def PotDisk(n, r):
"n: order of differentiation"
rratio = 0.5 * r / r0
if n == 1:
dPhidr_disk = (G * Md * r /
(2. * r0 * r0 * r0)) * (i0(rratio) * k0(rratio) -
i1(rratio) * k1(rratio))
return dPhidr_disk
elif n == 2:
d2Phidr2_disk = (G * Md / (4. * r0 * r0 * r0)) * (
-rratio * iv(2, rratio) * k1(rratio) + i0(rratio) *
(2 * k0(rratio) - 3. * rratio * k1(rratio)) + i1(rratio) *
(3. * rratio * k0(rratio) - 2. * k1(rratio) +
rratio * kn(2, rratio)))
return d2Phidr2_disk
elif n == 'v':
dPhidr_disk = (G * Md * r /
(2. * r0 * r0 * r0)) * (i0(rratio) * k0(rratio) -
i1(rratio) * k1(rratio))
Vc = np.sqrt(r * dPhidr_disk)
return Vc
else:
Phi_disk = (-G * Md * r / (2. * r0 * r0)) * (i0(rratio) * k1(rratio) -
i1(rratio) * k0(rratio))
return Phi_disk
def T6fun(i, x):
betah = (locs['a'])[i] * (locs['k'])[i] * locs['alpha'][i]
betam = par['am'] * par['km'] * par['alpha_m']
a = numpy.zeros([2, 2])
a[0, 0] = i0(locs['alpha'][i] * locs['rad'][i])
a[1, 0] = betah * i1(locs['alpha'][i] * locs['rad'][i])
a[0, 1] = -k0(par['alpha_m'] * locs['rad'][i])
a[1, 1] = betam * k1(par['alpha_m'] * locs['rad'][i])
## b is markedly different in zheng2009 and confirm.nb
## this is now the confirm.nb version
b = numpy.zeros([2, 1])
b[0] = (1.0 / par['cm']) - (locs['rad'][i] *
k0(locs['alpha'][i] * locs['rad'][i]) *
i1(locs['alpha'][i] * locs['rad'][i]) /
((locs['a'])[i] * locs['alpha'][i]))
b[1] = ((locs['k'])[i] * locs['rad'][i] *
k1(locs['alpha'][i] * locs['rad'][i]) *
i1(locs['alpha'][i] * locs['rad'][i]))
const = (inv(a)).dot(b)
qh = ((1.0 / (locs['c'])[i]) *
(1.0 - locs['alpha'][i] * locs['rad'][i] * i0(locs['alpha'][i] * x) *
k1(locs['alpha'][i] * locs['rad'][i])))
if (x < locs['rad'][i]):
return const[0] * i0(locs['alpha'][i] * x) + qh
else:
return const[1] * k0(par['alpha_m'] * x) + (1.0 / par['cm'])
def integr_kaiser1(x, *arg):
c, alpha, gamma, a, rabi_frequency, rotation_angle = arg
integral, err = quad(
lambda t: i1(gamma * sqrt(1 - (t / x)**2)) /
(i1(gamma) * sqrt(1 - (t / x)**2)), 0, x)
return rotation_angle / rabi_frequency - integral
def test_bessel_i1(self):
x_single = np.arange(-3, 3).reshape(1, 3, 2).astype(np.float32)
x_double = np.arange(-3, 3).reshape(1, 3, 2).astype(np.float64)
try:
from scipy import special # pylint: disable=g-import-not-at-top
self.assertAllClose(special.i1(x_single),
self.evaluate(special_math_ops.bessel_i1(x_single)))
self.assertAllClose(special.i1(x_double),
self.evaluate(special_math_ops.bessel_i1(x_double)))
except ImportError as e:
tf_logging.warn('Cannot test special functions: %s' % str(e))
def test_bessel_i1(self):
x_single = np.arange(-3, 3).reshape(1, 3, 2).astype(np.float32)
x_double = np.arange(-3, 3).reshape(1, 3, 2).astype(np.float64)
try:
from scipy import special # pylint: disable=g-import-not-at-top
self.assertAllClose(special.i1(x_single),
self.evaluate(special_math_ops.bessel_i1(x_single)))
self.assertAllClose(special.i1(x_double),
self.evaluate(special_math_ops.bessel_i1(x_double)))
except ImportError as e:
tf_logging.warn('Cannot test special functions: %s' % str(e))
def __init__(self, model = 'simple', scatt_alpha = 5.0/3.0, observer_screen_distance = 8.023*10**21, source_screen_distance = 1.790*10**22, theta_maj_mas_ref = 1.309, theta_min_mas_ref = 0.64, POS_ANG = 78, wavelength_reference_cm = 1.0, r_in = 10000*10**5, r_out = 10**20):
self.model = model
self.POS_ANG = POS_ANG #Major axis position angle [degrees, east of north]
self.observer_screen_distance = observer_screen_distance #cm
self.source_screen_distance = source_screen_distance #cm
M = observer_screen_distance/source_screen_distance
self.wavelength_reference = wavelength_reference_cm #Reference wavelength [cm]
self.r_in = r_in #inner scale [cm]
self.r_out = r_out #outer scale [cm]
self.scatt_alpha = scatt_alpha
if model == 'simple':
if r_in == 0.0:
print("Error! The 'simple' scattering model requires a finite inner scale.")
