Source code for astropy.stats.funcs

# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
This module contains simple statistical algorithms that are
straightforwardly implemented as a single python function (or family of
functions).

This module should generally not be used directly.  Everything in
__all__ is imported into astropy.stats, and hence that package
should be used for access.
"""

import math
import itertools

import numpy as np

from warnings import warn

from astropy.utils.decorators import deprecated_renamed_argument
from astropy.utils import isiterable
from . import _stats

__all__ = ['gaussian_fwhm_to_sigma', 'gaussian_sigma_to_fwhm',
'binom_conf_interval', 'binned_binom_proportion',
'signal_to_noise_oir_ccd', 'bootstrap', 'kuiper', 'kuiper_two',
'kuiper_false_positive_probability', 'cdf_from_intervals',
'interval_overlap_length', 'histogram_intervals', 'fold_intervals']

__doctest_skip__ = ['binned_binom_proportion']
__doctest_requires__ = {'binom_conf_interval': ['scipy'],
'poisson_conf_interval': ['scipy']}

gaussian_sigma_to_fwhm = 2.0 * np.sqrt(2.0 * np.log(2.0))
"""
Factor with which to multiply Gaussian 1-sigma standard deviation to
convert it to full width at half maximum (FWHM).
"""

gaussian_fwhm_to_sigma = 1. / gaussian_sigma_to_fwhm
"""
Factor with which to multiply Gaussian full width at half maximum (FWHM)
to convert it to 1-sigma standard deviation.
"""

# TODO Note scipy dependency
[docs]def binom_conf_interval(k, n, conf=0.68269, interval='wilson'): r"""Binomial proportion confidence interval given k successes, n trials. Parameters ---------- k : int or numpy.ndarray Number of successes (0 <= k <= n). n : int or numpy.ndarray Number of trials (n > 0). If both k and n are arrays, they must have the same shape. conf : float in [0, 1], optional Desired probability content of interval. Default is 0.68269, corresponding to 1 sigma in a 1-dimensional Gaussian distribution. interval : {'wilson', 'jeffreys', 'flat', 'wald'}, optional Formula used for confidence interval. See notes for details. The 'wilson' and 'jeffreys' intervals generally give similar results, while 'flat' is somewhat different, especially for small values of n. 'wilson' should be somewhat faster than 'flat' or 'jeffreys'. The 'wald' interval is generally not recommended. It is provided for comparison purposes. Default is 'wilson'. Returns ------- conf_interval : numpy.ndarray conf_interval and conf_interval correspond to the lower and upper limits, respectively, for each element in k, n. Notes ----- In situations where a probability of success is not known, it can be estimated from a number of trials (N) and number of observed successes (k). For example, this is done in Monte Carlo experiments designed to estimate a detection efficiency. It is simple to take the sample proportion of successes (k/N) as a reasonable best estimate of the true probability :math:\epsilon. However, deriving an accurate confidence interval on :math:\epsilon is non-trivial. There are several formulas for this interval (see _). Four intervals are implemented here: **1. The Wilson Interval.** This interval, attributed to Wilson _, is given by .. math:: CI_{\rm Wilson} = \frac{k + \kappa^2/2}{N + \kappa^2} \pm \frac{\kappa n^{1/2}}{n + \kappa^2} ((\hat{\epsilon}(1 - \hat{\epsilon}) + \kappa^2/(4n))^{1/2} where :math:\hat{\epsilon} = k / N and :math:\kappa is the number of standard deviations corresponding to the desired confidence interval for a *normal* distribution (for example, 1.0 for a confidence interval of 68.269%). For a confidence interval of 100(1 - :math:\alpha)%, .. math:: \kappa = \Phi^{-1}(1-\alpha/2) = \sqrt{2}{\rm erf}^{-1}(1-\alpha). **2. The Jeffreys Interval.** This interval is derived by applying Bayes' theorem to the binomial distribution with the noninformative Jeffreys prior _, _. The noninformative Jeffreys prior is the Beta distribution, Beta(1/2, 1/2), which has the density function .. math:: f(\epsilon) = \pi^{-1} \epsilon^{-1/2}(1-\epsilon)^{-1/2}. The justification for this prior is that it is invariant under reparameterizations of the binomial proportion. The posterior density function is also a Beta distribution: Beta(k + 1/2, N - k + 1/2). The interval is then chosen so that it is *equal-tailed*: Each tail (outside the interval) contains :math:\alpha/2 of the posterior probability, and the interval itself contains 1 - :math:\alpha. This interval must be calculated numerically. Additionally, when k = 0 the lower limit is set to 0 and when k = N the upper limit is set to 1, so that in these cases, there is only one tail containing :math:\alpha/2 and the interval itself contains 1 - :math:\alpha/2 rather than the nominal 1 - :math:\alpha. **3. A Flat prior.** This is similar to the Jeffreys interval, but uses a flat (uniform) prior on the binomial proportion over the range 0 to 1 rather than the reparametrization-invariant Jeffreys prior. The posterior density function is a Beta distribution: Beta(k + 1, N - k + 1). The same comments about the nature of the interval (equal-tailed, etc.) also apply to this option. **4. The Wald Interval.** This interval is given by .. math:: CI_{\rm Wald} = \hat{\epsilon} \pm \kappa \sqrt{\frac{\hat{\epsilon}(1-\hat{\epsilon})}{N}} The Wald interval gives acceptable results in some limiting cases. Particularly, when N is very large, and the true proportion :math:\epsilon is not "too close" to 0 or 1. However, as the later is not verifiable when trying to estimate :math:\epsilon, this is not very helpful. Its use is not recommended, but it is provided here for comparison purposes due to its prevalence in everyday practical statistics. References ---------- ..  Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001). "Interval Estimation for a Binomial Proportion". Statistical Science 16 (2): 101-133. doi:10.1214/ss/1009213286 ..  Wilson, E. B. (1927). "Probable inference, the law of succession, and statistical inference". Journal of the American Statistical Association 22: 209-212. ..  Jeffreys, Harold (1946). "An Invariant Form for the Prior Probability in Estimation Problems". Proc. R. Soc. Lond.. A 24 186 (1007): 453-461. doi:10.1098/rspa.1946.0056 ..  