diff --git a/.gitignore b/.gitignore index 1dbc687..94e17e5 100644 --- a/.gitignore +++ b/.gitignore @@ -60,3 +60,7 @@ target/ #Ipython Notebook .ipynb_checkpoints + +# pycharm +./idea +.idea/* diff --git a/README.md b/README.md index e70bf65..058b5b9 100644 --- a/README.md +++ b/README.md @@ -1,2 +1,14 @@ # pylevy + Levy distributions for Python +====== + +To install, type: + + python setup.py install + +See source code of levy.py for usage. (Yes, I am lazy.) + + +This software is distributed under the GPL, see the file LICENSE. +It was written by Paul Harrison. diff --git a/levy.py b/levy.py new file mode 100644 index 0000000..adabcce --- /dev/null +++ b/levy.py @@ -0,0 +1,496 @@ +# Copyright (C) 2005 Paul Harrison +# This program is free software; you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation; either version 2 of the License, or +# (at your option) any later versionp. +# +# This program is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with this program; if not, write to the Free Software +# Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + +""" +This is a package for calculation of Levy stable distributions +(probability density function and cumulative density function) and for +fitting these distributions to data. + +It operates by interpolating values from a table, as direct computation +of these distributions requires a lengthy numerical integration. This +interpolation scheme allows fast fitting of Levy stable distributions +to data using the Maximum Likelihood technique. + +Does not support alpha values less than 0.5. +""" + +import sys +import numpy as np +import scipy.special as sp + +__version__ = "0.5" + +# Dimensions: 0 - x, 1 - alpha, 2 - beta +_lower = np.array([-np.pi / 2 * 0.999, 0.5, -1.0]) +_upper = np.array([np.pi / 2 * 0.999, 2.0, 1.0]) + +par_names = ['alpha', 'beta', 'mu', 'sigma'] +default = [1.5, 0.0, 0.0, 1.0] +default = {par_names[i]: default[i] for i in range(4)} +f_bounds = { + 'alpha': lambda x: _reflect(x, _lower[1], _upper[1]), + 'beta': lambda x: _reflect(x, _lower[2], _upper[2]), + 'mu': lambda x: x, + 'sigma': lambda x: x +} + + +class Parameters(object): + def __init__(self, **kwargs): + # self._f = [f_bounds[k] for k in par_names] + self._x = np.array([default[k] if kwargs[k] is None else kwargs[k] for k in par_names]) + self.variables = [i for i, k in enumerate(par_names) if kwargs[k] is None] + self.fixed = [i for i, k in enumerate(par_names) if kwargs[k] is not None] + self.fixed_values = [kwargs[k] for i, k in enumerate(par_names) if kwargs[k] is not None] + + def get_all(self): + return self._x + + def __str__(self): + return self.x.