def lnlhood(self, param):
""" This is the function that evaluates the likelihood at each point in NDIM space
Parameters
----------
param :
Returns
-------
"""
likeit = 0
#loop over all the ions
for ii in range(self.nions):
thislike = 0
#parity check : make sure data and models are aligned
if (self.data[ii][0] != self.mod_colm_tag[ii]):
raise ValueError(
'Mismtach between observables and models. This is a big mistake!!!'
)
#now call the interpolator for models given current ion
mod_columns = self.interpol[ii](param)
#check if upper limit
if (self.data[ii][3] == -1):
#integrate the upper limit of a Gaussian - cumulative distribution
arg = ((self.data[ii][1] - mod_columns) /
(np.sqrt(2) * self.data[ii][2]))[0]
thislike = mmath.log(0.5 + 0.5 * mmath.erf(arg))
likeit = likeit + float(thislike)
#print self.data[ii][0], float(thislike), self.data[ii][1], mod_columns
#check if lower limit
elif (self.data[ii][3] == -2):
#integrate the lower limit of a Gaussian - Q function
arg = ((self.data[ii][1] - mod_columns) /
(np.sqrt(2) * self.data[ii][2]))[0]
thislike = mmath.log(0.5 - 0.5 * mmath.erf(arg))
likeit = likeit + float(thislike)
#print self.data[ii][0], float(thislike), self.data[ii][1], mod_columns
#if value, just eval Gaussian
else:
#add the likelihood for this ion
thislike = -1 * np.log(
np.sqrt(2 * np.pi) *
self.data[ii][2]) - (self.data[ii][1] - mod_columns)**2 / (
2 * self.data[ii][2]**2)
#mpmath equivalent
#thislike=-1*mmath.log(mmath.npdf(mod_columns[0],mu=self.data[ii][1],sigma=self.data[ii][2]))
likeit = likeit + float(thislike)
return likeit
def fl0xz(x0, seta2, seff2):
xm = x0 / np.sqrt(2. * seta2)
tau = lamda1 * seff2 / np.sqrt(2. * seta2)
ki = theta * (1. + lamda2 * seff2) / np.sqrt(2. * seta2)
intgr1 = erf(tau + ki - xm)
intgr2 = erf(tau + ki + xm)
return .5 * (intgr1 + intgr2)
def fl0xz(x0, seta2, seff2):
xm = x0/np.sqrt(2.*seta2)
tau = lamda1*seff2/np.sqrt(2.*seta2)
ki = theta*(1.+lamda2*seff2)/np.sqrt(2.*seta2)
intgr1 = erf(tau+ki-xm)
intgr2 = erf(tau+ki+xm)
return .5*(intgr1+intgr2)
def test_bounds_for_Q1():
"""
These bounds are from:
Jiangping Wang and Dapeng Wu,
"Tight bounds for the first order Marcum Q-function"
Wireless Communications and Mobile Computing,
Volume 12, Issue 4, March 2012, Pages 293-301.
"""
with mpmath.workdps(50):
sqrt2 = mpmath.sqrt(2)
sqrthalfpi = mpmath.sqrt(mpmath.pi / 2)
for b in [mpmath.mp.mpf(1), mpmath.mp.mpf(10)]:
for a in [b / 16, 0.5 * b, 0.875 * b, b]:
q1 = marcumq(1, a, b)
sinhab = mpmath.sinh(a * b)
i0ab = mpmath.besseli(0, a * b)
# lower bound when 0 <= a <= b
lb = (mpmath.sqrt(mpmath.pi / 8) * b * i0ab / sinhab *
(mpmath.erfc((b - a) / sqrt2) - mpmath.erfc(
(b + a) / sqrt2)))
