Feb 21, 2010

S-shaped Functions

The logit (logistic sigmoid) function:
$\sigma(x) = \frac{1}{1 + \exp(-x)}$

The tanh function:
$\tanh(x) = 2\sigma(x) - 1$

The probit function:
$\Phi(x) = \int_{-\infty}^x \mathcal{N}(\theta;0,1) \;\mathrm{d}\theta$
where
$\Phi\left(\sqrt{\frac{\pi}{8}}x\right) \approx \sigma(x)$
For more on this approximation, look at Figure 4.9 of Pattern Recognition and Machine Learning.