class GaussianEllipsePotential(LensProfileBase):
"""
this class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix
with ellipticity in the convergence
the calculation follows Glenn van de Ven et al. 2009
"""
param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'amp': 0,
'sigma': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'amp': 100,
'sigma': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.spherical = GaussianKappa()
self._diff = 0.000001
super(GaussianEllipsePotential, self).__init__()
def function(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
returns Gaussian
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = abs(1 - q)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_ = self.spherical.function(x_, y_, amp=amp, sigma=sigma)
return f_
def derivatives(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = abs(1 - q)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_x_prim, f_y_prim = self.spherical.derivatives(x_,
y_,
amp=amp,
sigma=sigma)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
alpha_ra, alpha_dec = self.derivatives(x, y, amp, sigma, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, amp, sigma,
e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, amp, sigma,
e1, e2, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
def density(self, r, amp, sigma, e1, e2):
"""
:param r:
:param amp:
:param sigma:
:return:
"""
return self.spherical.density(r, amp, sigma)
def density_2d(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
:param R:
:param am:
:param sigma_x:
:param sigma_y:
:return:
"""
return self.spherical.density_2d(x, y, amp, sigma, center_x, center_y)
def mass_2d(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma_x:
:param sigma_y:
:return:
"""
return self.spherical.mass_2d(R, amp, sigma)
def mass_3d(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma:
:param e1:
:param e2:
:return:
"""
return self.spherical.mass_3d(R, amp, sigma)
def mass_3d_lens(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma:
:param e1:
:param e2:
:return:
"""
return self.spherical.mass_3d_lens(R, amp, sigma)
def mass_2d_lens(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma_x:
:param sigma_y:
:return:
"""
return self.spherical.mass_2d_lens(R, amp, sigma)
class MultiGaussianKappa(object):
"""
"""
param_names = ['amp', 'sigma', 'center_x', 'center_y']
lower_limit_default = {
'amp': 0,
'sigma': 0,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'amp': 100,
'sigma': 100,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.gaussian_kappa = GaussianKappa()
def function(self,
x,
y,
amp,
sigma,
center_x=0,
center_y=0,
scale_factor=1):
"""
:param x:
:param y:
:param amp:
:param sigma:
:param center_x:
:param center_y:
:return:
"""
f_ = np.zeros_like(x, dtype=float)
for i in range(len(amp)):
f_ += self.gaussian_kappa.function(x,
y,
amp=scale_factor * amp[i],
sigma=sigma[i],
center_x=center_x,
center_y=center_y)
return f_
def derivatives(self,
x,
y,
amp,
sigma,
center_x=0,
center_y=0,
scale_factor=1):
"""
:param x:
:param y:
:param amp:
:param sigma:
:param center_x:
:param center_y:
:return:
"""
f_x, f_y = np.zeros_like(x, dtype=float), np.zeros_like(x, dtype=float)
for i in range(len(amp)):
f_x_i, f_y_i = self.gaussian_kappa.derivatives(x,
y,
amp=scale_factor *
amp[i],
sigma=sigma[i],