Jeffreys, Harold (1998). Theory of Probability. Oxford University Press, 3rd edition. ISBN 978-0198503682 Examples -------- Integer inputs return an array with shape (2,): >>> binom_conf_interval(4, 5, interval='wilson') array([0.57921724, 0.92078259]) Arrays of arbitrary dimension are supported. The Wilson and Jeffreys intervals give similar results, even for small k, N: >>> binom_conf_interval([0, 1, 2, 5], 5, interval='wilson') array([[0. , 0.07921741, 0.21597328, 0.83333304], [0.16666696, 0.42078276, 0.61736012, 1. ]]) >>> binom_conf_interval([0, 1, 2, 5], 5, interval='jeffreys') array([[0. , 0.0842525 , 0.21789949, 0.82788246], [0.17211754, 0.42218001, 0.61753691, 1. ]]) >>> binom_conf_interval([0, 1, 2, 5], 5, interval='flat') array([[0. , 0.12139799, 0.24309021, 0.73577037], [0.26422963, 0.45401727, 0.61535699, 1. ]]) In contrast, the Wald interval gives poor results for small k, N. For k = 0 or k = N, the interval always has zero length. >>> binom_conf_interval([0, 1, 2, 5], 5, interval='wald') array([[0. , 0.02111437, 0.18091075, 1. ], [0. , 0.37888563, 0.61908925, 1. ]]) For confidence intervals approaching 1, the Wald interval for 0 < k < N can give intervals that extend outside [0, 1]: >>> binom_conf_interval([0, 1, 2, 5], 5, interval='wald', conf=0.99) array([[ 0. , -0.26077835, -0.16433593, 1. ], [ 0. , 0.66077835, 0.96433593, 1. ]]) """ if conf < 0. or conf > 1.: raise ValueError('conf must be between 0. and 1.') alpha = 1. - conf k = np.asarray(k).astype(int) n = np.asarray(n).astype(int) if (n <= 0).any(): raise ValueError('n must be positive') if (k < 0).any() or (k > n).any(): raise ValueError('k must be in {0, 1, .., n}') if interval == 'wilson' or interval == 'wald': from scipy.special import erfinv kappa = np.sqrt(2.) * min(erfinv(conf), 1.e10) # Avoid overflows. k = k.astype(float) n = n.astype(float) p = k / n if interval == 'wilson': midpoint = (k + kappa ** 2 / 2.) / (n + kappa ** 2) halflength = (kappa * np.sqrt(n)) / (n + kappa ** 2) * \ np.sqrt(p * (1 - p) + kappa ** 2 / (4 * n)) conf_interval = np.array([midpoint - halflength, midpoint + halflength]) # Correct intervals out of range due to floating point errors. conf_interval[conf_interval < 0.] = 0. conf_interval[conf_interval > 1.] = 1. else: midpoint = p halflength = kappa * np.sqrt(p * (1. - p) / n) conf_interval = np.array([midpoint - halflength, midpoint + halflength]) elif interval == 'jeffreys' or interval == 'flat': from scipy.special import betaincinv if interval == 'jeffreys': lowerbound = betaincinv(k + 0.5, n - k + 0.5, 0.5 * alpha) upperbound = betaincinv(k + 0.5, n - k + 0.5, 1. - 0.5 * alpha) else: lowerbound = betaincinv(k + 1, n - k + 1, 0.5 * alpha) upperbound = betaincinv(k + 1, n - k + 1, 1. - 0.5 * alpha) # Set lower or upper bound to k/n when k/n = 0 or 1 # We have to treat the special case of k/n being scalars, # which is an ugly kludge if lowerbound.ndim == 0: if k == 0: lowerbound = 0. elif k == n: upperbound = 1. else: lowerbound[k == 0] = 0 upperbound[k == n] = 1 conf_interval = np.array([lowerbound, upperbound]) else: raise ValueError('Unrecognized interval: {0:s}'.format(interval)) return conf_interval
# TODO Note scipy dependency (needed in binom_conf_interval)
[docs]def binned_binom_proportion(x, success, bins=10, range=None, conf=0.68269, interval='wilson'): """Binomial proportion and confidence interval in bins of a continuous variable x. Given a set of datapoint pairs where the x values are continuously distributed and the success values are binomial ("success / failure" or "true / false"), place the pairs into bins according to x value and calculate the binomial proportion (fraction of successes) and confidence interval in each bin. Parameters ---------- x : list_like Values. success : list_like (bool) Success (True) or failure (False) corresponding to each value in x. Must be same length as x. bins : int or sequence of scalars, optional If bins is an int, it defines the number of equal-width bins in the given range (10, by default). If bins is a sequence, it defines the bin edges, including the rightmost edge, allowing for non-uniform bin widths (in this case, 'range' is ignored). range : (float, float), optional The lower and upper range of the bins. If None (default), the range is set to (x.min(), x.max()). Values outside the range are ignored. conf : float in [0, 1], optional Desired probability content in the confidence interval (p - perr, p + perr) in each bin. Default is 0.68269. interval : {'wilson', 'jeffreys', 'flat', 'wald'}, optional Formula used to calculate confidence interval on the binomial proportion in each bin. See binom_conf_interval for definition of the intervals. The 'wilson', 'jeffreys', and 'flat' intervals generally give similar results. 'wilson' should be somewhat faster, while 'jeffreys' and 'flat' are marginally superior, but differ in the assumed prior. The 'wald' interval is generally not recommended. It is provided for comparison purposes. Default is 'wilson'. Returns ------- bin_ctr : numpy.ndarray Central value of bins. Bins without any entries are not returned. bin_halfwidth : numpy.ndarray Half-width of each bin such that bin_ctr - bin_halfwidth and bin_ctr + bins_halfwidth give the left and right side of each bin, respectively. p : numpy.ndarray Efficiency in each bin. perr : numpy.ndarray 2-d array of shape (2, len(p)) representing the upper and lower uncertainty on p in each bin. See Also -------- binom_conf_interval : Function used to estimate confidence interval in each bin. Examples -------- Suppose we wish to estimate the efficiency of a survey in detecting astronomical sources as a function of magnitude (i.e., the probability of detecting a source given its magnitude). In a realistic case, we might prepare a large number of sources with randomly selected magnitudes, inject them into simulated images, and then record which were detected at the end of the reduction pipeline. As a toy example, we generate 100 data points with randomly selected magnitudes between 20 and 30 and "observe" them with a known