# 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 numpy as np import astropy.units as u from . import _stats __all__ = ['gaussian_fwhm_to_sigma', 'gaussian_sigma_to_fwhm', 'binom_conf_interval', 'binned_binom_proportion', 'poisson_conf_interval', 'median_absolute_deviation', 'mad_std', '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 * math.sqrt(2.0 * math.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. """ # NUMPY_LT_1_18 def _expand_dims(data, axis): """ Expand the shape of an array. Insert a new axis that will appear at the `axis` position in the expanded array shape. This function allows for tuple axis arguments. ``numpy.expand_dims`` currently does not allow that, but it will in numpy v1.18 (https://github.com/numpy/numpy/pull/14051). ``_expand_dims`` can be replaced with ``numpy.expand_dims`` when the minimum support numpy version is v1.18. Parameters ---------- data : array-like Input array. axis : int or tuple of int Position in the expanded axes where the new axis (or axes) is placed. A tuple of axes is now supported. Out of range axes as described above are now forbidden and raise an `AxisError`. Returns ------- result : ndarray View of ``data`` with the number of dimensions increased. """ if isinstance(data, np.matrix): data = np.asarray(data) else: data = np.asanyarray(data) if not isinstance(axis, (tuple, list)): axis = (axis,) out_ndim = len(axis) + data.ndim axis = np.core.numeric.normalize_axis_tuple(axis, out_ndim) shape_it = iter(data.shape) shape = [1 if ax in axis else next(shape_it) for ax in range(out_ndim)] return data.reshape(shape) def binom_conf_interval(k, n, confidence_level=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. confidence_level : float, optional Desired probability content of interval. Default is 0.68269, corresponding to 1 sigma in a 1-dimensional Gaussian distribution. Confidence level must be in range [0, 1]. 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 : ndarray ``conf_interval[0]`` and ``conf_interval[1]`` 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 [1]_). Four intervals are implemented here: **1. The Wilson Interval.** This interval, attributed to Wilson [2]_, 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 [3]_, [4]_. 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. This function requires ``scipy`` for all interval types. References ---------- .. [1] 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 .. [2] Wilson, E. B. (1927). "Probable inference, the law of succession, and statistical inference". Journal of the American Statistical Association 22: 209-212. .. [3] 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 .. [4] 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') # doctest: +FLOAT_CMP 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([1, 2], 5, interval='wilson') # doctest: +FLOAT_CMP array([[0.07921741, 0.21597328], [0.42078276, 0.61736012]]) >>> binom_conf_interval([1, 2,], 5, interval='jeffreys') # doctest: +FLOAT_CMP array([[0.0842525 , 0.21789949], [0.42218001, 0.61753691]]) >>> binom_conf_interval([1, 2], 5, interval='flat') # doctest: +FLOAT_CMP array([[0.12139799, 0.24309021], [0.45401727, 0.61535699]]) 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([1, 2], 5, interval='wald') # doctest: +FLOAT_CMP array([[0.02111437, 0.18091075], [0.37888563, 0.61908925]]) For confidence intervals approaching 1, the Wald interval for 0 < k < n can give intervals that extend outside [0, 1]: >>> binom_conf_interval([1, 2], 5, interval='wald', confidence_level=0.99) # doctest: +FLOAT_CMP array([[-0.26077835, -0.16433593], [ 0.66077835, 0.96433593]]) """ # noqa if confidence_level < 0. or confidence_level > 1.: raise ValueError('confidence_level must be between 0. and 1.') alpha = 1. - confidence_level 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(confidence_level), 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(f'Unrecognized interval: {interval:s}') return conf_interval def binned_binom_proportion(x, success, bins=10, range=None, confidence_level=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 : sequence Values. success : sequence of bool Success (`True`) or failure (`False`) corresponding to each value in ``x``. Must be same length as ``x``. bins : int or sequence of scalar, 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. confidence_level : float, optional Must be in range [0, 1]. Desired probability content in the confidence interval ``(p - perr[0], p + perr[1])`` 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 : ndarray Central value of bins. Bins without any entries are not returned. bin_halfwidth : 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 : ndarray Efficiency in each bin. perr : ndarray 2-d array of shape (2, len(p)) representing the upper and lower uncertainty on p in each bin. Notes ----- This function requires ``scipy`` for all interval types. 