center_x=center_x,
center_y=center_y)
f_x += f_x_i
f_y += f_y_i
return f_x, f_y
def hessian(self,
x,
y,
amp,
sigma,
center_x=0,
center_y=0,
scale_factor=1):
"""
:param x:
:param y:
:param amp:
:param sigma:
:param center_x:
:param center_y:
:return:
"""
f_xx, f_yy, f_xy = np.zeros_like(x, dtype=float), np.zeros_like(
x, dtype=float), np.zeros_like(x, dtype=float)
for i in range(len(amp)):
f_xx_i, f_yy_i, f_xy_i = self.gaussian_kappa.hessian(
x,
y,
amp=scale_factor * amp[i],
sigma=sigma[i],
center_x=center_x,
center_y=center_y)
f_xx += f_xx_i
f_yy += f_yy_i
f_xy += f_xy_i
return f_xx, f_yy, f_xy
def density(self, r, amp, sigma, scale_factor=1):
"""
:param r:
:param amp:
:param sigma:
:return:
"""
d_ = np.zeros_like(r, dtype=float)
for i in range(len(amp)):
d_ += self.gaussian_kappa.density(r, scale_factor * amp[i],
sigma[i])
return d_
def density_2d(self,
x,
y,
amp,
sigma,
center_x=0,
center_y=0,
scale_factor=1):
"""
:param R:
:param am:
:param sigma_x:
:param sigma_y:
:return:
"""
d_3d = np.zeros_like(x, dtype=float)
for i in range(len(amp)):
d_3d += self.gaussian_kappa.density_2d(x, y, scale_factor * amp[i],
sigma[i], center_x,
center_y)
return d_3d
def mass_3d_lens(self, R, amp, sigma, scale_factor=1):
"""
:param R:
:param amp:
:param sigma:
:return:
"""
mass_3d = np.zeros_like(R, dtype=float)
for i in range(len(amp)):
mass_3d += self.gaussian_kappa.mass_3d_lens(
R, scale_factor * amp[i], sigma[i])
return mass_3d
class GaussianEllipseKappa(LensProfileBase):
"""
This class contains functions to evaluate the derivative and hessian matrix
of the deflection potential for an elliptical Gaussian convergence.
The formulae are from Shajib (2019).
"""
param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'amp': 0,
'sigma': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'amp': 100,
'sigma': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self, use_scipy_wofz=True, min_ellipticity=1e-5):
"""
Setup which method to use the Faddeeva function and the
ellipticity limit for spherical approximation.
:param use_scipy_wofz: If ``True``, use ``scipy.special.wofz``.
:type use_scipy_wofz: ``bool``
:param min_ellipticity: Minimum allowed ellipticity. For ``q > 1 - min_ellipticity``, values for spherical case will be returned.
:type min_ellipticity: ``float``
"""
if use_scipy_wofz:
self.w_f = wofz
else:
self.w_f = self.w_f_approx
self.min_ellipticity = min_ellipticity
self.spherical = GaussianKappa()
super(GaussianEllipseKappa, self).__init__()
def function(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
Compute the potential function for elliptical Gaussian convergence.
:param x: x coordinate
:type x: ``float`` or ``numpy.array``
:param y: y coordinate
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian
:type sigma: ``float``
:param e1: Ellipticity parameter 1
:type e1: ``float``
:param e2: Ellipticity parameter 2
:type e2: ``float``
:param center_x: x coordinate of centroid
:type center_x: ``float``
:param center_y: y coordianate of centroid
:type center_y: ``float``
:return: Potential for elliptical Gaussian convergence
:rtype: ``float``, or ``numpy.array`` with shape equal to ``x.shape``
"""
phi_g, q = param_util.ellipticity2phi_q(e1, e2)
if q > 1 - self.min_ellipticity:
return self.spherical.function(x, y, amp, sigma, center_x,
center_y)
