par2-circadian 1.1.3

AR(2) eigenvalue analysis for gene expression time series

par2-circadian

AR(2) eigenvalue analysis for gene expression time series

Fits second-order autoregressive models to gene expression data and computes the eigenvalue modulus |λ|, a single number that quantifies how strongly a gene's past determines its future (temporal persistence). Discovers the three-layer hierarchy: Clock > Target > Background.

Installation

pip install par2-circadian

Or install from source:

git clone https://github.com/mickwh2764/par2-discovery-engine.git
cd par2-discovery-engine/par2-python-package
pip install .

Quick Start

Python API

import par2

# Single gene
result = par2.fit_ar2([1.2, 3.4, 2.1, 4.5, 3.2, 5.1, 2.8, 4.9, 3.5, 5.2, 2.9, 4.7])
print(f"|λ| = {result['eigenvalue']:.3f}, type = {result['root_type']}")

# Whole matrix from CSV
matrix, genes = par2.load_expression_matrix("my_data.csv")
results = par2.fit_ar2_batch(matrix, genes)

# Discover the three-layer hierarchy
hierarchy = par2.discover_hierarchy(results)
print(f"Clock median:  {hierarchy['clock_median']:.3f}")
print(f"Target median: {hierarchy['target_median']:.3f}")
print(f"Gearbox gap:   {hierarchy['gearbox_gap']:.3f}")
print(f"Health grade:  {hierarchy['health_grade']}")

# Save results
par2.save_results(results, "ar2_results.csv")

Command Line

# Analyse a CSV file (genes as rows, timepoints as columns)
par2 my_data.csv -o results.csv

# Show top 20 genes by eigenvalue
par2 my_data.csv --top 20

Example Dataset

An example dataset is included at data/example_circadian.csv (30 genes x 12 timepoints) with known clock, target, and background genes showing realistic circadian dynamics:

import par2

matrix, genes = par2.load_expression_matrix("data/example_circadian.csv")
results = par2.fit_ar2_batch(matrix, genes)
h = par2.discover_hierarchy(results)
print(f"Hierarchy preserved: {h['hierarchy_preserved']}")  # True
print(f"Health grade: {h['health_grade']}")  # A

Input Format

CSV file with:

• First row: header (timepoint labels)
• First column: gene names
• Remaining columns: expression values (minimum 6 timepoints)

Example:

Gene,ZT0,ZT2,ZT4,ZT6,ZT8,ZT10,ZT12,ZT14,ZT16,ZT18,ZT20,ZT22
Bmal1,12.3,14.1,15.8,16.2,14.5,11.2,9.8,8.1,7.5,8.9,10.2,11.5
Per2,8.1,7.2,6.5,7.8,10.2,13.5,15.1,14.8,13.2,10.5,9.1,8.5

Output

Per-Gene Results

Each gene gets:

• eigenvalue: |λ|, the eigenvalue modulus (0 to ~1). Higher = more persistent.
• phi1, phi2: AR(2) coefficients
• r2: goodness of fit
• root_type: 'Complex' (oscillatory) or 'Real' (monotone decay)
• half_life: persistence half-life in sampling intervals
• eigenperiod: intrinsic oscillation period (complex roots only)

Hierarchy Discovery

discover_hierarchy() returns:

• clock_median / target_median / background_median: layer-wise eigenvalue medians
• gearbox_gap: clock_median − target_median (the circadian health metric)
• hierarchy_preserved: True if clock > target > background
• health_grade: A (gap ≥ 0.15) through F (gap < 0.02)
• clock_genes / target_genes: per-gene eigenvalue lists

Interpreting |λ|

Range	Interpretation
0.8–1.0	Sustained oscillator (e.g., core clock genes)
0.5–0.8	Damped oscillator (e.g., clock-controlled targets)
0.3–0.5	Weak persistence (e.g., downstream effectors)
0.0–0.3	Rapidly decaying / noise-dominated

Method

The AR(2) model fits:

x(t) = φ₁·x(t-1) + φ₂·x(t-2) + ε

The characteristic equation r² − φ₁r − φ₂ = 0 yields eigenvalues whose modulus |λ| quantifies temporal persistence. Expression values are mean-centred before fitting.

For complex roots: |λ| = √(−φ₂) For real roots: |λ| = max(|r₁|, |r₂|)

The three-layer hierarchy emerges because clock genes (strong autonomous oscillation) have higher |λ| than clock-controlled target genes (driven oscillation), which in turn have higher |λ| than background genes (no circadian regulation).

See: Whiteside M (2026). "AR(2) eigenvalue modulus as a measure of temporal persistence in circadian gene expression." Research Square [Preprint]. doi:10.21203/rs.3.rs-9283100/v1

License

PolyForm Noncommercial License 1.0.0 — free for noncommercial use. Commercial use requires a separate commercial license (contact [e-mail]). See LICENSE for details. The PAR(2) methodology is the subject of a pending UK patent application, covering the methodology independently of this software license.

If you use this software in academic work, please cite:

Whiteside M (2026). "AR(2) eigenvalue modulus as a measure of temporal persistence in circadian gene expression." Research Square [Preprint]. doi:10.21203/rs.3.rs-9283100/v1