Turning Time Into Shapes: A Point-Cloud Framework With Chaotic Signatures for Time Series

Abstract

We propose a novel methodology for transforming financial time series into a geometric format via a sequence of point clouds, enabling richer modeling of nonstationary behavior. In this framework, volatility serves as a spatial directive to guide how overlapping temporal windows become connected in an adjacency tensor, capturing both local volatility relationships and temporal proximity. Spatial expansion then interpolates points of different connection strengths while gap filling ensures a regularized geometric structure. A subsequent relevance-weighted attention mechanism targets significant regions of each transformed window. To further illuminate underlying dynamics, we integrate the largest Lyapunov exponents directly into each point cloud, embedding a chaotic signature that quantifies local predictability. Unlike canonical CNN, RNN, or Transformer pipelines, this geometry-based representation makes it easier to detect abrupt changes, volatility clusters, and multiscale dependencies via explicit geometric and topological cues. Finally, an architecture incorporating graph-inspired components—along with point-cloud encoders and multihead attention—learns both short-term and long-term dynamics from the spatially enriched time series. The method's ability to harmonize volatility-driven structure, chaotic features, and temporal attention improves predictive performance in empirical testing on stock and cryptocurrency data, underscoring its potential for versatile financial analysis and risk-based applications.