Need a logistic like function with y=0 at x=0

Question

A logistic curve:

$$y=\frac{50}{1+e^{-k(x-10)}}$$

fits my exp data very well (having a maximum value 50, having a good trend). However, I hope it can return to zero when $x=0$. Is there an alternative function better than logistic function.

Answer 1

How about

$$\tilde{y}(x)=y(x)-y(0)e^{-x}$$

You have $\tilde{y}(0)=0$, $\lim\limits_{x\to\infty} \tilde{y}(x)=\lim\limits_{x\to\infty} y(x)$. You have continuity, and for $x$ being somewhat big enough, it behaves just like $y(x)$.

Answer 2

Logistic functions are a special case within the Richards family of asymptotic growth models with inflection points. Considerable attention has been given to this family with Tjørve and Tjørve (2017) unifying Richards models to two equivalent forms. Merging notations,

$$y = A \left(1 + \left(\left(\frac{y_0}{A}\right)^{1-d} - 1\right) \exp\left(\frac{-k_U \cdot x}{d^{\frac{d}{1 - d}}}\right)\right)^{\frac{1}{1 - d}}$$

where $A$ is the upper asymptote, $y_0$ the y intercept, $d$ controls inflection, and $k_U$ the relative maximum growth rate. In this case the choices $y_0 = 0$ and $A = 50$ are desired, leaving $k_U$ and $0 < d \le 1$ free for fitting. ($d = 1$ is the unified Gompertz, a special case, and $y_0 = 0$ results in $d > 1$ dividing by zero. $d < 0$ typically yields $y \in \mathbb{C}$.)

While unified logistic models have $d = -1$ it seems more likely $d$ in the vicinity of 0.85 would be of interest here, along with a $k_U$ of perhaps something like 0.1. However, it's been seven years since this question was asked.