nugap 0.4.0

Vinnicombe nu-gap metric and a pipeline for comparing time-course data across two conditions

# nugap

**Detect condition-specific changes in *dynamics* using the Vinnicombe ν-gap metric.**

[](https://pypi.org/project/nugap/)

[](LICENSE)

[](https://colab.research.google.com/github/comparativechrono/nugap/blob/main/tutorial\_nugap.ipynb)

nugap is a lightweight Python implementation of the **Vinnicombe ν-gap** — a

bounded (0–1) distance between linear dynamical systems from robust control

theory — together with the model identification and statistics needed to ask a

practical question of two-condition time-course data:

**Which variables, or which relationships between them, change their *dynamical

behaviour* between conditions?**

(for example wild type vs mutant, or untreated vs treated). Comparisons of

expression *level* answer a different question and are blind to changes in

timescale, gain or phase. The ν-gap is defined directly on dynamical models, so

it is not: two systems close in ν-gap behave the same way under feedback, and a

large ν-gap marks a genuine change in dynamics.

## Installation

pip install nugap

Requires Python ≥ 3.10. The core and pipelines depend only on **NumPy**, **SciPy**

and **pandas**; the plotting helpers additionally use **Matplotlib** and

**NetworkX**.

## Tutorial

A guided, runnable tutorial covers the metric, model fitting, and the two-condition

network comparison end to end:

- **Open in Colab:** https://colab.research.google.com/github/comparativechrono/nugap/blob/main/tutorial\_nugap.ipynb

- **View on GitHub:** [tutorial\_nugap.ipynb](https://github.com/comparativechrono/nugap/blob/main/tutorial\_nugap.ipynb)

## Quick start

The ν-gap between two systems (coefficients in descending powers of s):

from nugap import tf, nu\_gap



P1 = tf(\[1], \[1, 1])     # 1/(s+1)

P2 = tf(\[1], \[1, 3])     # 1/(s+3)

nu\_gap(P1, P2)           # -> 0.447   (0 = identical, 1 = maximally different)

Comparing a whole interaction network between two conditions, with replicate-based

significance:

from nugap import compare\_network



\# data\_A, data\_B: dict {variable\_name: array of shape (replicates, timepoints)}

\# t: the common time vector

edges = compare\_network(data\_A, data\_B, t, order=1, min\_r2=0.5)



edges.query("q\_global < 0.1")   # relationships rewired between conditions (FDR < 10%)

## What it provides

- **nu\_gap** — the Vinnicombe ν-gap for SISO systems in continuous or discrete

time, with an optional frequency-band restriction (band=) and a switchable

winding-number test (check\_winding=) for oscillatory data.

- **Model identification** — fit\_first\_order, fit\_model, fit\_arx,

fit\_prony, with simulation-based fit quality and an optional DC-gain floor

(min\_dc\_gain=); plus the dc\_gain helper.

- **compare\_conditions** — per-variable comparison of dynamics between two

conditions, with a fit-quality reliability flag.

- **compare\_network** — pairwise interaction-network comparison: a low-order

model is fitted to every ordered pair of variables in each condition, the ν-gap

is taken between conditions, and significance comes from a replicate-derived

empirical null with Benjamini–Hochberg FDR control.

- **Plotting** — nugap.viz (volcano plot, hub network, hub bar plot).

## How it works

For each variable or pairwise interaction, nugap fits a low-order linear

input–output model under each condition, then measures the ν-gap between the

fitted models. Because models are compared on mean-centred trajectories, the

metric reflects changes in the *relationship* — timescale, gain or phase — rather

than in absolute level. With biological replicates, the spread of within-condition

ν-gaps provides an empirical null and a per-edge noise floor, against which

between-condition changes are tested and FDR-controlled.

The models are single-input single-output, so an edge captures a pairwise

input–output relationship, not proven causation.

## Correctness

The ν-gap implementation is verified by several independent routes — exact

closed-form values, an algebraic invariance of the chordal metric, an independent

reference implementation, and the Vinnicombe robust-stability theorem — and is

cross-checked against MATLAB's gapmetric (Robust Control Toolbox), which it

reproduces to within ~10⁻⁶. The scripts are in [validation/](validation/), with

a fast subset run on every commit.

## Citing

If you use nugap in your work, please cite the software and the accompanying

methods paper:

Hearn, T. J. nugap: condition-specific changes in dynamical relationships via the

Vinnicombe ν-gap. Software archive: Zenodo, DOI: 10.5281/zenodo.XXXXXXX.



<methods paper reference — to be added>

## License

Released under the **MIT License**. Copyright © 2026 Tim Hearn. See

[LICENSE](LICENSE).