#Now, we need to solve for the effective parameters accounting for an inner scale
#By default, we will match the fitted Gaussian kernel as the wavelength goes to infinity
axial_ratio = theta_min_mas_ref/theta_maj_mas_ref #axial ratio of the scattering disk at long wavelengths (minor/major size < 1)
#self.C_scatt = (1.20488e-15*axial_ratio*self.r_in**(2.0 - self.scatt_alpha)*self.wavelength_reference**2)/self.scatt_alpha
self.C_scatt = 4.76299e-18*(1.0+M)**2 * np.pi**4 * r_in**(2.0 - scatt_alpha) * theta_maj_mas_ref * theta_min_mas_ref/(scatt_alpha * wavelength_reference_cm**2 * np.log(4.))
#Note: the prefactor is exactly equal to 1/209952000000000000
geometric_mean = (2.0*self.r_in**(2.0-self.scatt_alpha)/self.scatt_alpha/self.C_scatt)**0.5
self.r0_maj = geometric_mean*axial_ratio**0.5 #Phase coherence length at the reference wavelength [cm]
self.r0_min = geometric_mean/axial_ratio**0.5 #Phase coherence length at the reference wavelength [cm]
self.Qprefactor = self.C_scatt*(self.r0_maj*self.r0_min)**(self.scatt_alpha/2.0) # This accounts for the effects of a finite inner scale
elif model == 'power-law':
self.r0_maj = (2.0*np.log(2.0))**0.5/np.pi * wavelength_reference_cm/(theta_maj_mas_ref/1000.0/3600.0*np.pi/180.0) #Phase coherence length at the reference wavelength [cm]
self.r0_min = (2.0*np.log(2.0))**0.5/np.pi * wavelength_reference_cm/(theta_min_mas_ref/1000.0/3600.0*np.pi/180.0) #Phase coherence length at the reference wavelength [cm]
elif model == 'amph_von_Misses':
axial_ratio = theta_min_mas_ref/theta_maj_mas_ref #axial ratio of the scattering disk at long wavelengths (minor/major size < 1)
self.C_scatt = 4.76299e-18*(1.0+M)**2 * np.pi**4 * r_in**(2.0 - scatt_alpha) * theta_maj_mas_ref * theta_min_mas_ref/(scatt_alpha * wavelength_reference_cm**2 * np.log(4.))
#Note: the prefactor is exactly equal to 1/209952000000000000
geometric_mean = (2.0*self.r_in**(2.0-self.scatt_alpha)/self.scatt_alpha/self.C_scatt)**0.5
self.r0_maj = geometric_mean*axial_ratio**0.5 #Phase coherence length at the reference wavelength [cm]
self.r0_min = geometric_mean/axial_ratio**0.5 #Phase coherence length at the reference wavelength [cm]
self.r_in_p = 1.0/(sps.gamma(0.5 - self.scatt_alpha/2.0)*sps.gamma(1.0 + self.scatt_alpha)) * ((2.0**(self.scatt_alpha + 4.0) * np.pi)/(1.0 + np.cos(np.pi * self.scatt_alpha)))**0.5 * self.r_in
A = theta_maj_mas_ref/theta_min_mas_ref
self.kzeta = -0.17370 + 0.38067*A + 0.944246*A**2 # This is an approximate solution
self.zeta = 1.0 - 2.0*sps.i1(self.kzeta)/(self.kzeta * sps.i0(self.kzeta))
elif model == 'boxcar':
axial_ratio = theta_min_mas_ref/theta_maj_mas_ref #axial ratio of the scattering disk at long wavelengths (minor/major size < 1)
self.C_scatt = 4.76299e-18*(1.0+M)**2 * np.pi**4 * r_in**(2.0 - scatt_alpha) * theta_maj_mas_ref * theta_min_mas_ref/(scatt_alpha * wavelength_reference_cm**2 * np.log(4.))
#Note: the prefactor is exactly equal to 1/209952000000000000
geometric_mean = (2.0*self.r_in**(2.0-self.scatt_alpha)/self.scatt_alpha/self.C_scatt)**0.5
self.r0_maj = geometric_mean*axial_ratio**0.5 #Phase coherence length at the reference wavelength [cm]
self.r0_min = geometric_mean/axial_ratio**0.5 #Phase coherence length at the reference wavelength [cm]
self.r_in_p = 1.0/(sps.gamma(0.5 - self.scatt_alpha/2.0)*sps.gamma(1.0 + self.scatt_alpha)) * ((2.0**(self.scatt_alpha + 4.0) * np.pi)/(1.0 + np.cos(np.pi * self.scatt_alpha)))**0.5 * self.r_in
A = theta_maj_mas_ref/theta_min_mas_ref
self.kzeta = -0.17370 + 0.38067*A + 0.944246*A**2 # This is an approximate solution
self.kzeta_3 = 0.02987 + 0.28626*A # This is an approximate solution
self.zeta = 1.0 - 2.0*sps.i1(self.kzeta)/(self.kzeta * sps.i0(self.kzeta))
else:
print("Scattering Model Not Recognized!")
return
def region2_pressure(self):
self.kappa = np.sqrt(12*self.eps0*(1+self.nu)*(self.a/self.h)**2)
self.delta = self.gamma - self. alpha
self.g = -self.eps0/self.eps**2/self.kappa**2*(self.r-i1(self.kappa*self.r)/i1(self.kappa))
#self.beta = -6*(1-self.nu**2)/self.kappa**2*(self.p*self.a**3)/(self.E*self.h**3)*(self.r-i1(self.k*self.r)/i1(self.k))
self.beta = -6*(1-self.nu**2)/self.kappa**2*(self.p*self.a**3)/(self.E*self.h**3)*(self.r-i1(self.kappa*self.r)/i1(self.kappa))
self.numIntegrateBeta()
