"""
Module containing classical generalistic models
Gumbel:
To be used applying the Block Maxima approach
Generalised extreme value distribution (GEV):
To be used applying the Block Maxima approach
Generalised Pareto Distribution (GPD):
To be used applying the Peak-Over-Threshold approach
TODO
"""
from collections import OrderedDict
from scipy import stats as _st
from scipy import optimize as _op
from lmoments3 import distr as _lmdistr
import numpy as _np
import matplotlib.pyplot as _plt
import numdifftools as _ndt
from ..utils import bootstrap_ci as _bsci
from ..utils import gev_momfit as _gev_momfit
from ..utils import gum_momfit as _gum_momfit
class _Base:
def __init__(self, data, fit_method = 'mle',
ci = 0, ci_method = None,
return_periods = None, frec = 1):
# Data to be used for the fit
self.data = data
# Fit method to be used
if fit_method in ['mle', 'mom', 'lmoments']:
self.fit_method = fit_method
else:
raise ValueError(
("fit methods accepted are:\n"
" mle (Maximum Likelihood Estimation)\n"
" lmoments\n"
" mom (method of moments)\n")
)
# Calculate shape, location, scale and a frozen distribution
# with the calculated estimators (shape, location, scale)
self._fit()
# Check for calculations of return periods and return values.
self.frec = frec
if return_periods:
self.return_periods = _np.array(return_periods)
self.return_values = self.distr.isf(self.frec /
self.return_periods)
else:
self.return_periods = _np.array([])
self.return_values = _np.array([])
# Check for the estimation of confidence intervals
if ci == 0 or 0 < ci < 1:
self.ci = ci
else:
raise ValueError("ci should be a value in the interval 0 < ci < 1")
if self.ci:
if (ci_method and
fit_method == 'mle' and
ci_method in ['delta', 'bootstrap']):
self.ci_method = ci_method
self._ci()
elif (ci_method and
fit_method == 'lmoments' and
ci_method in ['bootstrap']):
self.ci_method = ci_method
self._ci()
elif (ci_method and
fit_method == 'mom' and
ci_method in ['bootstrap']):
self.ci_method = ci_method
self._ci()
else:
raise ValueError(
("You should provide a valid value for the confidence\n"
"interval calculation, 'ci_method'\n"))
def _fit(self):
# This is a base class and shouldn't be used as it.
# This method should be implemented in the subclass.
raise NotImplementedError("Subclasses should implement this!")
def _ci(self):
# This is a base class and shouldn't be used as it.
# This method should be implemented in the subclass.
raise NotImplementedError("Subclasses should implement this!")
def pdf(self, quantiles):
# A shortcut to the frozen distribution pdf as provided by scipy.
"""
Probability density function at x of the given frozen RV.
**Parameters**
x : array_like
quantiles
**Returns**
pdf : ndarray
Probability density function evaluated at x
"""
return self.distr.pdf(quantiles)
def cdf(self, quantiles):
# A shortcut to the frozen distribution cdf as provided by scipy.
"""
Cumulative distribution function of the given frozen RV.
**Parameters**
x : array_like
quantiles
**Returns**
cdf : ndarray
Cumulative distribution function evaluated at `x`
"""
return self.distr.cdf(quantiles)
def ppf(self, q):
# A shortcut to the frozen distribution ppf as provided by scipy.
"""
Percent point function (inverse of cdf) at q of the given frozen RV.
**Parameters**
q : array_like
lower tail probability
**Returns**
x : array_like
quantile corresponding to the lower tail probability q.
"""
return self.distr.ppf(q)
def stats(self, moments):
# A shortcut to the frozen distribution stats as provided by scipy.
"""
Some statistics of the given RV.
**Parameters**
moments : str, optional
composed of letters ['mvsk'] defining which moments to compute:
'm' = mean,
'v' = variance,
's' = (Fisher's) skew,
'k' = (Fisher's) kurtosis.
(default='mv')
**Returns**
stats : sequence
of requested moments.
"""
return self.distr.stats(moments)
def _plot(self, ax, title, xlabel, ylabel):
# helper function for:
# self.plot_density()
# self.plot_pp()
# self.plot_qq()
# self.plot_return_values()
# self.plot_summary()
ax.set_facecolor((0.95, 0.95, 0.95))
_plt.setp(ax.lines, linewidth = 2, color = 'magenta')
ax.set_title(title)
ax.set_xlabel(xlabel)
ax.set_ylabel(ylabel)
ax.grid(True)
return ax
def plot_density(self):
"""
Histogram of the empirical pdf data and the pdf plot of the
fitted distribution.
