Topological Data Analysis

AbstractIn developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

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Classification of apatite structures via topological data analysis: a framework for a ‘Materials Barcode’ representation of structure maps

Scientific Reports ◽

10.1038/s41598-021-90070-4 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Scott Broderick ◽

Ruhil Dongol ◽

Tianmu Zhang ◽

Krishna Rajan

Keyword(s):

Machine Learning ◽

Data Analysis ◽

Crystal Chemistry ◽

Persistent Homology ◽

Hierarchical Classification ◽

Topological Data Analysis ◽

Learning Tool ◽

Coordination Polyhedra ◽

Machine Learning Tool ◽

Topological Data

AbstractThis paper introduces the use of topological data analysis (TDA) as an unsupervised machine learning tool to uncover classification criteria in complex inorganic crystal chemistries. Using the apatite chemistry as a template, we track through the use of persistent homology the topological connectivity of input crystal chemistry descriptors on defining similarity between different stoichiometries of apatites. It is shown that TDA automatically identifies a hierarchical classification scheme within apatites based on the commonality of the number of discrete coordination polyhedra that constitute the structural building units common among the compounds. This information is presented in the form of a visualization scheme of a barcode of homology classifications, where the persistence of similarity between compounds is tracked. Unlike traditional perspectives of structure maps, this new “Materials Barcode” schema serves as an automated exploratory machine learning tool that can uncover structural associations from crystal chemistry databases, as well as to achieve a more nuanced insight into what defines similarity among homologous compounds.

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The Shape of Things: Topological Data Analysis

CHANCE ◽

10.1080/09332480.2021.1915036 ◽

2021 ◽

Vol 34 (2) ◽

pp. 59-64

Author(s):

Nicole Lazar ◽

Hyunnam Ryu

Keyword(s):

Data Analysis ◽

Topological Data Analysis ◽

Topological Data

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Empowering Advanced Driver-Assistance Systems from Topological Data Analysis

Mathematics ◽

10.3390/math9060634 ◽

2021 ◽

Vol 9 (6) ◽

pp. 634

Author(s):

Tarek Frahi ◽

Francisco Chinesta ◽

Antonio Falcó ◽

Alberto Badias ◽

Elias Cueto ◽

...

Keyword(s):

Data Analysis ◽

The State ◽

Sensor Data ◽

Topological Data Analysis ◽

Motion Sensors ◽

Driver Assistance Systems ◽

The Road ◽

Topological Features ◽

Recent Developments ◽

Topological Data

We are interested in evaluating the state of drivers to determine whether they are attentive to the road or not by using motion sensor data collected from car driving experiments. That is, our goal is to design a predictive model that can estimate the state of drivers given the data collected from motion sensors. For that purpose, we leverage recent developments in topological data analysis (TDA) to analyze and transform the data coming from sensor time series and build a machine learning model based on the topological features extracted with the TDA. We provide some experiments showing that our model proves to be accurate in the identification of the state of the user, predicting whether they are relaxed or tense.

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