detection function (here, the error function, with 50% detection probability at magnitude 25): >>> from scipy.special import erf >>> from scipy.stats.distributions import binom >>> def true_efficiency(x): ... return 0.5 - 0.5 * erf((x - 25.) / 2.) >>> mag = 20. + 10. * np.random.rand(100) >>> detected = binom.rvs(1, true_efficiency(mag)) >>> bins, binshw, p, perr = binned_binom_proportion(mag, detected, bins=20) >>> plt.errorbar(bins, p, xerr=binshw, yerr=perr, ls='none', marker='o', ... label='estimate') .. plot:: import numpy as np from scipy.special import erf from scipy.stats.distributions import binom import matplotlib.pyplot as plt from astropy.stats import binned_binom_proportion def true_efficiency(x): return 0.5 - 0.5 * erf((x - 25.) / 2.) np.random.seed(400) mag = 20. + 10. * np.random.rand(100) np.random.seed(600) detected = binom.rvs(1, true_efficiency(mag)) bins, binshw, p, perr = binned_binom_proportion(mag, detected, bins=20) plt.errorbar(bins, p, xerr=binshw, yerr=perr, ls='none', marker='o', label='estimate') X = np.linspace(20., 30., 1000) plt.plot(X, true_efficiency(X), label='true efficiency') plt.ylim(0., 1.) plt.title('Detection efficiency vs magnitude') plt.xlabel('Magnitude') plt.ylabel('Detection efficiency') plt.legend() plt.show() The above example uses the Wilson confidence interval to calculate the uncertainty perr in each bin (see the definition of various confidence intervals in binom_conf_interval). A commonly used alternative is the Wald interval. However, the Wald interval can give nonsensical uncertainties when the efficiency is near 0 or 1, and is therefore **not** recommended. As an illustration, the following example shows the same data as above but uses the Wald interval rather than the Wilson interval to calculate perr: >>> bins, binshw, p, perr = binned_binom_proportion(mag, detected, bins=20, ... interval='wald') >>> plt.errorbar(bins, p, xerr=binshw, yerr=perr, ls='none', marker='o', ... label='estimate') .. plot:: import numpy as np from scipy.special import erf from scipy.stats.distributions import binom import matplotlib.pyplot as plt from astropy.stats import binned_binom_proportion def true_efficiency(x): return 0.5 - 0.5 * erf((x - 25.) / 2.) np.random.seed(400) mag = 20. + 10. * np.random.rand(100) np.random.seed(600) detected = binom.rvs(1, true_efficiency(mag)) bins, binshw, p, perr = binned_binom_proportion(mag, detected, bins=20, interval='wald') plt.errorbar(bins, p, xerr=binshw, yerr=perr, ls='none', marker='o', label='estimate') X = np.linspace(20., 30., 1000) plt.plot(X, true_efficiency(X), label='true efficiency') plt.ylim(0., 1.) plt.title('The Wald interval can give nonsensical uncertainties') plt.xlabel('Magnitude') plt.ylabel('Detection efficiency') plt.legend() plt.show() """ x = np.ravel(x) success = np.ravel(success).astype(bool) if x.shape != success.shape: raise ValueError('sizes of x and success must match') # Put values into a histogram (n). Put "successful" values # into a second histogram (k) with identical binning. n, bin_edges = np.histogram(x, bins=bins, range=range) k, bin_edges = np.histogram(x[success], bins=bin_edges) bin_ctr = (bin_edges[:-1] + bin_edges[1:]) / 2. bin_halfwidth = bin_ctr - bin_edges[:-1] # Remove bins with zero entries. valid = n > 0 bin_ctr = bin_ctr[valid] bin_halfwidth = bin_halfwidth[valid] n = n[valid] k = k[valid] p = k / n bounds = binom_conf_interval(k, n, conf=conf, interval=interval) perr = np.abs(bounds - p) return bin_ctr, bin_halfwidth, p, perr
def _check_poisson_conf_inputs(sigma, background, conflevel, name): if sigma != 1: raise ValueError("Only sigma=1 supported for interval {0}" .format(name)) if background != 0: raise ValueError("background not supported for interval {0}" .format(name)) if conflevel is not None: raise ValueError("conflevel not supported for interval {0}" .format(name))
[docs]def poisson_conf_interval(n, interval='root-n', sigma=1, background=0, conflevel=None): r"""Poisson parameter confidence interval given observed counts Parameters ---------- n : int or numpy.ndarray Number of counts (0 <= n). interval : {'root-n','root-n-0','pearson','sherpagehrels','frequentist-confidence', 'kraft-burrows-nousek'}, optional Formula used for confidence interval. See notes for details. Default is 'root-n'. sigma : float, optional Number of sigma for confidence interval; only supported for the 'frequentist-confidence' mode. background : float, optional Number of counts expected from the background; only supported for the 'kraft-burrows-nousek' mode. This number is assumed to be determined from a large region so that the uncertainty on its value is negligible. conflevel : float, optional Confidence level between 0 and 1; only supported for the 'kraft-burrows-nousek' mode. Returns ------- conf_interval : numpy.ndarray conf_interval and conf_interval correspond to the lower and upper limits, respectively, for each element in n. Notes ----- The "right" confidence interval to use for Poisson data is a matter of debate. The CDF working group recommends <http://www-cdf.fnal.gov/physics/statistics/notes/pois_eb.txt>_ using root-n throughout, largely in the interest of comprehensibility, but discusses other possibilities. The ATLAS group also discusses <http://www.pp.rhul.ac.uk/~cowan/atlas/ErrorBars.pdf>_ several possibilities but concludes that no single representation is suitable for all cases. The suggestion has also been floated <http://adsabs.harvard.edu/abs/2012EPJP..127...24A>_ that error bars should be attached to theoretical predictions instead of observed data, which this function will not help with (but it's easy; then you really should use the square root of the theoretical prediction). The intervals implemented here are: **1. 'root-n'** This is a very widely used standard rule derived from the maximum-likelihood estimator for the mean of the Poisson process. While it produces questionable results for small n and outright wrong results for n=0, it is standard enough that people are (supposedly) used to interpreting these wonky values. The interval is .. math:: CI = (n-\sqrt{n}, n+\sqrt{n}) **2. 