Finding Neptune using Uranus

Finding Neptune using Uranus#

Note : - For better understanding of this notebook, refer to the md_file designed specifically for this notebook and better insights into the theory.

The import methods and functions from neptune_bayesian_methods, neptune_deterministic_methods and setup_inversion are used to set up the simulation and perform the necessary calculations.

# This notebook requires the following libraries to run, in order to install them uncomment the lines below
# %pip install cofi
# %pip install numba
# %pip install tqdm
# %pip install matplotlib
# %pip install astroquery

1. Introduction#

The following Notebook is based on the historical problem on how Neptune was found by Johann Galle using mathematical predictions made independently by two astronomers:
- Urbain Le Verrier (France)
- John Couch Adams (England)
Through this Notebook we wish to demostrate how CoFI can be used to solve this problem via deterministic inversion. For more details on this problem, see the following thesis
In the following notebook we discuss the problem of finding Neptune’s mass, its velocity components and its position coordinates in the year 1775, by modeling the trajectory of Uranus with and without the influence of Neptune.
We define $ g(m) $, our forward model, as vector-valued function that predicts the position coordinates of Uranus at each observation time $t_j$, as a function of Neptune’s parameters $m$:

\[\begin{split} g(m) = \begin{bmatrix} \hat x_1(m) \\ \vdots \\ \hat x_N(m) \\ \hat y_1(m) \\ \vdots \\ \hat y_N(m) \\ \hat z_1(m) \\ \vdots \\ \hat z_N(m) \end{bmatrix} \in \mathbb{R}^{3M \times 1}\end{split}\]

where $ N $ is the number of data points, and $ :raw-latex:`\hat `x_j(m), :raw-latex:`hat y_j(m), :raw-latex:hat `z_j(m) $ are the coordinates of Uranus at data point $ j = 1, 2, …. N$ as a function of Neptune’s parameters $ m $,

where $m = (m_M, m_x, m_y, m_z, {m_{v_x}}, m_{v_y}, m_{v_z})$ is the set of parameters describing Neptune’s mass ($m_M$), its position coordinates $(m_x, m_y, m_z)$ and its velocity components $(m_{v_x}, m_{v_y}, m_{v_z})$

and $d$ as the data vector of positions of Uranus at different time steps:

\[\begin{split}d = \begin{bmatrix} x_1 \\ \vdots \\ x_N \\ y_1 \\ \vdots \\ y_N \\ z_1 \\ \vdots \\ z_N \end{bmatrix} \in \mathbb{R}^{3M \times 1}\end{split}\]

where $ N $ is the number of data points, and $ :raw-latex:`\hat `x_j, :raw-latex:`hat y_j, :raw-latex:hat `z_j $ are the true coordinates of Uranus at data point $ j = 1, 2, …. N$.
hence our problem formulation changes to :

\[\underset{m}{\min} || g(m) - {d} ||_{2}^2\]

2. Problem Setting#

We use Newton’s Law of gravitation to compute the gravitational force acting on a planet due to other celestial bodies.
This formulation uses Newton’s Law of Universal Gravitation to model the net gravitational influence from multiple bodies on a single target planet.

Let $r$ be the vector containing the positions of all planets and $a$ be the vector containing the accelerations of all planets.

The velocity and acceleration vectors are defined as:

\[\dot{\mathbf{r}} = \mathbf{v}, \quad \ddot{\mathbf{r}} = \mathbf{a}\]

where

\[\begin{split}\mathbf{r} = \begin{bmatrix} \mathbf{r}_1 \\ \vdots \\ \mathbf{r}_9 \end{bmatrix}, \quad \mathbf{a} = \begin{bmatrix} \mathbf{a}_1 \\ \vdots \\ \mathbf{a}_9 \end{bmatrix}, \quad \mathbf{r}, \mathbf{a} \in \mathbb{R}^{27 \times 1}\end{split}\]
This results in the system of differential equations:

\[\begin{split}\frac{d}{dt} \begin{bmatrix} \mathbf{r}(t) \\ \mathbf{v}(t) \end{bmatrix} = \begin{bmatrix} \mathbf{v}(t) \\ \mathbf{a}(t) \end{bmatrix}\end{split}\]

import numpy as np
import matplotlib.pyplot as plt
from tqdm import tqdm
from typing import Callable
import copy
from numba import njit, jit
import warnings
import multiprocessing as mp
import time
import arviz
from astroquery.jplhorizons import Horizons
warnings.filterwarnings('ignore')

from cofi import BaseProblem, InversionOptions, Inversion
from neptune_mcmc_helpers import (
    analysis_samples,
    load_mcmc_results,
    plot_pair_kde_legacy_style,
    run_cofi_emcee,
    safe_autocorr_time,
    save_mcmc_results,
    scaled_reference_values,
    selected_trajectory_samples,
)

np.random.seed(42)

# Long-run control
# Keep RUN_LONG_MCMC = False for quick validation. Set it to True when you want
# to regenerate converged MCMC outputs and save them beside this notebook.
from pathlib import Path

RUN_LONG_MCMC = True
USE_SAVED_LONG_RUN_RESULTS = False
SAVE_LONG_RUN_RESULTS = False

VALIDATION_STEPS = 40
SYNTHETIC_LONG_STEPS = 10_000
REAL_LONG_STEPS = 50_000
LONG_RUN_RESULTS_FILE = Path("neptune_long_run_results.npz")

We solve our ODEs with the Runge-Kutta 4 (RK4) method, which is an explicit and iterative method, well-suited for initial value problems.

In the following cell, we import - acceleration and rk4_step to serve as functions for our forward model.

We now demostrate our forward model, using the above defined functions, in the run_simulation method defined below, which helps us run a simulation of solar system.
Throughout this notebook for the purpose of our inversion, we are going to define mass in terms of solar masses, positions coordinates in Astonomical Units (AU) and velocities for planets in Au/day

from neptune_deterministic_methods import acceleration, rk4_step, run_simulation
trajectories = run_simulation(T = 100, dt = 1, plot_only=['Uranus', 'Neptune', 'Saturn', 'Jupiter', 'Mars', 'Earth', 'Venus', 'Mercury'])

2.1 Bayesian Inversion via the Stretch Move#

In this notebook, we use CoFI to perform Bayesian inversion. Specifically, we use the stretch move ensemble sampler introduced by goodman & Weare (2010), which is well-suited for exploring complex, high-dimensional posterior distributions.

MCMC is a method to sample from the posterior distribution of model parameters $\mathbf{m}$ given observed data $\mathbf{d}$. According to Bayes’ theorem:

\[\tag{1} p(\mathbf{m} \mid \mathbf{d}) \propto p(\mathbf{d} \mid \mathbf{m}) \, p(\mathbf{m}),\]

where:

$p(\mathbf{m} \mid \mathbf{d})$ is the posterior probability of the model,
$p(\mathbf{d} \mid \mathbf{m})$ is the likelihood, and
$p(\mathbf{m})$ is the prior.

The likelihood measures how well a model explains the data.

In the stretch move algorithm, a population of walkers explores the posterior distribution simultaneously. Each walker proposes new positions based on the positions of other walkers, allowing for more efficient exploration of complex posteriors.

The sampling proceeds in the following steps:

Initialize a set of walkers in parameter space.
At each iteration, propose new positions for each walker using the stretch move.
Accept or reject the proposed positions based on the Metropolis criterion.
After a burn-in period, the samples from all walkers are collected to approximate the posterior.

This method yields not only the most likely model but also the full distribution over model space, capturing uncertainty in the inferred parameters.

3. Inversion on Synthetic Data#

We will first demonstrate bayesian inversion using CoFI on synthetic data.
The synthetic observations are generated by integrating our gravitational forward model with a fourth-order Runge-Kutta (RK4) solver to simulate Uranus’s trajectory under the influence of Neptune.

We simulate observational noise by sampling from zero-mean Gaussian distributions with specified variances for each coordinate:

\[x_\text{obs} = x + \epsilon_x, \quad y_\text{obs} = y + \epsilon_y, \quad z_\text{obs} = z + \epsilon_z\]

where
\[\epsilon_x \sim \mathcal{N}(0, \sigma_x^2), \quad \epsilon_y \sim \mathcal{N}(0, \sigma_y^2), \quad \epsilon_z \sim \mathcal{N}(0, \sigma_z^2)\]
with noise levels set as
\[\sigma_x = \sigma_y = 10^{-3}, \quad \sigma_z = 10^{-5}\]

The function below generates the synthetic data with the specified noise levels.

3.1 Generating synthetic data#

from neptune_deterministic_methods import generate_synthetic_data

T = 190 # time for which we want to generate synthetic data
z_scale_factor = 1
dt = 1

U_true = generate_synthetic_data(T = T,
                                 dt = dt,
                                 z_scaling = False,
                                 add_noise = True,
                                 noise_level = np.array([0.001, 0.001, 0.00001]))

Synthetic data ranges:
X: -18.298 to 20.084 (range: 38.383)
Y: -19.319 to 19.019 (range: 38.337)
Z: -0.272199 to 0.248322 (range: 0.520521)

Noisy synthetic data ranges:
X: -18.299 to 20.084 (range: 38.383)
Y: -19.321 to 19.020 (range: 38.340)
Z: -0.272212 to 0.248322 (range: 0.520534)

The cell below sets up the starting model for our walkers/chains in MCMC and some pre-defined scales to be used for scaling while running our inversion, to ensure that all parameters are roughly on the same scale or at least near to each other.
For our bayesian inversion we first scale all the parameters and the convert our mass to log mass. This ensures that all our parameters are in the same scale.

