[Paper Review] Topological Data Analysis Ball Mapper for Finance

Paweł Dłotko, Wanling Qiu+1 more|arXiv (Cornell University)|Jun 8, 2022

Topological and Geometric Data Analysis7 citations

TL;DR

This paper introduces the Topological Data Analysis Ball Mapper (BM) algorithm as a robust, visualization-driven tool for exploring high-dimensional financial data. By mapping data shape through adaptive ball clustering, BM reveals hidden patterns in credit risk and stock market direction forecasting, demonstrating consistent, interpretable insights across varying radii and outperforming traditional statistical methods in capturing non-linear relationships and data structure.

ABSTRACT

Finance is heavily influenced by data-driven decision-making. Meanwhile, our ability to comprehend the full informational content of data sets remains impeded by the tools we apply in analysis, especially where the data is high-dimensional. Presenting the Topological Data Analysis Ball Mapper algorithm this paper illuminates a new means of seeing the detail in data from data shape. With comparisons to existing approaches and illustrative examples, the value of the new tool is shown. Directions for employing Ball Mapper in practice are given and the benefits are reviewed.

Motivation & Objective

• To address the limitations of traditional statistical visualization in high-dimensional financial data by introducing a topology-based approach.
• To demonstrate how Ball Mapper (BM) enables intuitive, shape-aware exploration of complex financial datasets beyond two or three dimensions.
• To validate BM’s robustness and interpretability in real-world financial applications such as credit risk modeling and stock market direction prediction.
• To provide practical guidance for researchers and practitioners on applying BM in finance, including parameter sensitivity and visualization techniques.
• To establish BM as a stable, connectivity-aware alternative to clustering, emphasizing data shape and outcome mapping over group partitioning.

Proposed method

• The Ball Mapper (BM) algorithm constructs a topological map by covering a high-dimensional data cloud with overlapping balls of varying radii, where each ball represents a local region of data.
• Balls are centered on data points and grow to include neighboring points within a specified radius ε, forming a network where connected balls represent regions of high data density and continuity.
• The resulting graph structure preserves topological features such as connectivity, clustering, and correlation patterns, enabling visualization of complex data shapes in 2D.
• Outcome variables (e.g., S&P 500 return direction) are color-coded across balls to visualize how outcomes vary across the data manifold.
• Parameter sensitivity is analyzed by varying ε to assess stability of the map and identify consistent structural features across different scales.
• The method is applied to two financial datasets: one for credit risk and one for stock market direction forecasting using normalized macroeconomic and market indicators.

Experimental results

Research questions

• RQ1 How can Ball Mapper effectively reveal hidden, non-linear relationships in high-dimensional financial data that traditional statistical methods miss?
• RQ2 What is the impact of the ball radius ε on the stability and interpretability of the topological map in financial applications?
• RQ3 Can Ball Mapper provide consistent and actionable insights in credit risk modeling and stock market direction forecasting across different data scales?
• RQ4 How does the topological structure of financial data, as revealed by BM, compare to classical visualization and clustering approaches in terms of interpretability and information retention?
• RQ5 In what ways can outcome functions (e.g., market direction) be meaningfully visualized across the BM graph to support financial decision-making?

Key findings

• Ball Mapper successfully visualizes the shape of high-dimensional financial data, revealing non-linear dependencies and structural patterns invisible in standard scatter plots.
• The BM graph shows consistent topological structure across different ε values, with key features such as high-probability market rise regions remaining stable.
• In the stock market direction forecasting example, 70% or more of months with subsequent S&P 500 increases were concentrated in a single, well-defined region of the BM graph at ε = 0.3.
• Even at larger ε, the map retained meaningful outcome variation, with the lowest-probability region showing only 20% of subsequent increases, indicating no extreme predictability but strong signal in intermediate zones.
• The method outperforms summary statistics and linear models by capturing shape-based insights, as demonstrated by the contrast with Anscombe’s quartet and the datasaurus dataset.
• The BM approach enables identification of data regions with strong predictive power—such as the term spread and yield curve variables—without assuming linearity or parametric forms.