Source code for openquake.hmtk.seismicity.max_magnitude.kijko_nonparametric_gaussian

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'''
Module
:mod:`openquake.hmtk.seismicity.max_magnitude.kijko_nonparametric_gaussian`
implements the Non-Parametric Gaussian estimator of maximum magnitude
proposed by Kijko (2004)
'''
import numpy as np
from scipy.stats.mstats import mquantiles
from openquake.hmtk.seismicity.max_magnitude.base import (
    BaseMaximumMagnitude, MAX_MAGNITUDE_METHODS)


[docs]def check_config(config): '''Check config file inputs and overwrite bad values with the defaults''' essential_keys = ['number_earthquakes'] for key in essential_keys: if key not in config: raise ValueError('For Kijko Nonparametric Gaussian the key %s ' 'needs to be set in the configuation' % key) if config.get('tolerance', 0.0) <= 0.0: config['tolerance'] = 0.05 if config.get('maximum_iterations', 0) < 1: config['maximum_iterations'] = 100 if config.get('number_samples', 0) < 2: config['number_samples'] = 51 return config
def _get_exponential_spaced_values(mmin, mmax, number_samples): ''' Function to return a set of exponentially spaced values between mmin and mmax :param float mmin: Minimum value :param float mmax: Maximum value :param float number_samples: Number of exponentially spaced samples :return np.ndarray: Set of 'number_samples' exponentially spaced values ''' lhs = np.exp(mmin) + np.arange(0., number_samples - 1., 1.) *\ ((np.exp(mmax) - np.exp(mmin)) / (number_samples - 1.)) magval = np.hstack([lhs, np.exp(mmax)]) return np.log(magval)
[docs]@MAX_MAGNITUDE_METHODS.add( "get_mmax", number_earthquakes=np.float, number_samples=51, maximum_iterations=100, tolerance=0.05) class KijkoNonParametricGaussian(BaseMaximumMagnitude): ''' Class to implement non-parametric Gaussian methodology of Kijko (2004) '''
[docs] def get_mmax(self, catalogue, config): ''' Calculates maximum magnitude :param catalogue: Instance of :class: openquake.hmtk.seismicity.catalogue.Catalogue :param dict config: Configuration parameters - including: * 'number_earthquakes': Number of largest magnitudes to consider * 'number_samples' [optional]: Number of samples for integral {default=51} * 'maximum_iterations' [optional]: Maximum number of iterations {default=100} * 'tolerance' [optional]: Magnitude difference threshold for iterstor stability {default=0.05} :returns: Maximum magnitude and its uncertainty ''' config = check_config(config) # Unlike the exponential distributions, if the input mmax is # greater than the observed mmax the integral expands rapidly. # Therefore, only observed mmax is considered max_loc = np.argmax(catalogue.data['magnitude']) obsmax = catalogue.data['magnitude'][max_loc] if not(isinstance(catalogue.data['sigmaMagnitude'], np.ndarray)) or\ (len(catalogue.data['sigmaMagnitude']) == 0) or\ np.all(np.isnan(catalogue.data['sigmaMagnitude'])): obsmaxsig = 0. else: obsmaxsig = catalogue.data['sigmaMagnitude'][max_loc] # Find number_eqs largest events n_evts = np.shape(catalogue.data['magnitude'])[0] if n_evts <= config['number_earthquakes']: # Catalogue smaller than number of required events mag = np.copy(catalogue.data['magnitude']) neq = np.float(np.shape(mag)[0]) else: # Select number_eqs largest events mag = np.sort(catalogue.data['magnitude'], kind='quicksort') mag = mag[-config['number_earthquakes']:] neq = float(config['number_earthquakes']) mmin = np.min(mag) # Get smoothing factor hfact = self.h_smooth(mag) mmax = np.copy(obsmax) d_t = mmax.item() - 0. iterator = 0 while d_t > config['tolerance']: # Generate exponentially spaced samples magval = _get_exponential_spaced_values(mmin, mmax.item(), config['number_samples']) # Evaluate integral function using Simpson's method delta = self._kijko_npg_intfunc_simps(magval, mag, mmax.item(), hfact, neq) tmmax = obsmax + delta d_t = np.abs(tmmax - mmax.item()) mmax = np.copy(tmmax) iterator += 1 if iterator > config['maximum_iterations']: print('Kijko-Non-Parametric Gaussian estimator reached' 'maximum # of iterations') d_t = -np.inf return mmax.item(), np.sqrt(obsmaxsig ** 2. + (mmax.item() - obsmax) ** 2.)