# adjusting amplitude to make the notation compatible with the
# formulae given in Shajib (2019).
amp_ = amp / (2 * np.pi * sigma**2)
# converting ellipticity definition from q*x^2 + y^2/q to q^2*x^2 + y^2
sigma_ = sigma * np.sqrt(q) # * q
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_g)
sin_phi = np.sin(phi_g)
x_ = cos_phi * x_shift + sin_phi * y_shift
y_ = -sin_phi * x_shift + cos_phi * y_shift
_b = 1. / 2. / sigma_**2
_p = np.sqrt(_b * q**2 / (1. - q**2))
if isinstance(x_, int) or isinstance(x_, float):
return self._num_integral(x_, y_, amp_, sigma_, _p, q)
else:
f_ = []
for i in range(len(x_)):
f_.append(self._num_integral(x_[i], y_[i], amp_, sigma_, _p,
q))
return np.array(f_)
def _num_integral(self, x_, y_, amp_, sigma_, _p, q):
"""
:param x_:
:param y_:
:param _p:
:param q:
:return:
"""
def pot_real_line_integrand(_x):
sig_func_re, sig_func_im = self.sigma_function(_p * _x, 0, q)
alpha_x_ = amp_ * sigma_ * self.sgn(_x) * np.sqrt(
2 * np.pi / (1. - q**2)) * sig_func_re
return alpha_x_
def pot_imag_line_integrand(_y):
sig_func_re, sig_func_im = self.sigma_function(_p * x_, _p * _y, q)
alpha_y_ = -amp_ * sigma_ * self.sgn(x_ + 1j * _y) * np.sqrt(
2 * np.pi / (1. - q**2)) * sig_func_im
return alpha_y_
pot_on_real_line = quad(pot_real_line_integrand, 0, x_)[0]
pot_on_imag_parallel = quad(pot_imag_line_integrand, 0, y_)[0]
return (pot_on_real_line + pot_on_imag_parallel)
def derivatives(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
Compute the derivatives of function angles :math:`\partial
f/\partial x`, :math:`\partial f/\partial y` at :math:`x,\ y`.
:param x: x coordinate
:type x: ``float`` or ``numpy.array``
:param y: y coordinate
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian
:type sigma: ``float``
:param e1: Ellipticity parameter 1
:type e1: ``float``
:param e2: Ellipticity parameter 2
:type e2: ``float``
:param center_x: x coordinate of centroid
:type center_x: ``float``
:param center_y: y coordianate of centroid
:type center_y: ``float``
:return: Deflection angle :math:`\partial f/\partial x`, :math:`\partial f/\partial y` for elliptical Gaussian convergence.
:rtype: tuple ``(float, float)`` or ``(numpy.array, numpy.array)`` with each ``numpy.array``'s shape equal to ``x.shape``.
"""
phi_g, q = param_util.ellipticity2phi_q(e1, e2)
if q > 1 - self.min_ellipticity:
return self.spherical.derivatives(x, y, amp, sigma, center_x,
center_y)
# adjusting amplitude to make the notation compatible with the
# formulae given in Shajib (2019).
amp_ = amp / (2 * np.pi * sigma**2)
# converting ellipticity definition from q*x^2 + y^2/q to q^2*x^2 + y^2
sigma_ = sigma * np.sqrt(q) # * q
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_g)
sin_phi = np.sin(phi_g)
# rotated coordinates
x_ = cos_phi * x_shift + sin_phi * y_shift
y_ = -sin_phi * x_shift + cos_phi * y_shift
_p = q / sigma_ / np.sqrt(2 * (1. - q**2))
sig_func_re, sig_func_im = self.sigma_function(_p * x_, _p * y_, q)
alpha_x_ = amp_ * sigma_ * self.sgn(x_ + 1j * y_) * np.sqrt(
2 * np.pi / (1. - q**2)) * sig_func_re
alpha_y_ = -amp_ * sigma_ * self.sgn(x_ + 1j * y_) * np.sqrt(
2 * np.pi / (1. - q**2)) * sig_func_im
# rotate back to the original frame
f_x = alpha_x_ * cos_phi - alpha_y_ * sin_phi
f_y = alpha_x_ * sin_phi + alpha_y_ * cos_phi
return f_x, f_y
def hessian(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
Compute Hessian matrix of function :math:`\partial^2f/\partial x^2`,
:math:`\partial^2 f/\partial y^2`, :math:`\partial^2/\partial
x\partial y`.
:param x: x coordinate
:type x: ``float`` or ``numpy.array``
:param y: y coordinate
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian
:type sigma: ``float``
:param e1: Ellipticity parameter 1
:type e1: ``float``
:param e2: Ellipticity parameter 2
:type e2: ``float``
:param center_x: x coordinate of centroid
:type center_x: ``float``
:param center_y: y coordianate of centroid
:type center_y: ``float``
:return: Hessian :math:`A/(2 \pi \sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)` for elliptical Gaussian convergence.
