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Pattern recognition in complex systems via vector-field representations of spatio-temporal data

Ingrid Amaranta Membrillo Solis, Maria van Rossem, Tristan Madeleine, Tetiana Orlova, Nina Podoliak, Giampaolo D'Alessandro, Jacek Brodzki, Malgosia Kaczmarek

TL;DR

<3-5 sentence high-level summary>We introduce a geometric framework for complex-system analysis by treating spatio-temporal data as vector fields over discrete measure spaces and equipping them with a two-parameter $L^{p,q}$ metric family. By applying multidimensional scaling to pairwise $L^{p,q}$ distances, the method yields low-dimensional Euclidean embeddings that support pattern recognition, phase-space reconstruction, and attractor characterization across data types (images, gradients, graphs, and curved domains). The framework is validated on numerical simulations of the Ginzburg-Landau equation on flat domains and Gray-Scott Turing patterns on curved surfaces, demonstrating robust dimensionality reduction and the ability to distinguish chaotic from ordered dynamics without prior dynamical models. This data-driven approach offers a versatile tool for analyzing high-dimensional spatio-temporal data in complex systems where traditional modelling is impractical.</p>

Abstract

A complex system comprises multiple interacting entities whose interdependencies form a unified whole, exhibiting emergent behaviours not present in individual components. Examples include the human brain, living cells, soft matter, Earth's climate, ecosystems, and the economy. These systems exhibit high-dimensional, non-linear dynamics, making their modelling, classification, and prediction particularly challenging. Advances in information technology have enabled data-driven approaches to studying such systems. However, the sheer volume and complexity of spatio-temporal data often hinder traditional methods like dimensionality reduction, phase-space reconstruction, and attractor characterisation. This paper introduces a geometric framework for analysing spatio-temporal data from complex systems, grounded in the theory of vector fields over discrete measure spaces. We propose a two-parameter family of metrics suitable for data analysis and machine learning applications. The framework supports time-dependent images, image gradients, and real- or vector-valued functions defined on graphs and simplicial complexes. We validate our approach using data from numerical simulations of biological and physical systems on flat and curved domains. Our results show that the proposed metrics, combined with multidimensional scaling, effectively address key analytical challenges. They enable dimensionality reduction, mode decomposition, phase-space reconstruction, and attractor characterisation. Our findings offer a robust pathway for understanding complex dynamical systems, especially in contexts where traditional modelling is impractical but abundant experimental data are available.

Pattern recognition in complex systems via vector-field representations of spatio-temporal data

TL;DR

<3-5 sentence high-level summary>We introduce a geometric framework for complex-system analysis by treating spatio-temporal data as vector fields over discrete measure spaces and equipping them with a two-parameter metric family. By applying multidimensional scaling to pairwise distances, the method yields low-dimensional Euclidean embeddings that support pattern recognition, phase-space reconstruction, and attractor characterization across data types (images, gradients, graphs, and curved domains). The framework is validated on numerical simulations of the Ginzburg-Landau equation on flat domains and Gray-Scott Turing patterns on curved surfaces, demonstrating robust dimensionality reduction and the ability to distinguish chaotic from ordered dynamics without prior dynamical models. This data-driven approach offers a versatile tool for analyzing high-dimensional spatio-temporal data in complex systems where traditional modelling is impractical.</p>

Abstract

A complex system comprises multiple interacting entities whose interdependencies form a unified whole, exhibiting emergent behaviours not present in individual components. Examples include the human brain, living cells, soft matter, Earth's climate, ecosystems, and the economy. These systems exhibit high-dimensional, non-linear dynamics, making their modelling, classification, and prediction particularly challenging. Advances in information technology have enabled data-driven approaches to studying such systems. However, the sheer volume and complexity of spatio-temporal data often hinder traditional methods like dimensionality reduction, phase-space reconstruction, and attractor characterisation. This paper introduces a geometric framework for analysing spatio-temporal data from complex systems, grounded in the theory of vector fields over discrete measure spaces. We propose a two-parameter family of metrics suitable for data analysis and machine learning applications. The framework supports time-dependent images, image gradients, and real- or vector-valued functions defined on graphs and simplicial complexes. We validate our approach using data from numerical simulations of biological and physical systems on flat and curved domains. Our results show that the proposed metrics, combined with multidimensional scaling, effectively address key analytical challenges. They enable dimensionality reduction, mode decomposition, phase-space reconstruction, and attractor characterisation. Our findings offer a robust pathway for understanding complex dynamical systems, especially in contexts where traditional modelling is impractical but abundant experimental data are available.

Paper Structure

This paper contains 21 sections, 2 theorems, 15 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The geometry of spatio-temporal data
  3. Discrete measure spaces
  4. Examples
  5. Vector field representation of spatio-temporal data
  6. Examples
  7. $\mathcal{L}^{p,q}$ spaces
  8. Examples
  9. Low-dimensional Euclidean representations of $\mathcal{L}^{p,q}$ spaces
  10. Pattern recognition in low-dimensional representations of $\mathcal{L}^{p,q}$ spaces
  11. Applications to complex systems dynamics
  12. Ginzburg-Landau equation on a flat domain
  13. Data generation
  14. Time dynamics analysis and phase space reconstruction
  15. Dimensionality reduction
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $(\mathcal{M},\mu)$ be a discrete measure space, $V$ be a vector space of rank $d$ and $1\leq p,q\in \mathbb R$

Figures (10)

  • Figure 1: Examples of discrete measure spaces: (a) cubical lattice; (b) simplicial complex; (c) Voronoi diagram on the plane.
  • Figure 2: Examples of vector fields over discrete measure spaces: (a,b) image, (c,d) image gradient and (e) weather data. The images in panels a to d are data sets on squared lattices, in which each pixel is an element of the lattice, while (e) are data sets on simplicial complexes. Plots (c,d) are obtained by computing the gradient of the intensity of images (a) and (b), respectively and highlight regions of high contrast.
  • Figure 3: Pipeline for the application of the spaces of vector fields of rank $d$ to complex systems dynamics.
  • Figure 4: Time-evolution of the frozen states of the Ginzburg-Landau equation.(a,b) Distance matrices using the $L^p$-metric for $p=1,2$. The corresponding 3-dimensional embedding for the cases (c) $p=1$ and (d) $p=2$.
  • Figure 5: Time-evolution of the gradient field of frozen states of the Ginzburg-Landau equation.(a) $L^{1,1}$ and (b) $L^{2,2}$ distance matrices for the evolution of the gradient field of frozen states. Embeddings of the (c) $L^{1,1}$ and (d) $L^{2,2}$ distance metrics via classical multidimensional scaling
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 2.2: Low-rank approximation via principal coordinales
  • proof