# True/reference parameters for Neptune [mass, x, y, z, vx, vy, vz]
from setup_inversion import get_inversion_indices, set_true_m, get_param_bounds, get_param_scales, set_initial_conditions, get_starting_points, PARAM_NAMES

m_0 = set_true_m()
initial_conditions = set_initial_conditions()

PARAM_BOUNDS = get_param_bounds()
PARAM_SCALES = get_param_scales()
INVERT_INDICES = get_inversion_indices()
STARTING_POINTS = get_starting_points()

from setup_inversion import scale_param, unscale_param, validate_config

validate_config()

Inverting for: ['mass', 'x', 'y', 'z', 'vx', 'vy', 'vz']
Starting points: [5.129494593915012e-05, -29.91585533802299, 3.145551080953517, 0.1050744172524112, -0.0002804329678559355, -0.00311021549555015, 4.187313350941659e-05]

names = list(initial_conditions.keys())
n_bodies = len(names)
uranus_idx = names.index("Uranus")

from neptune_bayesian_methods import predict_U

m_start_scaled = scale_param(m_0)

if 0 in INVERT_INDICES:
    m_start_scaled[0] = np.log10(m_start_scaled[0])  # Convert mass to log scale

if len(INVERT_INDICES) == 1:
    m_start_scaled = m_start_scaled.item() if hasattr(m_start_scaled, 'item') else m_start_scaled

print(f"\nStarting points (unscaled): {STARTING_POINTS}")
print(f"Starting points (scaled): {m_start_scaled}")

print("\nTesting forward function...")
try:
    pred_test = predict_U(m_start_scaled, T = T, dt = dt, z_scale_factor = z_scale_factor)
    residual_test = pred_test - U_true
    print(f"Initial residual norm: {np.linalg.norm(residual_test):.6f}")
    print(f"Residual by component:")
    print(f"  X component: {np.linalg.norm(residual_test[:T]):.6f}")
    print(f"  Y component: {np.linalg.norm(residual_test[T:2*T]):.6f}")
    print(f"  Z component: {np.linalg.norm(residual_test[2*T:]):.6f}")

    # Match the archived notebook's pre-MCMC state exactly. In the archive,
    # predict_U/build_neptune_vector mutated m_start_scaled[0] in place from
    # log10(scaled_mass) back to scaled_mass during this forward-function test.
    # The current predictor copies inputs, so we reproduce that side effect
    # explicitly before constructing the emcee walkers.
    if 0 in INVERT_INDICES:
        m_start_scaled[0] = 10**m_start_scaled[0]
    print(f"Archive-compatible m_start_scaled before MCMC: {m_start_scaled}")
except Exception as e:
    print(f"Forward function test failed: {e}")
    import traceback
    traceback.print_exc()

Starting points (unscaled): [5.129494593915012e-05, -29.91585533802299, 3.145551080953517, 0.1050744172524112, -0.0002804329678559355, -0.00311021549555015, 4.187313350941659e-05]
Starting points (scaled): [-3.73377762 -4.61564393  0.42847056  0.40820822 -0.27220634 -3.73292208
  0.00484093]

Testing forward function...
Initial residual norm: 0.019639
Residual by component:
  X component: 0.013658
  Y component: 0.014112
  Z component: 0.000148
Archive-compatible m_start_scaled before MCMC: [ 1.84596041e-04 -4.61564393e+00  4.28470556e-01  4.08208224e-01
 -2.72206336e-01 -3.73292208e+00  4.84093052e-03]

3.2 Running the Inversion on Synthetic Data#

The Bayesian setup below keeps the original likelihood and prior formulation: a bounded uniform prior on the scaled parameters and a diagonal Gaussian data-error model, implemented as -0.5 * sum((d - g(m))**2 * inv_variances). The notebook defaults to a short validation run. Set RUN_LONG_MCMC = True in the configuration cell to run the original 10,000-step synthetic-data chain and save converged upload outputs into neptune_long_run_results.npz.

Defined below are our log prior and log likelihood functions.
We set a uniform prior on log of our mass and all the other parameters (Neptune’s position coordinates and velocity components).

param_bounds_lower = np.array([bound[0] for bound in PARAM_BOUNDS])
param_bounds_upper = np.array([bound[1] for bound in PARAM_BOUNDS])

bounds_lower_scaled = scale_param(param_bounds_lower)
bounds_upper_scaled = scale_param(param_bounds_upper)

bounds_lower_scaled[0] = np.log10(bounds_lower_scaled[0])  # Convert mass back to log scale
bounds_upper_scaled[0] = np.log10(bounds_upper_scaled[0])  # Convert mass back to log scale

sigma_x = 8e-3          # Standard deviation for X position in uranus orbit data, taken one order higher than the noise added
sigma_y = 8e-3          # Standard deviation for Y position in uranus orbit data, taken one order higher than the noise added
sigma_z = 1e-4           # Standard deviation for Z position in uranus orbit data, taken two orders higher than the noise added

diagonal_values = [1/sigma_x**2]*T + [1/sigma_y**2]*T + [1/sigma_z**2]*T
inv_variances = np.array(diagonal_values)

def my_log_likelihood(m_scaled : np.ndarray, U_true : np.ndarray) -> float:

    """
    Log-likelihood function for Bayesian inversion

    Parameters
    ----------
    m_scaled : np.array
        Scaled parameters being inverted.
    U_true : np.array
        True Uranus positions in the format [x, y, z] scaled by z_scale_factor.
    Returns
    -------
    log_likelihood : float
        The log-likelihood value, or -inf if the prediction fails or residuals are not finite.
    """
    m_scaled = np.atleast_1d(m_scaled).astype(np.float64)

    try:
        y_hat = predict_U(m_scaled, T = T, dt = dt, z_scale_factor = z_scale_factor)
        if not np.isfinite(y_hat).all():
            return -np.inf
        residuals = U_true - y_hat
        if residuals.shape != U_true.shape:
            return -np.inf
        return -0.5 * np.sum(residuals**2 * inv_variances)

    except (ValueError, ArithmeticError):
        return -np.inf
    except Exception as e:
        print(f"Error in log-likelihood calculation: {e}")
        return -np.inf

def my_log_prior(m_scaled : np.ndarray) -> float:
    """
    Log-prior function for Bayesian inversion

    Parameters
    ----------
    m_scaled : np.ndarray
        Scaled parameters being inverted.

    Returns
    -------
    float
        The log prior value, or -inf if the parameters are out of bounds.
    """
    m_scaled = np.atleast_1d(m_scaled)
    if np.any(m_scaled < bounds_lower_scaled) or np.any(m_scaled > bounds_upper_scaled):
        return -np.inf
    return 0

n_walkers = 20
n_dim = len(INVERT_INDICES)
nsteps = SYNTHETIC_LONG_STEPS if RUN_LONG_MCMC else VALIDATION_STEPS
use_pool = True
show_progress = RUN_LONG_MCMC
run_mode = "long" if RUN_LONG_MCMC else "validation"

print(f"Optimized setup: {len(inv_variances)} inverse variance values")
print(f"Running {run_mode} MCMC chain: {nsteps} steps, {n_walkers} walkers")
print("Set RUN_LONG_MCMC = True in the configuration cell to run the original 10,000-step synthetic chain.")

print("Parameter bounds (unscaled):")
print("  Lower:", param_bounds_lower)
print("  Upper:", param_bounds_upper)
print("Scaled bounds:")
print("  Lower:", bounds_lower_scaled)
print("  Upper:", bounds_upper_scaled)

print(f"Number of dimensions: {n_dim}")

# Reset the MCMC starting model to the scaled/log parameterization.
# The archive-compatibility forward test above intentionally mutates
# m_start_scaled[0] from log10(scaled_mass) back to scaled_mass. That
# state is useful for apples-to-apples archive checks, but it is wrong
# for tight walker initialization because it clips mass to the upper bound.
m_start_scaled = scale_param(m_0).astype(float)
if 0 in INVERT_INDICES:
    m_start_scaled[0] = np.log10(m_start_scaled[0])
print("MCMC start (scaled/log):", m_start_scaled)

# starting_lower_bounds = m_start_scaled - 10
# starting_upper_bounds = m_start_scaled + 10

# walkers_start = np.random.uniform(
#     starting_lower_bounds,
#     starting_upper_bounds,
#     size=(n_walkers, n_dim)
# )

rng = np.random.default_rng(42)

walker_scales = 0.075 * (bounds_upper_scaled - bounds_lower_scaled)

# Keep mass a bit tighter if desired
walker_scales[0] = 0.075

walkers_start = m_start_scaled + rng.normal(
    scale=walker_scales,
    size=(n_walkers, n_dim),
)

walkers_start = np.clip(walkers_start, bounds_lower_scaled, bounds_upper_scaled)

print("Initial state shape:", walkers_start.shape)
print(
    "Initial walker mass min/max/std:",
    f"{walkers_start[:, 0].min():.6g}",
    f"{walkers_start[:, 0].max():.6g}",
    f"{walkers_start[:, 0].std():.6g}",
)

for i, w in enumerate(walkers_start):
    ll = my_log_likelihood(w, U_true)
    if not np.isfinite(ll):
        print(f"Invalid log-likelihood at walker {i}: {w}, ll = {ll}")

for i, w in enumerate(walkers_start):
    walkers_start[i] = np.clip(w, bounds_lower_scaled, bounds_upper_scaled)
    if not np.array_equal(w, walkers_start[i]):
        print(f"Walker {i} clipped from {w} to {walkers_start[i]}")

print("All walkers are now within bounds.")

for i, w in enumerate(walkers_start):
    if not np.isfinite(my_log_prior(w)):
        print(f"ERROR: Walker {i} still outside prior bounds: {w}")
    else:
        ll = my_log_likelihood(w, U_true)
        if not np.isfinite(ll):
            print(f"WARNING: Walker {i} has invalid log-likelihood: {ll}")

inv_problem = BaseProblem()
inv_problem.name = "Neptune Orbit Determination - Config Driven"
inv_problem.set_data(U_true)
inv_problem.set_forward(predict_U)
inv_problem.set_initial_model(np.atleast_1d(m_start_scaled))
inv_problem.set_log_prior(my_log_prior)
inv_problem.set_log_likelihood(my_log_likelihood, args=(U_true,))
inv_problem.set_model_shape(n_dim)

import emcee
stretch_move = emcee.moves.StretchMove(a=1.2)

inv_result, elapsed = run_cofi_emcee(
    inv_problem,
    walkers_start,
    n_walkers,
    nsteps,
    InversionOptions,
    Inversion,
    progress=show_progress,
    use_pool=use_pool,
    n_threads=10,
    moves=stretch_move,
    skip_initial_state_check=True,
)

print(f"OPTIMIZED Inversion completed in {elapsed:.2f} seconds using `emcee`.")
print("The inversion result from `emcee`:")
inv_result.summary()

Optimized setup: 570 inverse variance values
Running long MCMC chain: 10000 steps, 20 walkers
Set RUN_LONG_MCMC = True in the configuration cell to run the original 10,000-step synthetic chain.
Parameter bounds (unscaled):
  Lower: [ 1.e-08 -5.e+01 -5.e+01 -1.e+01 -1.e-03 -1.e-02 -1.e-03]
  Upper: [1.e-03 5.e+01 5.e+01 1.e+01 1.e-03 1.e-02 1.e-03]
Scaled bounds:
  Lower: [ -7.44566917  -7.67844126  -6.8918876   -6.49935872  -0.79949471
 -12.03003616  -0.06734446]
  Upper: [-2.44566917  7.67844126  6.8918876   6.49935872  0.79949471 12.03003616
  0.06734446]
Number of dimensions: 7
MCMC start (scaled/log): [-3.73377762 -4.61564393  0.42847056  0.40820822 -0.27220634 -3.73292208
  0.00484093]
Initial state shape: (20, 7)
Initial walker mass min/max/std: -3.82945 -3.62403 0.0504557
All walkers are now within bounds.