:rtype: tuple ``(float, float, float)`` , or ``(numpy.array, numpy.array, numpy.array)`` with each ``numpy.array``'s shape equal to ``x.shape``.
"""
phi_g, q = param_util.ellipticity2phi_q(e1, e2)
if q > 1 - self.min_ellipticity:
return self.spherical.hessian(x, y, amp, sigma, center_x, center_y)
# adjusting amplitude to make the notation compatible with the
# formulae given in Shajib (2019).
amp_ = amp / (2 * np.pi * sigma**2)
# converting ellipticity definition from q*x^2 + y^2/q to q^2*x^2 + y^2
sigma_ = sigma * np.sqrt(q) # * q
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_g)
sin_phi = np.sin(phi_g)
# rotated coordinates
x_ = cos_phi * x_shift + sin_phi * y_shift
y_ = -sin_phi * x_shift + cos_phi * y_shift
_p = q / sigma_ / np.sqrt(2 * (1. - q**2))
sig_func_re, sig_func_im = self.sigma_function(_p * x_, _p * y_, q)
kappa = amp_ * np.exp(-(q**2 * x_**2 + y_**2) / 2 / sigma_**2)
shear = -1 / (1 - q * q) * (
(1 + q**2) * kappa - 2 * q * amp_ +
np.sqrt(2 * np.pi) * q * q * amp_ *
(x_ - 1j * y_) / sigma_ / np.sqrt(1 - q * q) *
(sig_func_re - 1j * sig_func_im))
# in rotated frame
f_xx_ = kappa + shear.real
f_yy_ = kappa - shear.real
f_xy_ = shear.imag
# rotate back to the original frame
f_xx = f_xx_ * cos_phi**2 + f_yy_ * sin_phi**2 \
- 2 * sin_phi * cos_phi * f_xy_
f_yy = f_xx_ * sin_phi**2 + f_yy_ * cos_phi**2 \
+ 2 * sin_phi * cos_phi * f_xy_
f_xy = sin_phi * cos_phi * (f_xx_ - f_yy_) \
+ (cos_phi**2 - sin_phi**2) * f_xy_
return f_xx, f_xy, f_xy, f_yy
def density_2d(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
Compute the density of elliptical Gaussian :math:`A/(2 \pi
\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)`.
:param x: x coordinate.
:type x: ``float`` or ``numpy.array``
:param y: y coordinate.
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian.
:type sigma: ``float``
:param e1: Ellipticity parameter 1.
:type e1: ``float``
:param e2: Ellipticity parameter 2.
:type e2: ``float``
:param center_x: x coordinate of centroid.
:type center_x: ``float``
:param center_y: y coordianate of centroid.
:type center_y: ``float``
:return: Density :math:`\kappa` for elliptical Gaussian convergence.
:rtype: ``float``, or ``numpy.array`` with shape = ``x.shape``.
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, amp, sigma, e1, e2,
center_x, center_y)
return (f_xx + f_yy) / 2
@staticmethod
def sgn(z):
"""
Compute the sign function :math:`\mathrm{sgn}(z)` factor for
deflection as sugggested by Bray (1984). For current implementation, returning 1 is sufficient.
:param z: Complex variable :math:`z = x + \mathrm{i}y`
:type z: ``complex``
:return: :math:`\mathrm{sgn}(z)`
:rtype: ``float``
"""
return 1.
# np.sqrt(z*z)/z #np.sign(z.real*z.imag)
#return np.sign(z.real)
#if z.real != 0:
# return np.sign(z.real)
#else:
# return np.sign(z.imag)
#return np.where(z.real == 0, np.sign(z.real), np.sign(z.imag))
def sigma_function(self, x, y, q):
r"""
Compute the function :math:`\varsigma (z; q)` from equation (4.12)
of Shajib (2019).
