About
The logit transform is a S-shaped curve that applies a softer function. It's a soft function of a step function:
- Never below 0,
- never above 1
- and a smooth transition in between.
<MATH> \begin{array}{rrrl} Logit(x) & = & \frac{\displaystyle e^{x}}{\displaystyle 1+ e^{x}} \\ \end{array} </MATH>
where:
- <math> e \approx 2:71828</math> is the scientific constant, the exponential. Euler's number
The values have to lie between 0 and 1 because:
- e to anything is positive.
- As the denominator is bigger than the numerator, it's always got to be bigger than 0.
- When <math>x</math> gets very large, this approaches 1.
Used to normalize?
The natural log of the odds is call the log-odds or logit.
Logistic function
The logistic function (= logit ?) asymptotically approaches 0 as the input approaches negative infinity and 1 as the input approaches positive infinity. Since the results are bounded by 0 and 1, it can be directly interpreted as a probability
The logistic function <MATH> \frac{1}{1 + \exp^{-z}} </MATH>