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OPTIMIZED Inversion completed in 1536.32 seconds using `emcee`.
The inversion result from `emcee`:
============================
Summary for inversion result
============================
SUCCESS
----------------------------
sampler: <emcee.ensemble.EnsembleSampler object>
blob_names: ['log_likelihood', 'log_prior']

3.3 Getting the acceptance rates and auto-correlation times#

print("\n" + "="*50)
print("RESULTS SUMMARY")
print("="*50)

param_names = PARAM_NAMES
param_labels = ["m" if idx == 0 else PARAM_NAMES[idx] for idx in INVERT_INDICES]

sampler = inv_result.sampler
reference_values_scaled = scaled_reference_values(scale_param, m_0)

n_trajectory_samples = 10_000
discard_burn = 0
thin_interval = 10

saved_result = None
if (not RUN_LONG_MCMC) and USE_SAVED_LONG_RUN_RESULTS:
    saved_result = load_mcmc_results(LONG_RUN_RESULTS_FILE, "synthetic")

if saved_result is not None:
    analysis_chain = saved_result["chain"]
    flat_samples = analysis_chain[discard_burn::thin_interval].reshape(-1, analysis_chain.shape[-1])
    acceptance_rates = saved_result["acceptance_fraction"]
    autocorr_times = saved_result["autocorr_times"]
    sample_source = f"saved long-run result ({LONG_RUN_RESULTS_FILE})"
    print(f"Loaded synthetic long-run results from {LONG_RUN_RESULTS_FILE}")
else:
    analysis_chain, flat_samples, sample_source = analysis_samples(
        sampler,
        reference_values_scaled,
        bounds_lower_scaled,
        bounds_upper_scaled,
        thin=thin_interval,
        burn=discard_burn,
        seed=42,
    )
    acceptance_rates = sampler.acceptance_fraction
    autocorr_times = safe_autocorr_time(sampler, fallback_chain=analysis_chain, discard=300)

selected_samples = selected_trajectory_samples(flat_samples, n_trajectory_samples, seed=42)

mean_acceptance = np.mean(acceptance_rates)
max_autocorr = np.max(autocorr_times)
mean_autocorr = np.mean(autocorr_times)

if RUN_LONG_MCMC and SAVE_LONG_RUN_RESULTS:
    save_mcmc_results(
        LONG_RUN_RESULTS_FILE,
        "synthetic",
        analysis_chain,
        flat_samples,
        acceptance_rates,
        autocorr_times,
        nsteps,
        n_walkers,
        param_labels,
    )
    print(f"Saved synthetic long-run results to {LONG_RUN_RESULTS_FILE}")

print(f"Posterior samples for plots: {sample_source} ({len(flat_samples):,} samples)")
print(f"Mean acceptance rate: {mean_acceptance:.3f}")
print("Acceptance rate per walker:")
for i, acc in enumerate(acceptance_rates):
    print(f"  Walker {i:2d}: {acc:.3f}")

print("\nAcceptance rate statistics:")
print(f"  Range: {np.min(acceptance_rates):.3f} - {np.max(acceptance_rates):.3f}")
print(f"  Std dev: {np.std(acceptance_rates):.3f}")

print("Autocorrelation times by parameter:")
for i, (label, tau) in enumerate(zip(param_labels, autocorr_times)):
    print(f"  {label:>3s}: {tau:6.1f} steps")

print("\nAutocorrelation statistics:")
print(f"  Mean τ: {mean_autocorr:.1f} steps")
print(f"  Max τ:  {max_autocorr:.1f} steps")

==================================================
RESULTS SUMMARY
==================================================
Posterior samples for plots: emcee chain (20,000 samples)
Mean acceptance rate: 0.265
Acceptance rate per walker:
  Walker  0: 0.461
  Walker  1: 0.456
  Walker  2: 0.455
  Walker  3: 0.001
  Walker  4: 0.460
  Walker  5: 0.048
  Walker  6: 0.015
  Walker  7: 0.457
  Walker  8: 0.003
  Walker  9: 0.002
  Walker 10: 0.454
  Walker 11: 0.457
  Walker 12: 0.002
  Walker 13: 0.462
  Walker 14: 0.106
  Walker 15: 0.450
  Walker 16: 0.467
  Walker 17: 0.044
  Walker 18: 0.449
  Walker 19: 0.047

Acceptance rate statistics:
  Range: 0.001 - 0.467
  Std dev: 0.214
Autocorrelation times by parameter:
    m:  718.9 steps
    x:  839.4 steps
    y:  794.0 steps
    z:  743.8 steps
   vx:  844.9 steps
   vy:  757.6 steps
   vz:  799.6 steps

Autocorrelation statistics:
  Mean τ: 785.5 steps
  Max τ:  844.9 steps

3.4 Plotting the Posteriors#

We now plot the posteriors and along with that, we alsomark the true parameter values of Neptune using a black ‘X’ mark.
Note that all our parameters are scaled.

reference_values_scaled = scaled_reference_values(scale_param, m_0)
param_labels = ["m" if idx == 0 else PARAM_NAMES[idx] for idx in INVERT_INDICES]
ref_vals = [reference_values_scaled[i] for i in range(n_dim)]

print("Creating corner plot with KDE contours and reference values...")

fig = plot_pair_kde_legacy_style(
    flat_samples,
    param_labels,
    ref_vals,
    figsize=(16, 14),
    textsize=10,
    hdi_probs=(0.3, 0.6, 0.9),
)

plt.suptitle(f'MCMC Parameter Posterior Distributions\n'
            f'(N_eff = {len(flat_samples):,}, burn-in = {discard_burn}, thin = {thin_interval})',
            fontsize=14, y=0.98)

plt.tight_layout()
plt.subplots_adjust(top=0.94)
plt.show()

Creating corner plot with KDE contours and reference values...

3.5 Plotting the Chains#

fig, axes = plt.subplots(3, 3, figsize=(15, 12))
axes = axes.flatten()

chain_full = analysis_chain
labels = ["m" if idx == 0 else PARAM_NAMES[idx] for idx in INVERT_INDICES]

for i in range(n_dim):
    ax = axes[i]

    for walker in range(min(14, chain_full.shape[1])):
        ax.plot(chain_full[:, walker, i], alpha=0.6, linewidth=0.5)

    ax.axhline(reference_values_scaled[i], color='black', linestyle='--', alpha=0.8,
            linewidth=2, label='Reference Value' if i == 0 else '')
    ax.set_xlabel('Step')
    ax.set_ylabel(f'{labels[i]}')
    ax.set_title(f'Trace: {labels[i]}')
    ax.grid(True, alpha=0.3)

    if i == 0:
        ax.legend()

for i in range(n_dim, len(axes)):
    axes[i].remove()

plt.suptitle('MCMC Trace Plots (All Walkers)', fontsize=16)
plt.tight_layout()
plt.show()

4. Real Data Inversion#

We then apply our deterministic inversion via CoFI on actual observational data obtained from NASA JPL Horizons.
The data consists of geometric Cartesian position and velocity vectors of Uranus, relative to the Solar System Barycenter, with the following settings:
- Target body: Uranus (799)
- Center body: Solar System Barycenter (0)
- Reference frame: Ecliptic of J2000.0
- Time span: A.D. 1775-Jan-01 to 2125-Jan-02
- Step size: 1 calendar year
- Output format: Cartesian position and velocity (AU, AU/day)
- Output type: GEOMETRIC states
- Calendar mode: Mixed Julian/Gregorian
- Ephemeris source: ura183_merged (Uranus), DE441 (Solar System)

This dataset provides real-world observations to test the robustness of our inversion pipeline.

#uncomment to install the astroquery package
# !pip install astroquery

4.1 Getting the actual data from NASA#

from neptune_deterministic_methods import get_actual_data
U_true = get_actual_data(z_scaling = False, T = 190)

Original data ranges:
X: -18.297 to 20.085 (range: 38.382)
Y: -19.319 to 19.013 (range: 38.332)
Z: -0.272231 to 0.248352 (range: 0.520583)

Z scaling is disabled, using original Z values.

4.2 Setting the Inversion and Running it#

The real-data section uses the same Bayesian objective as the synthetic section, but calibrates the likelihood with the effective per-coordinate residual scales obtained from the deterministic Morozov analysis. The MCMC walkers are initialized near that deterministic real-data solution rather than across the whole prior box.

param_bounds_lower = np.array([bound[0] for bound in PARAM_BOUNDS])
param_bounds_upper = np.array([bound[1] for bound in PARAM_BOUNDS])

bounds_lower_scaled = scale_param(param_bounds_lower)
bounds_upper_scaled = scale_param(param_bounds_upper)

bounds_lower_scaled[0] = np.log10(bounds_lower_scaled[0])  # Convert mass back to log scale
bounds_upper_scaled[0] = np.log10(bounds_upper_scaled[0])  # Convert mass back to log scale

# Effective real-data residual scales from the deterministic Morozov analysis.
# These include observational noise plus model mismatch for this forward model.
REAL_DETERMINISTIC_SIGMAS = np.array([7.8277e-02, 7.4224e-02, 9.7833e-04])
# sigma_x, sigma_y, sigma_z = REAL_DETERMINISTIC_SIGMAS
sigma_x, sigma_y, sigma_z = np.array([5e-1, 5e-1, 5e-3])

diagonal_values = [1/sigma_x**2]*T + [1/sigma_y**2]*T + [1/sigma_z**2]*T
inv_variances = np.array(diagonal_values)

# Deterministic real-data solution from the Morozov run; used as the MCMC centre.
REAL_DETERMINISTIC_M_UNSCALED = np.array([
    5.129472e-05,
    -2.991824e+01,
    3.145458e+00,
    1.050744e-01,
    -2.804390e-04,
    -3.109318e-03,
    4.187314e-05,
])

real_mcmc_center_scaled = scale_param(REAL_DETERMINISTIC_M_UNSCALED).astype(float)
real_mcmc_center_scaled[0] = np.log10(real_mcmc_center_scaled[0])
# real_mcmc_center_scaled = scale_param(m_0).astype(float)
# real_mcmc_center_scaled[0] = np.log10(real_mcmc_center_scaled[0])

real_center_pred = predict_U(real_mcmc_center_scaled, T=T, dt=dt, z_scale_factor=z_scale_factor)
real_center_residual = U_true - real_center_pred
real_center_reduced_chi2 = np.sum(real_center_residual**2 * inv_variances) / len(inv_variances)

print("Real-data effective sigmas:")
print(f"  sigma_x = {sigma_x:.4e}")
print(f"  sigma_y = {sigma_y:.4e}")
print(f"  sigma_z = {sigma_z:.4e}")
print("Deterministic real-data MCMC centre (scaled/log):", real_mcmc_center_scaled)
print("Residual RMS at deterministic centre:")
print(f"  X: {np.sqrt(np.mean(real_center_residual[:T]**2)):.4e}")
print(f"  Y: {np.sqrt(np.mean(real_center_residual[T:2*T]**2)):.4e}")
print(f"  Z: {np.sqrt(np.mean(real_center_residual[2*T:]**2)):.4e}")
print(f"Reduced chi^2 at deterministic centre: {real_center_reduced_chi2:.4f}")

Real-data effective sigmas:
  sigma_x = 5.0000e-01
  sigma_y = 5.0000e-01
  sigma_z = 5.0000e-03
Deterministic real-data MCMC centre (scaled/log): [-3.73559650e+00 -4.59450897e+00  4.33562860e-01  6.82916218e-02
 -2.24209496e-01 -3.74052080e+00  2.81992383e-03]
Residual RMS at deterministic centre:
  X: 7.9632e-02
  Y: 7.4244e-02
  Z: 9.9564e-04
Reduced chi^2 at deterministic centre: 0.0290

n_walkers = 28
n_dim = len(INVERT_INDICES)
nsteps = REAL_LONG_STEPS if RUN_LONG_MCMC else VALIDATION_STEPS
use_pool = True
show_progress = RUN_LONG_MCMC
run_mode = "long" if RUN_LONG_MCMC else "validation"

print(f"Optimized setup: {len(inv_variances)} inverse variance values")
print(f"Running {run_mode} MCMC chain: {nsteps} steps, {n_walkers} walkers")
print("Set RUN_LONG_MCMC = True in the configuration cell to run the calibrated 80,000-step real-data chain.")

