/
analytic_covariance.py
400 lines (300 loc) · 17.2 KB
/
analytic_covariance.py
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from scipy.constants import c
from scipy import signal
from generaltools import symlog_bounds
from radiotelescope import beam_width
from plottools import colorbar
from generaltools import from_eta_to_k_par
from generaltools import from_u_to_k_perp
from generaltools import from_jansky_to_milikelvin
import numpy
import powerbox
import matplotlib.colors as colors
def sky_covariance(u, v, nu, S_low=0.1, S_mid=1, S_high=1):
gamma = 0.0
nn1, nn2 = numpy.meshgrid(nu, nu)
width_1_tile = beam_width(nn1)
width_2_tile = beam_width(nn2)
Sigma = width_1_tile ** 2 * width_2_tile ** 2 / (width_1_tile ** 2 + width_2_tile ** 2)
mu_2_r = moment_returner(2, S_low=S_low, S_mid=S_mid, S_high=S_high)
sky_covariance = 2 * numpy.pi * (nn1 * nn2 / numpy.min(nu) ** 2) ** -gamma * mu_2_r * Sigma * numpy.exp(
-2 * numpy.pi ** 2 * (u ** 2 + v ** 2) * (nn1 - nn2) ** 2 / numpy.min(nu) ** 2 * Sigma)
return sky_covariance
def beam_covariance(u, v, nu, dx=1):
x_offsets = numpy.array([-1.5, -0.5, 0.5, 1.5, -1.5, -0.5, 0.5, 1.5, -1.5,
-0.5, 0.5, 1.5, -1.5, -0.5, 0.5, 1.5], dtype=numpy.float32) * dx
y_offsets = numpy.array([1.5, 1.5, 1.5, 1.5, 0.5, 0.5, 0.5, 0.5, -0.5, -0.5,
-0.5, -0.5, -1.5, -1.5, -1.5, -1.5], dtype=numpy.float32) * dx
nn1, nn2, xx = numpy.meshgrid(nu, nu, x_offsets)
nn1, nn2, yy = numpy.meshgrid(nu, nu, y_offsets)
mu_1_r = moment_returner(1, S_low = 100e-3, S_high=1)
mu_2_r = moment_returner(2, S_low = 100e-3, S_high=1)
mu_1_m = moment_returner(1, S_low=1, S_high=10)
mu_2_m = moment_returner(2, S_low=1, S_high=10)
width_1_tile = numpy.sqrt(2) * beam_width(nn1)
width_2_tile = numpy.sqrt(2) * beam_width(nn2)
width_1_dipole = numpy.sqrt(2) * beam_width(nn1, diameter=1)
width_2_dipole = numpy.sqrt(2) * beam_width(nn2, diameter=1)
sigma_A = (width_1_tile * width_2_tile * width_1_dipole * width_2_dipole) ** 2 / (
width_2_tile ** 2 * width_1_dipole ** 2 * width_2_dipole ** 2 +
width_1_tile ** 2 * width_1_dipole ** 2 * width_2_dipole ** 2 +
width_1_tile ** 2 * width_2_tile ** 2 * width_1_dipole ** 2 +
width_1_tile ** 2 * width_2_tile ** 2 * width_2_dipole ** 2)
sigma_B = (width_1_tile * width_2_tile * width_2_dipole) ** 2 / (
2 * width_2_tile ** 2 * width_2_dipole ** 2 + width_1_tile ** 2 * width_2_dipole ** 2 +
width_1_tile ** 2 * width_2_tile ** 2)
sigma_C = (width_1_tile * width_2_tile * width_1_dipole) ** 2 / (
width_2_tile ** 2 * width_1_dipole ** 2 + 2 * width_1_tile ** 2 * width_1_dipole ** 2 +
width_1_tile ** 2 * width_2_tile ** 2)
sigma_D1 = width_1_tile ** 2 * width_1_dipole ** 2 / (width_1_tile ** 2 + width_1_dipole ** 2)
sigma_D2 = width_2_tile ** 2 * width_2_dipole ** 2 / (width_2_tile ** 2 + width_2_dipole ** 2)
A = 2 * numpy.pi * (mu_2_m + mu_2_r) / len(y_offsets) ** 3 * numpy.sum(
sigma_A * numpy.exp(-2 * numpy.pi ** 2 * sigma_A * (
(u / nu[0] + xx / c) ** 2 + (v / nu[0] + yy / c) ** 2) * (nn1 - nn2) ** 2.), axis=-1)
B = -2 * numpy.pi * mu_2_r / len(y_offsets) ** 2 * numpy.sum(sigma_B * numpy.exp(-2 * numpy.pi ** 2 * sigma_B * (
