$\varepsilon$-Attracting Basin

Tools for computing and visualizing $\varepsilon$-attracting basin as introduced in Filtrations Indexed by Attracting Levels and their Applications. Y. Imoto and T. Yokoyama. arXiv:2506.18250.

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Table of Contents

• Overview
• Installation
• Usage
• API Reference
• Requirements
• Examples
• License
• Contact

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Overview

• Inputs are
  • multiple sequence (time-series) data $Y=$ { $(y_1^{(i)}, y_2^{(i)}, \dots, y_{n_i}^{(i)}) \in \mathbb{R}^d\ | i=1,\dots,I$ }, where $i$ is a sequence index, $n_i$ is the number of sample (length) for the $i$ th sequence, and $d>0$ is the data dimension (assuming $y_{t+1}^{(i)} = F(y_{t}^{(i)})$ ),
  • target cluster $A\subset Y$.
• Compute
  • $\varepsilon$-attracting basin $A_{F,\varepsilon}$, i.e., the set of states from which the system governed by $F$ can be driven into cluster $A$ by applying control of magnitude at most $\varepsilon>0$ at each time step.
  • $-\varepsilon$-attracting basin $A_{F,-\varepsilon}$, i.e., the set of states in cluster $A$ from which the system cannot escape even if control of magnitude at most $\varepsilon>0$ is applied at each time step.
  • $\varepsilon_{\Sigma}$-attracting basins $A_{F,\varepsilon_{\Sigma}}$ and $A_{F,-\varepsilon_{\Sigma}}$, i.e., the analogous sets when the total control energy over the entire sequence is at most $\varepsilon>0$.

For more theoretical background, see Filtrations Indexed by Attracting Levels and their Applications. Yusuke Imoto and Tomoo Yokoyama. arXiv:2506.18250.

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Installation

You can install epsbasin either from PyPI or directly from the GitHub repository:

From PyPI:

pip install epsbasin

From GitHub:

git clone https://github.com/yusuke-imoto-lab/eps-attracting-basin.git
cd eps-attracting-basin
pip install -r requirements.txt

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Usage

Put epsbasin.py in the same directory as the executable file and import it:

import epsbasin

Input Data Structure

The functions expect an anndata (annoted data matrix) object with the following structure:

• .X
  A numeric data matrix (observations/samples × features) used for computing pairwise costs if no precomputed cost matrix exists.

• .obs
  A pandas DataFrame containing at least:

  • 'cluster': Categorical labels indicating cluster membership for each observation.
  • 'seq_id': Identifiers sequences (sequence index).

• .uns (optional)

  • 'cost_matrix': Optional precomputed pairwise cost matrix (array of shape n_obs × n_obs). If absent, functions will compute it from .X and store it here.

• .obsm (optional)

  • 'plot_data': 2-dimensional data for 2D visualiztion plotting instead of .X.

Example of input data

For example, consider input data with two sequences, each having three time points $x^{(i)}_{j}~(i=1,2,~j=1,2,3)$ , where the final point of one sequence belongs to the “good” cluster and the final point of the other belongs to the “bad” cluster, as shown below:

Then, the input anndata object adata should be constructed by

adara.X =

1	6
5	6
9	9
1	5
5	5
9	2

adata.obs =

index	seq_id	cluster
0	0	other
1	0	other
2	0	good
3	1	other
4	1	other
5	1	bad

import anndata
import panadas as pd 

adata = anndata.AnnData(np.array([
    [1, 6],
    [5, 6],
    [9, 9],
    [1, 5],
    [5, 5],
    [9, 2]
]))
adata.obs = pd.DataFrame({
    "seq_id": [0, 0, 0, 1, 1, 1],
    "cluster": ["other", "other", "good", "other", "other", "bad"]
})

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API Reference

$\varepsilon$-attracting basin

• Function: eps_attracting_basin(adata, cluster_key="cluster", target_cluster_key="target_cluster", cost_matrix_key="cost_matrix", output_key="eps_attracting_basin", epsilon_delta=0.01)
• Description: Compute the debut function for the target cluster $A$ (indexed by target_cluster_key) for input data adata.X. If the cost matrix has not been precomputed (no adata.uns[cost_matrix_key]), the Euclidean distance matrix is ​​calculated as the cost matrix.