:param x: Real part of complex variable, :math:`x = \mathrm{Re}(z)`
:type x: ``float`` or ``numpy.array``
:param y: Imaginary part of complex variable, :math:`y = \mathrm{Im}(z)`
:type y: ``float`` or ``numpy.array``
:param q: Axis ratio
:type q: ``float``
:return: real and imaginary part of :math:`\varsigma(z; q)` function
:rtype: tuple ``(type(x), type(x))``
"""
y_sign = np.sign(y)
y_ = deepcopy(y) * y_sign
z = x + 1j * y_
zq = q * x + 1j * y_ / q
w = self.w_f(z)
wq = self.w_f(zq)
# exponential factor in the 2nd term of eqn. (4.15) of Shajib (2019)
exp_factor = np.exp(-x * x * (1 - q * q) - y_ * y_ * (1 / q / q - 1))
sigma_func_real = w.imag - exp_factor * wq.imag
sigma_func_imag = (-w.real + exp_factor * wq.real) * y_sign
return sigma_func_real, sigma_func_imag
@staticmethod
def w_f_approx(z):
"""
Compute the Faddeeva function :math:`w_{\mathrm F}(z)` using the
approximation given in Zaghloul (2017).
:param z: complex number
:type z: ``complex`` or ``numpy.array(dtype=complex)``
:return: :math:`w_\mathrm{F}(z)`
:rtype: ``complex``
"""
sqrt_pi = 1 / np.sqrt(np.pi)
i_sqrt_pi = 1j * sqrt_pi
wz = np.empty_like(z)
z_imag2 = z.imag**2
abs_z2 = z.real**2 + z_imag2
reg1 = (abs_z2 >= 38000.)
if np.any(reg1):
wz[reg1] = i_sqrt_pi / z[reg1]
reg2 = (256. <= abs_z2) & (abs_z2 < 38000.)
if np.any(reg2):
t = z[reg2]
wz[reg2] = i_sqrt_pi * t / (t * t - 0.5)
reg3 = (62. <= abs_z2) & (abs_z2 < 256.)
if np.any(reg3):
t = z[reg3]
wz[reg3] = (i_sqrt_pi / t) * (1 + 0.5 / (t * t - 1.5))
reg4 = (30. <= abs_z2) & (abs_z2 < 62.) & (z_imag2 >= 1e-13)
if np.any(reg4):
t = z[reg4]
tt = t * t
wz[reg4] = (i_sqrt_pi * t) * (tt - 2.5) / (tt * (tt - 3.) + 0.75)
reg5 = (62. > abs_z2) & np.logical_not(reg4) & (abs_z2 > 2.5) & (
z_imag2 < 0.072)
if np.any(reg5):
t = z[reg5]
u = -t * t
f1 = sqrt_pi
f2 = 1
s1 = [1.320522, 35.7668, 219.031, 1540.787, 3321.99, 36183.31]
s2 = [
1.841439, 61.57037, 364.2191, 2186.181, 9022.228, 24322.84,
32066.6
]
for s in s1:
f1 = s - f1 * u
for s in s2:
f2 = s - f2 * u
wz[reg5] = np.exp(u) + 1j * t * f1 / f2
reg6 = (30.0 > abs_z2) & np.logical_not(reg5)
if np.any(reg6):
t3 = -1j * z[reg6]
f1 = sqrt_pi
f2 = 1
s1 = [
5.9126262, 30.180142, 93.15558, 181.92853, 214.38239, 122.60793
]
s2 = [
10.479857, 53.992907, 170.35400, 348.70392, 457.33448,
352.73063, 122.60793
]
for s in s1:
f1 = f1 * t3 + s
for s in s2:
f2 = f2 * t3 + s
wz[reg6] = f1 / f2
return wz
class TestGaussianKappaPot(object):
"""
test the Gaussian with Gaussian kappa
"""
def setup(self):
self.gaussian_kappa = GaussianKappa()
self.ellipse = GaussianEllipsePotential()
def test_function(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.function(x, y, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.function(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
def test_derivatives(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_x, f_y = self.ellipse.derivatives(x, y, amp, sigma, e1, e2)
f_x_sphere, f_y_sphere = self.gaussian_kappa.derivatives(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_x, f_x_sphere, decimal=8)
npt.assert_almost_equal(f_y, f_y_sphere, decimal=8)
def test_hessian(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_xx, f_xy, f_yx, f_yy = self.ellipse.hessian(x, y, amp, sigma, e1, e2)