# Making ends meet: compare caclulated beta from relation to g with direct formula. Is this appropriate??
if 0:
self.beta = self.eps**self.delta*self.g
self.numIntegrateBeta()
def Gamma(nw_rad, lay_ox, L_d, L_tf, eps_1, eps_2, eps_3):
fact1 = (nw_rad + lay_ox) / L_d
fact2 = nw_rad / L_tf
fact3 = fact1**(-1)
fact4 = (nw_rad + lay_ox) / nw_rad
num = eps_1 * k0(fact1) * (L_d / L_tf) * i1(fact2)
denom1 = k0(fact1) * fact3
denom2 = log(fact4) * k1(fact1) * (eps_3 / eps_2)
denom3 = (denom1 + denom2) * eps_1 * fact2 * i1(fact2)
denom = denom3 + eps_3 * k1(fact1) * i0(fact2)
gamma = num / denom
return gamma
def Gamma(nw_rad, lay_ox, L_d, L_tf, eps_1, eps_2, eps_3):
fact1 = (nw_rad + lay_ox)/L_d
fact2 = nw_rad/L_tf
fact3 = fact1**(-1)
fact4 = (nw_rad + lay_ox)/nw_rad
num = eps_1*k0(fact1)*(L_d/L_tf)*i1(fact2)
denom1 = k0(fact1)*fact3
denom2 = log(fact4)*k1(fact1)*(eps_3/eps_2)
denom3 = (denom1 + denom2)*eps_1*fact2*i1(fact2)
denom = denom3 + eps_3*k1(fact1)*i0(fact2)
gamma = num/denom
return gamma
def ZTransDSC(self):
if self.beam.gammarel == float('inf'):
ZTrans_DSC = np.zeros(len(self.f)) + 1.j * np.zeros(len(self.f))
return ZTrans_DSC
kbess = 2 * const.pi * self.f / (self.beam.betarel * const.c)
argbess0 = kbess * self.beam.test_beam_shift / self.beam.gammarel
argbess1 = kbess * self.chamber.pipe_rad_m / self.beam.gammarel
BessBeamT = (i1(argbess0) / self.beam.test_beam_shift)**2
BessBeamTDSC = k1(argbess0) / i1(argbess0)
ZTrans_DSC = -(1.j * Z0 * self.chamber.pipe_len_m * BessBeamT *
BessBeamTDSC /
(const.pi * self.beam.gammarel**2 * self.beam.betarel))
return ZTrans_DSC
def testDerivativeKappa(self):
"Test vonMisesKappaConjugate derivative by changing kappa"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
c = 10
R0 = 1
self.J = IMP.isd.vonMisesKappaConjugateRestraint(self.m, self.kappa, c, R0)
for i in range(100):
no = uniform(0.1, 100)
self.kappa.set_scale(no)
self.J.evaluate(True)
ratio = i1(no) / i0(no)
self.assertAlmostEqual(self.kappa.get_scale_derivative(), -R0 + c * i1(no) / i0(no), delta=0.001)
def Vd(R, Rd, sigma_0):
y = R / (2 * Rd)
return np.sqrt(
4 * np.pi * G * sigma_0 * Rd * y**2 *
[special.i0(y) * special.k0(y) - special.i1(y) * special.k1(y)][0])
def B2B1(s, H_g, kappa, r_Db):
numerator = kappa * math.sqrt(H_g * s / kappa) * sp.i1(r_Db * math.sqrt(H_g * s / kappa)) * sp.k0(
r_Db * math.sqrt(s)) + math.sqrt(s) * sp.i0(r_Db * math.sqrt(H_g * s / kappa)) * sp.k1(r_Db * math.sqrt(s))
denominator = kappa * math.sqrt(H_g * s / kappa) * sp.k1(r_Db * math.sqrt(H_g * s / kappa)) * sp.k0(
r_Db * math.sqrt(s)) - math.sqrt(s) * sp.k0(r_Db * math.sqrt(H_g * s / kappa)) * sp.k1(r_Db * math.sqrt(s))
rt = numerator / denominator
return rt
def invmievonmises_pdf(theta, kappa, nu, lambda_, loc):
alpha1 = i1(kappa) / i0(kappa)
# inverse transformation by Newton's method
inv_theta = inv_trans_APF(theta, loc, lambda_, nu)
C = (1 - nu * alpha1)
p = vonmises.pdf(inv_theta, loc=0, kappa=kappa) / C
return p
def exponential_velocity(r, rd, vmax):
"""
Velocity function for an exponential profile
r and rd must be in the same units
Parameters
----------
r : array
radial positions where the model is to be computed
rd : float
radius at which the maximum velocity is reached
vmax : float
Maximum velocity of the model
Returns
-------
Array with the same shape of r, containing the model velocity curve/map
"""
# disk scale length
rd2 = rd / 2.15
vr = np.zeros(np.shape(r))
# to prevent any problem in the center
q = np.where(r != 0)
vr[q] = r[q] / rd2 * vmax / 0.88 * np.sqrt(
i0(0.5 * r[q] / rd2) * k0(0.5 * r[q] / rd2) -
i1(0.5 * r[q] / rd2) * k1(0.5 * r[q] / rd2))
return vr
def estimate_distribution(hours):
"""
Estimate the parameters of the von Mises distribution for the data.
Arguments:
hours: A NumPy array holding incident times as floats between 0 and 24.
Returns:
mu: The distribution's center as a float between -pi and pi.
kappa: The distribution's measure of concentration as a float.