All parameters are predefined from the frozen fitted model and empirical
data available.
**Returns**
Density plot.
"""
fig, ax = _plt.subplots(figsize=(8, 6))
# data
x = _np.linspace(self.distr.ppf(0.001), self.distr.ppf(0.999), 100)
# plot
ax.plot(x, self.distr.pdf(x), label = 'Fitted', color = 'k')
ax.hist(self.data, density = True,
color = 'yellow', alpha = 0.75, label = "Empirical")
ax = self._plot(ax, 'Density Plot', 'x', 'f(x)')
ax.legend(loc='best', frameon=False)
def plot_pp(self):
"""
PP (probability) plot between empirical and fitted data.
All parameters are predefined from the frozen fitted model and empirical
data available.
**Returns**
PP plot.
"""
fig, ax = _plt.subplots(figsize=(8, 6))
# data
data = _np.sort(self.data)
N = len(data)
y = _np.arange(1, N + 1) / (N + 1)
x = self.distr.cdf(data)
# plot
ax.scatter(x, y, color = 'darkcyan')
ax.plot([0, 1], [0, 1])
ax.set_xlim(0, 1)
ax.set_ylim(0, 1)
ax = self._plot(ax, 'P-P Plot', 'model', 'empirical')
def plot_qq(self):
"""
QQ (Quantile-Quantile) plot between empirical and fitted data.
All parameters are predefined from the frozen fitted model and empirical
data available.
**Returns**
QQ plot.
"""
fig, ax = _plt.subplots(figsize=(8, 6))
# data
y = _np.sort(self.data)
N = len(y)
x = _np.arange(1, N + 1) / (N + 1)
x = self.distr.ppf(x)
# plot
ax = self._plot(ax, 'Q-Q Plot', 'model', 'empirical')
ax.scatter(x, y, color = 'forestgreen')
low_lim = _np.min([x, y]) * 0.95
high_lim = _np.max([x, y]) * 1.05
ax.plot([low_lim, high_lim], [low_lim, high_lim], c='k')
ax.set_xlim(low_lim, high_lim)
ax.set_ylim(low_lim, high_lim)
def plot_return_values(self):
"""
Return values and return periods of data. If confidence interval
information has been provided it will show the confidence interval
values.
**Returns**
Return values and return periods plot.
"""
fig, ax = _plt.subplots(figsize=(8, 6))
# data
T = _np.arange(0.1, 500.1, 0.1)
sT = self.distr.isf(self.frec * 1./T)
N = _np.r_[1:len(self.data)+1] * self.frec
Nmax = max(N)
# plot
ax = self._plot(ax, 'Return Level Plot', 'Return period', 'Return level')
ax.semilogx(T, sT)
ax.scatter(self.frec * Nmax/N, sorted(self.data)[::-1], color = 'orangered')
# plot confidence intervals if available
if self.ci:
#y1 = sT - st.norm.ppf(1 - self.ci / 2) * np.sqrt(self._ci_se)
#y2 = sT + st.norm.ppf(1 - self.ci / 2) * np.sqrt(self._ci_se)
ax.semilogx(T, self._ci_Td, '--')
ax.semilogx(T, self._ci_Tu, '--')
ax.fill_between(T, self._ci_Td, self._ci_Tu, color = '0.75', alpha = 0.5)
def plot_summary(self):
"""
Summary plot including PP plot, QQ plot, empirical and fitted pdf and
return values and periods.