[docs] def h_smooth(self, mag): ''' Function to calculate smoothing coefficient (h) for Gaussian Kernel estimation - based on Silverman (1986) formula :param numpy.ndarray mag: Magnitude vector :returns: Smoothing coefficient (h) (float) ''' neq = np.float(len(mag)) # Calculate inter-quartile range qtiles = mquantiles(mag, prob=[0.25, 0.75]) iqr = qtiles[1] - qtiles[0] hfact = 0.9 * np.min([np.std(mag), iqr / 1.34]) * (neq ** (-1. / 5.)) # Round h to 2 dp hfact = np.round(100. * hfact) / 100. return hfact
def _gauss_cdf_hastings(self, xval, barx=0.0, sigx=1.0): '''Function to implement Hasting's approximation of the normalised cumulative normal function - this is taken from Kijko's own code so I don't really know why this is here!!!!! :param np.ndarray xval: x variate :param float barx: Mean of the distribution :param float sigx: Standard Deviation :return float yval: Gaussian Cumulative Distribution ''' x_norm = (xval - barx) / sigx # Fixed distribution co-efficients a_1 = 0.196854 a_2 = -0.115194 a_3 = 0.000344 a_4 = 0.019527 x_a = np.abs(x_norm) yval = 1.0 - 0.5 * (1. + a_1 * x_a + (a_2 * (x_a ** 2.)) + (a_3 * (x_a ** 3.)) + (a_4 * (x_a ** 4.))) ** (-4.) # Finally to normalise yval[x_norm < 0.] = 1. - yval[x_norm < 0.] # To deal with precision errors for tail ends yval[x_norm < -5.] = 0. yval[x_norm > 5.] = 1. return yval def _kijko_npg_intfunc_simps(self, mval, mag, mmax, hfact, neq): '''Integral function for non-parametric Gaussuan assuming that Simpson's rule has been invoked for exponentially spaced samples :param numpy.ndarray mval: Target Magnitudes :param numpy.ndarray mag: Observed Magnitude values :param float mmax: Maximum magnitude for integral :param float hfact: Smoothing coefficient (output of h_smooth) :param float neq: Number of earthquakes (effectively the length of mag) :return float intfunc: Integral of non-Parametric Gaussian function ''' nmval = len(mval) # Mmin and Mmax must be arrays to allow for indexing in # _gauss_cdf_hastings mmin = np.min(mag) p_min = self._gauss_cdf_hastings((mmin - mag) / hfact) p_max = self._gauss_cdf_hastings((mmax - mag) / hfact) cdf_func = np.zeros(nmval) for ival, target_mag in enumerate(mval): # Calculate normalised magnitudes p_mag = self._gauss_cdf_hastings((target_mag - mag) / hfact) cdf_func[ival] = ((np.sum(p_mag) - np.sum(p_min)) / (np.sum(p_max) - np.sum(p_min))) ** neq # Now to perform integration via mid-point rule intfunc = 0.5 * cdf_func[0] * (mval[1] - mval[0]) for iloc in range(1, nmval - 1): intfunc = intfunc + (0.5 * cdf_func[iloc] * (mval[iloc + 1] - mval[iloc - 1])) intfunc = intfunc + (0.5 * cdf_func[-1] * (mval[-1] - mval[-2])) return intfunc