Convert WaveX to PLRedNoise and DMWaveX to PLDMNoise

CHANGELOG-unreleased.md

@@ -30,6 +30,8 @@ the released changes.
     - `Residuals.d_lnlikelihood_d_whitenoise_param` will throw a `NotImplementedError` when correlated noise is present.
     - `DownhillFitter._fit_noise()` doesn't use derivatives when correlated noise is present.
     - Documentation: Noise fitting example notebook.
+- `freeze_params` option in `wavex_setup` and `dmwavex_setup`
+- `plrednoise_from_wavex`, `pldmnoise_from_dmwavex`, and `find_optimal_nharms` functions
 ### Fixed
 - `MCMC_walkthrough` notebook now runs
 - Fixed runtime data README 

docs/examples/rednoise-fit-example.py

@@ -0,0 +1,374 @@
+# %% [markdown]
+# # Red noise and DM noise fitting examples
+#
+# This notebook provides an example on how to fit for red noise
+# and DM noise using PINT using simulated datasets.
+#
+# We will use the `PLRedNoise` and `PLDMNoise` models to generate
+# noise realizations (these models provide Fourier Gaussian process
+# descriptions of achromatic red noise and DM noise respectively).
+#
+# We will fit the generated datasets using the `WaveX` and `DMWaveX` models,
+# which provide deterministic Fourier representations of achromatic red noise
+# and DM noise respectively.
+#
+# Finally, we will convert the `WaveX`/`DMWaveX` amplitudes into spectral
+# parameters and compare them with the injected values.
+
+# %%
+from pint import DMconst
+from pint.models import get_model
+from pint.simulation import make_fake_toas_uniform
+from pint.logging import setup as setup_log
+from pint.fitter import WLSFitter
+from pint.utils import (
+    dmwavex_setup,
+    find_optimal_nharms,
+    wavex_setup,
+    plrednoise_from_wavex,
+    pldmnoise_from_dmwavex,
+)
+
+from io import StringIO
+import numpy as np
+import astropy.units as u
+from matplotlib import pyplot as plt
+from copy import deepcopy
+
+setup_log(level="WARNING")
+
+# %% [markdown]
+# ## Red noise fitting
+
+# %% [markdown]
+# ### Simulation
+# The first step is to generate a simulated dataset for demonstration.
+# Note that we are adding PHOFF as a free parameter. This is required
+# for the fit to work properly.
+
+# %%
+par_sim = """
+    PSR           SIM3
+    RAJ           05:00:00     1
+    DECJ          15:00:00     1
+    PEPOCH        55000
+    F0            100          1
+    F1            -1e-15       1 
+    PHOFF         0            1
+    DM            15           1
+    TNREDAMP      -13
+    TNREDGAM      3.5
+    TNREDC        30
+    TZRMJD        55000
+    TZRFRQ        1400 
+    TZRSITE       gbt
+    UNITS         TDB
+    EPHEM         DE440
+    CLOCK         TT(BIPM2019)
+"""
+
+m = get_model(StringIO(par_sim))
+
+# %%
+# Now generate the simulated TOAs.
+ntoas = 2000
+toaerrs = np.random.uniform(0.5, 2.0, ntoas) * u.us
+freqs = np.linspace(500, 1500, 8) * u.MHz
+
+t = make_fake_toas_uniform(
+    startMJD=53001,
+    endMJD=57001,
+    ntoas=ntoas,
+    model=m,
+    freq=freqs,
+    obs="gbt",
+    error=toaerrs,
+    add_noise=True,
+    add_correlated_noise=True,
+    name="fake",
+    include_bipm=True,
+    include_gps=True,
+    multi_freqs_in_epoch=True,
+)
+
+# %% [markdown]
+# ### Optimal number of harmonics
+# The optimal number of harmonics can be estimated by minimizing the
+# Akaike Information Criterion (AIC). This is implemented in the
+# `pint.utils.find_optimal_nharms` function.
+
+# %%
+m1 = deepcopy(m)
+m1.remove_component("PLRedNoise")
+
+nharm_opt, d_aics = find_optimal_nharms(m1, t, "WaveX", 30)
+
+print("Optimum no of harmonics = ", nharm_opt)
+
+# %%
+print(np.argmin(d_aics))
+
+# %%
+# The Y axis is plotted in log scale only for better visibility.
+plt.scatter(list(range(len(d_aics))), d_aics + 1)
+plt.axvline(nharm_opt, color="red", label="Optimum number of harmonics")