'root-n-0'** This is identical to the above except that where n is zero the interval returned is (0,1). **3. 'pearson'** This is an only-slightly-more-complicated rule based on Pearson's chi-squared rule (as explained <http://www-cdf.fnal.gov/physics/statistics/notes/pois_eb.txt>_ by the CDF working group). It also has the nice feature that if your theory curve touches an endpoint of the interval, then your data point is indeed one sigma away. The interval is .. math:: CI = (n+0.5-\sqrt{n+0.25}, n+0.5+\sqrt{n+0.25}) **4. 'sherpagehrels'** This rule is used by default in the fitting package 'sherpa'. The documentation <http://cxc.harvard.edu/sherpa4.4/statistics/#chigehrels>_ claims it is based on a numerical approximation published in Gehrels (1986) <http://adsabs.harvard.edu/abs/1986ApJ...303..336G>_ but it does not actually appear there. It is symmetrical, and while the upper limits are within about 1% of those given by 'frequentist-confidence', the lower limits can be badly wrong. The interval is .. math:: CI = (n-1-\sqrt{n+0.75}, n+1+\sqrt{n+0.75}) **5. 'frequentist-confidence'** These are frequentist central confidence intervals: .. math:: CI = (0.5 F_{\chi^2}^{-1}(\alpha;2n), 0.5 F_{\chi^2}^{-1}(1-\alpha;2(n+1))) where :math:F_{\chi^2}^{-1} is the quantile of the chi-square distribution with the indicated number of degrees of freedom and :math:\alpha is the one-tailed probability of the normal distribution (at the point given by the parameter 'sigma'). See Maxwell (2011) <http://adsabs.harvard.edu/abs/2011arXiv1102.0822M>_ for further details. **6. 'kraft-burrows-nousek'** This is a Bayesian approach which allows for the presence of a known background :math:B in the source signal :math:N. For a given confidence level :math:CL the confidence interval :math:[S_\mathrm{min}, S_\mathrm{max}] is given by: .. math:: CL = \int^{S_\mathrm{max}}_{S_\mathrm{min}} f_{N,B}(S)dS where the function :math:f_{N,B} is: .. math:: f_{N,B}(S) = C \frac{e^{-(S+B)}(S+B)^N}{N!} and the normalization constant :math:C: .. math:: C = \left[ \int_0^\infty \frac{e^{-(S+B)}(S+B)^N}{N!} dS \right] ^{-1} = \left( \sum^N_{n=0} \frac{e^{-B}B^n}{n!} \right)^{-1} See Kraft, Burrows, and Nousek (1991) <http://adsabs.harvard.edu/abs/1991ApJ...374..344K>_ for further details. These formulas implement a positive, uniform prior. Kraft, Burrows, and Nousek (1991) <http://adsabs.harvard.edu/abs/1991ApJ...374..344K>_ discuss this choice in more detail and show that the problem is relatively insensitive to the choice of prior. This function has an optional dependency: Either Scipy <https://www.scipy.org/>_ or mpmath <http://mpmath.org/>_ need to be available (Scipy works only for N < 100). Examples -------- >>> poisson_conf_interval(np.arange(10), interval='root-n').T array([[ 0. , 0. ], [ 0. , 2. ], [ 0.58578644, 3.41421356], [ 1.26794919, 4.73205081], [ 2. , 6. ], [ 2.76393202, 7.23606798], [ 3.55051026, 8.44948974], [ 4.35424869, 9.64575131], [ 5.17157288, 10.82842712], [ 6. , 12. ]]) >>> poisson_conf_interval(np.arange(10), interval='root-n-0').T array([[ 0. , 1. ], [ 0. , 2. ], [ 0.58578644, 3.41421356], [ 1.26794919, 4.73205081], [ 2. , 6. ], [ 2.76393202, 7.23606798], [ 3.55051026, 8.44948974], [ 4.35424869, 9.64575131], [ 5.17157288, 10.82842712], [ 6. , 12. ]]) >>> poisson_conf_interval(np.arange(10), interval='pearson').T array([[ 0. , 1. ], [ 0.38196601, 2.61803399], [ 1. , 4. ], [ 1.69722436, 5.30277564], [ 2.43844719, 6.56155281], [ 3.20871215, 7.79128785], [ 4. , 9. ], [ 4.8074176 , 10.1925824 ], [ 5.62771868, 11.37228132], [ 6.45861873, 12.54138127]]) >>> poisson_conf_interval(np.arange(10), ... interval='frequentist-confidence').T array([[ 0. , 1.84102165], [ 0.17275378, 3.29952656], [ 0.70818544, 4.63785962], [ 1.36729531, 5.91818583], [ 2.08566081, 7.16275317], [ 2.84030886, 8.38247265], [ 3.62006862, 9.58364155], [ 4.41852954, 10.77028072], [ 5.23161394, 11.94514152], [ 6.05653896, 13.11020414]]) >>> poisson_conf_interval(7, ... interval='frequentist-confidence').T array([ 4.41852954, 10.77028072]) >>> poisson_conf_interval(10, background=1.5, conflevel=0.95, ... interval='kraft-burrows-nousek').T # doctest: +FLOAT_CMP array([[ 3.47894005, 16.113329533]]) """ if not np.isscalar(n): n = np.asanyarray(n) if interval == 'root-n': _check_poisson_conf_inputs(sigma, background, conflevel, interval) conf_interval = np.array([n - np.sqrt(n), n + np.sqrt(n)]) elif interval == 'root-n-0': _check_poisson_conf_inputs(sigma, background, conflevel, interval) conf_interval = np.array([n - np.sqrt(n), n + np.sqrt(n)]) if np.isscalar(n): if n == 0: conf_interval = 1 else: conf_interval[1, n == 0] = 1 elif interval == 'pearson': _check_poisson_conf_inputs(sigma, background, conflevel, interval) conf_interval = np.array([n + 0.5 - np.sqrt(n + 0.25), n + 0.5 + np.sqrt(n + 0.25)]) elif interval == 'sherpagehrels': _check_poisson_conf_inputs(sigma, background, conflevel, interval) conf_interval = np.array([n - 1 - np.sqrt(n + 0.75), n + 1 + np.sqrt(n + 0.75)]) elif interval == 'frequentist-confidence': _check_poisson_conf_inputs(1., background, conflevel, interval) import scipy.stats alpha = scipy.stats.norm.sf(sigma) conf_interval = np.array([0.5 * scipy.stats.chi2(2 * n).ppf(alpha), 0.5 * scipy.stats.chi2(2 * n + 2).isf(alpha)]) if np.isscalar(n): if n == 0: conf_interval = 0 else: conf_interval[0, n == 0] = 0 elif interval == 'kraft-burrows-nousek': if conflevel is None: raise ValueError('Set conflevel for method {0}. (sigma is ' 'ignored.)'.format(interval)) conflevel = np.asanyarray(conflevel) if np.any(conflevel <= 0) or np.any(conflevel >= 1): raise ValueError('Conflevel must be a number between 0 and 1.') background = np.asanyarray(background) if np.any(background < 0): raise ValueError('Background must be >= 0.') conf_interval = np.vectorize(_kraft_burrows_nousek, cache=True)(n, background, conflevel) conf_interval = np.vstack(conf_interval) else: raise ValueError("Invalid method for Poisson confidence intervals: " "{}".format(interval)) return conf_interval
