numpy.quantile#

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

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

New in version 1.15.0.

Parameters:
aarray_like of real numbers

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

qarray_like of float

Probability or sequence of probabilities of the quantiles to compute. Values must be between 0 and 1 inclusive.

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

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

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 quantile. There are many different methods, some unique to NumPy. The recommended options, numbered as they appear in [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. For backward compatibility with previous versions of NumPy, the following discontinuous variations of the default ‘linear’ (7.) option are available:

  • ‘lower’

  • ‘higher’,

  • ‘midpoint’

  • ‘nearest’

See Notes for details.

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.

weightsarray_like, optional

An array of weights associated with the values in a. Each value in a contributes to the quantile 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:
quantilescalar or ndarray

If q is a single probability and axis=None, then the result is a scalar. If multiple probability levels are given, first axis of the result corresponds to the quantiles. 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
percentile

equivalent to quantile, but with q in the range [0, 100].

median

equivalent to quantile(..., 0.5)

nanquantile

Notes

Given a sample a from an underlying distribution, quantile provides a nonparametric estimate of the inverse cumulative distribution function.

By default, this is done by interpolating between adjacent elements in y, a sorted copy of a:

(1-g)*y[j] + g*y[j+1]

where the index j and coefficient g are the integral and fractional components of q * (n-1), and n is the number of elements in the sample.

This is a special case of Equation 1 of H&F [1]. More generally,

  • j = (q*n + m - 1) // 1, and

  • g = (q*n + m - 1) % 1,

where m may be defined according to several different conventions. The preferred convention may be selected using the method parameter:

method

number in H&F

m

interpolated_inverted_cdf

4

0

hazen

5

1/2

weibull

6

q

linear (default)

7

1 - q

median_unbiased

8

q/3 + 1/3

normal_unbiased

9

q/4 + 3/8

Note that indices j and j + 1 are clipped to the range 0 to n - 1 when the results of the formula would be outside the allowed range of non-negative indices. The - 1 in the formulas for j and g accounts for Python’s 0-based indexing.

The table above includes only the estimators from H&F that are continuous functions of probability q (estimators 4-9). NumPy also provides the three discontinuous estimators from H&F (estimators 1-3), where j is defined as above and m and g are defined as follows.

  1. inverted_cdf: m = 0 and g = int(q*n > 0)

  2. averaged_inverted_cdf: m = 0 and g = (1 + int(q*n > 0)) / 2

  3. closest_observation: m = -1/2 and 1 - int((g == 0) & (j%2 == 0))

For backward compatibility with previous versions of NumPy, quantile provides four additional discontinuous estimators. Like method='linear', all have m = 1 - q so that j = q*(n-1) // 1, but g is defined as follows.

  • lower: g = 0

  • midpoint: g = 0.5

  • higher: g = 1

  • nearest: g = (q*(n-1) % 1) > 0.5

Weighted quantiles: More formally, the quantile at probability 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

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

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

Some of the estimators above arise 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 coverage conditions. Methods that follow this approach are inverted_cdf and averaged_inverted_cdf.

For weighted quantiles, the 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)

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.quantile(a, 0.5)
3.5
>>> np.quantile(a, 0.5, axis=0)
array([6.5, 4.5, 2.5])
>>> np.quantile(a, 0.5, axis=1)
array([7.,  2.])
>>> np.quantile(a, 0.5, axis=1, keepdims=True)
array([[7.],
       [2.]])
>>> m = np.quantile(a, 0.5, axis=0)
>>> out = np.zeros_like(m)
>>> np.quantile(a, 0.5, axis=0, out=out)
array([6.5, 4.5, 2.5])
>>> m
array([6.5, 4.5, 2.5])
>>> b = a.copy()
>>> np.quantile(b, 0.5, axis=1, overwrite_input=True)
array([7.,  2.])
>>> assert not np.all(a == b)

See also numpy.percentile for a visualization of most methods.

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