assert lb < q1
# upper bound when 0 <= a <= b
ub = ((i0ab + 3) / (mpmath.exp(a * b) + 3) * (
mpmath.exp(-(b - a)**2 / 2) + a * sqrthalfpi * mpmath.erfc(
(b - a) / sqrt2) + 3 * mpmath.exp(-(a**2 + b**2) / 2)))
assert q1 < ub, ("marcumq(1, a, b) < ub for a = %s, b = %s" %
(a, b))
for a in [mpmath.mp.mpf(1), mpmath.mp.mpf(10)]:
for b in [b / 16, 0.5 * b, 0.875 * b, b]:
q1 = marcumq(1, a, b)
sinhab = mpmath.sinh(a * b)
i0ab = mpmath.besseli(0, a * b)
# lower bound when 0 <= b <= a
lb = (1 - sqrthalfpi * b * i0ab / sinhab *
(mpmath.erf(a / sqrt2) - mpmath.erf(
(a - b) / sqrt2) / 2 - mpmath.erf(
(a + b) / sqrt2) / 2))
assert lb < q1
# upper bound when 0 <= b <= a
ub = (
1 - i0ab / (mpmath.exp(a * b) + 3) *
(4 * mpmath.exp(-a**2 / 2) - mpmath.exp(-(b - a)**2 / 2) -
3 * mpmath.exp(-(a**2 + b**2) / 2) + a * sqrthalfpi *
(mpmath.erfc(-a / sqrt2) - mpmath.erfc((b - a) / sqrt2))))
assert q1 < ub, ("marcumq(1, a, b) < ub for a = %s, b = %s" %
(a, b))
def fq(x0, seta2, seff2):
coeff = 1./np.sqrt(np.pi)
xm = x0/np.sqrt(2.*seta2)
tau = lamda1*seff2/np.sqrt(2.*seta2)
psi = lamda2*seff2*x0/np.sqrt(2.*seta2)
intgr1 = coeff*(-tau+2*psi+xm)*exp(-(tau+xm)**2) +.5*(1.+2.*(tau-psi)**2)*erfc(tau+xm)
intgr2 = coeff*(-tau-2*psi-xm)*exp(-(tau-xm)**2) +.5*(1.+2.*(tau+psi)**2)*erfc(tau-xm)
intgr3 = .5*x0**2*(erf(tau+xm)+erf(tau-xm))
return (seta2/(1.+lamda2*seff2)**2)*(intgr1+intgr2)+intgr3
def fmt2_erfc(t, m, low=0):
tt = mpmath.sqrt(t)
low = mpmath.mpf(low)
low2 = low * low
f = mpmath.sqrt(mpmath.pi)/2. / tt * (mpmath.erf(tt) - mpmath.erf(low*tt))
e = mpmath.exp(-t)
e1 = mpmath.exp(-t*low2) * low
b = mpmath.mpf('.5') / t
out = [f]
for i in range(m):
f = b * ((2*i+1) * f - e + e1)
e1 *= low2
out.append(f)
return np.array(out)
def fq(x0, seta2, seff2):
coeff = 1. / np.sqrt(np.pi)
xm = x0 / np.sqrt(2. * seta2)
tau = lamda1 * seff2 / np.sqrt(2. * seta2)
psi = lamda2 * seff2 * x0 / np.sqrt(2. * seta2)
intgr1 = coeff * (-tau + 2 * psi + xm) * exp(-(tau + xm)**2) + .5 * (
1. + 2. * (tau - psi)**2) * erfc(tau + xm)
intgr2 = coeff * (-tau - 2 * psi - xm) * exp(-(tau - xm)**2) + .5 * (
1. + 2. * (tau + psi)**2) * erfc(tau - xm)
intgr3 = .5 * x0**2 * (erf(tau + xm) + erf(tau - xm))
return (seta2 / (1. + lamda2 * seff2)**2) * (intgr1 + intgr2) + intgr3
def lnlhood(self, param):
""" This is the function that evaluates the likelihood at each point in NDIM space
Parameters
----------
param :
Returns
-------
"""
likeit=0
#loop over all the ions
for ii in range(self.nions):
#parity check : make sure data and models are aligned
if(self.data[ii][0] != self.mod_colm_tag[ii]):
raise ValueError('Mismtach between observables and models. This is a big mistake!!!')