+plt.axvline(
+    int(m.TNREDC.value), color="black", ls="--", label="Injected number of harmonics"
+)
+plt.xlabel("Number of harmonics")
+plt.ylabel("AIC - AIC$_\\min{} + 1$")
+plt.legend()
+plt.yscale("log")
+# plt.savefig("sim3-aic.pdf")
+
+# %%
+# Now create a new model with the optimum number of harmonics
+m2 = deepcopy(m1)
+Tspan = t.get_mjds().max() - t.get_mjds().min()
+wavex_setup(m2, T_span=Tspan, n_freqs=nharm_opt, freeze_params=False)
+
+ftr = WLSFitter(t, m2)
+ftr.fit_toas(maxiter=10)
+m2 = ftr.model
+
+print(m2)
+
+# %% [markdown]
+# ### Estimating the spectral parameters from the WaveX fit.
+
+# %%
+# Get the Fourier amplitudes and powers and their uncertainties.
+idxs = np.array(m2.components["WaveX"].get_indices())
+a = np.array([m2[f"WXSIN_{idx:04d}"].quantity.to_value("s") for idx in idxs])
+da = np.array([m2[f"WXSIN_{idx:04d}"].uncertainty.to_value("s") for idx in idxs])
+b = np.array([m2[f"WXCOS_{idx:04d}"].quantity.to_value("s") for idx in idxs])
+db = np.array([m2[f"WXCOS_{idx:04d}"].uncertainty.to_value("s") for idx in idxs])
+print(len(idxs))
+
+P = (a**2 + b**2) / 2
+dP = ((a * da) ** 2 + (b * db) ** 2) ** 0.5
+
+f0 = (1 / Tspan).to_value(u.Hz)
+fyr = (1 / u.year).to_value(u.Hz)
+
+# %%
+# We can create a `PLRedNoise` model from the `WaveX` model.
+# This will estimate the spectral parameters from the `WaveX`
+# amplitudes.
+m3 = plrednoise_from_wavex(m2)
+print(m3)
+
+# %%
+# Now let us plot the estimated spectrum with the injected
+# spectrum.
+plt.subplot(211)
+plt.errorbar(
+    idxs * f0,
+    b * 1e6,
+    db * 1e6,
+    ls="",
+    marker="o",
+    label="$\\hat{a}_j$ (WXCOS)",
+    color="red",
+)
+plt.errorbar(
+    idxs * f0,
+    a * 1e6,
+    da * 1e6,
+    ls="",
+    marker="o",
+    label="$\\hat{b}_j$ (WXSIN)",
+    color="blue",
+)
+plt.axvline(fyr, color="black", ls="dotted")
+plt.axhline(0, color="grey", ls="--")
+plt.ylabel("Fourier coeffs ($\mu$s)")
+plt.xscale("log")
+plt.legend(fontsize=8)
+
+plt.subplot(212)
+plt.errorbar(
+    idxs * f0, P, dP, ls="", marker="o", label="Spectral power (PINT)", color="k"
+)
+P_inj = m.components["PLRedNoise"].get_noise_weights(t)[::2][:nharm_opt]
+plt.plot(idxs * f0, P_inj, label="Injected Spectrum", color="r")
+P_est = m3.components["PLRedNoise"].get_noise_weights(t)[::2][:nharm_opt]
+print(len(idxs), len(P_est))
+plt.plot(idxs * f0, P_est, label="Estimated Spectrum", color="b")
+plt.xscale("log")
+plt.yscale("log")
+plt.ylabel("Spectral power (s$^2$)")
+plt.xlabel("Frequency (Hz)")
+plt.axvline(fyr, color="black", ls="dotted", label="1 yr$^{-1}$")
+plt.legend()
+
+# %% [markdown]
+# Note the outlier in the 1 year^-1 bin. This is caused by the covariance with RA and DEC, which introduce a delay with the same frequency.
+
+# %% [markdown]
+# ## DM noise fitting
+# Let us now do a similar kind of analysis for DM noise.
+
+# %%
+par_sim = """
+    PSR           SIM4
+    RAJ           05:00:00     1
+    DECJ          15:00:00     1
+    PEPOCH        55000
+    F0            100          1
+    F1            -1e-15       1 
+    PHOFF         0            1
+    DM            15           1
+    TNDMAMP       -13
+    TNDMGAM       3.5
+    TNDMC         30
+    TZRMJD        55000
+    TZRFRQ        1400 
+    TZRSITE       gbt
+    UNITS         TDB
+    EPHEM         DE440
+    CLOCK         TT(BIPM2019)
+"""
+
+m = get_model(StringIO(par_sim))
+
+# %%
+# Generate the simulated TOAs.
+ntoas = 2000
+toaerrs = np.random.uniform(0.5, 2.0, ntoas) * u.us
+freqs = np.linspace(500, 1500, 8) * u.MHz
+
+t = make_fake_toas_uniform(
+    startMJD=53001,
+    endMJD=57001,
+    ntoas=ntoas,
+    model=m,
+    freq=freqs,
+    obs="gbt",
+    error=toaerrs,
+    add_noise=True,
+    add_correlated_noise=True,
+    name="fake",
+    include_bipm=True,
+    include_gps=True,
