scipy.stats.bootstrap

scipy.stats.bootstrap(data, statistic, *, vectorized=True, paired=False, axis=0, confidence_level=0.95, n_resamples=9999, batch=None, method='BCa', random_state=None)[source]

Compute a two-sided bootstrap confidence interval of a statistic.

When method is 'percentile', a bootstrap confidence interval is computed according to the following procedure.

  1. Resample the data: for each sample in data and for each of n_resamples, take a random sample of the original sample (with replacement) of the same size as the original sample.

  2. Compute the bootstrap distribution of the statistic: for each set of resamples, compute the test statistic.

  3. Determine the confidence interval: find the interval of the bootstrap distribution that is

    • symmetric about the median and

    • contains confidence_level of the resampled statistic values.

While the 'percentile' method is the most intuitive, it is rarely used in practice. Two more common methods are available, 'basic' (‘reverse percentile’) and 'BCa' (‘bias-corrected and accelerated’); they differ in how step 3 is performed.

If the samples in data are taken at random from their respective distributions \(n\) times, the confidence interval returned by bootstrap will contain the true value of the statistic for those distributions approximately confidence_level\(\, \times \, n\) times.

Parameters
datasequence of array-like

Each element of data is a sample from an underlying distribution.

statisticcallable

Statistic for which the confidence interval is to be calculated. statistic must be a callable that accepts len(data) samples as separate arguments and returns the resulting statistic. If vectorized is set True, statistic must also accept a keyword argument axis and be vectorized to compute the statistic along the provided axis.

vectorizedbool, default: True

If vectorized is set False, statistic will not be passed keyword argument axis, and is assumed to calculate the statistic only for 1D samples.

pairedbool, default: False

Whether the statistic treats corresponding elements of the samples in data as paired.

axisint, default: 0

The axis of the samples in data along which the statistic is calculated.

confidence_levelfloat, default: 0.95

The confidence level of the confidence interval.

n_resamplesint, default: 9999

The number of resamples performed to form the bootstrap distribution of the statistic.

batchint, optional

The number of resamples to process in each vectorized call to statistic. Memory usage is O(batch`*``n`), where n is the sample size. Default is None, in which case batch = n_resamples (or batch = max(n_resamples, n) for method='BCa').

method{‘percentile’, ‘basic’, ‘bca’}, default: 'BCa'

Whether to return the ‘percentile’ bootstrap confidence interval ('percentile'), the ‘reverse’ or the bias-corrected and accelerated bootstrap confidence interval ('BCa'). Note that only 'percentile' and 'basic' support multi-sample statistics at this time.

random_state{None, int, numpy.random.Generator,

If seed is None (or np.random), the numpy.random.RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a Generator or RandomState instance then that instance is used.

Pseudorandom number generator state used to generate resamples.

Returns
resBootstrapResult

An object with attributes:

confidence_intervalConfidenceInterval

The bootstrap confidence interval as an instance of collections.namedtuple with attributes low and high.

standard_errorfloat or ndarray

The bootstrap standard error, that is, the sample standard deviation of the bootstrap distribution

References

1

B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall/CRC, Boca Raton, FL, USA (1993)

2

Nathaniel E. Helwig, “Bootstrap Confidence Intervals”, http://users.stat.umn.edu/~helwig/notes/bootci-Notes.pdf

3

Bootstrapping (statistics), Wikipedia, https://en.wikipedia.org/wiki/Bootstrapping_(statistics)

Examples

Suppose we have sampled data from an unknown distribution.

>>> import numpy as np
>>> rng = np.random.default_rng()
>>> from scipy.stats import norm
>>> dist = norm(loc=2, scale=4)  # our "unknown" distribution
>>> data = dist.rvs(size=100, random_state=rng)

We are interested int the standard deviation of the distribution.

>>> std_true = dist.std()      # the true value of the statistic
>>> print(std_true)
4.0
>>> std_sample = np.std(data)  # the sample statistic
>>> print(std_sample)
3.9460644295563863

We can calculate a 90% confidence interval of the statistic using bootstrap.