#now call the interpolator for models given current ion
mod_columns=self.interpol[ii](param)
#check if upper limit
if(self.data[ii][3] == -1):
#integrate the upper limit of a Gaussian - cumulative distribution
arg=((self.data[ii][1]-mod_columns)/(np.sqrt(2)*self.data[ii][2]))[0]
thislike=mmath.log(0.5+0.5*mmath.erf(arg))
likeit=likeit+float(thislike)
#print self.data[ii][0], float(thislike), self.data[ii][1], mod_columns
#check if lower limit
elif(self.data[ii][3] == -2):
#integrate the lower limit of a Gaussian - Q function
arg=((self.data[ii][1]-mod_columns)/(np.sqrt(2)*self.data[ii][2]))[0]
thislike=mmath.log(0.5-0.5*mmath.erf(arg))
likeit=likeit+float(thislike)
#print self.data[ii][0], float(thislike), self.data[ii][1], mod_columns
#if value, just eval Gaussian
else:
#add the likelihood for this ion
thislike=-1*np.log(np.sqrt(2*np.pi)*self.data[ii][2])-(self.data[ii][1]-mod_columns)**2/(2*self.data[ii][2]**2)
likeit=likeit+thislike
return likeit
def fmt2(t, m, low=None, factor=mpmath.mpf(1)):
tt = mpmath.sqrt(t)
f = mpmath.pi**.5 / 2. / tt * mpmath.erf(tt)
e = mpmath.exp(-t)
b = mpmath.mpf('.5') / t
out = [f]
for i in range(m):
f = b * ((2 * i + 1) * f - e)
out.append(f)
return np.array(out)
def fmt2(t, m, low=None):
half = mpmath.mpf('.5')
tt = mpmath.sqrt(t)
f = mpmath.pi**half / 2 / tt * mpmath.erf(tt)
e = mpmath.exp(-t)
b = half / t
out = [f]
for i in range(m):
f = b * ((2 * i + 1) * f - e)
out.append(f)
return np.array(out)
def test_erf_complex():
# need to increase mpmath precision for this test
old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec
try:
mpmath.mp.dps = 70
x1, y1 = np.meshgrid(np.linspace(-10, 1, 11), np.linspace(-10, 1, 11))
x2, y2 = np.meshgrid(np.logspace(-10, 0.8, 11), np.logspace(-10, 0.8, 11))
points = np.r_[x1.ravel(), x2.ravel()] + 1j * np.r_[y1.ravel(), y2.ravel()]
# note that the global accuracy of our complex erf algorithm is limited
# roughly to 2e-8
assert_func_equal(sc.erf, lambda x: complex(mpmath.erf(x)), points, vectorized=False, rtol=2e-8)
finally:
mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec
def test_erf_complex():
# need to increase mpmath precision for this test
old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec
try:
mpmath.mp.dps = 70
x1, y1 = np.meshgrid(np.linspace(-10, 1, 31), np.linspace(-10, 1, 11))
x2, y2 = np.meshgrid(np.logspace(-80, .8, 31), np.logspace(-80, .8, 11))
points = np.r_[x1.ravel(),x2.ravel()] + 1j*np.r_[y1.ravel(),y2.ravel()]
assert_func_equal(sc.erf, lambda x: complex(mpmath.erf(x)), points,
vectorized=False, rtol=1e-13)
assert_func_equal(sc.erfc, lambda x: complex(mpmath.erfc(x)), points,
vectorized=False, rtol=1e-13)
finally:
mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec
def test_erf_complex():
# need to increase mpmath precision for this test
old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec
try:
mpmath.mp.dps = 70
x1, y1 = np.meshgrid(np.linspace(-10, 1, 11), np.linspace(-10, 1, 11))
x2, y2 = np.meshgrid(np.logspace(-10, .8, 11), np.logspace(-10, .8, 11))
points = np.r_[x1.ravel(),x2.ravel()] + 1j*np.r_[y1.ravel(),y2.ravel()]
# note that the global accuracy of our complex erf algorithm is limited
# roughly to 2e-8
assert_func_equal(sc.erf, lambda x: complex(mpmath.erf(x)), points,
vectorized=False, rtol=2e-8)
finally:
mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec
def sf(x, mu=0, sigma=1):
"""
Survival function of the Lévy distribution.
"""
if sigma <= 0:
raise ValueError('sigma must be positive.')
if x <= mu:
return mpmath.one
with mpmath.extradps(5):
x = mpmath.mpf(x)
mu = mpmath.mpf(mu)
sigma = mpmath.mpf(sigma)
arg = mpmath.sqrt(sigma / (2 * (x - mu)))
return mpmath.erf(arg)
def test_erf_complex():
# need to increase mpmath precision for this test
old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec
try:
mpmath.mp.dps = 70
x1, y1 = np.meshgrid(np.linspace(-10, 1, 31), np.linspace(-10, 1, 11))
x2, y2 = np.meshgrid(np.logspace(-80, .8, 31),
np.logspace(-80, .8, 11))
points = np.r_[x1.ravel(), x2.ravel()] + 1j * np.r_[y1.ravel(),
y2.ravel()]
assert_func_equal(sc.erf,
lambda x: complex(mpmath.erf(x)),
points,
vectorized=False,
rtol=1e-13)
assert_func_equal(sc.erfc,
lambda x: complex(mpmath.erfc(x)),
points,
vectorized=False,
rtol=1e-13)
finally:
mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec
def pval(x):
return -mpmath.log10(1 - 0.5 * (1 + mpmath.erf(x / mpmath.sqrt(2))))
def eval(self, z):
return mpmath.erf(z)
def Z(x): return (1.j * mp.sqrt(mp.pi) * mp.exp(- x**2) * (1 + mp.erf(1.j * x))) def getGrowthNakata(ky_list, Setup, init = -0.07 -.015j):
for m in range(0, n+1):
xn12 = mp.mpf(1.0)/mp.power(x, m+0.5)
cgam = mp.gammainc(m+0.5) # "Complete" gamma?