$\varepsilon_\Sigma$-attracting basin

• Function: eps_sum_attracting_basin(adata, cluster_key="cluster", target_cluster_key="target_cluster", cost_matrix_key="cost_matrix", output_key="eps_sum_attracting_basin", epsilon_delta=0.01)
• Description: Compute the debut function for the target cluster $A$ (indexed by target_cluster_key) for input data adata.X. If the cost matrix has not been precomputed (no adata.uns[cost_matrix_key]), the Euclidean distance matrix is ​​calculated as the cost matrix.

Sublevel set visualization

• Function: sublevel_set_visualization(adata, plot_key=None, eps_key="eps_attracting_basin", target_cluster_key="good", linewidth=0.5, color_name="_$\\varepsilon$", vmin=None, vmax=None, xlim=None, ylim=None, xlabel=None, ylabel=None, xticks=None, yticks=None, show_xticks=True, show_yticks=True, show_xlabel=True, show_ylabel=True, show_colorbar=True, show_colorbar_label=True, show_xticklabels=True, show_yticklabels=True, figsize=(10,8), levels=10, area_percentile=99.5, edge_percentile=99.5, save_fig=False, save_fig_dir=".", save_fig_name="sublevel_set_visualization")
• Description: Visualize the $\varepsilon$- or $\varepsilon_\Sigma$-attracting basin (tha values on points are the debut function or ) for the target cluster $A$ (indexed by target_cluster_key) on a 2D Delaunay triangulation generated from adata.X (or adata.obsm[plot_key]). Triangles with too large sides or areas can be removed by adjusting the parameters area_percentile and edge_percentile.

Plot attracting basin

• Function: plot_attracting_basin(adata, plot_key=None, eps_key="eps_attracting_basin", cluster_key="cluster", good_cluster_key="good", bad_cluster_key="bad", eps_threshold=0, background=None, figsize=(10,10), fontsize=14, pointsize=20, circ_scale=10, xlim=None, ylim=None, xlabel=None, ylabel=None, xticks=None, yticks=None, show_legend=True, show_label=True, show_ticks=True, show_title=True, title=None, label_clusters=["$G$","$B$"], label_basins=["$G_{F, \\varepsilon}$","$B_{F, \\varepsilon}$"], save_fig=False, save_fig_dir=".", save_fig_name="attracting_basin")
• Description: Plot the $\varepsilon$- or $\varepsilon_\Sigma$-attracting basin of good and bad clusters (indexed by good_cluster_key anc bad_cluster_key) by mapping 2D coordinates from adata.X or adata.obsm[plot_key].

For detailed usage, see espbasin.py.

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Examples

• example_simple.ipynb: A simple exmaple of simulation data with one bifurcation in 2-dimensional space.

a, Input data with one bifurcation. b, Velocities of data (direction vectors of time increment $F(x)$ ). c, d, Distribution of debut functions $\underline{\varepsilon}$ (c) and $\underline{\varepsilon}_{\Sigma}$ (d) for clusters $G$ (reft) and $B$ (right).

• example_Dolphin.ipynb: An application exmaple of ensemble weather forecast data for tropical cyclone (typhoon), Dolphin, generated by the Meso-scale Ensemble Prediction System (MEPS), provided by the Japan Meteorological Agency (JSA).

a, Best‐track trajectory (real orbit, blue line) overlaid with all forecast center positions (gray dots) of Dolphin. b, Selected ensemble forecast tracks at representative initial times. c, Clustering and assigning good and bad clusters. d, Distribution of debut functions $\underline{\varepsilon}$ (top) and $\underline{\varepsilon}_{\Sigma}$ (bottom) for clusters $G$ (reft) and $B$ (right).

• example_MEARI.ipynb: An application exmaple of ensemble weather forecast data for tropical cyclone (typhoon), MEARI, generated by the Meso-scale Ensemble Prediction System (MEPS), provided by the Japan Meteorological Agency (JSA).

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Requirements

• Python ≥ 3.8
• numpy
• pandas
• scipy
• scikit-learn
• matplotlib
• anndata

Install all dependencies via:

pip install -r requirements.txt

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License

MIT © 2025 Yusuke Imoto

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Contact

• Yusuke Imoto
• Email: imoto.yusuke.4e@kyoto-u.ac.jp
• GitHub: yusuke-imoto-lab/eps-attracting-basin