(u * (nn1 - nn2) / nu[0] + xx / c * nn2) ** 2 + (v * (nn1 - nn2) / nu[0] + yy / c * nn2) ** 2)),
axis=-1)
C = -2 * numpy.pi * mu_2_r / len(y_offsets) ** 2 * numpy.sum(sigma_C * numpy.exp(-2 * numpy.pi ** 2 * sigma_C * (
(u * (nn1 - nn2) / nu[0] + xx / c * nn1) ** 2 + (v * (nn1 - nn2) / nu[0] + yy / c * nn1) ** 2)),
axis=-1)
D = (mu_1_m ** 2 + 2 * mu_1_m * mu_1_r + mu_1_r ** 2) * 2 * numpy.pi * numpy.sum(
sigma_D1 * sigma_D2 / len(x_offsets) ** 3 * \
numpy.exp(-2 * numpy.pi ** 2 * sigma_D1 * (
(u * nn1 / nu[0] - xx / c * nn1) ** 2 + (v * nn1 / nu[0] - yy / c * nn1) ** 2)) * \
numpy.exp(-2 * numpy.pi ** 2 * sigma_D2 * (
(u * nn2 / nu[0] - xx / c * nn2) ** 2 + (v * nn2 / nu[0] - yy / c * nn2) ** 2)), axis=-1)
E = -(mu_1_m ** 2 + 2 * mu_1_m * mu_1_r + mu_1_r ** 2) * 2 * numpy.pi * numpy.sum(
sigma_D1 * sigma_D2 / len(x_offsets) ** 4 * \
numpy.exp(-2 * numpy.pi ** 2 * sigma_D1 * (
(u * nn1 / nu[0] - xx / c * nn1) ** 2 + (v * nn1 / nu[0] - yy / c * nn1) ** 2)), axis=-1) * \
numpy.sum(numpy.exp(-2 * numpy.pi ** 2 * sigma_D2 * ((u * nn2 / nu[0] - xx / c * nn2) ** 2 +
(v * nn2 / nu[0] - yy / c * nn2) ** 2)), axis=-1)
return A + B + C + D + E
def moment_returner(n_order, k1=4100, gamma1=1.59, k2=4100, gamma2=2.5, S_low=400e-3, S_mid=1, S_high=5.):
moment = k1 / (n_order + 1 - gamma1) * (S_mid ** (n_order + 1 - gamma1)) - S_low ** (n_order + 1 - gamma1) + \
k2 / (n_order + 1 - gamma2) * (S_high ** (n_order + 1 - gamma2)) - S_mid ** (n_order + 1 - gamma2)
return moment
def dft_matrix(nu):
dft = numpy.exp(-2 * numpy.pi * 1j / len(nu)) ** numpy.arange(0, len(nu), 1)
dftmatrix = numpy.vander(dft, increasing=True) / numpy.sqrt(len(nu))
eta = numpy.arange(0, len(nu), 1) / (nu.max() - nu.min())
return dftmatrix, eta
def blackman_harris_taper(frequency_range):
window = signal.blackmanharris(len(frequency_range))
return window
def compute_ps_variance(taper1, taper2, covariance, dft_matrix):
tapered_cov = covariance * taper1 * taper2
eta_cov = numpy.dot(numpy.dot(dft_matrix.conj().T, tapered_cov), dft_matrix)
variance = numpy.diag(numpy.real(eta_cov))
return variance
def calculate_beam_power_spectrum_averaged(u, nu):
uu, vv = numpy.meshgrid(u, u)
variance_cube = numpy.zeros((len(u), len(u), len(nu)))
window_function = blackman_harris_taper(nu)
taper1, taper2 = numpy.meshgrid(window_function, window_function)
dftmatrix, eta = dft_matrix(nu)
print("calculating all variances for all uv-cells")
for i in range(len(u)):
for j in range(len(u)):
nu_cov = beam_covariance(uu[i, j], vv[i, j], nu)
variance_cube[i, j, :] = compute_ps_variance(taper1, taper2, nu_cov, dftmatrix)
print("Taking circular average")
# Take circular average
power_spectrum_2d, u_bins = powerbox.tools.angular_average_nd(variance_cube, coords=[u, u, eta], bins=len(u), n=2)
return eta[:int(len(eta) / 2)], power_spectrum_2d[:, :int(len(eta) / 2)]
def calculate_beam_power_spectrum(u, nu, save=False, plot_name="beam_2D_ps.pdf"):
window_function = blackman_harris_taper(nu)
taper1, taper2 = numpy.meshgrid(window_function, window_function)
dftmatrix, eta = dft_matrix(nu)