More information about the von Mises distribution:
https://en.wikipedia.org/wiki/Von_Mises_distribution
"""
theta = hours_to_radians(hours)
z = np.vectorize(complex)(np.cos(theta), np.sin(theta))
n = len(z)
z_mean = np.mean(z)
mu = np.arctan2(z_mean.imag, z_mean.real)
r2 = np.mean(z.real)**2 + np.mean(z.imag)**2
re = np.sqrt(n/(n - 1)*(r2 - 1/n))
x = np.arange(0, 10, 1e-4)
y = np.abs(i1(x)/i0(x) - re)
kappa = x[np.argmin(y)]
return mu, kappa
def _evaluate(self, R, z, phi=0., t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
potential at (R,z)
HISTORY:
2012-12-26 - Written - Bovy (IAS)
"""
if self._new:
#if R > 6.: return self._kp(R,z)
if nu.fabs(z) < 10.**-6.:
y = 0.5 * self._alpha * R
return -nu.pi * R * (special.i0(y) * special.k1(y) -
special.i1(y) * special.k0(y))
kalphamax = 10.
ks = kalphamax * 0.5 * (self._glx + 1.)
weights = kalphamax * self._glw
sqrtp = nu.sqrt(z**2. + (ks + R)**2.)
sqrtm = nu.sqrt(z**2. + (ks - R)**2.)
evalInt = nu.arcsin(2. * ks / (sqrtp + sqrtm)) * ks * special.k0(
self._alpha * ks)
return -2. * self._alpha * nu.sum(weights * evalInt)
raise NotImplementedError(
"Not new=True not implemented for RazorThinExponentialDiskPotential"
)
def tetmConstants(self, ri, ro, neff, wl, EH, c, idx):
a = numpy.empty((2, 2))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = j0(u)
B2 = y0(u)
F1 = j0(urp) / B1
F2 = y0(urp) / B2
F3 = -j1(urp) / B1
F4 = -y1(urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = i0(u)
B2 = k0(u)
F1 = i0(urp) / B1
F2 = k0(urp) / B2
F3 = i1(urp) / B1
F4 = -k1(urp) / B2
c1 = -wl.k0 * ro / u
c3 = c * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 0] = F3 * c3
a[1, 1] = F4 * c3
return numpy.linalg.solve(a, EH.take(idx))
def testContinuedFraction(self):
# Check that the simplest continued fraction returns the golden ratio.
self.assertAllClose(
self.evaluate(
_compute_general_continued_fraction(
100, [], partial_numerator_fn=lambda _: 1.)),
scipy_constants.golden - 1.)
# Check the continued fraction constant is returned.
cf_constant_denominators = scipy_special.i1(2.) / scipy_special.i0(2.)
self.assertAllClose(self.evaluate(
_compute_general_continued_fraction(
100, [], partial_denominator_fn=lambda i: i, tolerance=1e-5)),
cf_constant_denominators,
rtol=1e-5)
cf_constant_numerators = np.sqrt(
2 / (np.e * np.pi)) / (scipy_special.erfc(np.sqrt(0.5))) - 1.
# Check that we can specify dtype and tolerance.
self.assertAllClose(self.evaluate(
_compute_general_continued_fraction(
100, [],
partial_numerator_fn=lambda i: i,
tolerance=1e-5,
dtype=tf.float64)),
cf_constant_numerators,
rtol=1e-5)
def _evaluate(self,R,z,phi=0.,t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
potential at (R,z)
HISTORY:
2012-12-26 - Written - Bovy (IAS)
"""
if self._new:
#if R > 6.: return self._kp(R,z)
if nu.fabs(z) < 10.**-6.:
y= 0.5*self._alpha*R
return -nu.pi*R*(special.i0(y)*special.k1(y)-special.i1(y)*special.k0(y))
kalphamax= 10.
ks= kalphamax*0.5*(self._glx+1.)
weights= kalphamax*self._glw
sqrtp= nu.sqrt(z**2.+(ks+R)**2.)
sqrtm= nu.sqrt(z**2.+(ks-R)**2.)
evalInt= nu.arcsin(2.*ks/(sqrtp+sqrtm))*ks*special.k0(self._alpha*ks)
return -2.*self._alpha*nu.sum(weights*evalInt)
raise NotImplementedError("Not new=True not implemented for RazorThinExponentialDiskPotential")
def tetmConstants(self, ri, ro, neff, wl, EH, c, idx):
a = numpy.empty((2, 2))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = j0(u)
B2 = y0(u)
F1 = j0(urp) / B1
F2 = y0(urp) / B2
F3 = -j1(urp) / B1
F4 = -y1(urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = i0(u)
B2 = k0(u)
F1 = i0(urp) / B1
F2 = k0(urp) / B2
F3 = i1(urp) / B1
F4 = -k1(urp) / B2
c1 = -wl.k0 * ro / u
c3 = c * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 0] = F3 * c3
a[1, 1] = F4 * c3
return numpy.linalg.solve(a, EH.take(idx))
def plotting_arai(file1):
"""
Main plotting function
"""
freqs, _, control, _, _, _, _ = np.loadtxt(file1,
delimiter=',',
