AGN X-ray Variability: State-Space Modelling

State-space modelling of X-ray spectral variability in Active Galactic Nuclei (AGNs) and X-ray binaries (XRBs). The framework links observed photon count time series to two latent physical parameters --- hydrogen column density NH(t) and coronal height h_c(t) --- each modelled as an AR(1) stochastic process. Posterior inference is performed using Hamiltonian Monte Carlo (HMC/NUTS) implemented in NumPyro.

---

Overview

X-ray variability in AGNs arises from two sources:

This pipeline models both simultaneously by embedding an AR(1) stochastic process for each parameter inside a full X-ray spectral forward model:

F(E, t) = ZPOWERLW(E; γ, φ(t)) × ZWABS(E; NH(t)) × WABS(E; NH_gal)

Photon counts are then predicted by folding F through the instrument ARF and RMF, and a Poisson likelihood connects the model to observed counts:

λᵢ(t) = Σⱼ F(Eⱼ, t) · Δt · ΔEⱼ · ARFⱼ · RMFⱼᵢ
Cᵢ(t) ~ Poisson(λᵢ(t))

See Buchner & Boorman (2023) for the theoretical background.

---

Repository Structure

.
├── src/
│   ├── photo_electric_absorption.py   # Morrison & McCammon (1983) cross-sections
│   ├── ar1_process_generator.py       # AR(1) stochastic process generators
│   ├── count_simulator.py             # X-ray count spectrum simulator
│   ├── generate_synthetic_data.py     # Grid-search synthetic data generation
│   ├── run_inference.py               # HMC/NUTS Bayesian inference
│   └── visualise_results.py           # Posterior trace and histogram plots
│
├── data/
│   ├── fake_count.npz                 # Pre-processed instrument response grid
│   ├── photo_electric_sigma_redshift_0108.npz   # Pre-computed cross-sections
│   └── SDSSJ0932+0405.*              # Chandra observation files (not tracked)
│
├── output/
│   ├── inference/                     # Saved posterior sample arrays (.npz)
│   ├── summary/                       # MCMC diagnostic CSVs
│   └── plots/                         # Saved figures (.png)
│
├── requirements.txt
└── README.md

---

Pipeline

The pipeline runs in three stages:

Stage 1 — Pre-compute cross-sections

python src/photo_electric_absorption.py

Computes photoelectric absorption cross-sections σ(E) using the Morrison & McCammon (1983) piecewise polynomial model at both rest-frame and redshifted energies, and saves them to data/photo_electric_sigma_redshift_<tag>.npz.

Stage 2 — Generate synthetic count spectra

python src/generate_synthetic_data.py

Loops over a grid of (NH_mean, τ, φ_mean) parameter combinations. For each combination it:

1. Draws two AR(1) time series in log space for NH(t) and φ(t).
2. Simulates 1000 independent Poisson count realisations via count_generator.
3. Saves each dataset to data/synthetic_count_NH_and_phi_spec_<tag>.npz.

Edit the CONFIGURATION block at the top of the script to change the parameter grid. Expected runtime: several hours for the full 54-combination grid. Run a single combination for a quick test by temporarily setting each list to one element.

Stage 3 — Run Bayesian inference

python src/run_inference.py

Selects one synthetic dataset (configured via NH_LABEL, TAU_LABEL, PHI_LABEL at the top of the script), runs NUTS with 2000 warm-up and 6000 posterior samples, and saves:

• output/inference/<tag>.npz — posterior samples and posterior predictive counts.
• output/summary/<tag>.csv — per-parameter R-hat, ESS, mean, std.

Expected runtime: 30–120 min depending on the time-series length and hardware.

Stage 4 — Visualise results

python src/visualise_results.py

Loads the inference output and plots:

• Trace comparison: posterior NH(t) and φ(t) trajectories vs ground truth, plus posterior predictive total counts.
• Posterior histograms: marginals for μ_NH and μ_φ with 95% credible intervals and true values marked.

---

Installation

conda create -n agn-state-space python=3.11
conda activate agn-state-space
pip install -r requirements.txt

xraystan is not on PyPI; install from source:

pip install git+https://github.com/JohannesBuchner/xraystan

---

Data

The Chandra observation files for SDSSJ0932+0405 (SDSSJ0932+0405.pi, .rmf, .corr.arf, _bkg.*) are not tracked in this repository (they are not freely redistributable). Place them in the data/ directory before running Stage 2 or 3.

The two pre-processed files that are tracked:

---

Key Parameters

---

References

• Morrison, R. & McCammon, D. (1983), ApJ, 270, 119. doi:10.1086/161108
• Buchner, J. & Boorman, P. (2023), arXiv:2309.05705. arXiv
• Phan, D. et al. (2019), arXiv:1912.11554 — NumPyro. arXiv
• XSPEC model documentation: https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node220.html

---

Authors

Nozhan Balafkan and Dr. Johannes Buchner (Max Planck Institute for Extraterrestrial Physics, Garching, Germany)