numpy.percentile#

numpy.percentile(a, q, axis=None, out=None, overwrite_input=False, method='linear', keepdims=False, *, weights=None, interpolation=None)[source]#

Compute the q-th percentile of the data along the specified axis.

Returns the q-th percentile(s) of the array elements.

Parameters:
aarray_like of real numbers

Input array or object that can be converted to an array.

qarray_like of float

Percentage or sequence of percentages for the percentiles to compute. Values must be between 0 and 100 inclusive.

axis{int, tuple of int, None}, optional

Axis or axes along which the percentiles are computed. The default is to compute the percentile(s) along a flattened version of the array.

Changed in version 1.9.0: A tuple of axes is supported

outndarray, optional

Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output, but the type (of the output) will be cast if necessary.

overwrite_inputbool, optional

If True, then allow the input array a to be modified by intermediate calculations, to save memory. In this case, the contents of the input a after this function completes is undefined.

methodstr, optional

This parameter specifies the method to use for estimating the percentile. There are many different methods, some unique to NumPy. See the notes for explanation. The options sorted by their R type as summarized in the H&F paper [1] are:

  1. ‘inverted_cdf’

  2. ‘averaged_inverted_cdf’

  3. ‘closest_observation’

  4. ‘interpolated_inverted_cdf’

  5. ‘hazen’

  6. ‘weibull’

  7. ‘linear’ (default)

  8. ‘median_unbiased’

  9. ‘normal_unbiased’

The first three methods are discontinuous. NumPy further defines the following discontinuous variations of the default ‘linear’ (7.) option:

  • ‘lower’

  • ‘higher’,

  • ‘midpoint’

  • ‘nearest’

Changed in version 1.22.0: This argument was previously called “interpolation” and only offered the “linear” default and last four options.

keepdimsbool, optional

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array a.

New in version 1.9.0.

weightsarray_like, optional

An array of weights associated with the values in a. Each value in a contributes to the percentile according to its associated weight. The weights array can either be 1-D (in which case its length must be the size of a along the given axis) or of the same shape as a. If weights=None, then all data in a are assumed to have a weight equal to one. Only method=”inverted_cdf” supports weights. See the notes for more details.

New in version 2.0.0.

interpolationstr, optional

Deprecated name for the method keyword argument.

Deprecated since version 1.22.0.

Returns:
percentilescalar or ndarray

If q is a single percentile and axis=None, then the result is a scalar. If multiple percentiles are given, first axis of the result corresponds to the percentiles. The other axes are the axes that remain after the reduction of a. If the input contains integers or floats smaller than float64, the output data-type is float64. Otherwise, the output data-type is the same as that of the input. If out is specified, that array is returned instead.

See also

mean
median

equivalent to percentile(..., 50)

nanpercentile
quantile

equivalent to percentile, except q in the range [0, 1].

Notes

In general, the percentile at percentage level \(q\) of a cumulative distribution function \(F(y)=P(Y \leq y)\) with probability measure \(P\) is defined as any number \(x\) that fulfills the coverage conditions

\[P(Y < x) \leq q/100 \quad\text{and} \quad P(Y \leq x) \geq q/100\]

with random variable \(Y\sim P\). Sample percentiles, the result of percentile, provide nonparametric estimation of the underlying population counterparts, represented by the unknown \(F\), given a data vector a of length n.

One type of estimators arises when one considers \(F\) as the empirical distribution function of the data, i.e. \(F(y) = \frac{1}{n} \sum_i 1_{a_i \leq y}\). Then, different methods correspond to different choices of \(x\) that fulfill the above inequalities. Methods that follow this approach are inverted_cdf and averaged_inverted_cdf.

A more general way to define sample percentile estimators is as follows. The empirical q-percentile of a is the n * q/100-th value of the way from the minimum to the maximum in a sorted copy of a. The values and distances of the two nearest neighbors as well as the method parameter will determine the percentile if the normalized ranking does not match the location of n * q/100 exactly. This function is the same as the median if q=50, the same as the minimum if q=0 and the same as the maximum if q=100.

The optional method parameter specifies the method to use when the desired percentile lies between two indexes i and j = i + 1. In that case, we first determine i + g, a virtual index that lies between i and j, where i is the floor and g is the fractional part of the index. The final result is, then, an interpolation of a[i] and a[j] based on g. During the computation of g, i and j are modified using correction constants alpha and beta whose choices depend on the method used. Finally, note that since Python uses 0-based indexing, the code subtracts another 1 from the index internally.

