calculating Gini coefficient in Python/numpy

Question

i\'m calculating Gini coefficient (similar to: Python - Gini coefficient calculation using Numpy) but i get an odd result. for a uniform distribution sampled from np.r

Answer 1

A quick note on the original methodology:

When calculating Gini coefficients directly from areas under curves with np.traps or another integration method, the first value of the Lorenz curve needs to be 0 so that the area between the origin and the second value is accounted for. The following changes to G(v) fix this:

yvals = [0]
for b in bins[1:]:

I also discussed this issue in this answer, where including the origin in those calculations provides an equivalent answer to using the other methods discussed here (which do not need 0 to be appended).

In short, when calculating Gini coefficients directly using integration, start from the origin. If using the other methods discussed here, then it's not needed.

Answer 2

This is to be expected. A random sample from a uniform distribution does not result in uniform values (i.e. values that are all relatively close to each other). With a little calculus, it can be shown that the expected value (in the statistical sense) of the Gini coefficient of a sample from the uniform distribution on [0, 1] is 1/3, so getting values around 1/3 for a given sample is reasonable.

You'll get a lower Gini coefficient with a sample such as v = 10 + np.random.rand(500). Those values are all close to 10.5; the relative variation is lower than the sample v = np.random.rand(500). In fact, the expected value of the Gini coefficient for the sample base + np.random.rand(n) is 1/(6*base + 3).

Here's a simple implementation of the Gini coefficient. It uses the fact that the Gini coefficient is half the relative mean absolute difference.

def gini(x):
    # (Warning: This is a concise implementation, but it is O(n**2)
    # in time and memory, where n = len(x).  *Don't* pass in huge
    # samples!)

    # Mean absolute difference
    mad = np.abs(np.subtract.outer(x, x)).mean()
    # Relative mean absolute difference
    rmad = mad/np.mean(x)
    # Gini coefficient
    g = 0.5 * rmad
    return g

Here's the Gini coefficient for several samples of the form v = base + np.random.rand(500):

In [80]: v = np.random.rand(500)

In [81]: gini(v)
Out[81]: 0.32760618249832563

In [82]: v = 1 + np.random.rand(500)

In [83]: gini(v)
Out[83]: 0.11121487509454202

In [84]: v = 10 + np.random.rand(500)

In [85]: gini(v)
Out[85]: 0.01567937753659053

In [86]: v = 100 + np.random.rand(500)

In [87]: gini(v)
Out[87]: 0.0016594595244509495

Answer 3

A slightly faster implementation (using numpy vectorization and only computing each difference once):

def gini_coefficient(x):
    """Compute Gini coefficient of array of values"""
    diffsum = 0
    for i, xi in enumerate(x[:-1], 1):
        diffsum += np.sum(np.abs(xi - x[i:]))
    return diffsum / (len(x)**2 * np.mean(x))

Note: x must be a numpy array.

Answer 4

Gini coefficient is the area under the Lorence curve, usually calculated for analyzing the distribution of income in population. https://github.com/oliviaguest/gini provides simple implementation for the same using python.