numpy:quantile

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, m is defined as follows, and g is a function of the real-valued index = q*n + m - 1 and j.

• inverted_cdf: m = 0 and g = int(index - j > 0)

• averaged_inverted_cdf: m = 0 and g = (1 + int(index - j > 0)) / 2

• closest_observation: m = -1/2 and g = 1 - int((index == j) & (j%2 == 1))

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 $F(y)=P(Y\le y)$ with probability measure $P$ is defined as any number $x$ that fulfills the coverage conditions

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. $F(y)=\frac{1}{n}{\sum }_i{1}_{a_i\le y}$. 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. $P(Y\le t)=\frac{1}{{\sum }_i{w}_i}{\sum }_i{w}_i{1}_{x_i\le t}$. Only method="inverted_cdf" supports weights.