unpack=True)
k = np.linspace(0, 3000, 10000)
sigma = 0.07
a = (2e-3) / 2
rho = 1000
w_squared = ((sigma * k) /
(rho * a**2)) * (1 - k**2 * a**2) * (i1(k * a) / i0(k * a))
sqrt_w = np.sqrt(w_squared)
v = arai_velocity(1551)
wavelength = v / freqs
wavenumber = 2 * np.pi / wavelength
savgol_control = savgol_filter(control, 1001, 2)
fig, ax = plt.subplots()
ax.plot(k * a, sqrt_w, label='Rayleigh')
ax.plot(wavenumber * a,
savgol_control,
label='Experimental (average velocity)')
ax.set_xlim(0, 7)
ax.set_ylim(0, 100)
ax.legend()
ax.set_xlabel('ka', fontsize=16)
ax.set_ylabel('$\\omega$', fontsize=16)
def compute_by_noise_pow(signal, n_pow):
global _window
global _G
global _prevGamma
global _alpha
global _prevAmp
global _ratio
global _constant
global _gamma15
s_spec = np.fft.fftpack.fft(signal * _window)
s_amp = np.absolute(s_spec)
s_phase = np.angle(s_spec)
#for idx in xrange(len(s_phase)):
# print(s_phase[idx])
gamma = _calc_aposteriori_snr(s_amp, n_pow)
xi = _calc_apriori_snr(gamma)
_prevGamma = gamma
nu = gamma * xi / (1.0 + xi)
_G = (_gamma15 * np.sqrt(nu) / gamma) * np.exp(-nu / 2.0) *\
((1.0 + nu) * spc.i0(nu / 2.0) + nu * spc.i1(nu / 2.0))
idx = np.less(s_amp**2.0, n_pow)
_G[idx] = _constant
idx = np.isnan(_G) + np.isinf(_G)
_G[idx] = xi[idx] / (xi[idx] + 1.0)
idx = np.isnan(_G) + np.isinf(_G)
_G[idx] = _constant
_G = np.maximum(_G, 0.0)
amp = _G * s_amp
amp = np.maximum(amp, 0.0)
amp2 = _ratio * amp + (1.0 - _ratio) * s_amp
_prevAmp = amp
spec = amp2 * np.exp(s_phase * 1j)
return np.real(np.fft.fftpack.ifft(spec))
def update_gradients_full(self, dL_dK, X, X2=None):
if X2 is None: X2 = X
trig_arg = (2 * np.pi / self.period) * (X - X2.T)
cos_term = np.cos(trig_arg)
sin_term = np.sin(trig_arg)
invL2 = 1 / self.lengthscale**2
if np.any(self.lengthscale > 1e4): # Limit for l -> infinity
dK_dV = cos_term # K / V
dK_dp = (self.variance / self.period) * trig_arg * sin_term
# This is 0 in the limit, but best to set it to a small non-0 value
dK_dl = 1e-4 / self.lengthscale
elif np.any(invL2 < 3.75):
bessel0 = i0(invL2)
bessel1 = i1(invL2)
eInvL2 = np.exp(invL2)
dInvL2_dl = -2 * invL2 / self.lengthscale # == -2 / l^3
denom = eInvL2 - bessel0
exp_term = np.exp(cos_term * invL2)
K_no_Var = (
exp_term - bessel0
) / denom # == K / V; here just for clarity of further expressions
dK_dV = K_no_Var
dK_dp = (self.variance / self.period
) * invL2 * trig_arg * sin_term * exp_term / denom
dK_dl = dInvL2_dl * self.variance * (
(cos_term * exp_term - bessel1) - K_no_Var *
(eInvL2 - bessel1)) / denom
else:
embi0 = self.embi0(invL2)
# embi1 = self.embi1(invL2)
# embi0min1 = embi0 - embi1
embi0min1 = self.embi0min1(invL2)
dInvL2_dl = -2 * invL2 / self.lengthscale # == -2 / l^3
denom = 1 - embi0
exp_term = np.exp((cos_term - 1) * invL2)
K_no_Var = (
exp_term - embi0
) / denom # == K / V; here just for clarity of further expressions
dK_dV = K_no_Var
dK_dp = (
self.variance / self.period
) * invL2 * trig_arg * sin_term * exp_term / denom # I.e. SAME as the above case at this abstraction level
dK_dl = dInvL2_dl * self.variance * (
(cos_term - 1) * exp_term + embi0min1 -
K_no_Var * embi0min1) / denom
self.variance.gradient = np.sum(dL_dK * dK_dV)
self.period.gradient = np.sum(dL_dK * dK_dp)
self.lengthscale.gradient = np.sum(dL_dK * dK_dl)
def kappa_to_stddev(kappa):
'''
Convert kappa to wrapped gaussian std dev
std = 1 - I_1(kappa)/I_0(kappa)
'''
# return 1.0 - scsp.i1(kappa)/scsp.i0(kappa)
return np.sqrt(-2.*np.log(scsp.i1(kappa)/scsp.i0(kappa)))
def C0(s, N_s, N_w, H_g, kappa, r_Db):
rt = 1.0 - 1.0 / (
1.0 - kappa * N_s / 2 / N_w * math.sqrt(H_g * s / kappa) *
(sp.i1(math.sqrt(H_g * s / kappa)) -
B2B1(s, H_g, kappa, r_Db) * sp.k1(math.sqrt(H_g * s / kappa))) /
(sp.i0(math.sqrt(H_g * s / kappa)) +
B2B1(s, H_g, kappa, r_Db) * sp.k0(math.sqrt(H_g * s / kappa))))
return rt
def w_minus(self, r):
if self.r0 == 1:
return r * 0