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, confidence_level=confidence_level, interval=interval) perr = np.abs(bounds - p) return bin_ctr, bin_halfwidth, p, perr def _check_poisson_conf_inputs(sigma, background, confidence_level, name): if sigma != 1: raise ValueError(f"Only sigma=1 supported for interval {name}") if background != 0: raise ValueError(f"background not supported for interval {name}") if confidence_level is not None: raise ValueError(f"confidence_level not supported for interval {name}") def poisson_conf_interval(n, interval='root-n', sigma=1, background=0, confidence_level=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. confidence_level : float, optional Confidence level between 0 and 1; only supported for the 'kraft-burrows-nousek' mode. Returns ------- conf_interval : ndarray ``conf_interval[0]`` and ``conf_interval[1]`` 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 `_ using root-n throughout, largely in the interest of comprehensibility, but discusses other possibilities. The ATLAS group also discusses several possibilities but concludes that no single representation is suitable for all cases. The suggestion has also been `floated `_ 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 `_ 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 `_ claims it is based on a numerical approximation published in `Gehrels (1986) `_ 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) `_ 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) `_ for further details. These formulas implement a positive, uniform prior. `Kraft, Burrows, and Nousek (1991) `_ 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 `_ or `mpmath `_ need to be available (Scipy works only for N < 100). This code is very intense numerically, which makes it much slower than the other methods, in particular for large count numbers (above 1000 even with ``mpmath``). Fortunately, some of the other methods or a Gaussian approximation usually work well in this regime. 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, confidence_level=0.95, ... interval='kraft-burrows-nousek').T # doctest: +FLOAT_CMP array([[ 3.47894005, 16.113329533]]) """ # noqa if not np.isscalar(n): n = np.asanyarray(n) if interval == 'root-n': _check_poisson_conf_inputs(sigma, background, confidence_level, interval) conf_interval = np.array([n - np.sqrt(n), n + np.sqrt(n)]) elif interval == 'root-n-0': _check_poisson_conf_inputs(sigma, background, confidence_level, interval) conf_interval = np.array([n - np.sqrt(n), n + np.sqrt(n)]) if np.isscalar(n): if n == 0: conf_interval[1] = 1 else: conf_interval[1, n == 0] = 1 elif interval == 'pearson': _check_poisson_conf_inputs(sigma, background, confidence_level, 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, confidence_level, 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, confidence_level, 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] = 0 else: conf_interval[0, n == 0] = 0 elif interval == 'kraft-burrows-nousek': # Deprecation warning in Python 3.9 when N is float, so we force int, # see https://github.com/astropy/astropy/issues/10832 if np.isscalar(n): if not isinstance(n, int): raise TypeError('Number of counts must be integer.') elif not issubclass(n.dtype.type, np.integer): raise TypeError('Number of counts must be integer.') if confidence_level is None: raise ValueError('Set confidence_level for method {}. (sigma is ' 'ignored.)'.format(interval)) confidence_level = np.asanyarray(confidence_level) if np.any(confidence_level <= 0) or np.any(confidence_level >= 1): raise ValueError('confidence_level 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, confidence_level) conf_interval = np.vstack(conf_interval) else: raise ValueError(f"Invalid method for Poisson confidence intervals: {interval}") return conf_interval 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 : None, int, or tuple of int, optional The axis or axes 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.default_rng(12345) >>> from numpy.random import randn >>> mad = median_absolute_deviation(rand.standard_normal(1000)) >>> print(mad) # doctest: +FLOAT_CMP 0.6829504282771885 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=True) elif ignore_nan: is_masked = False func = np.nanmedian else: is_masked = False func = np.median # drops units if result is NaN 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) # this conditional can be removed after this PR is merged: # https://github.com/astropy/astropy/issues/12165 if (isinstance(data, u.Quantity) and func is np.median and data_median.ndim == 0 and np.isnan(data_median)): data_median = data.__array_wrap__(data_median) # broadcast the median array before subtraction if axis is not None: data_median = _expand_dims(data_median, axis=axis) # NUMPY_LT_1_18 result = func(np.abs(data - data_median), axis=axis, overwrite_input=True) # this conditional can be removed after this PR is merged: # https://github.com/astropy/astropy/issues/12165 if (isinstance(data, u.Quantity) and func is np.median and result.ndim == 0 and np.isnan(result)): result = data.__array_wrap__(result) 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 def mad_std(data, axis=None, func=None, ignore_nan=False): r""" Calculate a robust standard deviation using the `median absolute deviation (MAD) `_. 