This is a logistic sigmoid function:
I know x. How can I calculate F(x) in Python now?
Let's say x = 0.458.
F(x) = ?
This should do it:
import math def sigmoid(x): return 1 / (1 + math.exp(-x))
And now you can test it by calling:
>>> sigmoid(0.458) 0.61253961344091512
Update: Note that the above was mainly intended as a straight one-to-one translation of the given expression into Python code. It is not tested or known to be a numerically sound implementation. If you know you need a very robust implementation, I'm sure there are others where people have actually given this problem some thought.
It is also available in scipy: http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.logistic.html
In : from scipy.stats import logistic In : logistic.cdf(0.458) Out: 0.61253961344091512
which is only a costly wrapper (because it allows you to scale and translate the logistic function) of another scipy function:
In : from scipy.special import expit In : expit(0.458) Out: 0.61253961344091512
If you are concerned about performances continue reading, otherwise just use
In : def sigmoid(x): ....: return 1 / (1 + math.exp(-x)) ....: In : %timeit -r 1 sigmoid(0.458) 1000000 loops, best of 1: 371 ns per loop In : %timeit -r 1 logistic.cdf(0.458) 10000 loops, best of 1: 72.2 µs per loop In : %timeit -r 1 expit(0.458) 100000 loops, best of 1: 2.98 µs per loop
logistic.cdf is (much) slower than
expit is still slower than the python
sigmoid function when called with a single value because it is a universal function written in C ( http://docs.scipy.org/doc/numpy/reference/ufuncs.html ) and thus has a call overhead. This overhead is bigger than the computation speedup of
expit given by its compiled nature when called with a single value. But it becomes negligible when it comes to big arrays:
In : import numpy as np In : x = np.random.random(1000000) In : def sigmoid_array(x): ....: return 1 / (1 + np.exp(-x)) ....:
(You'll notice the tiny change from
np.exp (the first one does not support arrays, but is much faster if you have only one value to compute))
In : %timeit -r 1 -n 100 sigmoid_array(x) 100 loops, best of 1: 34.3 ms per loop In : %timeit -r 1 -n 100 expit(x) 100 loops, best of 1: 31 ms per loop
But when you really need performance, a common practice is to have a precomputed table of the the sigmoid function that hold in RAM, and trade some precision and memory for some speed (for example: http://radimrehurek.com/2013/09/word2vec-in-python-part-two-optimizing/ )
Also, note that
expit implementation is numerically stable since version 0.14.0: https://github.com/scipy/scipy/issues/3385