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dff_mle_fsolve_2vM_half.py
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dff_mle_fsolve_2vM_half.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 28 19:55:11 2018
@author: df
"""
import numpy as np
from scipy import integrate
from scipy import optimize
from scipy import stats
from scipy.stats import rv_continuous
import math
from numpy import i0 # modified Bessel function of the first kind order 0, I_0
from scipy.special import iv # modified Bessel function of first kind, I-v
import matplotlib.pyplot as plt
#import dff_StatsTools as dfst
#import dff_dispersionCalculator as dC
# size of sample:
N = 1000
# parameters for the von Mises member:
p1 = 0.5 # weight contribution of the 1st von Mises
p2 = 1. - p1 # weight contribution of the 2nd von Mises
kappa1_ = np.array((12.0)) # concentration for the 1st von Mises member
kappa2_ = np.array((5.0)) # concentration for the 2nd von Mises member
loc1_ = -np.pi/3.0 # location for the 1st von Mises member
loc2_ = np.pi/3.0 # location for the 1st von Mises member
#loc_cs = np.array(( np.cos(loc_), np.sin(loc_) )) # cos and sin of location
#print('loc_cs = ',loc_cs)
#kappa, mu, ll
class vonMises_gen(rv_continuous):
"my von Mises distribution"
def _pdf(self, x, *args):
#ll = 2.
#num = 1.0*ll
#denom = (2*np.pi) * i0(kappa)
#const = num / denom
kappa = args[0]
mu = args[1]
ll = args[2]
# print(kappa, mu, ll)
mmm = vonMises_gen._vmf_normalize(kappa, ll) * \
np.exp(kappa * np.cos(ll*(x - mu)))
return mmm
def _logpdf(self, x, *args):
return np.log(self._pdf(x, *args))
def _cdf_single(self, x, *args):
return integrate.quad(self._pdf, self.a, x, args=args)[0]
def _vmf_normalize(kappa, ll):
"""
Compute normalization constant using built-in numpy/scipy Bessel
approximations:
c(kappa) = 1 / (2 * pi * I0(kappa))
"""
num = 1.0*ll
#denom = (2*np.pi) * _bessel(0, kappa)
denom = (2*np.pi) * i0(kappa)
return num / denom
x_ = np.linspace( -np.pi/2, np.pi/2, N )
myvonMistest = vonMises_gen(a=-np.pi/2, b=np.pi/2, name='myVonMises')
inargs = np.array([kappa1_, loc1_, 2.])
my_pdf_test = myvonMistest._pdf(x_, kappa1_, loc1_, 2.)
my_cdf_test = myvonMistest._cdf(x_, kappa1_, loc1_, 2.)
my_sf_test = myvonMistest._sf(x_, kappa1_, loc1_, 2.)
fig, ax = plt.subplots(1, 2, figsize=(9,3))
ax[0].plot(x_, my_pdf_test, label='PDF')
ax[0].legend()
#ax.set_title('Random sample from mixture of von Mises and Uniform distriburtions')
ax[1].plot(x_,my_cdf_test, label='CDF')
ax[1].plot(x_, my_sf_test, label='SF')
ax[1].legend()
def _vmf_rvs(kappa, loc, size=N):
"""
Simulates n random angles from a von Mises distribution
with preferred direction loc and concentration kappa.
"""
a = 1. + np.sqrt(1. + 4.*kappa**2)
b = (a - np.sqrt(2.*a))/(2.*kappa)
r = (1. + b**2)/(2.*b)
theta = np.zeros(N)
for j in np.arange(N):
while True:
u = np.random.rand(3)
z = np.cos(np.pi*u[0])
f = (1. + r*z)/(r + z)
c = kappa*(r - f)
if u[1] < c*(2. - c) or np.log(c) - np.log(u[1]) + 1. - c < 0:
break
theta[j] = loc + np.sign(u[2] - 0.5)*np.arccos(f)
return theta
def _bessel(p, x):
"""
Compute the Modified Bessel function of first kind and order p.
x: in place of kappa
p: the order, 0 for I0 and 1 for I1.