return (special.i1(np.sqrt(self.Pi) * r) * self.nu_1(r) / r +
special.k1(np.sqrt(self.Pi) * r) * self.nu_2(r) / r)
def _tmcoeq(self, v0, nu):
u1r1, u2r1, u2r2, s1, s2, n1sq, n2sq, n3sq = self.__params(v0)
if s1 == 0: # e
f11a, f11b = 2, 1
elif s1 > 0: # a, b, d
f11a, f11b = j0(u1r1) * u1r1, j1(u1r1)
else: # c
f11a, f11b = i0(u1r1) * u1r1, i1(u1r1)
if s2 > 0:
f22a, f22b = j0(u2r2), y0(u2r2)
f2a = j1(u2r1) * f22b - y1(u2r1) * f22a
f2b = j0(u2r1) * f22b - y0(u2r1) * f22a
else: # a
f22a, f22b = i0(u2r2), k0(u2r2)
f2a = i1(u2r1) * f22b + k1(u2r1) * f22a
f2b = i0(u2r1) * f22b - k0(u2r1) * f22a
return f11a * n2sq * f2a - f11b * n1sq * f2b * u2r1
def rotation_velocity(pos):
rho = (pos[0]**2 + pos[1]**2)**0.5
phi = np.arctan2(pos[1], pos[0])
y = rho/(2*Rd)
sigma0 = M_dm / (2*pi*Rd**2)
speed = (4*pi*G*sigma0*y**2*(i0(y)*k0(y) - i1(y)*k1(y)) +
(G*M_dm*rho)/(rho+a_dm)**2 + (G*M_bulge*rho)/(rho+a_bulge)**2)**0.5
return (-speed*sin(phi), speed*cos(phi), 0)
def gendata(X):
l = '%5s%23s%23s%23s%23s%23s%23s%23s%23s\n' % ('x', 'I0', 'I1', 'I2', 'I3',
'K0', 'K1', 'K2', 'K3')
for i, x in enumerate(X):
l += '%5.2f%23.15e%23.15e%23.15e%23.15e%23.15e%23.15e%23.15e%23.15e\n' % (
x, sp.i0(x), sp.i1(x), sp.iv(2, x), sp.iv(
3, x), sp.k0(x), sp.k1(x), sp.kn(2, x), sp.kn(3, x))
return l
def testVonMisesVariance(self):
locs_v = np.array([-3., -2., -1., 0.3, 2.3])
concentrations_v = np.array([0.0, 0.1, 1.0, 2.0, 10.0])
von_mises = tfd.VonMises(self.make_tensor(locs_v),
self.make_tensor(concentrations_v))
expected_vars = 1.0 - sp_special.i1(concentrations_v) / sp_special.i0(
concentrations_v)
self.assertAllClose(expected_vars, self.evaluate(von_mises.variance()))
def gradient(self,phases,log10_ens=3,free=False):
e,width,loc = self._make_p(log10_ens)
my_i0 = i0(1./width)
my_i1 = i1(1./width)
z = TWOPI*(phases-loc)
cz = np.cos(z)
sz = np.sin(z)
f = (np.exp(cz)/width)/my_i0
return np.asarray([-cz/width**2*f,TWOPI*(sz/width+my_i1/my_i0)*f])
def gradient(self,phases,log10_ens=3,free=False):
e,width,loc = self._make_p(log10_ens)
my_i0 = i0(1./width)
my_i1 = i1(1./width)
z = TWOPI*(phases-loc)
cz = np.cos(z)
sz = np.sin(z)
f = (np.exp(cz)/width)/my_i0
return np.asarray([-cz/width**2*f,TWOPI*(sz/width+my_i1/my_i0)*f])
def testDerivativeKappa(self):
"Test vonMisesKappaConjugate derivative by changing kappa"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
c = 10
R0 = 1
self.J = IMP.isd.vonMisesKappaConjugateRestraint(
self.m, self.kappa, c, R0)
for i in range(100):
no = uniform(0.1, 100)
self.kappa.set_scale(no)
self.J.evaluate(True)
ratio = i1(no) / i0(no)
self.assertAlmostEqual(self.kappa.get_scale_derivative(),
-R0 + c * i1(no) / i0(no),
delta=0.001)
def mmse_stsa(xi, gamma): nu = np.multiply(xi, np.divide(gamma, np.add(1, xi))) G = np.multiply(np.multiply(np.multiply(np.divide(np.sqrt(np.pi), 2), np.divide(np.sqrt(nu), gamma)), np.exp(np.divide(-nu,2))), np.add(np.multiply(np.add(1, nu), spsp.i0(np.divide(nu,2))), np.multiply(nu, spsp.i1(np.divide(nu, 2))))) # MMSE-STSA gain function. idx = np.isnan(G) | np.isinf(G) # replace by Wiener gain. G[idx] = np.divide(xi[idx], np.add(1, xi[idx])) # Wiener gain. return G
def get_vcirc_expo(R, Mgas=3e10, Rd=4.0):
sigma0 = Mgas / (2.0 * np.pi * Rd**2)
sigma = sigma0 * np.exp(-R / Rd)
y = R / (2.0 * Rd)
I0 = special.i0(y)
K0 = special.k0(y)
I1 = special.i1(y)
K1 = special.k1(y)
return np.sqrt(4.0 * np.pi * G * sigma0 * Rd * y**2 * (I0 * K0 - I1 * K1))
def rotation_velocity(pos):
rho = (pos[0]**2 + pos[1]**2)**0.5
phi = np.arctan2(pos[1], pos[0])
y = rho / (2 * Rd)
sigma0 = M_dm / (2 * pi * Rd**2)
speed = (4 * pi * G * sigma0 * y**2 * (i0(y) * k0(y) - i1(y) * k1(y)) +
(G * M_dm * rho) / (rho + a_dm)**2 + (G * M_bulge * rho) /
(rho + a_bulge)**2)**0.5
return (-speed * sin(phi), speed * cos(phi), 0)
def giddings(t, w, x):
print(w, x)
# w != 0
y = np.zeros(len(t))
y[t > 0] = (1. / w) * sqrt(x / t[t > 0]) * exp((t[t > 0] + x) / -w)
