of the parameters A, B, C , and D (for example if $B<1 $ the first bending disappear, or if $C$ is near zero ) but unfortunately I failed as this appears not to be valid graphically. Can some one give me some tips in this manner.
Thank you in advance.
Your curve is defined by the equation $$f(z) = D + \frac{A-D}{1 + (\frac{z}{C})^B} $$ The interesting part of it is the second term (ignoring the inconsequential, for our purposes, constants at the numerator), $$\frac{1}{ 1 + (\frac{z}{C})^B}$$ Let us rescale the $z$ variable for simplicity, $\frac{z}{C} \to z$, and now look at the second derivative of $ \frac{1}{ 1 + z^B}$, which reads
$$ \frac{\mathrm{d}^2}{\mathrm{d}x^2} \Big( \frac{1}{ 1 + z^B}\Big) =\frac{Bx^{B-2} (x^B + B(x^B -1) +1)}{(x^B+1)^3} $$
For $B=1$ the second derivative is easily checked to have a constant sign. For $0 \leq B \leq 1$, as in your case, this is also true. Indeed, the denominator is always positive, for $x>0$. The first term on the numerator s positive. One is then left with te term $$ x^B + B(x^B -1) +1$$ whose derivative $B(B+1)x^{B-1}$ is positive and whose value for $x=0$ equals $ 1-B$, so positive for $B < 1$. It seems to me that only for $B>1$ one gets a sigmoid shape, i.e. the second derivative changes sign.