How to get the cumulative distribution function with NumPy?

Question

I want to create a CDF with NumPy, my code is the next:

histo = np.zeros(4096, dtype = np.int32)
for x in range(0, width):
   for y in range(0, height):
      histo[data[x][y]] += 1
      q = 0 
   cdf = list()
   for i in histo:
      q = q + i
      cdf.append(q)

I am walking by the array but take a long time the program execution. There is a built function with this feature, isn't?

Solution 1

I'm not really sure what your code is doing, but if you have hist and bin_edges arrays returned by numpy.histogram you can use numpy.cumsum to generate a cumulative sum of the histogram contents.

>>> import numpy as np
>>> hist, bin_edges = np.histogram(np.random.randint(0,10,100), normed=True)
>>> bin_edges
array([ 0. ,  0.9,  1.8,  2.7,  3.6,  4.5,  5.4,  6.3,  7.2,  8.1,  9. ])
>>> hist
array([ 0.14444444,  0.11111111,  0.11111111,  0.1       ,  0.1       ,
        0.14444444,  0.14444444,  0.08888889,  0.03333333,  0.13333333])
>>> np.cumsum(hist)
array([ 0.14444444,  0.25555556,  0.36666667,  0.46666667,  0.56666667,
        0.71111111,  0.85555556,  0.94444444,  0.97777778,  1.11111111])

Solution 2

Using a histogram is one solution but it involves binning the data. This is not necessary for plotting a CDF of empirical data. Let F(x) be the count of how many entries are less than x then it goes up by one, exactly where we see a measurement. Thus, if we sort our samples then at each point we increment the count by one (or the fraction by 1/N) and plot one against the other we will see the "exact" (i.e. un-binned) empirical CDF.

A following code sample demonstrates the method

import numpy as np
import matplotlib.pyplot as plt

N = 100
Z = np.random.normal(size = N)
# method 1
H,X1 = np.histogram( Z, bins = 10, normed = True )
dx = X1[1] - X1[0]
F1 = np.cumsum(H)*dx
#method 2
X2 = np.sort(Z)
F2 = np.array(range(N))/float(N)

plt.plot(X1[1:], F1)
plt.plot(X2, F2)
plt.show()

It outputs the following

Solution 3

update for numpy version 1.9.0. user545424's answer does not work in 1.9.0. This works:

>>> import numpy as np
>>> arr = np.random.randint(0,10,100)
>>> hist, bin_edges = np.histogram(arr, density=True)
>>> hist = array([ 0.16666667,  0.15555556,  0.15555556,  0.05555556,  0.08888889,
    0.08888889,  0.07777778,  0.04444444,  0.18888889,  0.08888889])
>>> hist
array([ 0.1       ,  0.11111111,  0.11111111,  0.08888889,  0.08888889,
    0.15555556,  0.11111111,  0.13333333,  0.1       ,  0.11111111])
>>> bin_edges
array([ 0. ,  0.9,  1.8,  2.7,  3.6,  4.5,  5.4,  6.3,  7.2,  8.1,  9. ])
>>> np.diff(bin_edges)
array([ 0.9,  0.9,  0.9,  0.9,  0.9,  0.9,  0.9,  0.9,  0.9,  0.9])
>>> np.diff(bin_edges)*hist
array([ 0.09,  0.1 ,  0.1 ,  0.08,  0.08,  0.14,  0.1 ,  0.12,  0.09,  0.1 ])
>>> cdf = np.cumsum(hist*np.diff(bin_edges))
>>> cdf
array([ 0.15,  0.29,  0.43,  0.48,  0.56,  0.64,  0.71,  0.75,  0.92,  1.  ])
>>>

Solution 4

To complement Dan's solution. In the case where there are several identical values in your sample, you can use numpy.unique :

Z = np.array([1,1,1,2,2,4,5,6,6,6,7,8,8])
X, F = np.unique(Z, return_index=True)
F=F/X.size

plt.plot(X, F)

View more solutions

Comments

[hans_meine]

However, this introduces a binning step that would not be necessary for a cumulative distribution.

[offwhitelotus]

user12287, I feel weird editing other people's answers. Besides, different answers for different versions.

[ali_m]

That gives you values of F that are greater than 1. Perhaps you meant to use F = F / float(F.max()) (also bear in mind that integer division would cause problems for people using Python 2x).

[omar]

This answer is old, thank you for your comments and answers. I have seen in each answer my rudimentary approach from three years ago.

[Dan]

@Alex this is not quite correct since it should go up by more than 1/N for the entries which are there more than once. You're correct my solution will only be correct for the last one of such occurances but it will plot correctly.

[ArtificiallyIntelligence]

"This keyword, normed is deprecated in Numpy 1.6 due to confusing/buggy behavior. It will be removed in Numpy 2.0. "There is a bug in the code if bin is not in [0,1]. Add x=np.cumsum(hist); x=(x - x.min()) / x.ptp()

[Jthorpe]

In principle you are using counts, but python uses zero based indexing in F, so perhaps you meant (F + 1) / (F[-1] + 1)

[Oliver Prislan]

as per numpy.histogram docs : normed is equivalent to the density argument, but produces incorrect results for unequal bin widths. Changed in version 1.15.0: DeprecationWarnings are actually emitted.