scipy.stats.bootstrap#
- scipy.stats.bootstrap(data, statistic, *, n_resamples=9999, batch=None, vectorized=True, paired=False, axis=0, confidence_level=0.95, 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.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.
Compute the bootstrap distribution of the statistic: for each set of resamples, compute the test statistic.
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 setTrue
, statistic must also accept a keyword argument axis and be vectorized to compute the statistic along the provided axis.- 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 isNone
, in which casebatch = n_resamples
(orbatch = max(n_resamples, n)
formethod='BCa'
).- 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.
- 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
, numpy.random.RandomState
}, optionalPseudorandom number generator state used to generate resamples.
If random_state is
None
(or np.random), thenumpy.random.RandomState
singleton is used. If random_state is an int, a newRandomState
instance is used, seeded with random_state. If random_state is already aGenerator
orRandomState
instance then that instance is used.
- 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
- Warns
DegenerateDataWarning
Generated when
method='BCa'
and the bootstrap distribution is degenerate (e.g. all elements are identical).
Notes
Elements of the confidence interval may be NaN for
method='BCa'
if the bootstrap distribution is degenerate (e.g. all elements are identical). In this case, consider using another method or inspecting data for indications that other analysis may be more appropriate (e.g. all observations are identical).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_%28statistics%29
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 wrapscipy.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
usingpaired=True
. Also, sincemy_statistic
isn’t vectorized to calculate the statistic along a given axis, we pass invectorized=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)