**Returns**
4-panel plot including PP, QQ, pdf and return level plots
"""
fig, ((ax3, ax2), (ax4, ax1)) = _plt.subplots(2, 2, figsize=(8, 6))
# PDF plot
x = _np.linspace(self.distr.ppf(0.001),
self.distr.ppf(0.999),
100)
ax1.plot(x, self.distr.pdf(x), label = 'Fitted')
ax1.hist(self.data, density = True,
color = 'yellow', alpha = 0.75, label = "Empirical")
ax1 = self._plot(ax1, 'Density Plot', 'x', 'f(x)')
ax1.legend(loc='best', frameon=False)
# QQ plot
data = _np.sort(self.data)
N = len(data)
y = _np.arange(1, N + 1) / (N + 1)
x = self.distr.cdf(data)
ax2.plot([0, 1], [0, 1])
ax2.set_xlim(0, 1)
ax2.set_ylim(0, 1)
ax2 = self._plot(ax2, 'P-P Plot', 'model', 'empirical')
ax2.scatter(x, y, color = 'darkcyan')
# PP Plot
y = _np.sort(self.data)
N = len(y)
x = _np.arange(1, N + 1) / (N + 1)
x = self.distr.ppf(x)
ax3.scatter(x, y, color = 'forestgreen')
low_lim = _np.min([x, y]) * 0.95
high_lim = _np.max([x, y]) * 1.05
ax3.plot([low_lim, high_lim], [low_lim, high_lim])
ax3.set_xlim(low_lim, high_lim)
ax3.set_ylim(low_lim, high_lim)
ax3 = self._plot(ax3, 'Q-Q Plot', 'model', 'empirical')
# Return levels plot
T = _np.arange(0.1, 500.1, 0.1)
sT = self.distr.isf(self.frec/T)
N = _np.r_[1:len(self.data)+1] * self.frec
Nmax=max(N)
ax4 = self._plot(ax4, 'Return Level Plot',
'Return period',
'Return level')
ax4.semilogx(T, sT, 'k')
ax4.scatter(self.frec * Nmax/N, sorted(self.data)[::-1], color = 'orangered')
if self.ci:
#y1 = sT - st.norm.ppf(1 - self.ci / 2) * np.sqrt(self._ci_se)
#y2 = sT + st.norm.ppf(1 - self.ci / 2) * np.sqrt(self._ci_se)
ax4.semilogx(T, self._ci_Td, '--')
ax4.semilogx(T, self._ci_Tu, '--')
ax4.fill_between(T, self._ci_Td, self._ci_Tu, color = '0.75', alpha = 0.5)
# I love matplotlib for stuff like this, thanks, guys!!!
_plt.tight_layout()
return fig, ax1, ax2, ax3, ax4
[docs]class GEV(_Base):
"""
Class to fit data to a Generalised extreme value (GEV) distribution.
**Parameters**
data : array_like
1D array_like with the extreme values to be considered
fit_method : str
String indicating the method used to fit the distribution.
Availalable values are 'mle' (default value), 'mom' and 'lmoments'.
ci : float (optional)
Float indicating the value to be used for the calculation of the
confidence interval. The returned values are (ci/2, 1-ci/2)
percentile confidence intervals. E.g., a value of 0.05 will
return confidence intervals at 0.025 and 0.975 percentiles.
ci_method : str (optional)
String indicating the method to be used to calculate the
confidence intervals. If ``ci`` is not supplied this parameter will
be ignored. Possible values depend of the fit method chosen. If
the fit method is 'mle' possible values for ci_method are
'delta' and 'bootstrap', if the fit method is 'mom' or
'lmoments' possible value for ci_method is 'bootstrap'.
'delta' is for delta method.
'bootstrap' is for parametric bootstrap.
return_period : array_like (optional)
1D array_like of values for the *return period*. Values indicate
**years**.
frec : int or float
Value indicating the frecuency of events per year. If frec is
not provided the data will be treated as yearly data (1 value per
year).
**Attributes and Methods**
params : OrderedDict
Ordered dictionary with the values of the *shape*, *location* and
*scale* parameters of the distribution.
c : flt
Float value for the *shape* parameter of the distribution.
loc : flt
Float value for the *location* parameter of the distribution.
scale : flt
Float value for the *scale* parameter of the distribution.
distr : object
Frozen RV object with the same methods of a continuous scipy
distribution but holding the given *shape*, *location*, and *scale*
fixed. See http://docs.scipy.org/doc/scipy/reference/stats.html
for more info.
data : array_like
Input data used for the fit
fit_method : str
String indicating the method used to fit the distribution,
values can be 'mle', 'mom' or 'lmoments'.
"""
def _fit(self):
# Fit can be made using Maximum Likelihood Estimation (mle) or using
# l-moments.
# L-moments is fast and accurate most of the time for the GEV
# distribution.
# MLE FIT
# In the case of the mle estimation, sometimes we get unstable values
# if we don't provide an initial guess of the parameters. Loc and scale
# are more or less stable but shape can be quite unstable depending the
# input data. This is why we are using lmoments to obtain start values
# for the mle optimization. For mle we are using fmin_bfgs as it is
# faster than others and with the first guess provide accurate results.
if self.fit_method == 'mle':
