# Source code for astropy.stats.jackknife

```# Licensed under a 3-clause BSD style license - see LICENSE.rst

import numpy as np

__all__ = ["jackknife_resampling", "jackknife_stats"]
__doctest_requires__ = {"jackknife_stats": ["scipy"]}

[docs]
def jackknife_resampling(data):
"""Performs jackknife resampling on numpy arrays.

Jackknife resampling is a technique to generate 'n' deterministic samples
of size 'n-1' from a measured sample of size 'n'. Basically, the i-th
sample, (1<=i<=n), is generated by means of removing the i-th measurement
of the original sample. Like the bootstrap resampling, this statistical
technique finds applications in estimating variance, bias, and confidence
intervals.

Parameters
----------
data : ndarray
Original sample (1-D array) from which the jackknife resamples will be
generated.

Returns
-------
resamples : ndarray
The i-th row is the i-th jackknife sample, i.e., the original sample
with the i-th measurement deleted.

References
----------
.. [1] McIntosh, Avery. "The Jackknife Estimation Method".
<https://arxiv.org/abs/1606.00497>

.. [2] Efron, Bradley. "The Jackknife, the Bootstrap, and other
Resampling Plans". Technical Report No. 63, Division of Biostatistics,
Stanford University, December, 1980.

.. [3] Jackknife resampling <https://en.wikipedia.org/wiki/Jackknife_resampling>
"""
n = data.shape[0]
if n <= 0:
raise ValueError("data must contain at least one measurement.")

resamples = np.empty([n, n - 1])

for i in range(n):
resamples[i] = np.delete(data, i)

return resamples

[docs]
def jackknife_stats(data, statistic, confidence_level=0.95):
"""Performs jackknife estimation on the basis of jackknife resamples.

This function requires `SciPy <https://www.scipy.org/>`_ to be installed.

Parameters
----------
data : ndarray
Original sample (1-D array).
statistic : function
Any function (or vector of functions) on the basis of the measured
data, e.g, sample mean, sample variance, etc. The jackknife estimate of
this statistic will be returned.
confidence_level : float, optional
Confidence level for the confidence interval of the Jackknife estimate.
Must be a real-valued number in (0,1). Default value is 0.95.

Returns
-------
estimate : float or `~numpy.ndarray`
The i-th element is the bias-corrected "jackknifed" estimate.

bias : float or `~numpy.ndarray`
The i-th element is the jackknife bias.

std_err : float or `~numpy.ndarray`
The i-th element is the jackknife standard error.

conf_interval : ndarray
If ``statistic`` is single-valued, the first and second elements are
the lower and upper bounds, respectively. If ``statistic`` is
vector-valued, each column corresponds to the confidence interval for
each component of ``statistic``. The first and second rows contain the
lower and upper bounds, respectively.

Examples
--------
1. Obtain Jackknife resamples:

>>> import numpy as np
>>> from astropy.stats import jackknife_resampling
>>> from astropy.stats import jackknife_stats
>>> data = np.array([1,2,3,4,5,6,7,8,9,0])
>>> resamples = jackknife_resampling(data)
>>> resamples
array([[2., 3., 4., 5., 6., 7., 8., 9., 0.],
[1., 3., 4., 5., 6., 7., 8., 9., 0.],
[1., 2., 4., 5., 6., 7., 8., 9., 0.],
[1., 2., 3., 5., 6., 7., 8., 9., 0.],
[1., 2., 3., 4., 6., 7., 8., 9., 0.],
[1., 2., 3., 4., 5., 7., 8., 9., 0.],
[1., 2., 3., 4., 5., 6., 8., 9., 0.],
[1., 2., 3., 4., 5., 6., 7., 9., 0.],
[1., 2., 3., 4., 5., 6., 7., 8., 0.],
[1., 2., 3., 4., 5., 6., 7., 8., 9.]])
>>> resamples.shape
(10, 9)

2. Obtain Jackknife estimate for the mean, its bias, its standard error,
and its 95% confidence interval:

>>> test_statistic = np.mean
>>> estimate, bias, stderr, conf_interval = jackknife_stats(
...     data, test_statistic, 0.95)
>>> estimate
4.5
>>> bias
0.0
>>> stderr  # doctest: +FLOAT_CMP
0.95742710775633832
>>> conf_interval
array([2.62347735,  6.37652265])

3. Example for two estimates

>>> test_statistic = lambda x: (np.mean(x), np.var(x))
>>> estimate, bias, stderr, conf_interval = jackknife_stats(
...     data, test_statistic, 0.95)
>>> estimate
array([4.5       ,  9.16666667])
>>> bias
array([ 0.        , -0.91666667])
>>> stderr
array([0.95742711,  2.69124476])
>>> conf_interval
array([[ 2.62347735,   3.89192387],
[ 6.37652265,  14.44140947]])

IMPORTANT: Note that confidence intervals are given as columns
"""
# jackknife confidence interval
if not (0 < confidence_level < 1):
raise ValueError("confidence level must be in (0, 1).")

# make sure original data is proper
n = data.shape[0]
if n <= 0:
raise ValueError("data must contain at least one measurement.")

# Only import scipy if inputs are valid
from scipy.special import erfinv

resamples = jackknife_resampling(data)

stat_data = statistic(data)
jack_stat = np.apply_along_axis(statistic, 1, resamples)
mean_jack_stat = np.mean(jack_stat, axis=0)

# jackknife bias
bias = (n - 1) * (mean_jack_stat - stat_data)

# jackknife standard error
std_err = np.sqrt(
(n - 1)
* np.mean((jack_stat - mean_jack_stat) * (jack_stat - mean_jack_stat), axis=0)
)

# bias-corrected "jackknifed estimate"
estimate = stat_data - bias

z_score = np.sqrt(2.0) * erfinv(confidence_level)
conf_interval = estimate + z_score * np.array((-std_err, std_err))

return estimate, bias, std_err, conf_interval

```