f_xx_sphere, f_xy_sphere, f_yx_sphere, f_yy_sphere = self.gaussian_kappa.hessian(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_xx, f_xx_sphere, decimal=5)
npt.assert_almost_equal(f_yy, f_yy_sphere, decimal=5)
npt.assert_almost_equal(f_xy, f_xy_sphere, decimal=5)
npt.assert_almost_equal(f_xy, f_yx, decimal=8)
def test_density_2d(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.density_2d(x, y, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.density_2d(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
def test_mass_2d(self):
r = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.mass_2d(r, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.mass_2d(r, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
def test_mass_2d_lens(self):
r = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.mass_2d_lens(r, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.mass_2d_lens(r, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
class TestGaussianKappa(object):
"""
test the Gaussian with Gaussian kappa
"""
def setup(self):
self.gaussian_kappa = GaussianKappa()
self.gaussian = Gaussian()
def test_derivatives(self):
x = np.linspace(0, 5, 10)
y = np.linspace(0, 5, 10)
amp = 1. * 2 * np.pi
center_x = 0.
center_y = 0.
sigma = 1.
f_x, f_y = self.gaussian_kappa.derivatives(x, y, amp, sigma, center_x,
center_y)
npt.assert_almost_equal(f_x[2], 0.63813558702212059, decimal=8)
npt.assert_almost_equal(f_y[2], 0.63813558702212059, decimal=8)
def test_hessian(self):
x = np.linspace(0, 5, 10)
y = np.linspace(0, 5, 10)
amp = 1. * 2 * np.pi
center_x = 0.
center_y = 0.
sigma = 1.
f_xx, f_yy, f_xy = self.gaussian_kappa.hessian(x, y, amp, sigma,
center_x, center_y)
kappa = 1. / 2 * (f_xx + f_yy)
kappa_true = self.gaussian.function(x, y, amp, sigma, sigma, center_x,
center_y)
print(kappa_true)
print(kappa)
npt.assert_almost_equal(kappa[0], kappa_true[0], decimal=5)
npt.assert_almost_equal(kappa[1], kappa_true[1], decimal=5)
def test_density_2d(self):
x = np.linspace(0, 5, 10)
y = np.linspace(0, 5, 10)
amp = 1. * 2 * np.pi
center_x = 0.
center_y = 0.
sigma = 1.
f_xx, f_yy, f_xy = self.gaussian_kappa.hessian(x, y, amp, sigma,
center_x, center_y)
kappa = 1. / 2 * (f_xx + f_yy)
amp_3d = self.gaussian_kappa._amp2d_to_3d(amp, sigma, sigma)
density_2d = self.gaussian_kappa.density_2d(x, y, amp_3d, sigma,
center_x, center_y)
npt.assert_almost_equal(kappa[1], density_2d[1], decimal=5)
npt.assert_almost_equal(kappa[2], density_2d[2], decimal=5)
def test_3d_2d_convention(self):
x = np.linspace(0, 5, 10)
y = np.linspace(0, 5, 10)
amp = 1. * 2 * np.pi
center_x = 0.
center_y = 0.
sigma = 1.
amp_3d = self.gaussian_kappa._amp2d_to_3d(amp, sigma, sigma)
density_2d_gauss = self.gaussian_kappa.density_2d(
x, y, amp_3d, sigma, center_x, center_y)
density_2d = self.gaussian.function(x, y, amp, sigma, sigma, center_x,
center_y)
npt.assert_almost_equal(density_2d_gauss[1], density_2d[1], decimal=5)
class TestGaussianEllipseKappa(object):
"""
This class tests the methods for elliptical Gaussian convergence.
"""
def setup(self):
"""
:return:
:rtype:
"""
self.gaussian_kappa = GaussianKappa()
self.gaussian_kappa_ellipse = GaussianEllipseKappa()
def test_function(self):
"""
Test the `function()` method at the spherical limit.