# Initial guess to make the fit of GEV more stable
# To do the initial guess we are using lmoments...
_params0 = _lmdistr.gev.lmom_fit(self.data)
# The mle fit will start with the initial estimators obtained
# with lmoments above
_params = _st.genextreme.fit(self.data, _params0['c'],
loc = _params0['loc'],
scale = _params0['scale'],
optimizer = _op.fmin_bfgs)
self.params = OrderedDict()
# For the shape parameter the value provided by scipy
# is defined as negative as that obtained from other
# packages in R, some textbooks, wikipedia,... ¿?
self.params["shape"] = _params[0]
self.params["location"] = _params[1]
self.params["scale"] = _params[2]
# L-MOMENTS FIT
if self.fit_method == 'lmoments':
_params = _lmdistr.gev.lmom_fit(self.data)
self.params = OrderedDict()
# For the shape parameter the value provided by lmoments3
# is defined as negative as that obtained from other
# packages in R, some textbooks, wikipedia,... ¿?
self.params["shape"] = _params['c']
self.params["location"] = _params['loc']
self.params["scale"] = _params['scale']
# METHOD OF MOMENTS FIT
if self.fit_method == 'mom':
_params = _gev_momfit(self.data)
self.params = OrderedDict()
self.params["shape"] = _params[0]
self.params["location"] = _params[1]
self.params["scale"] = _params[2]
# Estimators and a frozen distribution for the estimators
self.c = self.params['shape'] # shape
self.loc = self.params['location'] # location
self.scale = self.params['scale'] # scale
self.distr = _st.genextreme(self.c, # frozen distribution
loc = self.loc,
scale = self.scale)
def _nnlf(self, theta):
# This is used to calculate the variance-covariance matrix using the
# Hessian from numdifftools
# see self._ci_delta() method below
x = self.data
# Here we provide code for the GEV distribution and for the special
# case when shape parameter is 0 (Gumbel distribution).
if len(theta) == 3:
c = theta[0]
loc = theta[1]
scale = theta[2]
if len(theta) == 2:
c = 0
loc = theta[0]
scale = theta[1]
if c != 0:
expr = 1. + c * ((x - loc) / scale)
return (len(x) * _np.log(scale) +
(1. + 1. / c) * _np.sum(_np.log(expr)) +
_np.sum(expr ** (-1. / c)))
else:
expr = (x - loc) / scale
return (len(x) * _np.log(scale) +
_np.sum(expr) +
_np.sum(_np.exp(-expr)))
def _ci_delta(self):
# Calculate the variance-covariance matrix using the
# hessian from numdifftools
# This is used to obtain confidence intervals for the estimators and
# the return values for several return values.
#
# More info about the delta method can be found on:
# - Coles, Stuart: "An Introduction to Statistical Modeling of
# Extreme Values", Springer (2001)
# - https://en.wikipedia.org/wiki/Delta_method
# data
c = -self.c # We negate the shape to avoid inconsistency problems!?
loc = self.loc
scale = self.scale
hess = _ndt.Hessian(self._nnlf)
T = _np.arange(0.1, 500.1, 0.1)
sT = -_np.log(1.-self.frec/T)
sT2 = self.distr.isf(self.frec/T)
# VarCovar matrix and confidence values for estimators and return values
# Confidence interval for return values (up values and down values)
ci_Tu = _np.zeros(sT.shape)
ci_Td = _np.zeros(sT.shape)
if c: # If c then we are calculating GEV confidence intervals
varcovar = _np.linalg.inv(hess([c, loc, scale]))
self.params_ci = OrderedDict()
se = _np.sqrt(_np.diag(varcovar))
self._se = se
self.params_ci['shape'] = (self.c - _st.norm.ppf(1 - self.ci / 2) * se[0],
self.c + _st.norm.ppf(1 - self.ci / 2) * se[0])
self.params_ci['location'] = (self.loc - _st.norm.ppf(1 - self.ci / 2) * se[1],
self.loc + _st.norm.ppf(1 - self.ci / 2) * se[1])
self.params_ci['scale'] = (self.scale - _st.norm.ppf(1 - self.ci / 2) * se[2],
self.scale + _st.norm.ppf(1 - self.ci / 2) * se[2])
for i, val in enumerate(sT2):
gradZ = [scale * (c**-2) * (1 - sT[i] ** (-c)) - scale * (c**-1) * (sT[i]**-c) * _np.log(sT[i]),