variance = numpy.zeros((len(u), len(nu)))
print(f"Calculating covariances for all baselines")
for i in range(len(u)):
nu_cov = beam_covariance(u[i], v=0, nu=nu)
variance[i, :] = compute_ps_variance(taper1, taper2, nu_cov, dftmatrix)
return eta[:int(len(eta) / 2)], variance[:, :int(len(eta) / 2)]
def calculate_total_power_spectrum(u, nu, full=False, save=False, plot=True, plot_name="total_ps.pdf"):
window_function = blackman_harris_taper(nu)
taper1, taper2 = numpy.meshgrid(window_function, window_function)
dftmatrix, eta = dft_matrix(nu)
variance = numpy.zeros((len(u), len(nu)))
if full:
uu, vv = numpy.meshgrid(u, u)
variance_cube = numpy.zeros((len(u), len(u), len(nu)))
for i in range(len(u)):
for j in range(len(u)):
nu_cov = sky_covariance(u[i], 0, nu) + beam_covariance(uu[i, j], vv[i, j], nu)
variance_cube[i, j, :] = compute_ps_variance(taper1, taper2, nu_cov, dftmatrix)
variance[:], u_bins = powerbox.tools.angular_average_nd(variance_cube, coords=[u, u, eta], bins=len(u),
n=2)
else:
for i in range(len(u)):
nu_cov = sky_covariance(u[i], 0, nu) + beam_covariance(u[i], v=0, nu=nu)
variance[i, :] = compute_ps_variance(taper1, taper2, nu_cov, dftmatrix)
return eta[:int(len(eta) / 2)], variance[:, :int(len(eta) / 2)]
def calculate_sky_power_spectrum(u_range, nu_range, title="Sky", save=False, plot_name="sky_ps.pdf"):
window_function = blackman_harris_taper(nu_range)
taper1, taper2 = numpy.meshgrid(window_function, window_function)
dftmatrix, eta = dft_matrix(nu_range)
sky_variance = numpy.zeros((len(u_range), len(nu_range)))
for i in range(len(u_range)):
nu_cov = sky_covariance(u_range[i], 0, nu_range)
sky_variance[i, :] = compute_ps_variance(taper1, taper2, nu_cov, dftmatrix)
return eta[:int(len(eta) / 2)], sky_variance[:, :int(len(eta) / 2)]
def plot_PS(u_bins, eta_bins, nu, PS, cosmological=False, ratio=False, title=None, save=False, axes=None,
save_name="plot.pdf", axes_label_font=20, tickfontsize=15, xlabel_show=False, ylabel_show=False,
zlabel_show=False):
if axes is None:
figure, axes = pyplot.subplots(1, 1)
if cosmological:
central_frequency = nu[int(len(nu) / 2)]
x_values = from_u_to_k_perp(u_bins, central_frequency)
y_values = from_eta_to_k_par(eta_bins, central_frequency)
if ratio:
z_values = PS
else:
z_values = from_jansky_to_milikelvin(PS, nu)
x_label = r"$k_{\perp}$ [Mpc$^{-1}$]"
y_label = r"$k_{\parallel}$ [Mpc$^{-1}$]"
z_label = r"Variance [mK$^2$ Mpc$^3$ ]"
axes.set_xlim(1e-3, 1e-1)
axes.set_ylim(9e-3, 5e-1)
else:
x_values = u_bins
y_values = eta_bins
z_values = PS
x_label = r"|u|"
y_label = r"$\eta$ [MHz$^{-1}$]"
z_label = r"Variance [Jy$^2$ Hz$^2$]"
axes.set_xlim(xmin=1, xmax=200)
axes.set_ylim(eta_bins[1], eta_bins.max())
if PS.min() < -3e-5:
print(f"SymLog Norm scaled Data: {PS.min()} to {PS.max()}")
symlog_min, symlog_max, symlog_threshold, symlog_scale = symlog_bounds(numpy.real(z_values))
norm = colors.SymLogNorm(linthresh=symlog_threshold, linscale=1, vmin=-symlog_max, vmax=symlog_max)
colormap = "coolwarm"
else:
z_values[PS < 0] = numpy.abs(z_values[PS < 0])
print(z_values.min(), z_values.max())