[docs]@deprecated_renamed_argument('a', 'data', '2.0') def median_absolute_deviation(data, axis=None, func=None, ignore_nan=False): """ Calculate the median absolute deviation (MAD). The MAD is defined as median(abs(a - median(a))). Parameters ---------- data : array-like Input array or object that can be converted to an array. axis : {int, sequence of int, None}, optional Axis along which the MADs are computed. The default (None) is to compute the MAD of the flattened array. func : callable, optional The function used to compute the median. Defaults to numpy.ma.median for masked arrays, otherwise to numpy.median. ignore_nan : bool Ignore NaN values (treat them as if they are not in the array) when computing the median. This will use numpy.ma.median if axis is specified, or numpy.nanmedian if axis==None and numpy's version is >1.10 because nanmedian is slightly faster in this case. Returns ------- mad : float or ~numpy.ndarray The median absolute deviation of the input array. If axis is None then a scalar will be returned, otherwise a ~numpy.ndarray will be returned. Examples -------- Generate random variates from a Gaussian distribution and return the median absolute deviation for that distribution:: >>> import numpy as np >>> from astropy.stats import median_absolute_deviation >>> rand = np.random.RandomState(12345) >>> from numpy.random import randn >>> mad = median_absolute_deviation(rand.randn(1000)) >>> print(mad) # doctest: +FLOAT_CMP 0.65244241428454486 See Also -------- mad_std """ if func is None: # Check if the array has a mask and if so use np.ma.median # See https://github.com/numpy/numpy/issues/7330 why using np.ma.median # for normal arrays should not be done (summary: np.ma.median always # returns an masked array even if the result should be scalar). (#4658) if isinstance(data, np.ma.MaskedArray): is_masked = True func = np.ma.median if ignore_nan: data = np.ma.masked_where(np.isnan(data), data, copy=False) elif ignore_nan: is_masked = False func = np.nanmedian else: is_masked = False func = np.median else: is_masked = None data = np.asanyarray(data) # np.nanmedian has keepdims, which is a good option if we're not allowing # user-passed functions here data_median = func(data, axis=axis) # broadcast the median array before subtraction if axis is not None: if isiterable(axis): for ax in sorted(list(axis)): data_median = np.expand_dims(data_median, axis=ax) else: data_median = np.expand_dims(data_median, axis=axis) result = func(np.abs(data - data_median), axis=axis, overwrite_input=True) if axis is None and np.ma.isMaskedArray(result): # return scalar version result = result.item() elif np.ma.isMaskedArray(result) and not is_masked: # if the input array was not a masked array, we don't want to return a # masked array result = result.filled(fill_value=np.nan) return result
[docs]def mad_std(data, axis=None, func=None, ignore_nan=False): r""" Calculate a robust standard deviation using the median absolute deviation (MAD) <https://en.wikipedia.org/wiki/Median_absolute_deviation>_. The standard deviation estimator is given by: .. math:: \sigma \approx \frac{\textrm{MAD}}{\Phi^{-1}(3/4)} \approx 1.4826 \ \textrm{MAD} where :math:\Phi^{-1}(P) is the normal inverse cumulative distribution function evaluated at probability :math:P = 3/4. Parameters ---------- data : array-like Data array or object that can be converted to an array. axis : {int, sequence of int, None}, optional Axis along which the robust standard deviations are computed. The default (None) is to compute the robust standard deviation of the flattened array. func : callable, optional The function used to compute the median. Defaults to numpy.ma.median for masked arrays, otherwise to numpy.median. ignore_nan : bool Ignore NaN values (treat them as if they are not in the array) when computing the median. This will use numpy.ma.median if axis is specified, or numpy.nanmedian if axis=None and numpy's version is >1.10 because nanmedian is slightly faster in this case. Returns ------- mad_std : float or ~numpy.ndarray The robust standard deviation of the input data. If axis is None then a scalar will be returned, otherwise a ~numpy.ndarray will be returned. Examples -------- >>> import numpy as np >>> from astropy.stats import mad_std >>> rand = np.random.RandomState(12345) >>> madstd = mad_std(rand.normal(5, 2, (100, 100))) >>> print(madstd) # doctest: +FLOAT_CMP 2.0232764659422626 See Also -------- biweight_midvariance, biweight_midcovariance, median_absolute_deviation """ # NOTE: 1. / scipy.stats.norm.ppf(0.75) = 1.482602218505602 MAD = median_absolute_deviation( data, axis=axis, func=func, ignore_nan=ignore_nan) return MAD * 1.482602218505602
[docs]def signal_to_noise_oir_ccd(t, source_eps, sky_eps, dark_eps, rd, npix, gain=1.0): """Computes the signal to noise ratio for source being observed in the optical/IR using a CCD. Parameters ---------- t : float or numpy.ndarray CCD integration time in seconds source_eps : float Number of electrons (photons) or DN per second in the aperture from the source. Note that this should already have been scaled by the filter transmission and the quantum efficiency of the CCD. If the input is in DN, then be sure to set the gain to the proper value for the CCD. If the input is in electrons per second, then keep the gain as its default of 1.0. sky_eps : float Number of electrons (photons) or DN per second per pixel from the sky background. Should already be scaled by filter transmission and QE. This must be in the same units as source_eps for the calculation to make sense. dark_eps : float Number of thermal electrons per second per pixel. If this is given in DN or ADU, then multiply by the gain to get the value in electrons. rd : float Read noise of the CCD in electrons. If this is given in DN or ADU, then multiply by the gain to get the value in electrons. npix : float Size of the aperture in pixels gain : float, optional Gain of the CCD. In units of electrons per DN. Returns ---------- SNR : float or numpy.ndarray Signal to noise ratio calculated from the inputs """ signal = t * source_eps * gain noise = np.sqrt(t * (source_eps * gain + npix * (sky_eps * gain + dark_eps)) + npix * rd ** 2) return signal / noise