The following formula determines the virtual index i + g, the location of the percentile in the sorted sample:

\[i + g = (q / 100) * ( n - alpha - beta + 1 ) + alpha\]

The different methods then work as follows

inverted_cdf:

method 1 of H&F [1]. This method gives discontinuous results:

  • if g > 0 ; then take j

  • if g = 0 ; then take i

averaged_inverted_cdf:

method 2 of H&F [1]. This method gives discontinuous results:

  • if g > 0 ; then take j

  • if g = 0 ; then average between bounds

closest_observation:

method 3 of H&F [1]. This method gives discontinuous results:

  • if g > 0 ; then take j

  • if g = 0 and index is odd ; then take j

  • if g = 0 and index is even ; then take i

interpolated_inverted_cdf:

method 4 of H&F [1]. This method gives continuous results using:

  • alpha = 0

  • beta = 1

hazen:

method 5 of H&F [1]. This method gives continuous results using:

  • alpha = 1/2

  • beta = 1/2

weibull:

method 6 of H&F [1]. This method gives continuous results using:

  • alpha = 0

  • beta = 0

linear:

method 7 of H&F [1]. This method gives continuous results using:

  • alpha = 1

  • beta = 1

median_unbiased:

method 8 of H&F [1]. This method is probably the best method if the sample distribution function is unknown (see reference). This method gives continuous results using:

  • alpha = 1/3

  • beta = 1/3

normal_unbiased:

method 9 of H&F [1]. This method is probably the best method if the sample distribution function is known to be normal. This method gives continuous results using:

  • alpha = 3/8

  • beta = 3/8

lower:

NumPy method kept for backwards compatibility. Takes i as the interpolation point.

higher:

NumPy method kept for backwards compatibility. Takes j as the interpolation point.

nearest:

NumPy method kept for backwards compatibility. Takes i or j, whichever is nearest.

midpoint:

NumPy method kept for backwards compatibility. Uses (i + j) / 2.

For weighted percentiles, the above coverage conditions still hold. The empirical cumulative distribution is simply replaced by its weighted version, i.e. \(P(Y \leq t) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{x_i \leq t}\). Only method="inverted_cdf" supports weights.

References

[1] (1,2,3,4,5,6,7,8,9,10)

R. J. Hyndman and Y. Fan, “Sample quantiles in statistical packages,” The American Statistician, 50(4), pp. 361-365, 1996

Examples

>>> a = np.array([[10, 7, 4], [3, 2, 1]])
>>> a
array([[10,  7,  4],
       [ 3,  2,  1]])
>>> np.percentile(a, 50)
3.5
>>> np.percentile(a, 50, axis=0)
array([6.5, 4.5, 2.5])
>>> np.percentile(a, 50, axis=1)
array([7.,  2.])
>>> np.percentile(a, 50, axis=1, keepdims=True)
array([[7.],
       [2.]])

>>> m = np.percentile(a, 50, axis=0)
>>> out = np.zeros_like(m)
>>> np.percentile(a, 50, axis=0, out=out)
array([6.5, 4.5, 2.5])
>>> m
array([6.5, 4.5, 2.5])

>>> b = a.copy()
>>> np.percentile(b, 50, axis=1, overwrite_input=True)
array([7.,  2.])
>>> assert not np.all(a == b)

The different methods can be visualized graphically:

import matplotlib.pyplot as plt

a = np.arange(4)
p = np.linspace(0, 100, 6001)
ax = plt.gca()
lines = [
    ('linear', '-', 'C0'),
    ('inverted_cdf', ':', 'C1'),
    # Almost the same as `inverted_cdf`:
    ('averaged_inverted_cdf', '-.', 'C1'),
    ('closest_observation', ':', 'C2'),
    ('interpolated_inverted_cdf', '--', 'C1'),
    ('hazen', '--', 'C3'),
    ('weibull', '-.', 'C4'),
    ('median_unbiased', '--', 'C5'),
    ('normal_unbiased', '-.', 'C6'),
    ]
for method, style, color in lines:
    ax.plot(
        p, np.percentile(a, p, method=method),
        label=method, linestyle=style, color=color)
ax.set(
    title='Percentiles for different methods and data: ' + str(a),
    xlabel='Percentile',
    ylabel='Estimated percentile value',
    yticks=a)
ax.legend(bbox_to_anchor=(1.03, 1))
plt.tight_layout()
plt.show()
../../_images/numpy-percentile-1.png
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