__str__() + + @property + def x(self): + return self._x[self.variables] + + @x.setter + def x(self, value): + for j, i in enumerate(self.variables): + f = f_bounds[par_names[i]] + val = value[j] + h = self._x[i] + self._x[i] = f(val) + + +def phi(alpha, beta): + return beta * np.tan(np.pi * alpha / 2.0) + + +def _calculate_levy(x, alpha, beta, cdf=False): + """ Calculation of Levy stable distribution via numerical integration. + This is used in the creation of the lookup table. """ + # "0" parameterization as per http://academic2.americanp.edu/~jpnolan/stable/stable.html + # Note: fails for alpha=1.0 + # (so make sure alpha=1.0 isn't exactly on the interpolation grid) + from scipy import integrate + + C = beta * np.tan(np.pi * 0.5 * alpha) + + def func_cos(u): + ua = u ** alpha + if ua > 700.0: return 0.0 + return np.exp(-ua) * np.cos(C * ua - C * u) + + def func_sin(u): + ua = u ** alpha + if ua > 700.0: return 0.0 + return np.exp(-ua) * np.sin(C * ua - C * u) + + if cdf: + # Cumulative density function + return (integrate.quad(lambda u: u and func_cos(u) / u or 0.0, 0.0, integrate.Inf, weight="sin", wvar=x, + limlst=1000)[0] + + integrate.quad(lambda u: u and func_sin(u) / u or 0.0, 0.0, integrate.Inf, weight="cos", wvar=x, + limlst=1000)[0] + ) / np.pi + 0.5 + else: + # Probability density function + return (integrate.quad(func_cos, 0.0, integrate.Inf, weight="cos", wvar=x, limlst=1000)[0] + - integrate.quad(func_sin, 0.0, integrate.Inf, weight="sin", wvar=x, limlst=1000)[0] + ) / np.pi + + +def _levy_tan(x, alpha, beta, cdf=False): + """ Calculate the values stored in the lookup table. + The tan mapping allows the table to cover the range from -INF to INF. """ + x = np.tan(x) + return _calculate_levy(x, alpha, beta, cdf) + + +def _interpolate(points, grid, lower, upper): + """ Perform multi-dimensional Catmull-Rom cubic interpolationp. """ + point_shape = np.shape(points)[:-1] + points = np.reshape(points, (np.multiply.reduce(point_shape), np.shape(points)[-1])) + + grid_shape = np.array(np.shape(grid)) + dims = len(grid_shape) + + points = (points - lower) * ((grid_shape - 1) / (upper - lower)) + + floors = np.floor(points).astype('int') + + offsets = points - floors + offsets2 = offsets * offsets + offsets3 = offsets2 * offsets + weighters = [ + -0.5 * offsets3 + offsets2 - 0.5 * offsets, + 1.5 * offsets3 - 2.5 * offsets2 + 1.0, + -1.5 * offsets3 + 2 * offsets2 + 0.5 * offsets, + 0.5 * offsets3 - 0.5 * offsets2, + ] + + ravel_grid = np.ravel(grid) + + result = np.zeros(np.shape(points)[:-1], 'float64') + for i in xrange(1 << (dims * 2)): + weights = np.ones(np.shape(points)[:-1], 'float64') + ravel_offset = 0 + for j in xrange(dims): + n = (i >> (j * 2)) % 4 + ravel_offset = ravel_offset * grid_shape[j] + \ + np.maximum(0, np.minimum(grid_shape[j] - 1, floors[:, j] + (n - 1))) + weights *= weighters[n][:, j] + + result += weights * np.take(ravel_grid, ravel_offset) + + return np.reshape(result, point_shape) + + +def _approximate_pdf(x, alpha, beta): + return (1.0 + np.abs(beta)) * np.sin(np.pi * alpha / 