[docs]def bootstrap(data, bootnum=100, samples=None, bootfunc=None): """Performs bootstrap resampling on numpy arrays. Bootstrap resampling is used to understand confidence intervals of sample estimates. This function returns versions of the dataset resampled with replacement ("case bootstrapping"). These can all be run through a function or statistic to produce a distribution of values which can then be used to find the confidence intervals. Parameters ---------- data : numpy.ndarray N-D array. The bootstrap resampling will be performed on the first index, so the first index should access the relevant information to be bootstrapped. bootnum : int, optional Number of bootstrap resamples samples : int, optional Number of samples in each resample. The default None sets samples to the number of datapoints bootfunc : function, optional Function to reduce the resampled data. Each bootstrap resample will be put through this function and the results returned. If None, the bootstrapped data will be returned Returns ------- boot : numpy.ndarray If bootfunc is None, then each row is a bootstrap resample of the data. If bootfunc is specified, then the columns will correspond to the outputs of bootfunc. Examples -------- Obtain a twice resampled array: >>> from astropy.stats import bootstrap >>> import numpy as np >>> from astropy.utils import NumpyRNGContext >>> bootarr = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 0]) >>> with NumpyRNGContext(1): ... bootresult = bootstrap(bootarr, 2) ... >>> bootresult # doctest: +FLOAT_CMP array([[6., 9., 0., 6., 1., 1., 2., 8., 7., 0.], [3., 5., 6., 3., 5., 3., 5., 8., 8., 0.]]) >>> bootresult.shape (2, 10) Obtain a statistic on the array >>> with NumpyRNGContext(1): ... bootresult = bootstrap(bootarr, 2, bootfunc=np.mean) ... >>> bootresult # doctest: +FLOAT_CMP array([4. , 4.6]) Obtain a statistic with two outputs on the array >>> test_statistic = lambda x: (np.sum(x), np.mean(x)) >>> with NumpyRNGContext(1): ... bootresult = bootstrap(bootarr, 3, bootfunc=test_statistic) >>> bootresult # doctest: +FLOAT_CMP array([[40. , 4. ], [46. , 4.6], [35. , 3.5]]) >>> bootresult.shape (3, 2) Obtain a statistic with two outputs on the array, keeping only the first output >>> bootfunc = lambda x:test_statistic(x) >>> with NumpyRNGContext(1): ... bootresult = bootstrap(bootarr, 3, bootfunc=bootfunc) ... >>> bootresult # doctest: +FLOAT_CMP array([40., 46., 35.]) >>> bootresult.shape (3,) """ if samples is None: samples = data.shape # make sure the input is sane if samples < 1 or bootnum < 1: raise ValueError("neither 'samples' nor 'bootnum' can be less than 1.") if bootfunc is None: resultdims = (bootnum,) + (samples,) + data.shape[1:] else: # test number of outputs from bootfunc, avoid single outputs which are # array-like try: resultdims = (bootnum, len(bootfunc(data))) except TypeError: resultdims = (bootnum,) # create empty boot array boot = np.empty(resultdims) for i in range(bootnum): bootarr = np.random.randint(low=0, high=data.shape, size=samples) if bootfunc is None: boot[i] = data[bootarr] else: boot[i] = bootfunc(data[bootarr]) return boot
def _scipy_kraft_burrows_nousek(N, B, CL): '''Upper limit on a poisson count rate The implementation is based on Kraft, Burrows and Nousek ApJ 374, 344 (1991) <http://adsabs.harvard.edu/abs/1991ApJ...374..344K>_. The XMM-Newton upper limit server uses the same formalism. Parameters ---------- N : int Total observed count number B : float Background count rate (assumed to be known with negligible error from a large background area). CL : float Confidence level (number between 0 and 1) Returns ------- S : source count limit Notes ----- Requires scipy. This implementation will cause Overflow Errors for about N > 100 (the exact limit depends on details of how scipy was compiled). See ~astropy.stats.mpmath_poisson_upper_limit for an implementation that is slower, but can deal with arbitrarily high numbers since it is based on the mpmath <http://mpmath.org/>_ library. ''' from scipy.optimize import brentq from scipy.integrate import quad from math import exp def eqn8(N, B): n = np.arange(N + 1, dtype=np.float64) # Create an array containing the factorials. scipy.special.factorial # requires SciPy 0.14 (#5064) therefore this is calculated by using # numpy.cumprod. This could be replaced by factorial again as soon as # older SciPy are not supported anymore but the cumprod alternative # might also be a bit faster. factorial_n = np.ones(n.shape, dtype=np.float64) np.cumprod(n[1:], out=factorial_n[1:]) return 1. / (exp(-B) * np.sum(np.power(B, n) / factorial_n)) # The parameters of eqn8 do not vary between calls so we can calculate the # result once and reuse it. The same is True for the factorial of N. # eqn7 is called hundred times so "caching" these values yields a # significant speedup (factor 10). eqn8_res = eqn8(N, B) factorial_N = float(math.factorial(N)) def eqn7(S, N, B): SpB = S + B return eqn8_res * (exp(-SpB) * SpB**N / factorial_N) def eqn9_left(S_min, S_max, N, B): return quad(eqn7, S_min, S_max, args=(N, B), limit=500) def find_s_min(S_max, N, B): ''' Kraft, Burrows and Nousek suggest to integrate from N-B in both directions at once, so that S_min and S_max move similarly (see the article for details). Here, this is implemented differently: Treat S_max as the optimization parameters in func and then calculate the matching s_min that has has eqn7(S_max) = eqn7(S_min) here. ''' y_S_max = eqn7(S_max, N, B) if eqn7(0, N, B) >= y_S_max: return 0. else: return brentq(lambda x: eqn7(x, N, B) - y_S_max, 0, N - B) def func(s): s_min = find_s_min(s, N, B) out = eqn9_left(s_min, s, N, B) return out - CL S_max = brentq(func, N - B, 100) S_min = find_s_min(S_max, N, B) return S_min, S_max def _mpmath_kraft_burrows_nousek(N, B, CL): '''Upper limit on a poisson count rate The implementation is based on Kraft, Burrows and Nousek in ApJ 374, 344 (1991) <http://adsabs.harvard.edu/abs/1991ApJ...374..344K>_. The XMM-Newton upper limit server used the same formalism. Parameters ---------- N : int Total observed count number B : float Background count rate (assumed to be known with