+    multi_freqs_in_epoch=True,
+)
+
+# %%
+# Find the optimum number of harmonics by minimizing AIC.
+m1 = deepcopy(m)
+m1.remove_component("PLDMNoise")
+
+m2 = deepcopy(m1)
+
+nharm_opt, d_aics = find_optimal_nharms(m2, t, "DMWaveX", 30)
+print("Optimum no of harmonics = ", nharm_opt)
+
+# %%
+# The Y axis is plotted in log scale only for better visibility.
+plt.scatter(list(range(len(d_aics))), d_aics + 1)
+plt.axvline(nharm_opt, color="red", label="Optimum number of harmonics")
+plt.axvline(
+    int(m.TNDMC.value), color="black", ls="--", label="Injected number of harmonics"
+)
+plt.xlabel("Number of harmonics")
+plt.ylabel("AIC - AIC$_\\min{} + 1$")
+plt.legend()
+plt.yscale("log")
+# plt.savefig("sim3-aic.pdf")
+
+# %%
+# Now create a new model with the optimum number of
+# harmonics
+m2 = deepcopy(m1)
+
+Tspan = t.get_mjds().max() - t.get_mjds().min()
+dmwavex_setup(m2, T_span=Tspan, n_freqs=nharm_opt, freeze_params=False)
+
+ftr = WLSFitter(t, m2)
+ftr.fit_toas(maxiter=10)
+m2 = ftr.model
+
+print(m2)
+
+# %% [markdown]
+# ### Estimating the spectral parameters from the `DMWaveX` fit.
+
+# %%
+# Get the Fourier amplitudes and powers and their uncertainties.
+# Note that the `DMWaveX` amplitudes have the units of DM.
+# We multiply them by a constant factor to convert them to dimensions
+# of time so that they are consistent with `PLDMNoise`.
+scale = DMconst / (1400 * u.MHz) ** 2
+
+idxs = np.array(m2.components["DMWaveX"].get_indices())
+a = np.array(
+    [(scale * m2[f"DMWXSIN_{idx:04d}"].quantity).to_value("s") for idx in idxs]
+)
+da = np.array(
+    [(scale * m2[f"DMWXSIN_{idx:04d}"].uncertainty).to_value("s") for idx in idxs]
+)
+b = np.array(
+    [(scale * m2[f"DMWXCOS_{idx:04d}"].quantity).to_value("s") for idx in idxs]
+)
+db = np.array(
+    [(scale * m2[f"DMWXCOS_{idx:04d}"].uncertainty).to_value("s") for idx in idxs]
+)
+print(len(idxs))
+
+P = (a**2 + b**2) / 2
+dP = ((a * da) ** 2 + (b * db) ** 2) ** 0.5
+
+f0 = (1 / Tspan).to_value(u.Hz)
+fyr = (1 / u.year).to_value(u.Hz)
+
+# %%
+# We can create a `PLDMNoise` model from the `DMWaveX` model.
+# This will estimate the spectral parameters from the `DMWaveX`
+# amplitudes.
+m3 = pldmnoise_from_dmwavex(m2)
+print(m3)
+
+# %%
+# Now let us plot the estimated spectrum with the injected
+# spectrum.
+plt.subplot(211)
+plt.errorbar(
+    idxs * f0,
+    b * 1e6,
+    db * 1e6,
+    ls="",
+    marker="o",
+    label="$\\hat{a}_j$ (DMWXCOS)",
+    color="red",
+)
+plt.errorbar(
+    idxs * f0,
+    a * 1e6,
+    da * 1e6,
+    ls="",
+    marker="o",
+    label="$\\hat{b}_j$ (DMWXSIN)",
+    color="blue",
+)
+plt.axvline(fyr, color="black", ls="dotted")
+plt.axhline(0, color="grey", ls="--")
+plt.ylabel("Fourier coeffs ($\mu$s)")
+plt.xscale("log")
+plt.legend(fontsize=8)
+
+plt.subplot(212)
+plt.errorbar(
+    idxs * f0, P, dP, ls="", marker="o", label="Spectral power (PINT)", color="k"
+)
+P_inj = m.components["PLDMNoise"].get_noise_weights(t)[::2][:nharm_opt]
+plt.plot(idxs * f0, P_inj, label="Injected Spectrum", color="r")
+P_est = m3.components["PLDMNoise"].get_noise_weights(t)[::2][:nharm_opt]
+print(len(idxs), len(P_est))
+plt.plot(idxs * f0, P_est, label="Estimated Spectrum", color="b")
+plt.xscale("log")
+plt.yscale("log")
+plt.ylabel("Spectral power (s$^2$)")
+plt.xlabel("Frequency (Hz)")
+plt.axvline(fyr, color="black", ls="dotted", label="1 yr$^{-1}$")
+plt.legend()

docs/tutorials.rst

@@ -63,6 +63,7 @@ are not included in the default build because they take too long, but you can do
    examples/How_to_build_a_timing_model_component.ipynb
    examples/understanding_fitters.ipynb
    examples/noise-fitting-example.ipynb
+   examples/rednoise-fit-example.ipynb
    examples/WorkingWithFlags.ipynb
    examples/Wideband_TOA_walkthrough.ipynb
    examples/Simulate_and_make_MassMass.ipynb