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 : None, int, or tuple of int, optional The axis or axes 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.default_rng(12345) >>> madstd = mad_std(rand.normal(5, 2, (100, 100))) >>> print(madstd) # doctest: +FLOAT_CMP 1.984147963351707 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 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 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 : 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 : 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)[0] >>> 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[0] # 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[0], 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) `_. The XMM-Newton upper limit server uses the same formalism. Parameters ---------- N : int or np.int32/np.int64 Total observed count number B : float or np.float32/np.float64 Background count rate (assumed to be known with negligible error from a large background area). CL : float or np.float32/np.float64 Confidence level (number between 0 and 1) Returns ------- S : source count limit Notes ----- Requires :mod:`~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 `_ library. ''' from scipy.optimize import brentq from scipy.integrate import quad from scipy.special import factorial from math import exp def eqn8(N, B): n = np.arange(N + 1, dtype=np.float64) 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[0] - 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) `_. The XMM-Newton upper limit server used the same formalism. Parameters ---------- N : int or np.int32/np.int64 Total observed count number B : float or np.float32/np.float64 Background count rate (assumed to be known with negligible error from a large background area). CL : float or np.float32/np.float64 Confidence level (number between 0 and 1) Returns ------- S : source count limit Notes ----- Requires the `mpmath `_ 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 # We convert these values to float. Because for some reason, # mpmath.mpf cannot convert from numpy.int64 N = mpf(float(N)) B = mpf(float(B)) CL = mpf(float(CL)) tol = 1e-4 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 B > N, then N-B, the "most probable" values is < 0 # and thus s_min is certainly 0. # Note: For small N, s_max is also close to 0 and root finding # might find the wrong root, thus it is important to handle this # case here and return the analytical answer (s_min = 0). if (B >= N) or (eqn7(0, N, B) >= y_S_max): return 0. else: def eqn7ysmax(x): return eqn7(x, N, B) - y_S_max return findroot(eqn7ysmax, [0., N - B], solver='ridder', tol=tol) def func(s): s_min = find_s_min(s, N, B) out = eqn9_left(s_min, s, N, B) return out - CL # Several numerical problems were found prevent the solvers from finding # the roots unless the starting values are very close to the final values. # Thus, this primitive, time-wasting, brute-force stepping here to get # an interval that can be fed into the ridder solver. s_max_guess = max(N - B, 1.) while func(s_max_guess) < 0: s_max_guess += 1 S_max = findroot(func, [s_max_guess - 1, s_max_guess], solver='ridder', tol=tol) 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) `_. The XMM-Newton upper limit server used the same formalism. Parameters ---------- N : int or np.int32/np.int64 Total observed count number B : float or np.float32/np.float64 Background count rate (assumed to be known with negligible error from a large background area). CL : float or np.float32/np.float64 Confidence level (number between 0 and 1) Returns ------- S : source count limit Notes ----- This functions has an optional dependency: Either :mod:`scipy` or `mpmath `_ need to be available. (Scipy only works for N < 100). ''' from astropy.utils.compat.optional_deps import HAS_SCIPY, HAS_MPMATH 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.') 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 ---------- .. [1] 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. .. [2] 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) term1 = comb(N, t) term2 = (1 - D - t / N)**(N - t - 1) # term1 is formally finite, but is approximated by numpy as np.inf for # large values, so we set them to zero manually when they would be # multiplied by zero anyway term1[(term1 == np.inf) & (term2 == 0)] = 0. final_term = Tt * term1 * term2 return final_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 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 ---------- .. [1] 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) 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) 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 (3,) tuple For each tuple (ai,bi,wi); ai and bi are the limits of the interval, and wi is the weight to apply to the interval. Returns ------- breaks : (N,) array of float The endpoints of a set of intervals covering [0,1]; breaks[0]=0 and breaks[-1] = 1 weights : (N-1,) array of float 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 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 : (N,) array of float The boundaries of successive intervals. totals : (N-1,) array of float 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] != 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) def interval_overlap_length(i1, i2): """Compute the length of overlap of two intervals. Parameters ---------- i1, i2 : (float, float) The two intervals, (interval 1, interval 2). 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 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 : (N,) array of float Endpoints of the intervals in the PDF totals : (N-1,) array of float Probability densities in each bin Returns ------- h : array of float The average weight for each bin """ h = np.zeros(n) start = breaks[0] 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