# How to understand logistic trend parameters proposed by Prophet (Facebook)?

I read the paper and I saw that the logistic trend is defined as below : $$g(t) = \frac{C(t)}{1 + \exp{ -k(t) (t - m(t)) }}$$ Where $$k$$ and $$m$$ are respectively a growth rate and an offset parameter. $$C$$ denotes the carrying capacity of the trend. I am trying to understand those parameters, so I have some questions to ask.

First of all, how can one to set this carrying capacity $$C(t)$$ in Python. I read the documentation and I only noticed two parameters for saturating growth which are the cap and the floor. But nothing else. So, I assume that a parametric model of the capacity is already set so that one just have to give the cap and the floor. But, I am not sure, this assumption may be totally wrong.

Moreover, I do not understand the explanations done while defining the growth rate. Let me remind you something about it. Let's consider a set of change points, i.e. points from which the trend is allowed to change, $$\{ s_j , 1 \leq j \leq S \}$$. We define the growth rate $$k(t) = k_0 + a(t)^T \delta$$ where :

$$a(t) = [a_j(t)]_{1 \leq j \leq S}$$ a vector

$$a_j(t) = 1$$ if $$t \geq s_j$$ , $$0$$ otherwise

$$\delta = [\delta_j]_{1 \leq j \leq S}$$ a vector, i.e. the change-point rate vector.

Well, I do not understand what those equations mean and the need to offset the model with $$m(t)$$, assuming this definition of $$k(t)$$. The paper seems to be unclear for me. I do not understand it clearly, I do assume that setting $$k$$ and $$m$$ as such would allow a flexibility while setting the trend, I mean it is pretty obvious that the growth will not exactly look like an usual logistic function, but I want to have a deeper comprehension of those parameters.

Does someone help me to understand how the growth logistic is defined ? I would really appreciate any help.

Many thanks. Hope I am clear, so please let me know if I am unclear :)