"""
bintegrand = lambda t: np.exp(x*np.cos(t)) * np.cos(p*t) / np.pi
return integrate.quad(bintegrand, 0, np.pi)[0]
# ---------------------------------------------------------------------- #
# now generate the random sample:
# "Creating a mixture of probability distributions for sampling"
# a question on stackoverflow:
# https://stackoverflow.com/questions/47759577/
# creating-a-mixture-of-probability-distributions-for-sampling
u1_ = -np.pi/2
u2_ = np.pi
distributions = [
{ "type": stats.vonmises.rvs, "args": {"kappa":kappa1_, "loc":loc1_ }},
{ "type": stats.vonmises.rvs, "args": {"kappa":kappa2_, "loc":loc2_ }}
]
#distributions = [
# { "type": _vmf_rvs, "args": {"kappa":kappa1_, "loc":loc1_ }},
# { "type": _vmf_rvs, "args": {"kappa":kappa2_, "loc":loc2_ }}
#]
coefficients = np.array( [ p1, p2 ] ) # these are the weights
coefficients /= coefficients.sum() # in case these did not add up to 1
sample_size = N
num_distr = len(distributions)
data = np.zeros((sample_size, num_distr))
for idx, distr in enumerate(distributions):
data[:, idx] = distr["type"]( **distr["args"], size=(sample_size,))
random_idx = np.random.choice( np.arange(num_distr), \
size=(sample_size,), p=coefficients )
X_samples = data[ np.arange(sample_size), random_idx ]
fig, ax = plt.subplots(1, 1, figsize=(9,3))
ax.hist( X_samples, bins=100, density=True );
ax.set_title('Random sample from mixture of von Mises and Uniform distriburtions')
# ---------------------------------------------------------------------- #
# --------------------------------------------------------------- #
def _vmf_pdf(X, kappa, mu, ll):
"""
Computes the pdf(vM(X, kappa, mu) )using built-in numpy/scipy Bessel
approximations:
( c(kappa) * exp(kappa * mu * X) )
"""
# n = X.shape
return _vmf_normalize(kappa, ll) * np.exp(kappa * np.cos(ll*(X - mu)))
def _vmf_normalize(kappa, ll):
"""
Compute normalization constant using built-in numpy/scipy Bessel
approximations:
c(kappa) = 1 / (2 * pi * I0(kappa))
"""
num = 1.0*ll
denom = (2*np.pi) * _bessel(0, kappa)
# denom = (2*np.pi) * i0(kappa)
return num / denom
# ---------------------------------------------------------------------- #
def myF_2vM( theta, *params ):
"""
The first derivative of the log-likelihood function 'l' to be set equal to
zero in order to estimate the mixture parameters:
p1, p2: the weights of the two von Mises distributions
kappa2, kappa2: the concentrations of the two von Mises distributions
mu1, mu2: the locations of the two von Mises distributions
theta:= [ p1, kappa1, mu1, p2, kappa2, mu2 ]
params:= X_samples: the observations sample
The function returns a vector F with the derivatives of 'l' wrt the
components of theta.
This function is to be called with optimize.fsolve function of scipy:
roots_ = optimize.fsolve( myF_2vM, in_guess, args=my_par )
"""
ll = 2.0
x_ = np.array(params).T
# print(type(x_))
# print('x_=', x_.shape)
# the unknown parameters:
p1_ = theta[0]
kappa1_ = theta[1]
m1_ = theta[2]
kappa2_ = theta[3]
m2_ = theta[4]
p2_ = 1.0 - p1_
# the 1st von Mises distribution:
fvm1_ = _vmf_pdf( x_, kappa1_, m1_, ll )
# fvm1_ = stats.vonmises.pdf( x_, kappa1_, m1_ )
# print('fvm_=', fvm_.shape)
# the 2nd von Mises distribution:
fvm2_ = _vmf_pdf( x_, kappa2_, m2_, ll )
# fvm2_ = stats.vonmises.pdf( x_, kappa2_, m2_ )
# mixture distribution:
fm_ = p1_*fvm1_ + p2_*fvm2_
# print('fm_=', fm_.shape)
# first derivative wrt weight p1:
dldp1 = sum( np.divide( np.subtract( fvm1_, fvm2_ ), fm_ ) )
# print('dldp=', dldp.shape)
# first derivative wrt location mu1:=
dldm1 = 2.*kappa1_*p1_*sum( np.multiply( np.divide( fvm1_, fm_ ), \
np.sin(2.*(x_ - m1_)) ) )
# print('dldm=', dldm.shape)
# first derivative wrt location mu2:=
dldm2 = 2.*kappa2_*p2_*sum( np.multiply( np.divide( fvm2_, fm_ ), \
np.sin(2.*(x_ - m2_)) ) )
# first derivative wrt concentration kappa1:
Ak1 = ( _bessel(1, kappa1_) / _bessel(0, kappa1_) )
dldk1 = p1_*sum( np.multiply( np.divide( fvm1_, fm_ ), \
( np.cos(2.*(x_ - m1_)) - Ak1 ) ) )
# print('dldk=', dldk.shape)
# first derivative wrt concentration kappa1:
# Ak2 = ( iv(1.0, kappa2_) / i0(kappa2_) )
Ak2 = ( _bessel(1, kappa2_) / _bessel(0, kappa2_) )
dldk2 = p2_*sum( np.multiply( np.divide( fvm2_, fm_ ), \
( np.cos(2.*(x_ - m2_)) - Ak2 ) ) )
F = [ dldp1[0], dldk1[0], dldm1[0], dldk2[0], dldm2[0] ]
return F
# ---------------------------------------------------------------------- #
# obtain the solutions by calling fsolve:
my_par = X_samples # parameters needed inside F and J
# initial guesses for the parameters:
in_guess = [ p1, kappa1_, loc1_, kappa2_, loc2_ ]
#in_guess = [ 0.5, 11.0, 1.3, 0.5, 6.0, -0.4 ]
print('METHOD II: - - - - - - without providing the Jacobian - - - - - - ')
r_min_III = optimize.fsolve( myF_2vM, in_guess, args=my_par, \
full_output=True, xtol=1.49012e-8, maxfev=0 )
print('solution with METHOD II = ', r_min_III)
print('parameters with METHOD II = ', r_min_III[0])
sol_III = r_min_III[0]