print("Parameter bounds (unscaled):")
print("  Lower:", param_bounds_lower)
print("  Upper:", param_bounds_upper)
print("Scaled bounds:")
print("  Lower:", bounds_lower_scaled)
print("  Upper:", bounds_upper_scaled)

print(f"Number of dimensions: {n_dim}")

rng = np.random.default_rng(84)

walker_scales = 0.03 * (bounds_upper_scaled - bounds_lower_scaled)
walker_scales[0] = 0.05

walkers_start = real_mcmc_center_scaled + rng.normal(
    scale=walker_scales,
    size=(n_walkers, n_dim),
)

walkers_start = np.clip(walkers_start, bounds_lower_scaled, bounds_upper_scaled)

print("Initial state shape:", walkers_start.shape)
print(
    "Initial walker mass min/max/std:",
    f"{walkers_start[:, 0].min():.6g}",
    f"{walkers_start[:, 0].max():.6g}",
    f"{walkers_start[:, 0].std():.6g}",
)

for i, w in enumerate(walkers_start):
    ll = my_log_likelihood(w, U_true)
    if not np.isfinite(ll):
        print(f"Invalid log-likelihood at walker {i}: {w}, ll = {ll}")

for i, w in enumerate(walkers_start):
    walkers_start[i] = np.clip(w, bounds_lower_scaled, bounds_upper_scaled)
    if not np.array_equal(w, walkers_start[i]):
        print(f"Walker {i} clipped from {w} to {walkers_start[i]}")

print("All walkers are now within bounds.")

def my_log_prior(m_scaled):
    # The synthetic example can use a broad uniform prior because the data are
    # generated by this same simplified forward model. The real Horizons data
    # include full-Solar-System and ephemeris effects that this reduced model
    # cannot represent exactly. With only a uniform box prior, the sampler can
    # find unphysical Neptune parameters that overfit those model discrepancies.
    # This Gaussian prior is the Bayesian analogue of the deterministic
    # regularization term: it encodes external knowledge that Neptune's state is
    # near the MAP value determined in the deterministic inversion notebook
    # (finding_neptune_via_deterministic_inv) while still allowing uncertainty around it.
    if np.any(m_scaled < bounds_lower_scaled) or np.any(m_scaled > bounds_upper_scaled):
        return -np.inf

    prior_sigma_full = np.array([
        0.2,  # log mass, atry 0.1
        0.5,  # x
        0.5,  # y
        0.20,  # z
        0.1,  # vx
        0.1,  # vy
        0.20,  # vz
    ])

    prior_sigma = prior_sigma_full[INVERT_INDICES]

    dm = m_scaled - real_mcmc_center_scaled
    return -0.5 * np.sum((dm / prior_sigma)**2)


for i, w in enumerate(walkers_start):
    if not np.isfinite(my_log_prior(w)):
        print(f"ERROR: Walker {i} still outside prior bounds: {w}")
    else:
        ll = my_log_likelihood(w, U_true)
        if not np.isfinite(ll):
            print(f"WARNING: Walker {i} has invalid log-likelihood: {ll}")

inv_problem = BaseProblem()
inv_problem.name = "Neptune Orbit Determination - Config Driven"
inv_problem.set_data(U_true)
inv_problem.set_forward(predict_U)
inv_problem.set_initial_model(np.atleast_1d(real_mcmc_center_scaled))
inv_problem.set_log_prior(my_log_prior)
inv_problem.set_log_likelihood(my_log_likelihood, args=(U_true,))
inv_problem.set_model_shape(n_dim)

import emcee
stretch_move = emcee.moves.StretchMove(a=1.2)

inv_result, elapsed = run_cofi_emcee(
    inv_problem,
    walkers_start,
    n_walkers,
    nsteps,
    InversionOptions,
    Inversion,
    progress=show_progress,
    use_pool=use_pool,
    n_threads=14,
    moves=stretch_move,
    skip_initial_state_check=True,
)

print(f"OPTIMIZED Inversion completed in {elapsed:.2f} seconds using `emcee`.")
print("The inversion result from `emcee`:")
inv_result.summary()

Optimized setup: 570 inverse variance values
Running long MCMC chain: 50000 steps, 28 walkers
Set RUN_LONG_MCMC = True in the configuration cell to run the calibrated 80,000-step real-data chain.
Parameter bounds (unscaled):
  Lower: [ 1.e-08 -5.e+01 -5.e+01 -1.e+01 -1.e-03 -1.e-02 -1.e-03]
  Upper: [1.e-03 5.e+01 5.e+01 1.e+01 1.e-03 1.e-02 1.e-03]
Scaled bounds:
  Lower: [ -7.44566917  -7.67844126  -6.8918876   -6.49935872  -0.79949471
 -12.03003616  -0.06734446]
  Upper: [-2.44566917  7.67844126  6.8918876   6.49935872  0.79949471 12.03003616
  0.06734446]
Number of dimensions: 7
Initial state shape: (28, 7)
Initial walker mass min/max/std: -3.83146 -3.6507 0.0483306
All walkers are now within bounds.

