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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 :math:`n` times, the confidence interval returned by `bootstrap` will contain the true value of the statistic for those distributions approximately `confidence_level`:math:`\, \times \, n` times. Parameters ---------- data : sequence of array-like Each element of data is a sample from an underlying distribution. statistic : callable 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`. vectorized : bool, 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. paired : bool, default: ``False`` Whether the statistic treats corresponding elements of the samples in `data` as paired. axis : int, default: ``0`` The axis of the samples in `data` along which the `statistic` is calculated. confidence_level : float, default: ``0.95`` The confidence level of the confidence interval. n_resamples : int, default: ``9999`` The number of resamples performed to form the bootstrap distribution of the statistic. batch : int, 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`, `numpy.random.RandomState`}, optional Pseudorandom number generator state used to generate resamples. If `random_state` is ``None`` (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Returns ------- res : BootstrapResult An object with attributes: confidence_interval : ConfidenceInterval The bootstrap confidence interval as an instance of `collections.namedtuple` with attributes `low` and `high`. standard_error : float or ndarray The bootstrap standard error, that is, the sample standard deviation of the bootstrap distribution 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 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) Nr)r7r8rrrrrHrN)rrCrBr.cSstj||ddS)Nr)r;qr)rrA)r;rarrrpercentile_funsz!bootstrap..percentile_fundrO)Zddofr)r]r^) r\r)r*r9rIrrJrLrEr_rZstd)r!r#rVrWrrXr7r.rYr8r rBr0r1r2Zresampled_datarZresamplerCintervalrbZci_lZci_urKrrr bootstrapsF g     rer)NN)r>ZnumpyrZscipy._lib._utilrZ scipy.specialrrrZ dataclassesrZ_commonrZ_axis_nan_policyr RuntimeWarningr r'r6r9r=rErLr\Zfieldsr_rerrrrs,         P