>>> from scipy.stats import bootstrap
>>> data = (data,)  # samples must be in a sequence
>>> res = bootstrap(data, np.std, confidence_level=0.9,
...                 random_state=rng)
>>> print(res.confidence_interval)
ConfidenceInterval(low=3.57655333533867, high=4.382043696342881)

If we sample from the distribution 1000 times and form a bootstrap confidence interval for each sample, the confidence interval contains the true value of the statistic approximately 900 times.

>>> n_trials = 1000
>>> ci_contains_true_std = 0
>>> for i in range(n_trials):
...    data = (dist.rvs(size=100, random_state=rng),)
...    ci = bootstrap(data, np.std, confidence_level=0.9, n_resamples=1000,
...                   random_state=rng).confidence_interval
...    if ci[0] < std_true < ci[1]:
...        ci_contains_true_std += 1
>>> print(ci_contains_true_std)
875

Rather than writing a loop, we can also determine the confidence intervals for all 1000 samples at once.

>>> data = (dist.rvs(size=(n_trials, 100), random_state=rng),)
>>> res = bootstrap(data, np.std, axis=-1, confidence_level=0.9,
...                 n_resamples=1000, random_state=rng)
>>> ci_l, ci_u = res.confidence_interval

Here, ci_l and ci_u contain the confidence interval for each of the n_trials = 1000 samples.

>>> print(ci_l[995:])
[3.77729695 3.75090233 3.45829131 3.34078217 3.48072829]
>>> print(ci_u[995:])
[4.88316666 4.86924034 4.32032996 4.2822427  4.59360598]

And again, approximately 90% contain the true value, std_true = 4.

>>> print(np.sum((ci_l < std_true) & (std_true < ci_u)))
900

bootstrap can also be used to estimate confidence intervals of multi-sample statistics, including those calculated by hypothesis tests. scipy.stats.mood perform’s Mood’s test for equal scale parameters, and it returns two outputs: a statistic, and a p-value. To get a confidence interval for the test statistic, we first wrap scipy.stats.mood in a function that accepts two sample arguments, accepts an axis keyword argument, and returns only the statistic.

>>> from scipy.stats import mood
>>> def my_statistic(sample1, sample2, axis):
...     statistic, _ = mood(sample1, sample2, axis=-1)
...     return statistic

Here, we use the ‘percentile’ method with the default 95% confidence level.

>>> sample1 = norm.rvs(scale=1, size=100, random_state=rng)
>>> sample2 = norm.rvs(scale=2, size=100, random_state=rng)
>>> data = (sample1, sample2)
>>> res = bootstrap(data, my_statistic, method='basic', random_state=rng)
>>> print(mood(sample1, sample2)[0])  # element 0 is the statistic
-5.521109549096542
>>> print(res.confidence_interval)
ConfidenceInterval(low=-7.255994487314675, high=-4.016202624747605)

The bootstrap estimate of the standard error is also available.

>>> print(res.standard_error)
0.8344963846318795

Paired-sample statistics work, too. For example, consider the Pearson correlation coefficient.

>>> from scipy.stats import pearsonr
>>> n = 100
>>> x = np.linspace(0, 10, n)
>>> y = x + rng.uniform(size=n)
>>> print(pearsonr(x, y)[0])  # element 0 is the statistic
0.9962357936065914

We wrap pearsonr so that it returns only the statistic.

>>> def my_statistic(x, y):
...     return pearsonr(x, y)[0]

We call bootstrap using paired=True. Also, since my_statistic isn’t vectorized to calculate the statistic along a given axis, we pass in vectorized=False.

>>> res = bootstrap((x, y), my_statistic, vectorized=False, paired=True,
...                 random_state=rng)
>>> print(res.confidence_interval)
ConfidenceInterval(low=0.9950085825848624, high=0.9971212407917498)
scipy.stats.median_abs_deviation scipy.stats.cumfreq