negligible error from a large background area). CL : float Confidence level (number between 0 and 1) Returns ------- S : source count limit Notes ----- Requires the mpmath <http://mpmath.org/>_ library. See ~astropy.stats.scipy_poisson_upper_limit for an implementation that is based on scipy and evaluates faster, but runs only to about N = 100. ''' from mpmath import mpf, factorial, findroot, fsum, power, exp, quad N = mpf(N) B = mpf(B) CL = mpf(CL) def eqn8(N, B): sumterms = [power(B, n) / factorial(n) for n in range(int(N) + 1)] return 1. / (exp(-B) * fsum(sumterms)) eqn8_res = eqn8(N, B) factorial_N = factorial(N) def eqn7(S, N, B): SpB = S + B return eqn8_res * (exp(-SpB) * SpB**N / factorial_N) def eqn9_left(S_min, S_max, N, B): def eqn7NB(S): return eqn7(S, N, B) return quad(eqn7NB, [S_min, S_max]) def find_s_min(S_max, N, B): ''' Kraft, Burrows and Nousek suggest to integrate from N-B in both directions at once, so that S_min and S_max move similarly (see the article for details). Here, this is implemented differently: Treat S_max as the optimization parameters in func and then calculate the matching s_min that has has eqn7(S_max) = eqn7(S_min) here. ''' y_S_max = eqn7(S_max, N, B) if eqn7(0, N, B) >= y_S_max: return 0. else: def eqn7ysmax(x): return eqn7(x, N, B) - y_S_max return findroot(eqn7ysmax, (N - B) / 2.) def func(s): s_min = find_s_min(s, N, B) out = eqn9_left(s_min, s, N, B) return out - CL S_max = findroot(func, N - B, tol=1e-4) S_min = find_s_min(S_max, N, B) return float(S_min), float(S_max) def _kraft_burrows_nousek(N, B, CL): '''Upper limit on a poisson count rate The implementation is based on Kraft, Burrows and Nousek in ApJ 374, 344 (1991) <http://adsabs.harvard.edu/abs/1991ApJ...374..344K>_. The XMM-Newton upper limit server used the same formalism. Parameters ---------- N : int Total observed count number B : float Background count rate (assumed to be known with negligible error from a large background area). CL : float Confidence level (number between 0 and 1) Returns ------- S : source count limit Notes ----- This functions has an optional dependency: Either scipy or mpmath <http://mpmath.org/>_ need to be available. (Scipy only works for N < 100). ''' try: import scipy HAS_SCIPY = True except ImportError: HAS_SCIPY = False try: import mpmath HAS_MPMATH = True except ImportError: HAS_MPMATH = False if HAS_SCIPY and N <= 100: try: return _scipy_kraft_burrows_nousek(N, B, CL) except OverflowError: if not HAS_MPMATH: raise ValueError('Need mpmath package for input numbers this ' 'large.') if HAS_MPMATH: return _mpmath_kraft_burrows_nousek(N, B, CL) raise ImportError('Either scipy or mpmath are required.')
[docs]def kuiper_false_positive_probability(D, N): """Compute the false positive probability for the Kuiper statistic. Uses the set of four formulas described in Paltani 2004; they report the resulting function never underestimates the false positive probability but can be a bit high in the N=40..50 range. (They quote a factor 1.5 at the 1e-7 level.) Parameters ---------- D : float The Kuiper test score. N : float The effective sample size. Returns ------- fpp : float The probability of a score this large arising from the null hypothesis. Notes ----- Eq 7 of Paltani 2004 appears to incorrectly quote the original formula (Stephens 1965). This function implements the original formula, as it produces a result closer to Monte Carlo simulations. References ---------- ..  Paltani, S., "Searching for periods in X-ray observations using Kuiper's test. Application to the ROSAT PSPC archive", Astronomy and Astrophysics, v.240, p.789-790, 2004. ..  Stephens, M. A., "The goodness-of-fit statistic VN: distribution and significance points", Biometrika, v.52, p.309, 1965. """ try: from scipy.special import factorial, comb except ImportError: # Retained for backwards compatibility with older versions of scipy # (factorial appears to have moved here in 0.14) from scipy.misc import factorial, comb if D < 0. or D > 2.: raise ValueError("Must have 0<=D<=2 by definition of the Kuiper test") if D < 2. / N: return 1. - factorial(N) * (D - 1. / N)**(N - 1) elif D < 3. / N: k = -(N * D - 1.) / 2. r = np.sqrt(k**2 - (N * D - 2.)**2 / 2.) a, b = -k + r, -k - r return 1 - (factorial(N - 1) * (b**(N - 1) * (1 - a) - a**(N - 1) * (1 - b)) / N**(N - 2) / (b - a)) elif (D > 0.5 and N % 2 == 0) or (D > (N - 1.) / (2. * N) and N % 2 == 1): # NOTE: the upper limit of this sum is taken from Stephens 1965 t = np.arange(np.floor(N * (1 - D)) + 1) y = D + t / N Tt = y**(t - 3) * (y**3 * N - y**2 * t * (3 - 2 / N) + y * t * (t - 1) * (3 - 2 / N) / N - t * (t - 1) * (t - 2) / N**2) term = Tt * comb(N, t) * (1 - D - t / N)**(N - t - 1) return term.sum() else: z = D * np.sqrt(N) # When m*z>18.82 (sqrt(-log(finfo(double))/2)), exp(-2m**2z**2) # underflows. Cutting off just before avoids triggering a (pointless) # underflow warning if under="warn". ms = np.arange(1, 18.82 / z) S1 = (2 * (4 * ms**2 * z**2 - 1) * np.exp(-2 * ms**2 * z**2)).sum() S2 = (ms**2 * (4 * ms**2 * z**2 - 3) * np.exp(-2 * ms**2 * z**2)).sum() return S1 - 8 * D / 3 * S2
[docs]def kuiper(data, cdf=lambda x: x, args=()): """Compute the Kuiper statistic. Use the Kuiper statistic version of the Kolmogorov-Smirnov test to find the probability that a sample like data was drawn from the distribution whose CDF is given as cdf. .. warning:: This will not work correctly for distributions that are actually discrete (Poisson, for example). Parameters ---------- data : array-like The data values. cdf : callable A callable to evaluate the CDF of the distribution being tested against. Will be called with a vector of all values at once. The default is a uniform distribution. args : list-like, optional Additional arguments to be supplied to cdf. Returns ------- D : float The raw statistic. fpp : float The probability of a D this large arising with a sample drawn from the distribution whose CDF is cdf. Notes ----- The Kuiper statistic resembles the Kolmogorov-Smirnov test in that it is nonparametric and invariant under reparameterizations of the data. The Kuiper statistic, in addition, is equally sensitive throughout the domain, and it is also invariant under cyclic permutations (making it particularly appropriate for analyzing circular data). Returns (D, fpp), where D is the Kuiper D number and fpp is the probability that a value as large as D would occur if data was drawn from cdf. .. warning:: The fpp is calculated only approximately, and it can be as much as 1.5 times the true value. Stephens 1970 claims this is more effective than the KS at detecting changes in the variance of a distribution; the KS is (he claims) more sensitive at detecting changes in the mean. If cdf was obtained from data by fitting, then fpp is not correct and it will be necessary to do Monte Carlo simulations to interpret D. D should normally be independent of the shape of CDF. References ---------- ..  