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$$ \begin{equation} f(x)=\frac{L}{1+e^{-k(x-x_0)}} \end{equation} $$

Fig 1. (img) Logistic Function

$$ \begin{equation} S(x)= \frac{1}{1+e^{-t}} \end{equation} $$

Fig 2. (img) Sigmoid Function

What are the differences between Logistic Function and Sigmoid Function? Is the logistic function a generalized kind of sigmoid function where you could have a higher maximum value?

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  • $\begingroup$ $S$ is an increasing function in $t$, $f$ is increasing for values of $x \leq x_0$, decreasing for values of $x \geq x_0$ $\endgroup$
    – mic
    Commented Mar 30, 2016 at 7:25
  • $\begingroup$ @mic I don't think that's right. If $t>0$, $S$ (the OP's "sigmoid function") is monotonically increasing. If $k>0$, (the OP's logistic function) $f$ is monotonically increasing. Both functions are monotonically increasing or decreasing, none of them increase for some values of $x$ and decrease for other values of $x$. The derivative increases then decreases, as shown here, is that what you meant by $f$? $\endgroup$ Commented Jun 30, 2023 at 21:54

3 Answers 3

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Yes, the sigmoid function is a special case of the Logistic function when $L=1$, $k=1$, $x_0 =0$.

If you play around with the parameters (Wolfram Alpha), you will see that

  • $L$ controls the maximum value the function can take. $e^{-k(x-x_0)}$ is always greater or equal than 0, so the maximum point is achieved when $e^{-k(x-x_0)}$ is 0 so the logistic function has value $L/1$.

  • $x_0$ controls where on the $x$ axis the growth is centered, because if you put $x_0$ in the function, $x_0 - x_0$ cancel out and $e^0 = 1$, so you end up with $f(x_0) = L/2$, the midpoint of the growth.

  • $k$ controls how steep the growth from the minimum to the maximum value is.

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The logistic function is: $$ f(x) = \frac{K}{1+Ce^{-rx}} $$ where $C$ is the constant from integration, $r$ is the proportionality constant, and $K$ is the threshold limit.

Assuming the limits are between $0$ and $1$, we get $\frac{1}{1+e^{-x}}$ which is the sigmoid function.

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  • $\begingroup$ what is $C$ -- I don't see it in the OP's question. What does "constant from integration" mean? Do you have an example to cite, so we can learn more? $\endgroup$ Commented Jun 30, 2023 at 22:00
  • $\begingroup$ Trying to answer my question, this resource, ("A First Course on Time Series Analysis") says: "the function $\frac{β3}{1 + β2 exp(−β1t)}$ with $β1, β2, β3 ∈ \!R $ **is the widely used logistic function** Its $β2$ is same as your $C$. This video is also consistent with your definition. But I'm still trying to understand the significance of $C$, aka $β2$, and why Wikipedia doesn't mention it. $\endgroup$ Commented Jul 6, 2023 at 23:20
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I would like to say in opposite way to the answer "the sigmoid function is a special case of the Logistic function" into "The Logisitic function is a special case of the sigmoid function". All S shape curved monotonically increasing fuction being confined a and b are sigmoid functions.

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    $\begingroup$ That is also how Wikipedia describes the sigmoid functions. $\endgroup$
    – Jiho Noh
    Commented Mar 21, 2023 at 19:08
  • $\begingroup$ Interesting - I upvoted you -- Wikipedia does say that sigmoid functions is a general category of any "S-shaped curve", and gives logistic function as just one example. However, Wikipedia labels the "logistic function" example with the definition $f(x) = \frac{1}{1+e^{-x}}$ , which is what the OP calls the "sigmoid function". There are more parameters in what the OP calls the "logistic function". So you're right, but I also think the accepted answer is right in addressing the OP's definitions. $\endgroup$ Commented Jun 30, 2023 at 21:53

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