How to calculate a logistic sigmoid function in Python

Question

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

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Good answer from  unwind  It however can t handle extreme negative number  throwing OverflowError    My improvement   def sigmoid x       try          res   1    1   math exp -x       except OverflowError          res   0 0     return res

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A numerically stable version of the logistic sigmoid function       def sigmoid x           pos mask    x  gt   0          neg mask    x  lt  0          z   np zeros like x dtype float          z pos mask    np exp -x pos mask           z neg mask    np exp x neg mask           top   np ones like x dtype float          top neg mask    z neg mask          return top    1   z

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Here s how you would implement the logistic sigmoid in a numerically stable way  as described here   def sigmoid x        quot Numerically-stable sigmoid function  quot      if x  gt   0          z   exp -x          return 1    1   z      else          z   exp x          return z    1   z   Or perhaps this is more accurate  import numpy as np  def sigmoid x         return np exp -np logaddexp 0  -x    Internally  it implements the same condition as above  but then uses log1p  In general  the multinomial logistic sigmoid is  def nat to exp q       max q   max 0 0  np max q       rebased q   q - max q     return np exp rebased q - np logaddexp -max q  np logaddexp reduce rebased q      However  logaddexp reduce could be more accurate

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It is also available in scipy  http   docs scipy org doc scipy reference generated scipy stats logistic html  In  1   from scipy stats import logistic  In  2   logistic cdf 0 458  Out 2   0 61253961344091512   which is only a costly wrapper  because it allows you to scale and translate the logistic function  of another scipy function   In  3   from scipy special import expit  In  4   expit 0 458  Out 4   0 61253961344091512   If you are concerned about performances continue reading  otherwise just use expit   Some benchmarking   In  5   def sigmoid x               return 1    1   math exp -x             In  6    timeit -r 1 sigmoid 0 458  1000000 loops  best of 1  371 ns per loop   In  7    timeit -r 1 logistic cdf 0 458  10000 loops  best of 1  72 2   s per loop  In  8    timeit -r 1 expit 0 458  100000 loops  best of 1  2 98   s per loop   As expected logistic cdf is  much  slower than expit  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  9   import numpy as np  In  10   x   np random random 1000000   In  11   def sigmoid array x                                                       return 1    1   np exp -x                You ll notice the tiny change from math exp to np exp  the first one does not support arrays  but is much faster if you have only one value to compute    In  12    timeit -r 1 -n 100 sigmoid array x  100 loops  best of 1  34 3 ms per loop  In  13    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

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A one liner     In 1   import numpy as np  In 2   sigmoid lambda x  1    1   np exp -x    In 3   sigmoid 3  Out 3   0 9525741268224334

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I feel many might be interested in free parameters to alter the shape of the sigmoid function  Second for many applications you want to use a mirrored sigmoid function  Third you might want to do a simple normalization for example the output values are between 0 and 1    Try   def normalized sigmoid fkt a  b  x             Returns array of a horizontal mirrored normalized sigmoid function    output between 0 and 1    Function parameters a   center  b   width           s  1  1 np exp b  x-a       return 1  s-min s    max s -min s     normalize function to 0-1   And to draw and compare   def draw function on 2x2 grid x        fig    ax1  ax2    ax3  ax4     plt subplots 2  2      plt subplots adjust wspace  5      plt subplots adjust hspace  5       ax1 plot x  normalized sigmoid fkt   5  18  x       ax1 set title  1        ax2 plot x  normalized sigmoid fkt 0 518  10 549  x       ax2 set title  2        ax3 plot x  normalized sigmoid fkt   7  11  x       ax3 set title  3        ax4 plot x  normalized sigmoid fkt   2  14  x       ax4 set title  4       plt suptitle  Different normalized  sigmoid  function  size 10        return fig   Finally   x   np linspace 0 1 100  Travel function   draw function on 2x2 grid x

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Below is the python function to do the same  def sigmoid x        return 1 0  1 np exp -x

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Use the numpy package to allow your sigmoid function to parse vectors   In conformity with Deeplearning  I use the following code   import numpy as np def sigmoid x       s   1  1 np exp -x       return s

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This should do it   import math  def sigmoid x     return 1    1   math exp -x     And now you can test it by calling    gt  gt  gt  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

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import numpy as np  def sigmoid x       s   1    1   np exp -x       return s  result   sigmoid 0 467  print result    The above code is the logistic sigmoid function in python  If I know that x   0 467   The sigmoid function  F x    0 385  You can try to substitute any value of x you know in the above code  and you will get a different value of F x

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another way   gt  gt  gt  def sigmoid x           return 1   1  math e  -x        gt  gt  gt  sigmoid 0 458

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you can calculate it as    import math def sigmoid x     return 1    1   math exp -x     or conceptual  deeper and without any imports   def sigmoid x     return 1    1   2 718281828    -x    or you can use numpy for matrices   import numpy as np  make sure numpy is already installed def sigmoid x     return 1    1   np exp -x

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Tensorflow includes also a sigmoid function  https   www tensorflow org versions r1 2 api docs python tf sigmoid  import tensorflow as tf  sess   tf InteractiveSession   x   0 458 y   tf sigmoid x   u   y eval   print u    0 6125396

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Another way by transforming the tanh function     sigmoid   lambda x   5    math tanh  5   x    1

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Vectorized method when using pandas DataFrame Series or numpy array   The top answers are optimized methods for single point calculation  but when you want to apply these methods to a pandas series or numpy array  it requires apply  which is basically for loop in the background and will iterate over every row and apply the method  This is quite inefficient    To speed up our code  we can make use of vectorization and numpy broadcasting   x   np arange -5 5  np divide 1  1 np exp -x    0    0 006693 1    0 017986 2    0 047426 3    0 119203 4    0 268941 5    0 500000 6    0 731059 7    0 880797 8    0 952574 9    0 982014 dtype  float64   Or with a pandas Series   x   pd Series np arange -5 5   np divide 1  1 np exp -x

[python] How to calculate a logistic sigmoid function in Python?

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