# TODO: "overflow encountered in i1"
# y[t > 0] *= i1(2. * sqrt(x * t[t > 0]) / w)
# trying to keep the shape, but not allow such high numbers?
y[t > 0] *= i1(np.linspace(2, 10, sum(t > 0)))
return y
def giddings(t, w, x):
print(w, x)
# w != 0
y = np.zeros(len(t))
y[t > 0] = (1. / w) * sqrt(x / t[t > 0]) * exp((t[t > 0] + x) / -w)
# TODO: "overflow encountered in i1"
# y[t > 0] *= i1(2. * sqrt(x * t[t > 0]) / w)
# trying to keep the shape, but not allow such high numbers?
y[t > 0] *= i1(np.linspace(2, 10, sum(t > 0)))
return y
def cutoffHE1(b):
a = rho * b
i = ivp(1, u1*a) / (u1*a * i1(u1*a))
X = (1 / (u1*a)**2 + 1 / (u2*a)**2)
P = j1(u2*a) * y1(u2*b) - y1(u2*a) * j1(u2*b)
Ps = (jvp(1, u2*a) * y1(u2*b) - yvp(1, u2*a) * j1(u2*b)) / (u2 * a)
return (i * P + Ps) * (n12 * i * P + n22 * Ps) - n32 * X * X * P * P
def testValueP(self):
"Test vonMisesKappaJeffreys probability"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
for i in range(100):
no = uniform(0.1, 100)
self.kappa.set_scale(no)
ratio = i1(no) / i0(no)
self.assertAlmostEqual(
self.J.get_probability(), sqrt(ratio * (no - ratio - no * ratio * ratio)), delta=0.001
)
def testEvaluateDKappa(self):
"tests vonMises.evaluate_derivative_kappa"
try:
from scipy.special import i0,i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
for i in xrange(100):
randno = [uniform(-4*pi,4*pi), uniform(-pi,pi),
uniform(0.1,100)]
fn=vonMises(*randno)
self.assertAlmostEqual(fn.evaluate_derivative_kappa(),
i1(randno[2])/i0(randno[2]) - cos(randno[0]-randno[1]),
delta=0.001)
def testVonMisesVariance(self):
locs_v = np.array([-3., -2., -1., 0.3, 2.3])
concentrations_v = np.array([0.0, 0.1, 1.0, 2.0, 10.0])
von_mises = tfd.VonMises(
self.make_tensor(locs_v), self.make_tensor(concentrations_v))
try:
from scipy import special # pylint:disable=g-import-not-at-top
except ImportError:
tf.logging.warn("Skipping scipy-dependent tests")
return
expected_vars = 1.0 - special.i1(concentrations_v) / special.i0(
concentrations_v)
self.assertAllClose(expected_vars, self.evaluate(von_mises.variance()))
def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
R2deriv
PURPOSE:
evaluate R2 derivative
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
-d K_R (R,z) d R
HISTORY:
2012-12-27 - Written - Bovy (IAS)
"""
if self._new:
if nu.fabs(z) < 10.**-6.:
y= 0.5*self._alpha*R
return nu.pi*self._alpha*(special.i0(y)*special.k0(y)-special.i1(y)*special.k1(y)) \
+nu.pi/4.*self._alpha**2.*R*(special.i1(y)*(3.*special.k0(y)+special.kn(2,y))-special.k1(y)*(3.*special.i0(y)+special.iv(2,y)))
raise AttributeError("'R2deriv' for RazorThinExponentialDisk not implemented for z =/= 0")
def testValueE(self):
"Test if vonMisesKappaJeffreys score is log(scale)"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
for i in range(100):
no = uniform(0.1, 100)
self.kappa.set_scale(no)
ratio = i1(no) / i0(no)
self.assertAlmostEqual(
self.J.unprotected_evaluate(None), -0.5 * log(ratio * (no - ratio - no * ratio * ratio)), delta=0.001
)
def testDerivative(self):
"test the derivative of the restraint"
try:
from scipy.special import i0,i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
for i in xrange(100):
no=uniform(0.1,100)
self.kappa.set_scale(no)
self.m.evaluate(self.DA)
ratio=i1(no)/i0(no)
self.assertAlmostEqual(self.kappa.get_scale_derivative(),
0.5*(-1/ratio+3*ratio+1/no+1/(no-no**2/ratio+ratio*no**2)),
delta=0.001)
def testVonMisesStddev(self):
locs_v = np.array([-3., -2., -1., 0.3, 2.3]).reshape([1, -1])
concentrations_v = np.array([0.0, 0.1, 1.0, 2.0, 10.0]).reshape([-1, 1])
von_mises = tfd.VonMises(
self.make_tensor(locs_v), self.make_tensor(concentrations_v))
try:
from scipy import special # pylint:disable=g-import-not-at-top
except ImportError:
tf.logging.warn("Skipping scipy-dependent tests")
return
expected_stddevs = (np.sqrt(1.0 - special.i1(concentrations_v)
/ special.i0(concentrations_v))
+ np.zeros_like(locs_v))
self.assertAllClose(expected_stddevs, self.evaluate(von_mises.stddev()))
def testEvaluateDKappa(self):
"Test vonMisesSufficient.evaluate_derivative_kappa"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
for i in xrange(100):
x = uniform(-4 * pi, 4 * pi)
N = randint(1, 20)
R = randint(1, N)
chiexp = uniform(-pi, pi)
kappa = uniform(0.1, 100)
fn = vonMisesSufficient(x, N, R, chiexp, kappa)
self.assertAlmostEqual(
fn.evaluate_derivative_kappa(), N * i1(kappa) / i0(kappa) - R * cos(x - chiexp), delta=0.001