1,
-(1 - sT[i] ** (-c)) / c]
se = _np.dot(_np.dot(gradZ, varcovar), _np.array(gradZ).T)
ci_Tu[i] = val + _st.norm.ppf(1 - self.ci / 2) * _np.sqrt(se)
ci_Td[i] = val - _st.norm.ppf(1 - self.ci / 2) * _np.sqrt(se)
else: # else then we are calculating Gumbel confidence intervals
varcovar = _np.linalg.inv(hess([loc, scale]))
self.params_ci = OrderedDict()
se = _np.sqrt(_np.diag(varcovar))
self._se = se
self.params_ci['shape'] = (0, 0)
self.params_ci['location'] = (self.loc - _st.norm.ppf(1 - self.ci / 2) * se[0],
self.loc + _st.norm.ppf(1 - self.ci / 2) * se[0])
self.params_ci['scale'] = (self.scale - _st.norm.ppf(1 - self.ci / 2) * se[1],
self.scale + _st.norm.ppf(1 - self.ci / 2) * se[1])
for i, val in enumerate(sT2):
gradZ = [1, -_np.log(sT[i])]
se = _np.dot(_np.dot(gradZ, varcovar), _np.array(gradZ).T)
ci_Tu[i] = val + _st.norm.ppf(1 - self.ci / 2) * _np.sqrt(se)
ci_Td[i] = val - _st.norm.ppf(1 - self.ci / 2) * _np.sqrt(se)
self._ci_Tu = ci_Tu
self._ci_Td = ci_Td
def _ci_bootstrap(self):
# Calculate confidence intervals using parametric bootstrap and the
# percentil interval method
# This is used to obtain confidence intervals for the estimators and
# the return values for several return values.
# all the code in skextremes.utils.bootstrap_ci has been adapted and
# simplified from that on https://github.com/cgevans/scikits-bootstrap.
#
# More info about bootstrapping can be found on:
# - https://github.com/cgevans/scikits-bootstrap
# - Efron: "An Introduction to the Bootstrap", Chapman & Hall (1993)
# - https://en.wikipedia.org/wiki/Bootstrapping_%28statistics%29
# parametric bootstrap for return levels and parameters
# The function to bootstrap
def func(data):
sample = _st.genextreme.rvs(self.c,
loc = self.loc,
scale = self.scale,
size = len(self.data))
c, loc, scale = _st.genextreme.fit(sample, self.c,
loc = self.loc,
scale = self.scale,
optimizer = _op.fmin_bfgs)
T = _np.arange(0.1, 500.1, 0.1)
sT = _st.genextreme.isf(self.frec/T, c, loc = loc, scale = scale)
res = [c, loc, scale]
res.extend(sT.tolist())
return tuple(res)
# the calculations itself
out = _bsci(self.data, statfunction = func, n_samples = 500)
self._ci_Td = out[0, 3:]
self._ci_Tu = out[1, 3:]
self.params_ci = OrderedDict()
self.params_ci['shape'] = (out[0,0], out[1,0])
self.params_ci['location'] = (out[0,1], out[1,1])
self.params_ci['scale'] = (out[0,2], out[1,3])
def _ci(self):
# Method called internally to calculate confidence intervals if
# required. To see more info about available methods see comments on
# self._ci_delta and self._ci_bootstrap methods.
if self.ci_method == "delta":
self._ci_delta()
if self.ci_method == "bootstrap":
self._ci_bootstrap()
[docs]class Gumbel(GEV):
__doc__ = GEV.__doc__.replace("Generalised extreme value (GEV) distribution.",
("Gumbel distribution. Note that this is a "
"special case of the ``GEV`` class where "
"the 'shape' is fixed to 0."))
def _fit(self):
if self.fit_method == 'mle':
_params = _st.gumbel_r.fit(self.data)
self.params = OrderedDict()
self.params["shape"] = 0
self.params["location"] = _params[0]
self.params["scale"] = _params[1]
if self.fit_method == 'lmoments':
_params = _lmdistr.gum.lmom_fit(self.data)
self.params = OrderedDict()
self.params["shape"] = 0
self.params["location"] = _params['loc']
self.params["scale"] = _params['scale']
# METHOD OF MOMENTS FIT
if self.fit_method == 'mom':
_params = _gum_momfit(self.data)
self.params = OrderedDict()
self.params["shape"] = _params[0]
self.params["location"] = _params[1]
self.params["scale"] = _params[2]
self.c = self.params['shape']
self.loc = self.params['location']
self.scale = self.params['scale']
self.distr = _st.gumbel_r(loc = self.loc,
scale = self.scale)