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# Copyright (C) 2010-2021 GEM Foundation, G. Weatherill, M. Pagani, D. Monelli
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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