# symlog_min, symlog_max, symlog_threshold, symlog_scale = symlog_bounds(numpy.real(z_values))
norm = colors.LogNorm(vmin=numpy.real(z_values).min(), vmax=numpy.real(z_values).max())
colormap = "viridis"
if title is not None:
axes.set_title(title)
# psplot = axes.pcolor(x_values, y_values, z_values.T, norm=norm, cmap=colormap, rasterized = True)
psplot = axes.pcolor(x_values, y_values, z_values.T, cmap=colormap, rasterized=True, norm=norm)
cax = colorbar(psplot)
axes.set_xscale('log')
axes.set_yscale('log')
if xlabel_show:
axes.set_xlabel(x_label, fontsize=axes_label_font)
if ylabel_show:
axes.set_ylabel(y_label, fontsize=axes_label_font)
if zlabel_show:
cax.set_label(z_label, fontsize=axes_label_font)
axes.tick_params(axis='both', which='major', labelsize=tickfontsize)
cax.ax.tick_params(axis='both', which='major', labelsize=tickfontsize)
if save:
figure.savefig(save_name)
else:
pass
return
def test_dft_on_signal():
# make a sinusoidal signal
time = numpy.linspace(0, 100, 101)
f1 = 1 / 5
f2 = 1 / 4
sample_rate = 1 / (time.max() - time.min())
signal = numpy.sin(2 * numpy.pi * f1 * time) + numpy.sin(2 * numpy.pi * f2 * time)
taper = blackman_harris_taper(time)
dftmatrix = dft_matrix(time)
frequencies = numpy.arange(0, len(time), 1) * sample_rate
ft_signal = numpy.dot(dftmatrix, taper * signal)
inverse_ft_signal = numpy.dot(dftmatrix.conj().T, ft_signal)
fig = pyplot.figure()
axes1 = fig.add_subplot(131)
axes2 = fig.add_subplot(132)
axes3 = fig.add_subplot(133)
axes1.plot(time, taper * signal)
axes2.plot(frequencies[:int(len(frequencies) / 2)], numpy.abs(ft_signal[:int(len(frequencies) / 2)]))
axes3.plot(inverse_ft_signal)
pyplot.show()
return
def gain_error_covariance(u_range, frequency_range, residuals='both', weights=None, broken_baseline_weight = 1):
model_variance = numpy.diag(sky_covariance(0, 0, frequency_range, S_low=1, S_high=10))
model_normalisation = numpy.sqrt(numpy.outer(model_variance, model_variance))
gain_error_covariance = numpy.zeros((len(u_range), len(frequency_range), len(frequency_range)))
# Compute all residual to model ratios at different u scales
for u_index in range(len(u_range)):
if residuals == "sky":
residual_covariance = sky_covariance(u_range[u_index], v=0, nu=frequency_range)
elif residuals == "beam":
residual_covariance = broken_baseline_weight **2*beam_covariance(u_range[u_index], v=0, nu=frequency_range)
elif residuals == 'both':
residual_covariance = sky_covariance(u_range[u_index], v=0, nu=frequency_range) + \
broken_baseline_weight **2*beam_covariance(u_range[u_index], v=0, nu=frequency_range)
gain_error_covariance[u_index, :, :] = residual_covariance / model_normalisation
if weights is None:
gain_averaged_covariance = numpy.sum(gain_error_covariance, axis=0) * (1/(127*len(u_range))) ** 2
else:
gain_averaged_covariance = gain_error_covariance.copy()
for u_index in range(len(u_range)):
u_weight_reshaped = numpy.tile(weights[u_index, :].flatten(), (len(frequency_range), len(frequency_range), 1)).T
gain_averaged_covariance[u_index, ...] = numpy.sum(gain_error_covariance * u_weight_reshaped, axis=0)
return gain_averaged_covariance