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  3%|▎         | 1712/50000 [04:47<2:05:28,  6.41it/s]
  3%|▎         | 1713/50000 [04:47<2:08:31,  6.26it/s]
  3%|▎         | 1714/50000 [04:47<2:06:05,  6.38it/s]
  3%|▎         | 1715/50000 [04:47<2:05:26,  6.42it/s]
  3%|▎         | 1716/50000 [04:48<2:06:19,  6.37it/s]
  3%|▎         | 1717/50000 [04:48<2:05:17,  6.42it/s]
  3%|▎         | 1718/50000 [04:48<2:05:16,  6.42it/s]
  3%|▎         | 1719/50000 [04:48<2:08:20,  6.27it/s]
  3%|▎         | 1720/50000 [04:48<2:14:56,  5.96it/s]
  3%|▎         | 1721/50000 [04:48<2:14:40,  5.97it/s]
  3%|▎         | 1722/50000 [04:49<2:11:45,  6.11it/s]
  3%|▎         | 1723/50000 [04:49<2:09:02,  6.24it/s]
  3%|▎         | 1724/50000 [04:49<2:09:54,  6.19it/s]
  3%|▎         | 1725/50000 [04:49<2:18:28,  5.81it/s]
  3%|▎         | 1726/50000 [04:49<2:16:01,  5.92it/s]
  3%|▎         | 1727/50000 [04:49<2:11:29,  6.12it/s]
  3%|▎         | 1728/50000 [04:50<2:12:48,  6.06it/s]
  3%|▎         | 1729/50000 [04:50<2:14:37,  5.98it/s]
  3%|▎         | 1730/50000 [04:50<2:14:55,  5.96it/s]
  3%|▎         | 1731/50000 [04:50<2:22:15,  5.66it/s]
  3%|▎         | 1732/50000 [04:50<2:20:44,  5.72it/s]
  3%|▎         | 1733/50000 [04:50<2:25:12,  5.54it/s]
  3%|▎         | 1734/50000 [04:51<2:24:19,  5.57it/s]
  3%|▎         | 1735/50000 [04:51<2:18:14,  5.82it/s]
  3%|▎         | 1736/50000 [04:51<2:20:52,  5.71it/s]
  3%|▎         | 1737/50000 [04:51<2:25:32,  5.53it/s]
  3%|▎         | 1738/50000 [04:51<2:25:36,  5.52it/s]
  3%|▎         | 1739/50000 [04:52<2:24:45,  5.56it/s]
  3%|▎         | 1740/50000 [04:52<2:24:08,  5.58it/s]
  3%|▎         | 1741/50000 [04:52<2:23:52,  5.59it/s]
  3%|▎         | 1742/50000 [04:52<2:20:49,  5.71it/s]
  3%|▎         | 1743/50000 [04:52<2:21:48,  5.67it/s]
  3%|▎         | 1744/50000 [04:52<2:27:38,  5.45it/s]
  3%|▎         | 1745/50000 [04:53<2:26:15,  5.50it/s]
  3%|▎         | 1746/50000 [04:53<2:20:00,  5.74it/s]
  3%|▎         | 1747/50000 [04:53<2:16:13,  5.90it/s]
  3%|▎         | 1748/50000 [04:53<2:11:28,  6.12it/s]
  3%|▎         | 1749/50000 [04:53<2:10:07,  6.18it/s]
  4%|▎         | 1750/50000 [04:53<2:11:31,  6.11it/s]
  4%|▎         | 1751/50000 [04:54<2:09:44,  6.20it/s]
  4%|▎         | 1752/50000 [04:54<2:07:48,  6.29it/s]
  4%|▎         | 1753/50000 [04:54<2:14:53,  5.96it/s]
  4%|▎         | 1754/50000 [04:54<2:14:37,  5.97it/s]
  4%|▎         | 1755/50000 [04:54<2:16:57,  5.87it/s]
  4%|▎         | 1756/50000 [04:54<2:12:25,  6.07it/s]
  4%|▎         | 1757/50000 [04:55<2:13:36,  6.02it/s]
  4%|▎         | 1758/50000 [04:55<2:16:48,  5.88it/s]
  4%|▎         | 1759/50000 [04:55<2:12:59,  6.05it/s]
  4%|▎         | 1760/50000 [04:55<2:13:23,  6.03it/s]
  4%|▎         | 1761/50000 [04:55<2:09:39,  6.20it/s]
  4%|▎         | 1762/50000 [04:55<2:09:40,  6.20it/s]
  4%|▎         | 1763/50000 [04:56<2:12:58,  6.05it/s]
  4%|▎         | 1764/50000 [04:56<2:10:36,  6.16it/s]
  4%|▎         | 1765/50000 [04:56<2:08:36,  6.25it/s]
  4%|▎         | 1766/50000 [04:56<2:05:50,  6.39it/s]
  4%|▎         | 1767/50000 [04:56<2:03:38,  6.50it/s]
  4%|▎         | 1768/50000 [04:56<2:03:22,  6.52it/s]
  4%|▎         | 1769/50000 [04:56<2:03:54,  6.49it/s]
  4%|▎         | 1770/50000 [04:57<2:03:36,  6.50it/s]
  4%|▎         | 1771/50000 [04:57<2:08:06,  6.27it/s]
  4%|▎         | 1772/50000 [04:57<2:10:51,  6.14it/s]
  4%|▎         | 1773/50000 [04:57<2:09:20,  6.21it/s]
  4%|▎         | 1774/50000 [04:57<2:11:09,  6.13it/s]
  4%|▎         | 1775/50000 [04:57<2:17:12,  5.86it/s]
  4%|▎         | 1776/50000 [04:58<2:21:23,  5.68it/s]
  4%|▎         | 1777/50000 [04:58<2:20:47,  5.71it/s]
  4%|▎         | 1778/50000 [04:58<2:18:21,  5.81it/s]
  4%|▎         | 1779/50000 [04:58<2:17:22,  5.85it/s]
  4%|▎         | 1780/50000 [04:58<2:17:28,  5.85it/s]
  4%|▎         | 1781/50000 [04:59<2:16:49,  5.87it/s]
  4%|▎         | 1782/50000 [04:59<2:16:32,  5.89it/s]
  4%|▎         | 1783/50000 [04:59<2:13:33,  6.02it/s]
  4%|▎         | 1784/50000 [04:59<2:17:41,  5.84it/s]
  4%|▎         | 1785/50000 [04:59<2:17:01,  5.86it/s]
  4%|▎         | 1786/50000 [04:59<2:15:58,  5.91it/s]
  4%|▎         | 1787/50000 [05:00<2:21:43,  5.67it/s]
  4%|▎         | 1788/50000 [05:00<2:17:59,  5.82it/s]
  4%|▎         | 1789/50000 [05:00<2:17:09,  5.86it/s]
  4%|▎         | 1790/50000 [05:00<2:20:49,  5.71it/s]
  4%|▎         | 1791/50000 [05:00<2:28:01,  5.43it/s]
  4%|▎         | 1792/50000 [05:00<2:31:54,  5.29it/s]
  4%|▎         | 1793/50000 [05:01<2:30:44,  5.33it/s]
  4%|▎         | 1794/50000 [05:01<2:30:27,  5.34it/s]
  4%|▎         | 1795/50000 [05:01<2:26:29,  5.48it/s]
  4%|▎         | 1796/50000 [05:01<2:23:02,  5.62it/s]
  4%|▎         | 1797/50000 [05:01<2:28:24,  5.41it/s]
  4%|▎         | 1798/50000 [05:02<2:26:54,  5.47it/s]
  4%|▎         | 1799/50000 [05:02<2:27:25,  5.45it/s]
  4%|▎         | 1800/50000 [05:02<2:29:20,  5.38it/s]
  4%|▎         | 1801/50000 [05:02<2:36:35,  5.13it/s]
  4%|▎         | 1802/50000 [05:02<2:34:48,  5.19it/s]
  4%|▎         | 1803/50000 [05:03<2:35:23,  5.17it/s]
  4%|▎         | 1804/50000 [05:03<2:32:23,  5.27it/s]
  4%|▎         | 1805/50000 [05:03<2:40:50,  4.99it/s]
  4%|▎         | 1806/50000 [05:03<2:43:14,  4.92it/s]
  4%|▎         | 1807/50000 [05:03<2:44:23,  4.89it/s]
  4%|▎         | 1808/50000 [05:04<2:42:40,  4.94it/s]
  4%|▎         | 1809/50000 [05:04<2:48:45,  4.76it/s]
  4%|▎         | 1810/50000 [05:04<2:51:02,  4.70it/s]
  4%|▎         | 1811/50000 [05:04<2:46:23,  4.83it/s]
  4%|▎         | 1812/50000 [05:04<2:36:18,  5.14it/s]
  4%|▎         | 1813/50000 [05:05<2:42:28,  4.94it/s]
  4%|▎         | 1814/50000 [05:05<2:38:08,  5.08it/s]
  4%|▎         | 1815/50000 [05:05<2:45:08,  4.86it/s]
  4%|▎         | 1816/50000 [05:05<2:39:58,  5.02it/s]
  4%|▎         | 1817/50000 [05:05<2:35:34,  5.16it/s]
  4%|▎         | 1818/50000 [05:06<2:33:14,  5.24it/s]
  4%|▎         | 1819/50000 [05:06<2:30:39,  5.33it/s]
  4%|▎         | 1820/50000 [05:06<2:31:31,  5.30it/s]
  4%|▎         | 1821/50000 [05:06<2:29:12,  5.38it/s]
  4%|▎         | 1822/50000 [05:06<2:28:40,  5.40it/s]
  4%|▎         | 1823/50000 [05:06<2:31:44,  5.29it/s]
  4%|▎         | 1824/50000 [05:07<2:43:54,  4.90it/s]
  4%|▎         | 1825/50000 [05:07<2:32:12,  5.28it/s]
  4%|▎         | 1826/50000 [05:07<2:23:40,  5.59it/s]
  4%|▎         | 1827/50000 [05:07<2:18:49,  5.78it/s]
  4%|▎         | 1828/50000 [05:07<2:15:38,  5.92it/s]
  4%|▎         | 1829/50000 [05:08<2:12:58,  6.04it/s]
  4%|▎         | 1830/50000 [05:08<2:10:31,  6.15it/s]
  4%|▎         | 1831/50000 [05:08<2:10:04,  6.17it/s]
  4%|▎         | 1832/50000 [05:08<2:10:12,  6.17it/s]
  4%|▎         | 1833/50000 [05:08<2:14:13,  5.98it/s]
  4%|▎         | 1834/50000 [05:08<2:14:53,  5.95it/s]
  4%|▎         | 1835/50000 [05:09<2:15:02,  5.94it/s]
  4%|▎         | 1836/50000 [05:09<2:18:35,  5.79it/s]
  4%|▎         | 1837/50000 [05:09<2:17:22,  5.84it/s]
  4%|▎         | 1838/50000 [05:09<2:18:30,  5.80it/s]
  4%|▎         | 1839/50000 [05:09<2:18:48,  5.78it/s]
  4%|▎         | 1840/50000 [05:09<2:22:08,  5.65it/s]
  4%|▎         | 1841/50000 [05:10<2:22:08,  5.65it/s]
  4%|▎         | 1842/50000 [05:10<2:19:51,  5.74it/s]
  4%|▎         | 1843/50000 [05:10<2:24:44,  5.55it/s]
  4%|▎         | 1844/50000 [05:10<2:19:07,  5.77it/s]
  4%|▎         | 1845/50000 [05:10<2:20:32,  5.71it/s]
  4%|▎         | 1846/50000 [05:10<2:24:28,  5.55it/s]
  4%|▎         | 1847/50000 [05:11<2:21:35,  5.67it/s]
  4%|▎         | 1848/50000 [05:11<2:23:06,  5.61it/s]
  4%|▎         | 1849/50000 [05:11<2:34:12,  5.20it/s]
  4%|▎         | 1850/50000 [05:11<2:36:56,  5.11it/s]
  4%|▎         | 1851/50000 [05:11<2:31:38,  5.29it/s]
  4%|▎         | 1852/50000 [05:12<2:31:51,  5.28it/s]
  4%|▎         | 1853/50000 [05:12<2:34:27,  5.20it/s]
  4%|▎         | 1854/50000 [05:12<2:31:39,  5.29it/s]
  4%|▎         | 1855/50000 [05:12<2:30:41,  5.32it/s]
  4%|▎         | 1856/50000 [05:12<2:28:53,  5.39it/s]
  4%|▎         | 1857/50000 [05:13<2:26:10,  5.49it/s]
  4%|▎         | 1858/50000 [05:13<2:24:31,  5.55it/s]
  4%|▎         | 1859/50000 [05:13<2:23:38,  5.59it/s]
  4%|▎         | 1860/50000 [05:13<2:23:05,  5.61it/s]
  4%|▎         | 1861/50000 [05:13<2:21:32,  5.67it/s]
  4%|▎         | 1862/50000 [05:13<2:21:30,  5.67it/s]
  4%|▎         | 1863/50000 [05:14<2:23:32,  5.59it/s]
  4%|▎         | 1864/50000 [05:14<2:30:49,  5.32it/s]
  4%|▎         | 1865/50000 [05:14<2:34:46,  5.18it/s]
  4%|▎         | 1866/50000 [05:14<2:31:53,  5.28it/s]
  4%|▎         | 1867/50000 [05:14<2:31:37,  5.29it/s]
  4%|▎         | 1868/50000 [05:15<2:27:07,  5.45it/s]
  4%|▎         | 1869/50000 [05:15<2:26:42,  5.47it/s]
  4%|▎         | 1870/50000 [05:15<2:24:46,  5.54it/s]
  4%|▎         | 1871/50000 [05:15<2:21:26,  5.67it/s]
  4%|▎         | 1872/50000 [05:15<2:18:22,  5.80it/s]
  4%|▎         | 1873/50000 [05:15<2:19:09,  5.76it/s]
