arXiv subjects

Pawel Dlotko

Publications and source records attributed to Pawel Dlotko.

12 recordsLinked to original sources

arXiv2025

Data Driven Perspectives on Knot Theory

Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a way to extend mathematical exposition of various invariants of knots towards understanding their relations in statistical and cumulative way. Paper included examples illustrating how topological data analysis can illuminate structure and relations between knot invariants, state new hypothesis, and gain new insides into long standing conjectures.

math.GT↗

arXiv2023

Persistence Norms and the Datasaurus

Topological Data Analysis (TDA) provides a toolkit for the study of the shape of high dimensional and complex data. While operating on a space of persistence diagrams is cumbersome, persistence norms provide a simple real value measure of multivariate data which is seeing greater adoption within finance. A growing literature seeks links between persistence norms and the summary statistics of the data being analysed. This short note targets the demonstration of differences in the persistence norms of the Datasaurus datasets of Matejka and Fitzmaurice. We show that persistence norms can be used as additional measures that often discriminate datasets with the same collection of summary statistics. Treating each of the data sets as a point cloud we construct the $L_1$ and $L_2$ persistence norms in dimensions 0 and 1. We show multivariate distributions with identical covariance and correlation matrices can have considerably different persistence norms. Through the example, we remind users of persistence norms of the importance of checking the distribution of the point clouds from which the norms are constructed.

stat.ME↗

arXiv2022

Topological Data Analysis Ball Mapper for Finance

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.

q-fin.GN↗

arXiv2021

An Economic Topology of the Brexit vote

A desire to understand the decision of the UK to leave the European Union, Brexit, in the referendum of June 2016 has continued to occupy academics, the media and politicians. Using topological data analysis ball mapper we extract information from multi-dimensional datasets gathered on Brexit voting and regional socio-economic characteristics. While we find broad patterns consistent with extant empirical work, we also evidence that support for Leave drew from a far more homogenous demographic than Remain. Obtaining votes from this concise set was more straightforward for Leave campaigners than was Remain's task of mobilising a diverse group to oppose Brexit.

econ.EM↗

arXiv2021

Uncertainty, volatility and the persistence norms of financial time series

Norms of Persistent Homology introduced in topological data analysis are seen as indicators of system instability, analogous to the changing predictability that is captured in financial market uncertainty indexes. This paper demonstrates norms from the financial markets are significant in explaining financial uncertainty, whilst macroeconomic uncertainty is only explainable by market volatility. Meanwhile, volatility is insignificant in the determination of norms when uncertainty enters the regression. Persistence norms therefore have potential as a further tool in asset pricing, and also as a means of capturing signals from financial time series beyond volatility.

q-fin.GN↗

arXiv2020

Topological Data Analysis: Concepts, Computation, and Applications in Chemical Engineering

A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional spaces that are composed of elementary geometric objects. Key property of these elementary objects (also known as topological features) are that they persist at different scales and that they are stable under perturbations (e.g., noise, stretching, twisting, and bending). In this work, we review key mathematical concepts and methods of TDA and present different applications in chemical engineering.

math.AT↗

arXiv2020

Refining Understanding of Corporate Failure through a Topological Data Analysis Mapping of Altman's Z-Score Model

Corporate failure resonates widely leaving practitioners searching for understanding of default risk. Managers seek to steer away from trouble, credit providers to avoid risky loans and investors to mitigate losses. Applying Topological Data Analysis tools this paper explores whether failing firms from the United States organise neatly along the five predictors of default proposed by the Z-score models. Firms are represented as a point cloud in a five dimensional space, one axis for each predictor. Visualising that cloud using Ball Mapper reveals failing firms are not often neighbours. As new modelling approaches vie to better predict firm failure, often using black boxes to deliver potentially over-fitting models, a timely reminder is sounded on the importance of evidencing the identification process. Value is added to the understanding of where in the parameter space failure occurs, and how firms might act to move away from financial distress. Further, lenders may find opportunity amongst subsets of firms that are traditionally considered to be in danger of bankruptcy but actually sit in characteristic spaces where failure has not occurred.

q-fin.GN↗

arXiv2020

Visualising the Evolution of English Covid-19 Cases with Topological Data Analysis Ball Mapper

Understanding disease spread through data visualisation has concentrated on trends and maps. Whilst these are helpful, they neglect important multi-dimensional interactions between characteristics of communities. Using the Topological Data Analysis Ball Mapper algorithm we construct an abstract representation of NUTS3 level economic data, overlaying onto it the confirmed cases of Covid-19 in England. In so doing we may understand how the disease spreads on different socio-economical dimensions. It is observed that some areas of the characteristic space have quickly raced to the highest levels of infection, while others close by in the characteristic space, do not show large infection growth. Likewise, we see patterns emerging in very different areas that command more monitoring. A strong contribution for Topological Data Analysis, and the Ball Mapper algorithm especially, in comprehending dynamic epidemic data is signposted.

physics.soc-ph↗

arXiv2019

Topologically Mapping the Macroeconomy

An understanding of the economic landscape in a world of ever increasing data necessitates representations of data that can inform policy, deepen understanding and guide future research. Topological Data Analysis offers a set of tools which deliver on all three calls. Abstract two-dimensional snapshots of multi-dimensional space readily capture non-monotonic relationships, inform of similarity between points of interest in parameter space, mapping such to outcomes. Specific examples show how some, but not all, countries have returned to Great Depression levels, and reappraise the links between real private capital growth and the performance of the economy. Theoretical and empirical expositions alike remind on the dangers of assuming monotonic relationships and discounting combinations of factors as determinants of outcomes; both dangers Topological Data Analysis addresses. Policy-makers can look at outcomes and target areas of the input space where such are not satisfactory, academics may additionally find evidence to motivate theoretical development, and practitioners can gain a rapid and robust base for decision making.

econ.EM↗

arXiv2019

Financial ratios and stock returns reappraised through a topological data analysis lens

Firm financials are well established as return predictors, being the inspiration for a large set of anomalies in the asset pricing literature. Employing topological data analysis we revisit the question of association between seven of the most commonly studied financial ratios and stock returns. Specifically the TDA Ball Mapper algorithm is applied to visualise the point cloud of financial ratios as an abstract two-dimensional graph readily allowing for identification of interdependencies between factors. These relationships are seldom monotonic, opportunities for investors to profitably exploit this knowledge provided by TDA abound. Clear potential offered by the tools of TDA to shed new light on asset pricing models is demonstrated. Scope for benefit is limited only by the availability of information to the analyst.

q-fin.ST↗

arXiv2019

Persistence Codebooks for Topological Data Analysis

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence codebooks, a novel expressive and discriminative fixed-size vectorized representation of PDs. To this end, we adapt bag-of-words (BoW), vectors of locally aggregated descriptors (VLAD) and Fischer vectors (FV) for the quantization of PDs. Persistence codebooks represent PDs in a convenient way for machine learning and statistical analysis and have a number of favorable practical and theoretical properties including 1-Wasserstein stability. We evaluate the presented representations on several heterogeneous datasets and show their (high) discriminative power. Our approach achieves state-of-the-art performance and beyond in much less time than alternative approaches.

stat.ML↗

arXiv2015

A persistence landscapes toolbox for topological statistics

Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning. We give efficient algorithms for calculating persistence landscapes, their averages, and distances between such averages. We discuss an implementation of these algorithms and some related procedures. These are intended to facilitate the combination of statistics and machine learning with topological data analysis. We present an experiment showing that the low-dimensional persistence landscapes of points sampled from spheres (and boxes) of varying dimensions differ.

cs.CG↗