# Thread: How to simply sigmoid function?

1. ## How to simply sigmoid function?

Is it possible to simply the combination of two sigmoid functions

$$\sigma\big(k(t_1-t)\big) - \sigma\big(k(t_2-t)\big)$$

2. ## Re: How to simply sigmoid function?

I'm not sure what you are asking. Simply is not a verb. If you are asking if it is possible to simplify, it depends on which sigmoid function you are using and what you mean by simplify. I'm gonna go with probably not.

3. ## Re: How to simply sigmoid function?

Sorry for the typo, yes I meant simplify.

Is it possible to express

$$x = \sigma\big(k(t_1-t)\big) - \sigma\big(k(t_1-t)\big)$$

in a simpler form?

4. ## Re: How to simply sigmoid function?

Originally Posted by brianx
Sorry for the typo, yes I meant simplify.

Is it possible to express

$$x = \sigma\big(k(t_1-t)\big) - \sigma\big(k(t_1-t)\big)$$

in a simpler form?
Yes. $x = 0.$

$$x = \sigma\big(k(t_1-t)\big) - \sigma\big(k(t_1-t)\big) = 0$$ is simpler.

5. ## Re: How to simply sigmoid function?

Originally Posted by brianx
Sorry for the typo, yes I meant simplify.

Is it possible to express

$$x = \sigma\big(k(t_1-t)\big) - \sigma\big(k(t_1-t)\big)$$

in a simpler form?
An example sigmoid function is: $\sigma(z) = \dfrac{1}{1+e^z}$. If we were to plug in:

$\dfrac{1}{1+e^{k(t_1-t)}} - \dfrac{1}{1+e^{k(t_2-t)}} = \dfrac{e^{k(t_2-t)}-e^{k(t_1-t)}}{\left( 1+e^{k(t_1-t)}\right) \left( 1+e^{k(t_2-t)} \right)}$

I do not see that as simpler. So, it depends on which sigmoid function you are using and what you mean by "simpler".