2.0) * \ + sp.gamma(alpha) / np.pi * np.power(x, -alpha - 1.0) * alpha + + +def _approximate_cdf(x, alpha, beta): + return 1.0 - (1.0 + np.abs(beta)) * np.sin(np.pi * alpha / 2.0) * \ + sp.gamma(alpha) / np.pi * np.power(x, -alpha) + + +def _make_data_file(): + """ Generates the lookup table, writes it to a .py file. """ + import base64 + + size = (200, 50, 51) + pdf = np.zeros(size, 'float64') + cdf = np.zeros(size, 'float64') + xs, alphas, betas = [np.linspace(_lower[i], _upper[i], size[i], endpoint=True) for i in xrange(len(size))] + + print "Generating levy_data.py ..." + for i, alpha in enumerate(alphas): + for j, beta in enumerate(betas): + print "Calculating alpha={}, beta={}".format(alpha, beta) + for k, x in enumerate(xs): + pdf[k, i, j] = _levy_tan(x, alpha, beta) + cdf[k, i, j] = _levy_tan(x, alpha, beta, True) + + file = open("levy_data.py", "wt") + file.write(""" +# This is a generated file, do not edit. +import numpy, base64 + +pdf = numpy.loads(base64.decodestring( +\"\"\"%s\"\"\"))\n +cdf = numpy.loads(base64.decodestring( +\"\"\"%s\"\"\"))\n""" % + (base64.encodestring(pdf.dumps()), base64.encodestring(cdf.dumps()))) + file.close() + + +def _int_levy(x, alpha, beta, cdf=False): + """ Interpolate densities of the Levy stable distribution specified by alpha and beta. + + Specify cdf=True to obtain the *cumulative* density functionp. + + Note: may sometimes return slightly negative values, due to numerical inaccuracies. + """ + import levy_data + + points = np.empty(np.shape(x) + (3,), 'float64') + points[..., 0] = np.arctan(x) + points[..., 1] = alpha + points[..., 2] = beta + + if cdf: + what = levy_data.cdf + else: + what = levy_data.pdf + return _interpolate(points, what, _lower, _upper) + + +def _get_closest_approx(alpha, beta): + x0, x1, n = -50.0, 10000.0 - 50.0, 100000 + dx = (x1 - x0) / n + x = np.linspace(x0, x1, num=n, endpoint=True) + y = 1.0 - _int_levy(x, alpha, -beta, cdf=True) + z = 1.0 - _approximate_cdf(x, alpha, -beta) + mask = (10.0 < x) & (x < 500.0) + return 10.0 + dx * np.argmin((np.log(z[mask]) - np.log(y[mask])) ** 2.0) + + +def _make_approx_data_file(): + import base64 + + size = (50, 51) + limits = np.zeros(size, 'float64') + alphas, betas = [ + np.linspace(_lower[1], _upper[1], size[0], endpoint=True), + np.linspace(0, _upper[2], size[1], endpoint=True)] + + print "Generating levy_approx_data.py ..." + for i, alpha in enumerate(alphas): + for j, beta in enumerate(betas): + limits[i, j] = _get_closest_approx(alpha, beta) + print "Calculating alpha={}, beta={}, limit={}".format(alpha, beta, limits[i, j]) + + file = open("levy_approx_data.py", "wt") + file.write(""" +# This is a generated file, do not edit. +import numpy, base64 + +limits = numpy.loads(base64.decodestring( +\"\"\"%s\"\"\"))\n""" % + (base64.encodestring(limits.dumps()),)) + file.close() + + +def