Stephens, M. A., "Use of the Kolmogorov-Smirnov, Cramer-Von Mises and Related Statistics Without Extensive Tables", Journal of the Royal Statistical Society. Series B (Methodological), Vol. 32, No. 1. (1970), pp. 115-122. """ data = np.sort(data) cdfv = cdf(data, *args) N = len(data) D = (np.amax(cdfv - np.arange(N) / float(N)) + np.amax((np.arange(N) + 1) / float(N) - cdfv)) return D, kuiper_false_positive_probability(D, N)
[docs]def kuiper_two(data1, data2): """Compute the Kuiper statistic to compare two samples. Parameters ---------- data1 : array-like The first set of data values. data2 : array-like The second set of data values. Returns ------- D : float The raw test statistic. fpp : float The probability of obtaining two samples this different from the same distribution. .. warning:: The fpp is quite approximate, especially for small samples. """ data1 = np.sort(data1) data2 = np.sort(data2) n1, = data1.shape n2, = data2.shape common_type = np.find_common_type([], [data1.dtype, data2.dtype]) if not (np.issubdtype(common_type, np.number) and not np.issubdtype(common_type, np.complexfloating)): raise ValueError('kuiper_two only accepts real inputs') # nans, if any, are at the end after sorting. if np.isnan(data1[-1]) or np.isnan(data2[-1]): raise ValueError('kuiper_two only accepts non-nan inputs') D = _stats.ks_2samp(np.asarray(data1, common_type), np.asarray(data2, common_type)) Ne = len(data1) * len(data2) / float(len(data1) + len(data2)) return D, kuiper_false_positive_probability(D, Ne)
[docs]def fold_intervals(intervals): """Fold the weighted intervals to the interval (0,1). Convert a list of intervals (ai, bi, wi) to a list of non-overlapping intervals covering (0,1). Each output interval has a weight equal to the sum of the wis of all the intervals that include it. All intervals are interpreted modulo 1, and weights are accumulated counting multiplicity. This is appropriate, for example, if you have one or more blocks of observation and you want to determine how much observation time was spent on different parts of a system's orbit (the blocks should be converted to units of the orbital period first). Parameters ---------- intervals : list of three-element tuples (ai,bi,wi) The intervals to fold; ai and bi are the limits of the interval, and wi is the weight to apply to the interval. Returns ------- breaks : array of floats length N The endpoints of a set of intervals covering [0,1]; breaks=0 and breaks[-1] = 1 weights : array of floats of length N-1 The ith element is the sum of number of times the interval breaks[i],breaks[i+1] is included in each interval times the weight associated with that interval. """ r = [] breaks = set() tot = 0 for (a, b, wt) in intervals: tot += (np.ceil(b) - np.floor(a)) * wt fa = a % 1 breaks.add(fa) r.append((0, fa, -wt)) fb = b % 1 breaks.add(fb) r.append((fb, 1, -wt)) breaks.add(0.) breaks.add(1.) breaks = sorted(breaks) breaks_map = dict([(f, i) for (i, f) in enumerate(breaks)]) totals = np.zeros(len(breaks) - 1) totals += tot for (a, b, wt) in r: totals[breaks_map[a]:breaks_map[b]] += wt return np.array(breaks), totals
[docs]def cdf_from_intervals(breaks, totals): """Construct a callable piecewise-linear CDF from a pair of arrays. Take a pair of arrays in the format returned by fold_intervals and make a callable cumulative distribution function on the interval (0,1). Parameters ---------- breaks : array of floats of length N The boundaries of successive intervals. totals : array of floats of length N-1 The weight for each interval. Returns ------- f : callable A cumulative distribution function corresponding to the piecewise-constant probability distribution given by breaks, weights """ if breaks != 0 or breaks[-1] != 1: raise ValueError("Intervals must be restricted to [0,1]") if np.any(np.diff(breaks) <= 0): raise ValueError("Breaks must be strictly increasing") if np.any(totals < 0): raise ValueError( "Total weights in each subinterval must be nonnegative") if np.all(totals == 0): raise ValueError("At least one interval must have positive exposure") b = breaks.copy() c = np.concatenate(((0,), np.cumsum(totals * np.diff(b)))) c /= c[-1] return lambda x: np.interp(x, b, c, 0, 1)
[docs]def interval_overlap_length(i1, i2): """Compute the length of overlap of two intervals. Parameters ---------- i1, i2 : pairs of two floats The two intervals. Returns ------- l : float The length of the overlap between the two intervals. """ (a, b) = i1 (c, d) = i2 if a < c: if b < c: return 0. elif b < d: return b - c else: return d - c elif a < d: if b < d: return b - a else: return d - a else: return 0
[docs]def histogram_intervals(n, breaks, totals): """Histogram of a piecewise-constant weight function. This function takes a piecewise-constant weight function and computes the average weight in each histogram bin. Parameters ---------- n : int The number of bins breaks : array of floats of length N Endpoints of the intervals in the PDF totals : array of floats of length N-1 Probability densities in each bin Returns ------- h : array of floats The average weight for each bin """ h = np.zeros(n) start = breaks for i in range(len(totals)): end = breaks[i + 1] for j in range(n): ol = interval_overlap_length((float(j) / n, float(j + 1) / n), (start, end)) h[j] += ol / (1. / n) * totals[i] start = end return h