:return:
:rtype:
"""
# almost spherical case
x = 1.
y = 1.
e1, e2 = 5e-5, 0.
sigma = 1.
amp = 2.
f_ = self.gaussian_kappa_ellipse.function(x, y, amp, sigma, e1, e2)
r2 = x*x + y*y
f_sphere = amp/(2.*np.pi*sigma**2) * sigma**2 * (np.euler_gamma -
expi(-r2/2./sigma**2) + np.log(r2/2./sigma**2))
npt.assert_almost_equal(f_, f_sphere, decimal=4)
# spherical case
e1, e2 = 0., 0.
f_ = self.gaussian_kappa_ellipse.function(x, y, amp, sigma, e1, e2)
npt.assert_almost_equal(f_, f_sphere, decimal=4)
def test_derivatives(self):
"""
Test the `derivatives()` method at the spherical limit.
:return:
:rtype:
"""
# almost spherical case
x = 1.
y = 1.
e1, e2 = 5e-5, 0.
sigma = 1.
amp = 2.
f_x, f_y = self.gaussian_kappa_ellipse.derivatives(x, y, amp, sigma,
e1, e2)
f_x_sphere, f_y_sphere = self.gaussian_kappa.derivatives(x, y, amp=amp,
sigma=sigma)
npt.assert_almost_equal(f_x, f_x_sphere, decimal=4)
npt.assert_almost_equal(f_y, f_y_sphere, decimal=4)
# spherical case
e1, e2 = 0., 0.
f_x, f_y = self.gaussian_kappa_ellipse.derivatives(x, y, amp, sigma,
e1, e2)
npt.assert_almost_equal(f_x, f_x_sphere, decimal=4)
npt.assert_almost_equal(f_y, f_y_sphere, decimal=4)
def test_hessian(self):
"""
Test the `hessian()` method at the spherical limit.
:return:
:rtype:
"""
# almost spherical case
x = 1.
y = 1.
e1, e2 = 5e-5, 0.
sigma = 1.
amp = 2.
f_xx, f_yy, f_xy = self.gaussian_kappa_ellipse.hessian(x, y, amp,
sigma, e1, e2)
f_xx_sphere, f_yy_sphere, f_xy_sphere = self.gaussian_kappa.hessian(x,
y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_xx, f_xx_sphere, decimal=4)
npt.assert_almost_equal(f_yy, f_yy_sphere, decimal=4)
npt.assert_almost_equal(f_xy, f_xy_sphere, decimal=4)
# spherical case
e1, e2 = 0., 0.
f_xx, f_yy, f_xy = self.gaussian_kappa_ellipse.hessian(x, y, amp,
sigma, e1, e2)
npt.assert_almost_equal(f_xx, f_xx_sphere, decimal=4)
npt.assert_almost_equal(f_yy, f_yy_sphere, decimal=4)
npt.assert_almost_equal(f_xy, f_xy_sphere, decimal=4)
def test_density_2d(self):
"""
Test the `density_2d()` method at the spherical limit.
:return:
:rtype:
"""
# almost spherical case
x = 1.
y = 1.
e1, e2 = 5e-5, 0.
sigma = 1.
amp = 2.
f_ = self.gaussian_kappa_ellipse.density_2d(x, y, amp, sigma, e1, e2)
f_sphere = amp / (2.*np.pi*sigma**2) * np.exp(-(x*x+y*y)/2./sigma**2)
npt.assert_almost_equal(f_, f_sphere, decimal=4)
def test_w_f_approx(self):
"""
Test the `w_f_approx()` method with values computed using
`scipy.special.wofz()`.