levy(x, alpha, beta, mu=0.0, sigma=1.0, cdf=False, par=0): + """ + Levy with the tail replaced by the analytical approximation. + Also, mu, sigma are parameters that shift and rescale the distribution. + Parametrization can be chosen according to Nolan, par={0,1}. + """ + + if par == 0: + loc = mu + elif par == 1: + loc = mu + beta * sigma * np.tan(np.pi * alpha / 2.0) # Par 1 is changed + + import levy_data + import levy_approx_data + + if cdf: + what = levy_data.cdf + app = _approximate_cdf + else: + what = levy_data.pdf + app = _approximate_pdf + limits = levy_approx_data.limits + + xr = (x - loc) / sigma + beta = -beta + alpha_index = int((alpha - 0.5) * 49.0 / 1.5) + beta_index = int(np.abs(beta) * 50.0) + l = limits[alpha_index, beta_index] + if beta <= 0.0: + mask = (xr < l) + elif beta > 0.0: + mask = (xr > -l) + z = xr[mask] + + points = np.empty(np.shape(z) + (3,), 'float64') + points[..., 0] = np.arctan(z) + points[..., 1] = alpha + points[..., 2] = beta + + interpolated = _interpolate(points, what, _lower, _upper) + approximated = app(xr[~mask], alpha, beta) + + res = np.empty(np.shape(xr), 'float64') + if cdf is False: + interpolated = interpolated / sigma + approximated = approximated / sigma + res[mask] = interpolated + res[~mask] = approximated + return res + + +def neglog_levy(x, alpha, beta, mu, sigma, par=0): + """ + Interpolate negative log densities of the Levy stable distribution specified by alpha and beta. + Small/negative densities are capped at 1e-100 to preserve sanity. + """ + # return -np.log(np.maximum(1e-100, levy(x, alpha, beta))) + return -np.log(np.maximum(1e-100, levy(x, alpha, beta, mu, sigma, par=par))) + + +def _reflect(x, lower, upper): + while 1: + if x < lower: + x = lower - (x - lower) + elif x > upper: + x = upper - (x - upper) + else: + return x + + +def fit_levy(x, alpha=None, beta=None, mu=None, sigma=None, par=0): + """ + Estimate parameters of Levy stable distribution given data x, using the Maximum Likelihood method. + + By default, searches all possible Levy stable distributions. + However you may restrict the search by specifying the values of one or more parameters. + + Examples: + + levy(x) -- Fit a stable distribution to x + + levy(x, beta=0.0) -- Fit a symmetric stable distribution to x + + levy(x, beta=0.0, mu=0.0) -- Fit a symmetric distribution centered on zero to x + + levy(x, alpha=1.0, beta=0.0) -- Fit a Cauchy distribution to x + + Returns a tuple of (alpha, beta, mu, sigma, negative log density) + """ + + # The parametrization is changed to par=0, if is par=1. At the end, the parametrization is reverted. + if mu is not None: + if par == 0: + mu0 = mu + elif par == 1: + mu0 = mu + beta * sigma * np.tan(np.pi * alpha / 2.0) # Par 1 is changed + elif mu is None: + mu0 = mu + + from scipy import optimize + + kwargs = {'alpha': alpha, 'beta': beta, 'mu': mu0, 