  4%|▎         | 1874/50000 [05:16<2:17:24,  5.84it/s]
  4%|▍         | 1875/50000 [05:16<2:14:00,  5.99it/s]
  4%|▍         | 1876/50000 [05:16<2:15:19,  5.93it/s]
  4%|▍         | 1877/50000 [05:16<2:23:28,  5.59it/s]
  4%|▍         | 1878/50000 [05:16<2:19:40,  5.74it/s]
  4%|▍         | 1879/50000 [05:16<2:19:33,  5.75it/s]
  4%|▍         | 1880/50000 [05:17<2:24:59,  5.53it/s]
  4%|▍         | 1881/50000 [05:17<2:24:42,  5.54it/s]
  4%|▍         | 1882/50000 [05:17<2:19:28,  5.75it/s]
  4%|▍         | 1883/50000 [05:17<2:14:10,  5.98it/s]
  4%|▍         | 1884/50000 [05:17<2:10:22,  6.15it/s]
  4%|▍         | 1885/50000 [05:17<2:07:59,  6.27it/s]
  4%|▍         | 1886/50000 [05:18<2:06:25,  6.34it/s]
  4%|▍         | 1887/50000 [05:18<2:05:09,  6.41it/s]
  4%|▍         | 1888/50000 [05:18<2:04:03,  6.46it/s]
  4%|▍         | 1889/50000 [05:18<2:02:48,  6.53it/s]
  4%|▍         | 1890/50000 [05:18<2:02:16,  6.56it/s]
  4%|▍         | 1891/50000 [05:18<2:14:52,  5.95it/s]
  4%|▍         | 1892/50000 [05:19<2:18:47,  5.78it/s]
  4%|▍         | 1893/50000 [05:19<2:18:16,  5.80it/s]
  4%|▍         | 1894/50000 [05:19<2:20:51,  5.69it/s]
  4%|▍         | 1895/50000 [05:19<2:18:32,  5.79it/s]
  4%|▍         | 1896/50000 [05:19<2:15:58,  5.90it/s]
  4%|▍         | 1897/50000 [05:19<2:11:22,  6.10it/s]
  4%|▍         | 1898/50000 [05:20<2:08:16,  6.25it/s]
  4%|▍         | 1899/50000 [05:20<2:08:00,  6.26it/s]
  4%|▍         | 1900/50000 [05:20<2:08:09,  6.26it/s]
  4%|▍         | 1901/50000 [05:20<2:18:14,  5.80it/s]
  4%|▍         | 1902/50000 [05:20<2:16:48,  5.86it/s]
  4%|▍         | 1903/50000 [05:20<2:13:22,  6.01it/s]
  4%|▍         | 1904/50000 [05:21<2:11:48,  6.08it/s]
  4%|▍         | 1905/50000 [05:21<2:08:44,  6.23it/s]
  4%|▍         | 1906/50000 [05:21<2:05:35,  6.38it/s]
  4%|▍         | 1907/50000 [05:21<2:06:10,  6.35it/s]
  4%|▍         | 1908/50000 [05:21<2:07:00,  6.31it/s]
  4%|▍         | 1909/50000 [05:21<2:05:41,  6.38it/s]
  4%|▍         | 1910/50000 [05:21<2:03:31,  6.49it/s]
  4%|▍         | 1911/50000 [05:22<2:02:09,  6.56it/s]
  4%|▍         | 1912/50000 [05:22<2:02:23,  6.55it/s]
  4%|▍         | 1913/50000 [05:22<2:02:11,  6.56it/s]
  4%|▍         | 1914/50000 [05:22<2:01:54,  6.57it/s]
  4%|▍         | 1915/50000 [05:22<2:02:27,  6.54it/s]
  4%|▍         | 1916/50000 [05:22<2:02:49,  6.52it/s]
  4%|▍         | 1917/50000 [05:23<2:07:37,  6.28it/s]
  4%|▍         | 1918/50000 [05:23<2:05:13,  6.40it/s]
  4%|▍         | 1919/50000 [05:23<2:03:59,  6.46it/s]
  4%|▍         | 1920/50000 [05:23<2:12:23,  6.05it/s]
  4%|▍         | 1921/50000 [05:23<2:13:37,  6.00it/s]
  4%|▍         | 1922/50000 [05:23<2:23:15,  5.59it/s]
  4%|▍         | 1923/50000 [05:24<2:19:45,  5.73it/s]
  4%|▍         | 1924/50000 [05:24<2:16:57,  5.85it/s]
  4%|▍         | 1925/50000 [05:24<2:25:42,  5.50it/s]
  4%|▍         | 1926/50000 [05:24<2:31:20,  5.29it/s]
  4%|▍         | 1927/50000 [05:24<2:21:30,  5.66it/s]
  4%|▍         | 1928/50000 [05:24<2:15:03,  5.93it/s]
  4%|▍         | 1929/50000 [05:25<2:16:45,  5.86it/s]
  4%|▍         | 1930/50000 [05:25<2:12:29,  6.05it/s]
  4%|▍         | 1931/50000 [05:25<2:08:18,  6.24it/s]
  4%|▍         | 1932/50000 [05:25<2:06:37,  6.33it/s]
  4%|▍         | 1933/50000 [05:25<2:05:16,  6.40it/s]
  4%|▍         | 1934/50000 [05:25<2:04:03,  6.46it/s]
  4%|▍         | 1935/50000 [05:26<2:02:37,  6.53it/s]
  4%|▍         | 1936/50000 [05:26<2:05:29,  6.38it/s]
  4%|▍         | 1937/50000 [05:26<2:13:02,  6.02it/s]
  4%|▍         | 1938/50000 [05:26<2:12:33,  6.04it/s]
  4%|▍         | 1939/50000 [05:26<2:08:32,  6.23it/s]
  4%|▍         | 1940/50000 [05:26<2:11:49,  6.08it/s]
  4%|▍         | 1941/50000 [05:27<2:14:40,  5.95it/s]
  4%|▍         | 1942/50000 [05:27<2:14:53,  5.94it/s]
  4%|▍         | 1943/50000 [05:27<2:19:20,  5.75it/s]
  4%|▍         | 1944/50000 [05:27<2:18:38,  5.78it/s]
  4%|▍         | 1945/50000 [05:27<2:19:17,  5.75it/s]
  4%|▍         | 1946/50000 [05:27<2:23:27,  5.58it/s]
  4%|▍         | 1947/50000 [05:28<2:21:49,  5.65it/s]
  4%|▍         | 1948/50000 [05:28<2:15:25,  5.91it/s]
  4%|▍         | 1949/50000 [05:28<2:16:03,  5.89it/s]
  4%|▍         | 1950/50000 [05:28<2:14:36,  5.95it/s]
  4%|▍         | 1951/50000 [05:28<2:23:10,  5.59it/s]
  4%|▍         | 1952/50000 [05:29<2:20:41,  5.69it/s]
  4%|▍         | 1953/50000 [05:29<2:21:48,  5.65it/s]
  4%|▍         | 1954/50000 [05:29<2:22:58,  5.60it/s]
  4%|▍         | 1955/50000 [05:29<2:18:34,  5.78it/s]
  4%|▍         | 1956/50000 [05:29<2:14:31,  5.95it/s]
  4%|▍         | 1957/50000 [05:29<2:10:18,  6.14it/s]
  4%|▍         | 1958/50000 [05:29<2:09:13,  6.20it/s]
  4%|▍         | 1959/50000 [05:30<2:07:45,  6.27it/s]
  4%|▍         | 1960/50000 [05:30<2:06:11,  6.35it/s]
  4%|▍         | 1961/50000 [05:30<2:04:22,  6.44it/s]
  4%|▍         | 1962/50000 [05:30<2:02:57,  6.51it/s]
  4%|▍         | 1963/50000 [05:30<2:06:38,  6.32it/s]
  4%|▍         | 1964/50000 [05:30<2:06:58,  6.31it/s]
  4%|▍         | 1965/50000 [05:31<2:07:45,  6.27it/s]
  4%|▍         | 1966/50000 [05:31<2:14:34,  5.95it/s]
  4%|▍         | 1967/50000 [05:31<2:15:51,  5.89it/s]
  4%|▍         | 1968/50000 [05:31<2:18:11,  5.79it/s]
  4%|▍         | 1969/50000 [05:31<2:24:11,  5.55it/s]
  4%|▍         | 1970/50000 [05:32<2:23:35,  5.57it/s]
  4%|▍         | 1971/50000 [05:32<2:19:45,  5.73it/s]
  4%|▍         | 1972/50000 [05:32<2:26:00,  5.48it/s]
  4%|▍         | 1973/50000 [05:32<2:17:37,  5.82it/s]
  4%|▍         | 1974/50000 [05:32<2:16:14,  5.87it/s]
  4%|▍         | 1975/50000 [05:32<2:20:19,  5.70it/s]
  4%|▍         | 1976/50000 [05:33<2:17:45,  5.81it/s]
  4%|▍         | 1977/50000 [05:33<2:19:36,  5.73it/s]
  4%|▍         | 1978/50000 [05:33<2:17:45,  5.81it/s]
  4%|▍         | 1979/50000 [05:33<2:15:05,  5.92it/s]
  4%|▍         | 1980/50000 [05:33<2:13:42,  5.99it/s]
  4%|▍         | 1981/50000 [05:33<2:22:00,  5.64it/s]
  4%|▍         | 1982/50000 [05:34<2:21:45,  5.65it/s]
  4%|▍         | 1983/50000 [05:34<2:21:30,  5.66it/s]
  4%|▍         | 1984/50000 [05:34<2:27:09,  5.44it/s]
  4%|▍         | 1985/50000 [05:34<2:22:00,  5.64it/s]
  4%|▍         | 1986/50000 [05:34<2:19:21,  5.74it/s]
  4%|▍         | 1987/50000 [05:34<2:19:32,  5.73it/s]
  4%|▍         | 1988/50000 [05:35<2:14:34,  5.95it/s]
  4%|▍         | 1989/50000 [05:35<2:10:57,  6.11it/s]
  4%|▍         | 1990/50000 [05:35<2:07:11,  6.29it/s]
  4%|▍         | 1991/50000 [05:35<2:10:37,  6.13it/s]
  4%|▍         | 1992/50000 [05:35<2:11:22,  6.09it/s]
  4%|▍         | 1993/50000 [05:35<2:20:05,  5.71it/s]
  4%|▍         | 1994/50000 [05:36<2:20:24,  5.70it/s]
  4%|▍         | 1995/50000 [05:36<2:17:43,  5.81it/s]
  4%|▍         | 1996/50000 [05:36<2:17:15,  5.83it/s]
  4%|▍         | 1997/50000 [05:36<2:17:35,  5.81it/s]
  4%|▍         | 1998/50000 [05:36<2:21:26,  5.66it/s]
  4%|▍         | 1999/50000 [05:37<2:22:40,  5.61it/s]
  4%|▍         | 2000/50000 [05:37<2:20:11,  5.71it/s]
  4%|▍         | 2001/50000 [05:37<2:13:29,  5.99it/s]
  4%|▍         | 2002/50000 [05:37<2:21:47,  5.64it/s]
  4%|▍         | 2003/50000 [05:37<2:20:50,  5.68it/s]
  4%|▍         | 2004/50000 [05:37<2:21:05,  5.67it/s]
  4%|▍         | 2005/50000 [05:38<2:24:26,  5.54it/s]
  4%|▍         | 2006/50000 [05:38<2:22:53,  5.60it/s]
  4%|▍         | 2007/50000 [05:38<2:29:34,  5.35it/s]
  4%|▍         | 2008/50000 [05:38<2:27:21,  5.43it/s]
  4%|▍         | 2009/50000 [05:38<2:32:47,  5.23it/s]
  4%|▍         | 2010/50000 [05:39<2:27:50,  5.41it/s]
  4%|▍         | 2011/50000 [05:39<2:25:57,  5.48it/s]
  4%|▍         | 2012/50000 [05:39<2:19:25,  5.74it/s]
  4%|▍         | 2013/50000 [05:39<2:14:08,  5.96it/s]
  4%|▍         | 2014/50000 [05:39<2:15:56,  5.88it/s]
  4%|▍         | 2015/50000 [05:39<2:18:55,  5.76it/s]
  4%|▍         | 2016/50000 [05:40<2:27:23,  5.43it/s]
  4%|▍         | 2017/50000 [05:40<2:24:08,  5.55it/s]
  4%|▍         | 2018/50000 [05:40<2:16:59,  5.84it/s]
  4%|▍         | 2019/50000 [05:40<2:13:03,  6.01it/s]
  4%|▍         | 2020/50000 [05:40<2:13:56,  5.97it/s]
  4%|▍         | 2021/50000 [05:40<2:10:35,  6.12it/s]
  4%|▍         | 2022/50000 [05:41<2:07:55,  6.25it/s]
  4%|▍         | 2023/50000 [05:41<2:10:54,  6.11it/s]
  4%|▍         | 2024/50000 [05:41<2:07:27,  6.27it/s]
  4%|▍         | 2025/50000 [05:41<2:04:59,  6.40it/s]
  4%|▍         | 2026/50000 [05:41<2:09:02,  6.20it/s]
  4%|▍         | 2027/50000 [05:41<2:07:06,  6.29it/s]
  4%|▍         | 2028/50000 [05:41<2:05:44,  6.36it/s]
  4%|▍         | 2029/50000 [05:42<2:10:59,  6.10it/s]
  4%|▍         | 2030/50000 [05:42<2:12:30,  6.03it/s]
  4%|▍         | 2031/50000 [05:42<2:14:27,  5.95it/s]
  4%|▍         | 2032/50000 [05:42<2:10:55,  6.11it/s]
  4%|▍         | 2033/50000 [05:42<2:11:24,  6.08it/s]
  4%|▍         | 2034/50000 [05:42<2:07:35,  6.27it/s]
  4%|▍         | 2035/50000 [05:43<2:07:35,  6.27it/s]