:return:
:rtype:
"""
x = np.logspace(-3., 3., 100)
y = np.logspace(-3., 3., 100)
X, Y = np.meshgrid(x, y)
w_f_app = self.gaussian_kappa_ellipse.w_f_approx(X+1j*Y)
w_f_scipy = wofz(X+1j*Y)
npt.assert_allclose(w_f_app.real, w_f_scipy.real, rtol=4e-5, atol=0)
npt.assert_allclose(w_f_app.imag, w_f_scipy.imag, rtol=4e-5, atol=0)
# check `derivatives()` method with and without `scipy.special.wofz()`
x = 1.
y = 1.
e1, e2 = 5e-5, 0
sigma = 1.
amp = 2.
# with `scipy.special.wofz()`
gauss_scipy = GaussianEllipseKappa(use_scipy_wofz=True)
f_x_sp, f_y_sp = gauss_scipy.derivatives(x, y, amp, sigma, e1, e2)
# with `GaussEllipseKappa.w_f_approx()`
gauss_approx = GaussianEllipseKappa(use_scipy_wofz=False)
f_x_ap, f_y_ap = gauss_approx.derivatives(x, y, amp, sigma, e1, e2)
npt.assert_almost_equal(f_x_sp, f_x_ap, decimal=4)
npt.assert_almost_equal(f_y_sp, f_y_ap, decimal=4)
class MultiGaussian_kappa(object):
"""
"""
def __init__(self):
self.gaussian_kappa = GaussianKappa()
def function(self, x, y, amp, sigma, center_x=0, center_y=0):
"""
:param x:
:param y:
:param amp:
:param sigma:
:param center_x:
:param center_y:
:return:
"""
f_ = np.zeros_like(x)
for i in range(len(amp)):
f_ += self.gaussian_kappa.function(x,
y,
amp=amp[i],
sigma_x=sigma[i],
sigma_y=sigma[i],
center_x=center_x,
center_y=center_y)
return f_
def derivatives(self, x, y, amp, sigma, center_x=0, center_y=0):
"""
:param x:
:param y:
:param amp:
:param sigma:
:param center_x:
:param center_y:
:return:
"""
f_x, f_y = np.zeros_like(x), np.zeros_like(x)
for i in range(len(amp)):
f_x_i, f_y_i = self.gaussian_kappa.derivatives(x,
y,
amp=amp[i],
sigma_x=sigma[i],
sigma_y=sigma[i],
center_x=center_x,
center_y=center_y)
f_x += f_x_i
f_y += f_y_i
return f_x, f_y
def hessian(self, x, y, amp, sigma, center_x=0, center_y=0):
"""
:param x:
:param y:
:param amp:
:param sigma:
:param center_x:
:param center_y:
:return:
"""
f_xx, f_yy, f_xy = np.zeros_like(x), np.zeros_like(x), np.zeros_like(x)
for i in range(len(amp)):
f_xx_i, f_yy_i, f_xy_i = self.gaussian_kappa.hessian(
x,
y,
amp=amp[i],
sigma_x=sigma[i],
sigma_y=sigma[i],
center_x=center_x,
center_y=center_y)
f_xx += f_xx_i
f_yy += f_yy_i
f_xy += f_xy_i
return f_xx, f_yy, f_xy
def density(self, r, amp, sigma):
"""
:param r:
:param amp:
:param sigma:
:return:
"""
d_ = np.zeros_like(r)
for i in range(len(amp)):
d_ += self.gaussian_kappa.density(r, amp[i], sigma[i], sigma[i])
return d_
def density_2d(self, x, y, amp, sigma, center_x=0, center_y=0):
"""
:param R:
:param am:
:param sigma_x:
:param sigma_y:
:return:
"""
d_3d = np.zeros_like(x)
for i in range(len(amp)):
d_3d += self.gaussian_kappa.density_2d(x, y, amp[i], sigma[i],
sigma[i], center_x,
center_y)
return d_3d
def mass_3d_lens(self, R, amp, sigma):
"""
:param R:
:param amp:
:param sigma:
:return:
"""
mass_3d = np.zeros_like(R)
for i in range(len(amp)):
mass_3d += self.gaussian_kappa.mass_3d_lens(
R, amp[i], sigma[i], sigma[i])
return mass_3d