def compute_weights(u_cells, u, v):
u_bin_edges = numpy.zeros(len(u_cells) + 1)
baseline_lengths = numpy.sqrt(u**2 + v**2)
log_steps = numpy.diff(numpy.log10(u_cells))
u_bin_edges[1:] = 10**(numpy.log10(u_cells) + 0.5*log_steps[0])
u_bin_edges[0] = 10**(numpy.log10(u_cells[0] - 0.5*log_steps[0]))
counts, bin_edges = numpy.histogram(baseline_lengths, bins=u_bin_edges)
prime, unprime = numpy.meshgrid(counts/len(baseline_lengths), counts/len(baseline_lengths))
weights = prime*unprime*(2/127)**2
return weights
def residual_ps_error(u_range, frequency_range, residuals='both', broken_baselines_weight = 1, weights = None, path="./", plot=True):
cal_variance = numpy.zeros((len(u_range), len(frequency_range)))
raw_variance = numpy.zeros((len(u_range), len(frequency_range)))
window_function = blackman_harris_taper(frequency_range)
taper1, taper2 = numpy.meshgrid(window_function, window_function)
dftmatrix, eta = dft_matrix(frequency_range)
gain_averaged_covariance = gain_error_covariance(u_range, frequency_range, residuals=residuals, weights= weights,
broken_baseline_weight = broken_baselines_weight)
# Compute the gain corrected residuals at all u scales
if residuals == "sky":
residual_variance = sky_covariance(0, 0, frequency_range)
elif residuals == "beam":
residual_variance = broken_baselines_weight **2*beam_covariance(0, v=0, nu=frequency_range)
elif residuals == 'both':
residual_variance = sky_covariance(0, 0, frequency_range) + \
broken_baselines_weight **2*beam_covariance(0, v=0, nu=frequency_range)
gain = residual_variance / sky_covariance(0, 0, frequency_range)
for i in range(len(u_range)):
if residuals == "sky":
residual_covariance = sky_covariance(u_range[i], 0, frequency_range)
blaah = 0
elif residuals == "beam":
residual_covariance = broken_baselines_weight**2*beam_covariance(u_range[i], v=0, nu=frequency_range)
blaah = 0
elif residuals == 'both':
residual_covariance = sky_covariance(u_range[i], 0, frequency_range) + \
broken_baselines_weight**2*beam_covariance(u_range[i], v=0, nu=frequency_range)
blaah = 0
model_covariance = sky_covariance(u_range[i], 0, frequency_range, S_low=1, S_high=10)
scale = numpy.diag(numpy.zeros_like(frequency_range) ) + 1 + blaah
if weights is None:
nu_cov = 2*gain_averaged_covariance*model_covariance + \
(scale + 2*gain_averaged_covariance)*residual_covariance
else:
nu_cov = 2*gain_averaged_covariance[i, ...]*model_covariance + \
(scale + 2*gain_averaged_covariance[i, ...])*residual_covariance
cal_variance[i, :] = compute_ps_variance(taper1, taper2, nu_cov, dftmatrix)
raw_variance[i, :] = compute_ps_variance(taper1, taper2, residual_covariance, dftmatrix)
return eta[:int(len(eta) / 2)], raw_variance[:, :int(len(eta) / 2)], cal_variance[:, :int(len(eta) / 2)]
if __name__ == "__main__":
from matplotlib import pyplot
u = numpy.logspace(-1, 2.5, 100)
nu = numpy.linspace(140, 160, 101) * 1e6
output_folder = "../../Plots/Analytic_Covariance/"
#beam_covariance(u =0, v=0, nu=nu, dx=1)
# calculate_sky_power_spectrum(u, nu, save=False, plot_name=output_folder + "sky_ps.pdf")
# pyplot.show()
compute_weights(u, 0, 0)