  4%|▍         | 2036/50000 [05:43<2:07:40,  6.26it/s]
  4%|▍         | 2037/50000 [05:43<2:09:03,  6.19it/s]
  4%|▍         | 2038/50000 [05:43<2:06:46,  6.31it/s]
  4%|▍         | 2039/50000 [05:43<2:04:28,  6.42it/s]
  4%|▍         | 2040/50000 [05:43<2:03:40,  6.46it/s]
  4%|▍         | 2041/50000 [05:44<2:02:29,  6.53it/s]
  4%|▍         | 2042/50000 [05:44<2:02:31,  6.52it/s]
  4%|▍         | 2043/50000 [05:44<2:04:10,  6.44it/s]
  4%|▍         | 2044/50000 [05:44<2:11:14,  6.09it/s]
  4%|▍         | 2045/50000 [05:44<2:19:57,  5.71it/s]
  4%|▍         | 2046/50000 [05:44<2:14:58,  5.92it/s]
  4%|▍         | 2047/50000 [05:45<2:10:47,  6.11it/s]
  4%|▍         | 2048/50000 [05:45<2:08:55,  6.20it/s]
  4%|▍         | 2049/50000 [05:45<2:06:11,  6.33it/s]
  4%|▍         | 2050/50000 [05:45<2:04:31,  6.42it/s]
  4%|▍         | 2051/50000 [05:45<2:02:41,  6.51it/s]
  4%|▍         | 2052/50000 [05:45<2:02:16,  6.54it/s]
  4%|▍         | 2053/50000 [05:45<2:02:30,  6.52it/s]
  4%|▍         | 2054/50000 [05:46<2:01:58,  6.55it/s]
  4%|▍         | 2055/50000 [05:46<2:02:38,  6.52it/s]
  4%|▍         | 2056/50000 [05:46<2:02:31,  6.52it/s]
  4%|▍         | 2057/50000 [05:46<2:02:33,  6.52it/s]
  4%|▍         | 2058/50000 [05:46<2:02:24,  6.53it/s]
  4%|▍         | 2059/50000 [05:46<2:01:43,  6.56it/s]
  4%|▍         | 2060/50000 [05:47<2:01:58,  6.55it/s]
  4%|▍         | 2061/50000 [05:47<2:01:50,  6.56it/s]
  4%|▍         | 2062/50000 [05:47<2:01:21,  6.58it/s]
  4%|▍         | 2063/50000 [05:47<2:00:50,  6.61it/s]
  4%|▍         | 2064/50000 [05:47<2:00:50,  6.61it/s]
  4%|▍         | 2065/50000 [05:47<2:01:38,  6.57it/s]
  4%|▍         | 2066/50000 [05:47<2:01:49,  6.56it/s]
  4%|▍         | 2067/50000 [05:48<2:01:10,  6.59it/s]
  4%|▍         | 2068/50000 [05:48<2:01:54,  6.55it/s]
  4%|▍         | 2069/50000 [05:48<2:02:01,  6.55it/s]
  4%|▍         | 2070/50000 [05:48<2:01:45,  6.56it/s]
  4%|▍         | 2071/50000 [05:48<2:02:57,  6.50it/s]
  4%|▍         | 2072/50000 [05:48<2:02:20,  6.53it/s]
  4%|▍         | 2073/50000 [05:49<2:01:44,  6.56it/s]
  4%|▍         | 2074/50000 [05:49<2:03:39,  6.46it/s]
  4%|▍         | 2075/50000 [05:49<2:02:25,  6.52it/s]
  4%|▍         | 2076/50000 [05:49<2:01:33,  6.57it/s]
  4%|▍         | 2077/50000 [05:49<2:02:39,  6.51it/s]
  4%|▍         | 2078/50000 [05:49<2:02:12,  6.54it/s]
  4%|▍         | 2079/50000 [05:49<2:01:40,  6.56it/s]
  4%|▍         | 2080/50000 [05:50<2:06:54,  6.29it/s]
  4%|▍         | 2081/50000 [05:50<2:04:50,  6.40it/s]
  4%|▍         | 2082/50000 [05:50<2:03:22,  6.47it/s]
  4%|▍         | 2083/50000 [05:50<2:04:06,  6.43it/s]
  4%|▍         | 2084/50000 [05:50<2:02:33,  6.52it/s]
  4%|▍         | 2085/50000 [05:50<2:06:26,  6.32it/s]
  4%|▍         | 2086/50000 [05:51<2:04:45,  6.40it/s]
  4%|▍         | 2087/50000 [05:51<2:02:46,  6.50it/s]
  4%|▍         | 2088/50000 [05:51<2:05:39,  6.35it/s]
  4%|▍         | 2089/50000 [05:51<2:09:56,  6.14it/s]
  4%|▍         | 2090/50000 [05:51<2:08:19,  6.22it/s]
  4%|▍         | 2091/50000 [05:51<2:19:37,  5.72it/s]
  4%|▍         | 2092/50000 [05:52<2:18:34,  5.76it/s]
  4%|▍         | 2093/50000 [05:52<2:17:28,  5.81it/s]
  4%|▍         | 2094/50000 [05:52<2:13:57,  5.96it/s]
  4%|▍         | 2095/50000 [05:52<2:14:53,  5.92it/s]
  4%|▍         | 2096/50000 [05:52<2:13:00,  6.00it/s]
  4%|▍         | 2097/50000 [05:52<2:16:37,  5.84it/s]
  4%|▍         | 2098/50000 [05:53<2:19:14,  5.73it/s]
  4%|▍         | 2099/50000 [05:53<2:26:35,  5.45it/s]
  4%|▍         | 2100/50000 [05:53<2:22:46,  5.59it/s]
  4%|▍         | 2101/50000 [05:53<2:22:54,  5.59it/s]
  4%|▍         | 2102/50000 [05:53<2:19:44,  5.71it/s]
  4%|▍         | 2103/50000 [05:53<2:17:43,  5.80it/s]
  4%|▍         | 2104/50000 [05:54<2:14:34,  5.93it/s]
  4%|▍         | 2105/50000 [05:54<2:12:41,  6.02it/s]
  4%|▍         | 2106/50000 [05:54<2:11:42,  6.06it/s]
  4%|▍         | 2107/50000 [05:54<2:11:18,  6.08it/s]
  4%|▍         | 2108/50000 [05:54<2:10:13,  6.13it/s]
  4%|▍         | 2109/50000 [05:54<2:08:28,  6.21it/s]
  4%|▍         | 2110/50000 [05:55<2:07:21,  6.27it/s]
  4%|▍         | 2111/50000 [05:55<2:07:36,  6.25it/s]
  4%|▍         | 2112/50000 [05:55<2:07:16,  6.27it/s]
  4%|▍         | 2113/50000 [05:55<2:06:05,  6.33it/s]
  4%|▍         | 2114/50000 [05:55<2:06:24,  6.31it/s]
  4%|▍         | 2115/50000 [05:55<2:06:10,  6.32it/s]
  4%|▍         | 2116/50000 [05:56<2:15:48,  5.88it/s]
  4%|▍         | 2117/50000 [05:56<2:13:53,  5.96it/s]
  4%|▍         | 2118/50000 [05:56<2:11:22,  6.07it/s]
  4%|▍         | 2119/50000 [05:56<2:09:30,  6.16it/s]
  4%|▍         | 2120/50000 [05:56<2:07:56,  6.24it/s]
  4%|▍         | 2121/50000 [05:56<2:07:41,  6.25it/s]
  4%|▍         | 2122/50000 [05:57<2:07:01,  6.28it/s]
  4%|▍         | 2123/50000 [05:57<2:05:52,  6.34it/s]
  4%|▍         | 2124/50000 [05:57<2:04:52,  6.39it/s]
  4%|▍         | 2125/50000 [05:57<2:04:42,  6.40it/s]
  4%|▍         | 2126/50000 [05:57<2:05:55,  6.34it/s]
  4%|▍         | 2127/50000 [05:57<2:05:34,  6.35it/s]
  4%|▍         | 2128/50000 [05:57<2:05:42,  6.35it/s]
  4%|▍         | 2129/50000 [05:58<2:05:08,  6.38it/s]
  4%|▍         | 2130/50000 [05:58<2:04:56,  6.39it/s]
  4%|▍         | 2131/50000 [05:58<2:05:13,  6.37it/s]
  4%|▍         | 2132/50000 [05:58<2:05:44,  6.35it/s]
  4%|▍         | 2133/50000 [05:58<2:05:11,  6.37it/s]
  4%|▍         | 2134/50000 [05:58<2:04:35,  6.40it/s]
  4%|▍         | 2135/50000 [05:59<2:05:07,  6.38it/s]
  4%|▍         | 2136/50000 [05:59<2:10:16,  6.12it/s]
  4%|▍         | 2137/50000 [05:59<2:14:57,  5.91it/s]
  4%|▍         | 2138/50000 [05:59<2:14:51,  5.92it/s]
  4%|▍         | 2139/50000 [05:59<2:16:47,  5.83it/s]
  4%|▍         | 2140/50000 [05:59<2:13:37,  5.97it/s]
  4%|▍         | 2141/50000 [06:00<2:12:04,  6.04it/s]
  4%|▍         | 2142/50000 [06:00<2:15:24,  5.89it/s]
  4%|▍         | 2143/50000 [06:00<2:11:53,  6.05it/s]
  4%|▍         | 2144/50000 [06:00<2:09:52,  6.14it/s]
  4%|▍         | 2145/50000 [06:00<2:13:49,  5.96it/s]
  4%|▍         | 2146/50000 [06:00<2:11:20,  6.07it/s]
  4%|▍         | 2147/50000 [06:01<2:09:15,  6.17it/s]
  4%|▍         | 2148/50000 [06:01<2:07:20,  6.26it/s]
  4%|▍         | 2149/50000 [06:01<2:10:14,  6.12it/s]
  4%|▍         | 2150/50000 [06:01<2:06:44,  6.29it/s]
  4%|▍         | 2151/50000 [06:01<2:05:38,  6.35it/s]
  4%|▍         | 2152/50000 [06:01<2:10:42,  6.10it/s]
  4%|▍         | 2153/50000 [06:02<2:08:20,  6.21it/s]
  4%|▍         | 2154/50000 [06:02<2:06:51,  6.29it/s]
  4%|▍         | 2155/50000 [06:02<2:11:19,  6.07it/s]
  4%|▍         | 2156/50000 [06:02<2:08:16,  6.22it/s]
  4%|▍         | 2157/50000 [06:02<2:07:48,  6.24it/s]
  4%|▍         | 2158/50000 [06:02<2:12:57,  6.00it/s]
  4%|▍         | 2159/50000 [06:03<2:10:18,  6.12it/s]
  4%|▍         | 2160/50000 [06:03<2:07:53,  6.23it/s]
  4%|▍         | 2161/50000 [06:03<2:12:23,  6.02it/s]
  4%|▍         | 2162/50000 [06:03<2:12:30,  6.02it/s]
  4%|▍         | 2163/50000 [06:03<2:10:11,  6.12it/s]
  4%|▍         | 2164/50000 [06:03<2:13:31,  5.97it/s]
  4%|▍         | 2165/50000 [06:03<2:09:19,  6.16it/s]
  4%|▍         | 2166/50000 [06:04<2:16:47,  5.83it/s]
  4%|▍         | 2167/50000 [06:04<2:22:10,  5.61it/s]
  4%|▍         | 2168/50000 [06:04<2:21:12,  5.65it/s]
  4%|▍         | 2169/50000 [06:04<2:16:02,  5.86it/s]
  4%|▍         | 2170/50000 [06:04<2:23:26,  5.56it/s]
  4%|▍         | 2171/50000 [06:05<2:30:51,  5.28it/s]
  4%|▍         | 2172/50000 [06:05<2:27:19,  5.41it/s]
  4%|▍         | 2173/50000 [06:05<2:28:11,  5.38it/s]
  4%|▍         | 2174/50000 [06:05<2:26:25,  5.44it/s]
  4%|▍         | 2175/50000 [06:05<2:23:40,  5.55it/s]
  4%|▍         | 2176/50000 [06:06<2:23:55,  5.54it/s]
  4%|▍         | 2177/50000 [06:06<2:20:55,  5.66it/s]
  4%|▍         | 2178/50000 [06:06<2:26:42,  5.43it/s]
  4%|▍         | 2179/50000 [06:06<2:28:00,  5.38it/s]
  4%|▍         | 2180/50000 [06:06<2:30:29,  5.30it/s]
  4%|▍         | 2181/50000 [06:06<2:29:14,  5.34it/s]
  4%|▍         | 2182/50000 [06:07<2:30:19,  5.30it/s]
  4%|▍         | 2183/50000 [06:07<2:26:40,  5.43it/s]
  4%|▍         | 2184/50000 [06:07<2:21:32,  5.63it/s]
  4%|▍         | 2185/50000 [06:07<2:21:57,  5.61it/s]
  4%|▍         | 2186/50000 [06:07<2:16:47,  5.83it/s]
  4%|▍         | 2187/50000 [06:07<2:13:05,  5.99it/s]
  4%|▍         | 2188/50000 [06:08<2:16:30,  5.84it/s]
  4%|▍         | 2189/50000 [06:08<2:12:34,  6.01it/s]
  4%|▍         | 2190/50000 [06:08<2:10:22,  6.11it/s]
  4%|▍         | 2191/50000 [06:08<2:13:26,  5.97it/s]
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OPTIMIZED Inversion completed in 8764.15 seconds using `emcee`.
The inversion result from `emcee`:
============================
Summary for inversion result
============================
SUCCESS
----------------------------
sampler: <emcee.ensemble.EnsembleSampler object>
blob_names: ['log_likelihood', 'log_prior']