)
def testValuePR0(self):
"Test vonMisesKappaConjugate probability by changing R0"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
c = 10.0
no = 1.0
self.kappa.set_scale(no)
for i in range(100):
R0 = uniform(0.0, 10.0)
self.J = IMP.isd.vonMisesKappaConjugateRestraint(self.m, self.kappa, c, R0)
ratio = i1(no) / i0(no)
py = exp(no * R0) / i0(no) ** c
cpp = self.J.get_probability()
self.assertAlmostEqual(cpp, py, delta=0.001)
def estimate_distribution(hours):
theta = hours_to_radians(hours)
z = np.vectorize(complex)(np.cos(theta), np.sin(theta))
n = len(z)
z_mean = np.mean(z)
mu = np.arctan2(z_mean.imag, z_mean.real)
r2 = np.mean(z.real)**2 + np.mean(z.imag)**2
re = np.sqrt(n/(n - 1)*(r2 - 1/n))
x = np.arange(0, 10, 1e-4)
y = np.abs(i1(x)/i0(x) - re)
kappa = x[np.argmin(y)]
return mu, kappa
def testValueEc(self):
"Test vonMisesKappaConjugate energy by changing c"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
R0 = 1
no = 1.0
self.kappa.set_scale(no)
for i in range(100):
c = uniform(1.0, 100)
self.J = IMP.isd.vonMisesKappaConjugateRestraint(self.m, self.kappa, c, R0)
ratio = i1(no) / i0(no)
py = -no * R0 + c * log(i0(no))
cpp = self.J.evaluate(False)
self.assertAlmostEqual(cpp, py, delta=0.001)
def K1RI1r(self, rin):
rv = np.zeros(len(self.lab))
if self.islarge.any():
index = (self.R - rin) / self.labbig < 10
if index.any():
r = rin / self.labbig[index]
R = self.R / self.labbig[index]
rv[self.islarge * index] = np.sqrt(1 / (4 * r * R)) * np.exp(r - R) * \
(1 + 3 / (8 * R) - 15 / (128 * R ** 2) + 315 / (3072 * R ** 3)) * \
(1 - 3 / (8 * r) - 15 / (128 * r ** 2) - 315 / (3072 * r ** 3))
if ~self.islarge.any():
index = (self.R - rin) / self.labsmall < 10
if index.any():
r = rin / self.labsmall[index]
rv[~self.islarge * index] = self.k1Roverlab[index] * i1(r)
return rv
def testValueEKappa(self):
"Test vonMisesKappaConjugate energy by changing kappa"
try:
from scipy.special import i0,i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
c=10
R0=1
self.J = IMP.isd.vonMisesKappaConjugateRestraint(self.kappa,c,R0)
self.m.add_restraint(self.J)
for i in xrange(100):
no=uniform(0.1,100)
self.kappa.set_scale(no)
ratio=i1(no)/i0(no)
py=-no*R0 + c*log(i0(no))
cpp=self.J.evaluate(None)
self.assertAlmostEqual(cpp,py,delta=0.001)
def testDerivative(self):
"Test the derivative of vonMisesKappaJeffreysRestraint"
try:
from scipy.special import i0, i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
sf = IMP.core.RestraintsScoringFunction([self.J])
for i in range(100):
no = uniform(0.1, 100)
self.kappa.set_scale(no)
sf.evaluate(True)
ratio = i1(no) / i0(no)
self.assertAlmostEqual(
self.kappa.get_scale_derivative(),
0.5 * (-1 / ratio + 3 * ratio + 1 / no + 1 / (no - no ** 2 / ratio + ratio * no ** 2)),
delta=0.001,
)
def testValueDKappa1(self):
"""test derivatives for kappa by varying kappa"""
try:
from scipy.special import i0,i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
self.setup_restraint()
self.p3.set_coordinates(IMP.algebra.Vector3D(0,1,-1))
for i in xrange(100):
kappa = uniform(0.1,10)
self.kappa.set_scale(kappa)
self.talos.evaluate(self.DA)
py=self.N*i1(kappa)/i0(kappa) - self.R*cos(pi/2-self.chiexp)
cpp=self.kappa.get_scale_derivative()
if py == 0:
self.assertEqual(cpp,0)
else:
self.assertAlmostEqual(cpp/py,1.0,delta=1e-6)
def testValuePc(self):
"test probability by changing c"
try:
from scipy.special import i0,i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
R0=1.0
no=1.0
self.kappa.set_scale(no)
for i in xrange(100):
c=uniform(2.0,75)
self.J = IMP.isd.vonMisesKappaConjugateRestraint(self.kappa,c,R0)
self.m.add_restraint(self.J)
ratio=i1(no)/i0(no)
py=exp(no*R0)/i0(no)**c
cpp=self.J.get_probability()
self.assertAlmostEqual(cpp,py,delta=0.001)
self.m.remove_restraint(self.J)
def testValueDKappa2(self):
"""Test TALOS derivatives for kappa by varying the angle"""
try:
from scipy.special import i0,i1
except ImportError:
self.skipTest("this test requires the scipy Python module")
self.setup_restraint()
for i in xrange(100):
x=i/(2*pi)
self.p3.set_coordinates(IMP.algebra.Vector3D(
cos(2*pi-x),1,sin(2*pi-x)))
kappa = self.kappa.get_scale()
self.talos.evaluate(self.DA)
py=self.N*i1(kappa)/i0(kappa) - self.R*cos(x-self.chiexp)
cpp=self.kappa.get_scale_derivative()
if py == 0:
self.assertEqual(cpp,0)
else:
self.assertAlmostEqual(cpp/py,1.0,delta=1e-6)
