numpy.quantile — NumPy v2.0 Manual (2024)

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 for 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 isto compute the quantile(s) along a flattened version of the array.

outndarray, optional

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

overwrite_inputbool, optional

If True, then allow the input array a to be modified byintermediate calculations, to save memory. In this case, thecontents of the input a after this function completes isundefined.

methodstr, optional

This parameter specifies the method to use for estimating thequantile. There are many different methods, some unique to NumPy.See the notes for explanation. The options sorted by their R typeas 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 thefollowing discontinuous variations of the default ‘linear’ (7.) option:

• ‘lower’

• ‘higher’,

• ‘midpoint’

• ‘nearest’

  See Also

  numpy.quantile — NumPy v2.0 Manualnumpy.quantile() [de]Numpy Quantile() Explained With Examplesnumpy.quantile — NumPy v1.23 Manual

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

keepdimsbool, optional

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

weightsarray_like, optional

An array of weights associated with the values in a. Each value ina contributes to the quantile according to its associated weight.The weights array can either be 1-D (in which case its length must bethe 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 aweight 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 resultis a scalar. If multiple probability levels are given, first axisof the result corresponds to the quantiles. The other axes arethe axes that remain after the reduction of a. If the inputcontains integers or floats smaller than float64, the outputdata-type is float64. Otherwise, the output data-type is thesame as that of the input. If out is specified, that array isreturned instead.

Notes

In general, the quantile at probability level \(q\) of a cumulativedistribution function \(F(y)=P(Y \leq y)\) with probability measure\(P\) is defined as any number \(x\) that fulfills thecoverage 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 nonparametricestimation of the underlying population counterparts, represented by theunknown \(F\), given a data vector a of length n.

One type of estimators arises when one considers \(F\) as the empiricaldistribution 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 \(x\) thatfulfill the above inequalities. Methods that follow this approach areinverted_cdf and averaged_inverted_cdf.

A more general way to define sample quantile estimators is as follows.The empirical q-quantile of a is the n * q-th value of theway from the minimum to the maximum in a sorted copy of a. The valuesand distances of the two nearest neighbors as well as the methodparameter will determine the quantile if the normalized ranking does notmatch the location of n * q exactly. This function is the same asthe median if q=0.5, the same as the minimum if q=0.0 and the sameas the maximum if q=1.0.

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

The following formula determines the virtual index i + g, the locationof the quantile in the sorted sample:

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

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 sampledistribution 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 sampledistribution 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.

Weighted quantiles:For weighted quantiles, the above coverage conditions still hold. Theempirical cumulative distribution is simply replaced by its weightedversion, 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

Examples

>>> a = np.array([[10, 7, 4], [3, 2, 1]])>>> aarray([[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])>>> marray([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.