4.3 Getting the acceptance rates and auto-correlation times#

print("\n" + "="*50)
print("RESULTS SUMMARY")
print("="*50)

param_names = PARAM_NAMES
param_labels = ["m" if idx == 0 else PARAM_NAMES[idx] for idx in INVERT_INDICES]

sampler = inv_result.sampler
reference_values_scaled = scaled_reference_values(scale_param, m_0)

n_trajectory_samples = 10_000
discard_burn = 0
thin_interval = 1

saved_result = None
if (not RUN_LONG_MCMC) and USE_SAVED_LONG_RUN_RESULTS:
    saved_result = load_mcmc_results(LONG_RUN_RESULTS_FILE, "real")

if saved_result is not None:
    analysis_chain = saved_result["chain"]
    flat_samples = analysis_chain[discard_burn::thin_interval].reshape(-1, analysis_chain.shape[-1])
    acceptance_rates = saved_result["acceptance_fraction"]
    autocorr_times = saved_result["autocorr_times"]
    sample_source = f"saved long-run result ({LONG_RUN_RESULTS_FILE})"
    print(f"Loaded real-data long-run results from {LONG_RUN_RESULTS_FILE}")
else:
    analysis_chain, flat_samples, sample_source = analysis_samples(
        sampler,
        reference_values_scaled,
        bounds_lower_scaled,
        bounds_upper_scaled,
        thin=thin_interval,
        burn=discard_burn,
        seed=84,
    )
    acceptance_rates = sampler.acceptance_fraction
    autocorr_times = safe_autocorr_time(sampler, fallback_chain=analysis_chain, discard=300)

selected_samples = selected_trajectory_samples(flat_samples, n_trajectory_samples, seed=84)

mean_acceptance = np.mean(acceptance_rates)
max_autocorr = np.max(autocorr_times)
mean_autocorr = np.mean(autocorr_times)

if RUN_LONG_MCMC and SAVE_LONG_RUN_RESULTS:
    save_mcmc_results(
        LONG_RUN_RESULTS_FILE,
        "real",
        analysis_chain,
        flat_samples,
        acceptance_rates,
        autocorr_times,
        nsteps,
        n_walkers,
        param_labels,
    )
    print(f"Saved real-data long-run results to {LONG_RUN_RESULTS_FILE}")

print(f"Posterior samples for plots: {sample_source} ({len(flat_samples):,} samples)")
print(f"Mean acceptance rate: {mean_acceptance:.3f}")
print("Acceptance rate per walker:")
for i, acc in enumerate(acceptance_rates):
    print(f"  Walker {i:2d}: {acc:.3f}")

print("\nAcceptance rate statistics:")
print(f"  Range: {np.min(acceptance_rates):.3f} - {np.max(acceptance_rates):.3f}")
print(f"  Std dev: {np.std(acceptance_rates):.3f}")

print("Autocorrelation times by parameter:")
for i, (label, tau) in enumerate(zip(param_labels, autocorr_times)):
    print(f"  {label:>3s}: {tau:6.1f} steps")

print("\nAutocorrelation statistics:")
print(f"  Mean τ: {mean_autocorr:.1f} steps")
print(f"  Max τ:  {max_autocorr:.1f} steps")

==================================================
RESULTS SUMMARY
==================================================
Posterior samples for plots: emcee chain (1,400,000 samples)
Mean acceptance rate: 0.637
Acceptance rate per walker:
  Walker  0: 0.615
  Walker  1: 0.642
  Walker  2: 0.669
  Walker  3: 0.631
  Walker  4: 0.662
  Walker  5: 0.651
  Walker  6: 0.623
  Walker  7: 0.619
  Walker  8: 0.659
  Walker  9: 0.602
  Walker 10: 0.640
  Walker 11: 0.651
  Walker 12: 0.637
  Walker 13: 0.599
  Walker 14: 0.639
  Walker 15: 0.598
  Walker 16: 0.633
  Walker 17: 0.627
  Walker 18: 0.648
  Walker 19: 0.651
  Walker 20: 0.650
  Walker 21: 0.647
  Walker 22: 0.622
  Walker 23: 0.652
  Walker 24: 0.649
  Walker 25: 0.658
  Walker 26: 0.597
  Walker 27: 0.654

Acceptance rate statistics:
  Range: 0.597 - 0.669
  Std dev: 0.020
Autocorrelation times by parameter:
    m:  578.7 steps
    x:  726.2 steps
    y:  785.4 steps
    z:  616.0 steps
   vx:  581.9 steps
   vy:  622.1 steps
   vz:  826.7 steps

Autocorrelation statistics:
  Mean τ: 676.7 steps
  Max τ:  826.7 steps

4.4 Plotting the Posteriors#

We now plot the posteriors and along with that, we alsomark the true parameter values of Neptune using a black ‘X’ mark.
Note that all our parameters are scaled.

reference_values_scaled = scaled_reference_values(scale_param, m_0)
param_labels = ["m" if idx == 0 else PARAM_NAMES[idx] for idx in INVERT_INDICES]
ref_vals = [reference_values_scaled[i] for i in range(n_dim)]

print("Creating corner plot with KDE contours and reference values...")

fig = plot_pair_kde_legacy_style(
    flat_samples,
    param_labels,
    ref_vals,
    figsize=(16, 14),
    textsize=10,
    hdi_probs=(0.3, 0.6, 0.9),
)

plt.suptitle(f'MCMC Parameter Posterior Distributions\n'
            f'(N_eff = {len(flat_samples):,}, burn-in = {discard_burn}, thin = {thin_interval})',
            fontsize=14, y=0.98)

plt.tight_layout()
plt.subplots_adjust(top=0.94)
plt.show()

Creating corner plot with KDE contours and reference values...

4.5 Plotting the Chains#

fig, axes = plt.subplots(3, 3, figsize=(15, 12))
axes = axes.flatten()

chain_full = analysis_chain
labels = ["m" if idx == 0 else PARAM_NAMES[idx] for idx in INVERT_INDICES]

for i in range(n_dim):
    ax = axes[i]

    for walker in range(min(14, chain_full.shape[1])):
        ax.plot(chain_full[:, walker, i], alpha=0.6, linewidth=0.5)

    ax.axhline(reference_values_scaled[i], color='black', linestyle='--', alpha=0.8,
            linewidth=2, label='Reference Value' if i == 0 else '')
    ax.set_xlabel('Step')
    ax.set_ylabel(f'{labels[i]}')
    ax.set_title(f'Trace: {labels[i]}')
    ax.grid(True, alpha=0.3)

    if i == 0:
        ax.legend()

for i in range(n_dim, len(axes)):
    axes[i].remove()

plt.suptitle('MCMC Trace Plots (All Walkers)', fontsize=16)
plt.tight_layout()
plt.show()

true_scaled = scale_param(m_0).astype(float)
true_scaled[0] = np.log10(true_scaled[0])

det_scaled = real_mcmc_center_scaled

flat = sampler.get_chain(discard=10000, thin=100, flat=True)
logp = sampler.get_log_prob(discard=10000, thin=100, flat=True)

map_scaled = flat[np.argmax(logp)]

print("log posterior at true:", my_log_prior(true_scaled) + my_log_likelihood(true_scaled, U_true))
print("log posterior at deterministic centre:", my_log_prior(det_scaled) + my_log_likelihood(det_scaled, U_true))
print("log posterior at chain MAP:", my_log_prior(map_scaled) + my_log_likelihood(map_scaled, U_true))

print("true:", true_scaled)
print("det :", det_scaled)
print("MAP :", map_scaled)

log posterior at true: -9.975970292104922
log posterior at deterministic centre: -8.271254785759565
log posterior at chain MAP: -2.398142782038685
true: [-3.73377762 -4.61564393  0.42847056  0.40820822 -0.27220634 -3.73292208
  0.00484093]
det : [-3.73559650e+00 -4.59450897e+00  4.33562860e-01  6.82916218e-02
 -2.24209496e-01 -3.74052080e+00  2.81992383e-03]
MAP : [-3.57053498e+00 -3.70900076e+00  6.27588934e-01  9.68876835e-02
 -2.24811733e-01 -3.69971220e+00  3.81295991e-04]

5. Watermark#

For version of libraries used.

import numpy as np
import matplotlib.pyplot as plt
import tqdm
import numba
import arviz
import astroquery
import cofi

def print_versions():
    print("Library Versions:")
    print(f"{'numpy':<15}: {np.__version__}")
    print(f"{'matplotlib':<15}: {plt.matplotlib.__version__}")
    print(f"{'tqdm':<15}: {tqdm.__version__}")
    print(f"{'numba':<15}: {numba.__version__}")
    print(f"{'arviz':<15}: {arviz.__version__}")
    print(f"{'astroquery':<15}: {astroquery.__version__}")
    print(f"{'cofi':<15}: {cofi.__version__}")

print_versions()

Library Versions:
numpy          : 2.2.6
matplotlib     : 3.10.9
tqdm           : 4.67.3
numba          : 0.65.1
arviz          : 1.1.0
astroquery     : 0.4.11
cofi           : 0.2.11+68.gc319bc9

sphinx_gallery_thumbnail_number = -1

Total running time of the script: (244 minutes 28.758 seconds)

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