# Incomplete gamma via continued fractions and Lentz method
lgam = LentzIncGamma(m+0.5, x, maxiter)
# put it together
F.append(mp.mpf(0.5) * (xn12 * cgam - ex*lgam))
return F
X = 0.01
N = 1
maxiter = 500
oldF = BoysValue(N, X)
newF = BoysValue_Gamma(N, X, maxiter)
sx = mp.sqrt(X)
fac = mp.sqrt(mp.pi)/2.0
print("ERF: {}".format(mp.nstr(fac/sx * mp.erf(sx), 16)))
print("OLD: {}".format(mp.nstr(oldF[0], 16)))
print("NEW: {}".format(mp.nstr(newF[0], 16)))
def test_erf_complex(self):
assert_mpmath_equal(sc.erf,
lambda z: mpmath.erf(z), [ComplexArg()],
n=200)
def test_erf(self):
assert_mpmath_equal(sc.erf, lambda z: mpmath.erf(z), [Arg()])
def Erf(x):
return(float(mpmath.erf(x)))
def ps0_lp (κ, s): k = σ * κ return (1 - mpmath.exp(k**2/2)/2 * ( mpmath.exp(+κ*x[i]) * mpmath.erfc((b+k)/sqrt2) + mpmath.exp(-κ*x[i]) * (1+mpmath.erf((b-k)/sqrt2)) ) ) / s
def test_erf(self):
assert_mpmath_equal(sc.erf,
lambda z: mpmath.erf(z),
[Arg()])
def normal_cdf(x, mu=0, sigma=1):
return (1 + mpmath.erf((x - mu) / mpmath.sqrt(2) / sigma)) / 2
def pval(x):
return float(1 - 0.5 * (1 + mpmath.erf(x / mpmath.sqrt(2))))
def test_erf_complex(self):
assert_mpmath_equal(sc.erf,
lambda z: mpmath.erf(z),
[ComplexArg()], n=200)
def f46(x):
# erf
return mpmath.erf(x)
def eval(self, z):
return mpmath.erf(z)
"""
fun = lambda t: f(t + x)
npoints = len(intrv)
df = self.__dip__(p, npoints - 1, fun, intrv, eps)
if not self.r:
warnings.warn("Derivation failed to reach precision goal.",
RuntimeWarning)
return df
if __name__ == "__main__":
# function to test
f = lambda t: 6 * mp.exp(-6 * t) + t + mp.log(0.2 * t + 4) - 2**t - mp.erf(
t)
# Real derivatives, for testing
f_d = []
f_d.append(lambda t: 1 / (5 * (t / 5 + 4)) - 36 * mp.exp(-6 * t) - 2**t *
mp.log(2) - (2 * mp.exp(-t**2)) / mp.pi**(1 / 2) + 1)
f_d.append(lambda t: 216 * mp.exp(-6 * t) - 1 / (25 * (t / 5 + 4)**2) - 2**
t * mp.log(2)**2 + (4 * t * mp.exp(-t**2)) / mp.pi**(1 / 2))
f_d.append(lambda t: 2.0 / (125 * (t / 5 + 4)**3) - 1296 * mp.exp(-6 * t) +
(4 * mp.exp(-t**2)) / mp.pi**(1 / 2) - 2**t * mp.log(2)**3 -
(8 * t**2 * mp.exp(-t**2)) / mp.pi**(1 / 2))
f_d.append(lambda t: 7776 * mp.exp(-6 * t) - 6.0 / (625 * (t / 5 + 4)**4) -
2**t * mp.log(2)**4 - (24 * t * mp.exp(-t**2)) / mp.pi**
(1 / 2) + (16 * t**3 * mp.exp(-t**2)) / mp.pi**(1 / 2))
nPoints = 100 # points around 0
