Statistics - (Logit|Logistic) (Function|Transformation)

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>