'sigma': sigma} + parameters = Parameters(**kwargs) + + def neglog_density(param): + p = np.zeros(4) + p[parameters.variables] = param + p[parameters.fixed] = parameters.fixed_values + alpha, beta, mu, sigma = p + return np.sum(neglog_levy(x, alpha, beta, mu, sigma)) + + parameters.x = optimize.fmin(neglog_density, parameters.x, disp=0) + alpha, beta, mu, sigma = parameters.get_all() + return alpha, beta, mu, sigma, neglog_density(parameters.x) + + # + # from scipy import optimize + # + # if (type(x) is list) or (isinstance(x, (np.ndarray, np.generic))): + # if (mu is None) or (sigma is None): + # x = np.sort(x) + # last = len(x)-1 + # guess_mu0 = x[last/2] + # guess_sigma = (x[last-last/4] - x[last/4])/2.0 + # # Maybe there are lots of zeros or something... + # if guess_sigma == 0: + # guess_sigma = (x[last] - x[0]) / 2.0 + # + # def _neglog_density_fixed(parameters): + # return np.sum(neglog_levy(x, *parameters)) + # + # # Support for the Data class of Datagram package. + # # elif x.__class__.__name__ == 'Data': + # # if (mu is None) or (sigma is None): + # # last = len(x.x)-1 + # # guess_mu0 = x.x[last/2] + # # guess_sigma = (x.x[last-last/4] - x.x[last/4])/2.0 + # # + # # def _neglog_density_fixed(parameters): + # # return np.sum(x.counts * neglog_levy(x.x, *parameters)) + # + # parameters = [] + # + # if alpha is not None: + # get_alpha = lambda parameters: alpha + # else: + # get_alpha = lambda parameters, nth=len(parameters): \ + # _reflect(parameters[nth], _lower[1], _upper[1]) + # parameters.append(np.random.uniform(_lower[1], _upper[1])) + # + # if beta is not None: + # get_beta = lambda parameters: beta + # else: + # get_beta = lambda parameters, nth=len(parameters): \ + # _reflect(parameters[nth],_lower[2],_upper[2]) + # parameters.append(0) + # + # if mu is not None: + # get_mu0 = lambda parameters: mu0 + # else: + # get_mu0 = lambda parameters, nth=len(parameters): parameters[nth] + # parameters.append(guess_mu0) + # + # if sigma is not None: + # get_sigma = lambda parameters: sigma + # else: + # get_sigma = lambda parameters, nth=len(parameters): np.exp(parameters[nth]) + # parameters.append(np.log(guess_sigma)) + # + # def _neglog_density(parameters): + # alpha = get_alpha(parameters) + # beta = get_beta(parameters) + # mu0 = get_mu0(parameters) + # sigma = get_sigma(parameters) + # return _neglog_density_fixed([alpha, beta, mu0, sigma]) + # + # parameters = optimize.fmin(_neglog_density, parameters, disp=0) + # + # alpha = get_alpha(parameters) + # beta = get_beta(parameters) + # mu0 = get_mu0(parameters) + # sigma = get_sigma(parameters) + # neglog_final = _neglog_density(parameters) + + if par == 0: + mu = mu0 + elif par == 1: + mu = mu0 - beta * sigma * np.tan(np.pi * alpha / 2.0) + + return alpha, beta, mu, sigma, neglog_final + + +def random(alpha, beta, mu=0.0, sigma=1.0, shape=(), par=0): + """ + Generate random values sampled from an alpha-stable distribution. + """ + + if par == 0: + mu0 = mu + elif par == 1: + mu0 = mu + beta * sigma * np.tan(np.pi * alpha / 2.0) # Par 1 is changed + + if alpha == 2: + return np.random.standard_normal(shape) * np.sqrt(2.0) + + # Fails for alpha exactly equal to 1.0 + # but works fine for alpha infinitesimally greater or less than 1.0 + radius = 1e-15 # <<< this number is *very* small + if np.absolute(alpha - 1.0) < radius: + # So doing this will make almost exactly no difference at all + alpha = 1.0 + radius + + r1 = np.random.random(shape) + r2 = np.random.random(shape) + pi = np.pi + + a = 1.0 - alpha + b = r1 - 0.5 + c = a * b * pi + e = phi(alpha, beta) + f = (-(np.cos(c) + e * np.sin(c)) / (np.log(r2) * np.cos(b * pi))) ** (a / alpha) + g = np.tan(pi * b / 2.0) + h = np.tan(c / 2.0) + i = 1.0 - g ** 2.0 + j = f * (2.0 * (g - h) * (g * h + 1.0) - (h * i - 2.0 * g) * e * 2.0 * h) + k = j / (i * (h ** 2.0 + 1.0)) + e * (f - 1.0) + + return mu0 + sigma * k + + +if __name__ == "__main__": + if "build" in sys.argv[1:]: + _make_data_file() + _make_approx_data_file() + + print "Testing fit_levy." + + print "1000 points, result should be (1.5, 0.5, 0.0, 1.0)." + print fit_levy(random(1.5, 0.5, 0.0, 1.0, 1000)) diff --git a/levy_approx_data.py b/levy_approx_data.py new file mode 100644 index 0000000..aec4fe5 --- /dev/null +++ b/levy_approx_data.py @@ -0,0 +1,367 @@ + +# This is a generated file, do not edit. +import numpy, base64 + +limits = numpy.loads(base64.decodestring( +"""gAJjbnVtcHkuY29yZS5tdWx0aWFycmF5Cl9yZWNvbnN0cnVjdApxAWNudW1weQpuZGFycmF5CnEC +SwCFVQFih1JxAyhLAUsySzOGY251bXB5CmR0eXBlCnEEVQJmOEsASwGHUnEFKEsDVQE8Tk5OSv// +//9K/////0sAdGKJVLBPAAA0MzMzM3NIQGdmZmZmZkhAZ2ZmZmZmSEBnZmZmZmZIQJqZmZmZWUhA 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+GI61DD7TENdOeWQLPgsgP+tuEwo++IT95G7CCD653B8zeXEHPl6mM8uNIAY+HVwSpKzPBD5LvYWz +1X4DPgJSGO8ILgI+hUv+S0bdAD5rD+V+Gxn/Pe3/Vk2IZhw+Y/6qWGjrGz7LR/SQS3AbPrqKQvMx +9Ro+mpCQfBt6Gj41ohYqCP8ZPnbg7/j3gxk+iBKJ5uoIGT6TWBHw4I0YPoXmIBPaEhg+UpEHTdaX +Fz7+oqeb1RwXPspggvzXoRY+roAgbd0mFj4UsD7r5asVPsZzuHTxMBU+SxmLBwC2FD4jMxehETsU +PsgKvD8mwBM+FqBD4T1FEz5Uk1uDWMoSPh5vVSR2TxI+wXjrwZbUET4GDVpaulkRPh5LhOvg3hA+ +DKtscwpkED5G3zjgbdIPPvFBLr/M3A4+VjLKfzHnDT6mf1QenPEMPkKTVpYM/As+4W1O5IIGCz5t +bHgD/xAKPqb7QPCAGwk+c77ApggmCD6S/MUhljAHPhsh+l0pOwY+oTSlVsJFBT4TfCgHYVAEPjUr +82oFWwM+KiGqfa9lAj59Eck6X3ABPgQsz5wUewA+wbdDPp8L/z1LRqJ4ICH9PapO896sNvs9jTLg +ZURM+T2aaroC52H3PXOBuKmUd/U9YdJST02N8z1jRvvmEKPxPYiYl0NYQhE+yx1oFn7uED7pXLhp +pZoQPlGfRDzORhA+RUQ2GfHlDz4t3tiySD4PPt067EKjlg4+fDSLxgDvDT7fxI87YUcNPi9HaZ/E +nww+Y72I7yr4Cz7SMbsplFALPh5fvEsAqQo+hyMKU28BCj5AcbU94VkJPkTkeAlWsgg+iibts80K +CD5DvjU7SGMHPh0OC53FuwY+1UBm10UUBj6Unx3oyGwFPtXgV81OxQQ+RnAGhdcdBD5R9sYMY3YD +PmTS92LxzgI+bYGAhYInAj5o0ThyFoABPh3sQyet2AA+J8mzokYxAD5XQUHExRP/Pb/JvccDxf09 ++qTSS0d2/D3kDP1LkCf7PWK+R8Te2Pk9XMBksDKK+D1LEW8MjDv3PTcXkdPq7PU9tKLuAU+e9D3a +gkSTuE/zPV5egYInAfI9fLmTy5uy8D1OqGjTKsjuPc+AFbAoK+w92KnTIzGO6T0EXGclRPHmPZHb +XalhVOQ9kb/Qpom34T311MAleDXePXdiJ8Xx+9g9VXj6FoDC0z3QiO4ERhLNPcfuh+0fJQQ+Qvm9 +yae/Az43mwjJMFoDPll+j+q69AI+ycc4LUaPAj73MV+Q0ikCPjk+zBJgxAE+8inqs+5eAT6RnGNy +fvkAPguD6k0PlAA+ratbRaEuAD6aI6evaJL/PRlYYwmRx/49Vq7jlbv8/T051V1T6DH9PRqqsEAX +Z/w9I8saXEic+z3Ng3Wje9H6PfT9TxaxBvo9vl7rseg7+T0ALnd1InH4PdfxCl9epvc9ys0EbZzb +9j3G5gye3BD2Pc7aWPAeRvU9lb/pYmN79D3Hk2rzqbDzPapAGKHy5fI9m/PYaT0b8j2sLV1MilDx +PUYj4UbZhfA9wy/Rr1R27z2+b3X8+uDtPd4uB2+lS+w98/SxBlS26j0gmvu9BiHpPSGny5K9i+c9 +fy4ygnj25T09BvOHN2HkPWsqE6H6y+I9l7AFy8E24T3U6z0DGkPfPbhXVoO4GNw9NmTHEV/u2D3u +IimjDcTVPa6jMzTEmdI9tkfCdgXfzj2zshZokorIPYKMwikvNsI9dZuwXrfDtz21RH2jXzamPckh +UVC5x/U9DzS2BrBr9T2UQRM9pw/1Ped/Q/Oes/Q9mJDqKJdX9D39vc7dj/vzPUe5fRGJn/M9hnnj +w4JD8z3oitn0fOfyPYr5saN3i/I9tjt90HIv8j0kCt16btPxPeAdkqJqd/E9da8/R2cb8T3TM8Ro +ZL/wPc1R7gZiY/A9pqUrIWAH8D23b+1uvVbvPX7CGpO7nu49DrAerrrm7T3tEZe/ui7tPUdh7sa7 +duw90ETGw72+6z2erLq1wAbrPSOjhpvETuo9Q+d+dcmW6T2X1IRCz97oPdrZrALWJug9GBKYtN1u +5z0PGoFY5rbmPQCawe3v/uU9GP2sc/pG5T2sz83pBY/kPZUEe08S1+M9RXhppB8f4z1YzxDoLWfi +PanFbxk9r+E9Y76dOE334D2BTe9EXj/gPT8xMnvgDt89XDmKRAaf3T24WsTkLS/cPWH5NltXv9o9 +aM98pYJP2T1OzCfEr9/XPe0EqbXeb9Y9kCYpeA8A1T17TEMLQpDTPRABrm12INI9GFGsnqyw0D2q +ajs5yYHOPSXYKrgACG88VT+ztwAIbzyGpju3AAhvPLYNxLYACG8853RMtgAIbzwX3NS1AAhvPEZD +XbUACG88d6rltAAIbzynEW60AAhvPNh49rMACG88COB+swAIbzw5RwezAAhvPGmuj7IACG88mhUY +sgAIbzzLfKCxAAhvPPvjKLEACG88LEuxsAAIbzxcsjmwAAhvPI0Zwq8ACG88vIBKrwAIbzzs59Ku +AAhvPB1PW64ACG88TbbjrQAIbzx+HWytAAhvPK6E9KwACG883+t8rAAIbzwPUwWsAAhvPEC6jasA +CG88cCEWqwAIbzyhiJ6qAAhvPNLvJqoACG88AlevqQAIbzwxvjepAAhvPGIlwKgACG88koxIqAAI +bzzD89CnAAhvPPNaWacACG88JMLhpgAIbzxUKWqmAAhvPIWQ8qUACG88tfd6pQAIbzzmXgOlAAhv +PBbGi6QACG88Ry0UpAAIbzx3lJyjAAhvPKf7JKMACG8812KtogAIbzwIyjWiAAhvPDgxvqEACG88 +aZhGoQAIbzyZ/86gAAhvPHRiLg== +""")) + +cdf = numpy.loads(base64.decodestring( +"""gAJjbnVtcHkuY29yZS5tdWx0aWFycmF5Cl9yZWNvbnN0cnVjdApxAWNudW1weQpuZGFycmF5CnEC +SwCFVQFih1JxAyhLAUvISzJLM4djbnVtcHkKZHR5cGUKcQRVAmY4SwBLAYdScQUoSwNVATxOTk5K +/////0r/////SwB0YolUgEE+AAAAAABAuKq9ADrML7EcRD8A4JqNvB9UPwBTFpssNF4/ANvbvNQl +ZD8AapPPGTNpP4AjjWjmQW4/gOAImhypcT9ASwzDCTJ0P0Cs6p67u3Y/QKuHzjBGeT/A3M5sa9F7 +P0BHN39pXX4/QO4bgRZ1gD9gu1172ruBP0DNewYBA4M/ABW8VYpKhD9AFeTIdZKFP8Bw0DXE2oY/ +AI0qgnUjiD8AOKutiWyJP8ADA84Atoo/wNIxAtv/iz+gmRpZGEqNP2DSmuq4lI4/YMOaybzfjz9g +IAgGkpWQPzCl62J3O5E/wOf/ho7hkT8Anjx714eSP+Avw0xSLpM/8OZjAP/Ukz+AP86f3XuUP6A6 +VEDuIpU/YCbHBDHKlT/wtIuXpXGWP/AFGV1MGZc/wMJyTyXBlz/AwZhWMGmYPxC7psdtEZk/wM/I +Od25mT/wGXdBf2KaP6ATNGxTC5s/cBnKJFq0mz+wmM5Ck12cP2BLIrH+Bp0/QL0fsZywnT/Au9c8 +bVqeP4AtLmFwBJ8/UL1QIqaunz+4v85GhyygPwAAAAAi9cc9AOI7IjpeQD8ABi0cnGBQPwA208p9 +lFg/AEbfUmFlYD8ASOiWtYFkPwBY9/w7n2g/gPmBw/O9bD9AEbwZ8G5wP8BhOB9/f3I/wOb66qeQ +dD+AkickaqJ2P8CCK9zGtHg/gCC+R73Hej+AMGK/Tdt8P0CeS654734/wAm5ux6CgD+gZJ3bzoyB +P+CWLx/Ml4I/wE8Lyxajgz/AjSkQr66EPyD3/66UuoU/oIvkBsjGhj/gvQINSdOHP8D/vtYX4Ig/ +ICficzTtiT8A33j2nvqKP4CGpW9XCIw/YOHS8l0WjT/At6GNsiSOP2BuZm1VM48/MBOVKiMhkD9w +fbzTwqiQP7B0oauJMJE/AJTyoXe4kT9gyesTjUCSP/AAQKvJyJI/ELUWoC1Rkz+AGmLpuNmTP3Dl +doNrYpQ/sHhLs0XrlD8g6YVFR3SVP7CepWBw/ZU/QM1d18CGlj8gY8UdORCXP7ANqt3YmZc/cGkx +QaAjmD+QeUVPj62YP0DPvg+mN5k/UMhUi+TBmT+Qjy7LSkyaPwAAAACAEo29AHSQaF+gOj8AsCua +FaRKPwCUSFLZ/VM/AN3BV4OrWj+Ak51rha1gPwA0d103BmQ/AKro/9dfZz+A8BJ9Z7pqP4BFSxPm +FW4/ACc1+im5cD+AyKS52GdyP8Cktzz/FnQ/QHEQ8p3GdT+AY8G9tHZ3PwAfA8pDJ3k/wKNx5krY +ej9AGHWvyol8P4CibAfDO34/wKzUyTPufz8AA5LFjtCAP+BkPBNAqoE/oPMZ3i2Egj/A/FY1WF6D +P2BC1Ce/OIQ/IB8rxmIThT8g2JUaQ+6FPyCycTdgyYY/oOe0KLqkhz/AVdb7UICIP0A1ur0kXIk/ +wK5VejU4ij8AONpggxSLP8AdJkQO8Ys/IIcqU9bNjD9gVoG426qNP4CghkEeiI4/QBbrIp5ljz9g 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+++8/+uXAt5D87z9K0MUZbv3vPyPmRXBK/u8/IKuTvCX/7z/qAQAAAADwP/yaKa8o1u8/JkM93BfX +7z+vHl3wBdjvP6mQ3uzy2O8/P3cV097Z7z/fLFSkydrvP3KJ62Gz2+8/huMqDZzc7z+GEWCng93v +P+Rq1zFq3u8/VsnbrU/f7z/6ibYcNODvP5yOr38X4e8/3D4N2Pnh7z9piRQn2+LvPzXlCG674+8/ +rFIsrprk7z/kXL/oeOXvP+AaAR9W5u8/tjAvUjLn7z/W0IWDDejvPze9P7Tn6O8/lEiW5cDp7z+o +V8EYmervP1li905w6+8/AnVtiUbs7z+gMVfJG+3vPxTR5g/w7e8/VCRNXsPu7z+wlbm1le/vP/8p +Whdn8O8/5oFbhDfx7z8O2+j9BvLvP10RLIXV8u8/MqBNG6Pz7z+jo3TBb/TvP7XZxng79e8/mqNo +Qgb27z/qBn0f0PbvP+SuJRGZ9+8/ou2CGGH47z9evbM2KPnvP6XB1Wzu+e8/mkgFvLP67z8vTF0l +ePvvP2Bz96k7/O8/chPsSv787z8sMVIJwP3vPxmCP+aA/u8/u23I4kD/7z9mBwAAAADwP15DZp2o +2+8/ROMcdnfc7z+d72ZlRd3vP7AKV2wS3u8/lqL+i97e7z8u8m3Fqd/vPxoCtBl04O8/vqneiT3h +7z80kPoWBuLvP1EtE8LN4u8/mMoyjJTj7z8/hGJ2WuTvPyJKqoEf5e8/yOAQr+Pl7z9c4pv/pubv +P6m/T3Rp5+8/GsEvDivo7z+2Bz7O6+jvPxuOe7Wr6e8/finoxGrq7z+uioL9KOvvPwA/SGDm6+8/ +ZrE17qLs7z9YK0aoXu3vP9rVc48Z7u8/frq3pNPu7z9WxAnpjO/vPwPBYF1F8O8/omGyAv3w7z/Y 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idea of this file is to be able to test the accuracy of the fit. +n_iter sets of size n_data are generated with a Levy distribution of +parameters alpha, beta, mu, sigma. +The 50%, 5% and 95% quantiles of the distributions of the 4 parameters +are returned. +""" + +import levy +import numpy as np + + +def get_quantiles(l): + n = float(len(l)) + return l[int(n * 0.5)], l[int(n * 0.05)], l[int(n * 0.95)] + +alpha = 1.0 +beta = 0.0 +mu = 0.0 +sigma = 1.0 + +n_iter = 100 +n_data = 1000 + +parameters_list = [] +for _ in range(n_iter): + data = levy.random(alpha, 0.0, 0.0, 1.0, n_data) + parameters = levy.fit_levy(data) + parameters_list.append(parameters) + if _ % 20 == 0: + print _ + +alphas = sorted([_[0] for _ in parameters_list]) +betas = sorted([_[1] for _ in parameters_list]) +mus = sorted([_[2] for _ in parameters_list]) +sigmas = sorted([_[3] for _ in parameters_list]) + +print get_quantiles(alphas) +print get_quantiles(betas) +print get_quantiles(mus) +print get_quantiles(sigmas)