Help on function percentile in module numpy:
percentile(a, q, axis=None, out=None, overwrite_input=False, method='linear', keepdims=False, *, interpolation=None)
Compute the q-th percentile of the data along the specified axis.
Returns the q-th percentile(s) of the array elements.
Parameters
----------
a : array_like
Input array or object that can be converted to an array.
q : array_like of float
Percentile or sequence of percentiles to compute, which 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.
.. versionchanged:: 1.9.0
A tuple of axes is supported
out : ndarray, 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_input : bool, 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.
method : str, 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 discontiuous. NumPy further defines the
following discontinuous variations of the default 'linear' (7.) option:
* 'lower'
* 'higher',
* 'midpoint'
* 'nearest'
.. versionchanged:: 1.22.0
This argument was previously called "interpolation" and only
offered the "linear" default and last four options.
keepdims : bool, 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`.
.. versionadded:: 1.9.0
interpolation : str, optional
Deprecated name for the method keyword argument.
.. deprecated:: 1.22.0
Returns
-------
percentile : scalar 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
-----
Given a vector ``V`` of length ``N``, the q-th percentile of ``V`` is
the value ``q/100`` of the way from the minimum to the maximum in a
sorted copy of ``V``. 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
``q`` 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``.
This optional `method` parameter specifies the method to use when the
desired quantile lies between two data points ``i < j``.
If ``g`` is the fractional part of the index surrounded by ``i`` and
alpha and beta are correction constants modifying i and j.
Below, 'q' is the quantile value, 'n' is the sample size and
alpha and beta are constants.
The following formula gives an interpolation "i + g" of where the quantile
would be in the sorted sample.
With 'i' being the floor and 'g' the fractional part of the result.
.. math::
i + g = (q - alpha) / ( n - alpha - beta + 1 )
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 give 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 give 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 give continuous results using:
* alpha = 0
* beta = 1
hazen:
method 5 of H&F [1]_.
This method give continuous results using:
* alpha = 1/2
* beta = 1/2
weibull:
method 6 of H&F [1]_.
This method give continuous results using:
* alpha = 0
* beta = 0
linear:
method 7 of H&F [1]_.
This method give 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 give 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 give 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``.
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:
.. plot::
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()
plt.show()
References
----------
.. [1] R. J. Hyndman and Y. Fan,
"Sample quantiles in statistical packages,"
The American Statistician, 50(4), pp. 361-365, 1996
