An introduction to persistent homology for time series

2021 ◽  
Vol 13 (3) ◽  
Author(s):  
Nalini Ravishanker ◽  
Renjie Chen
Keyword(s):  
Time Series ◽  
Persistent Homology

Stability Determination in Turning Using Persistent Homology and Time Series Analysis

2014 ◽  
Author(s):  
Firas A. Khasawneh ◽  
Elizabeth Munch
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Point Cloud ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Dimensional Euclidean Space ◽  
Stochastic Delay ◽  
Persistence Diagram ◽  
The Stability ◽  
Threshold Criteria

This paper describes a new approach for ascertaining the stability of autonomous stochastic delay equations in their parameter space by examining their time series using topological data analysis. We use a nonlinear model that describes the tool oscillations due to self-excited vibrations in turning. The time series is generated using Euler-Maruyama method and then is turned into a point cloud in a high dimensional Euclidean space using the delay embedding. The point cloud can then be analyzed using persistent homology. Specifically, in the deterministic case, the system has a stable fixed point while the loss of stability is associated with Hopf bifurcation whereby a limit cycle branches from the fixed point. Since periodicity in the signal translates into circularity in the point cloud, the persistence diagram associated to the periodic time series will have a high persistence point. This can be used to determine a threshold criteria that can automatically classify the system behavior based on its time series. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data.

scholarly journals A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification

2021 ◽  
Vol 12 ◽  
Author(s):  
Yu-Min Chung ◽  
Chuan-Shen Hu ◽  
Yu-Lun Lo ◽  
Hau-Tieng Wu
Keyword(s):  
Heart Rate ◽  
Time Series ◽  
Heart Rate Variability ◽  
Algebraic Topology ◽  
Persistent Homology ◽  
Heart Rate Variability Analysis ◽  
Analysis Tool ◽  
Classification Problems ◽  
Hrv Analysis ◽  
Persistence Diagrams

Persistent homology is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply persistent homology to study time series, particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects—the persistent homologies and hence persistence diagrams of its sub-level set and Taken's lag map. Second, we propose a systematic and computationally efficient approach to summarize persistence diagrams, which we coined persistence statistics. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets.

Persistent homology for time series and spatial data clustering

2015 ◽  
Vol 42 (15-16) ◽  
pp. 6026-6038 ◽  
Author(s):  
Cássio M.M. Pereira ◽  
Rodrigo F. de Mello
Keyword(s):  
Time Series ◽  
Spatial Data ◽  
Data Clustering ◽  
Persistent Homology

scholarly journals TimeCycle: Topology Inspired MEthod for the Detection of Cycling Transcripts in Circadian Time-Series Data

2020 ◽  
Author(s):  
Elan Ness-Cohn ◽  
Rosemary Braun
Keyword(s):  
Time Series ◽  
Time Series Data ◽  
Detection Method ◽  
Persistent Homology ◽  
Null Distribution ◽  
Algebraic Method ◽  
Optimal Choice ◽  
Biological Data ◽  
Series Data ◽  
Topological Features

AbstractThe circadian rhythm drives the oscillatory expression of thousands of genes across all tissues. The recent revolution in high-throughput transcriptomics, coupled with the significant implications of the circadian clock for human health, has sparked an interest in circadian profiling studies to discover genes under circadian control. Here we present TimeCycle: a topology-based rhythm detection method designed to identify cycling transcripts. For a given time-series, the method reconstructs the state space using time-delay embedding, a data transformation technique from dynamical systems. In the embedded space, Takens’ theorem proves that the dynamics of a rhythmic signal will exhibit circular patterns. The degree of circularity of the embedding is calculated as a persistence score using persistent homology, an algebraic method for discerning the topological features of data. By comparing the persistence scores to a bootstrapped null distribution, cycling genes are identified. Results in both synthetic and biological data highlight TimeCycle’s ability to identify cycling genes across a range of sampling schemes, number of replicates, and missing data. Comparison to competing methods highlights their relative strengths, providing guidance as to the optimal choice of cycling detection method.

The LDE-Type Flare Occurrence (1969-1993)

1994 ◽  
Vol 144 ◽  
pp. 279-282
Author(s):  
A. Antalová
Keyword(s):  
Time Series ◽  
Fourier Analysis ◽  
Sunspot Cycle ◽  
List Type ◽  
Type Number ◽  
Solar Cycles ◽  
Type Iv ◽  
Long Duration ◽  
Cycle Maximum ◽  
Flare Index

AbstractThe occurrence of LDE-type flares in the last three cycles has been investigated. The Fourier analysis spectrum was calculated for the time series of the LDE-type flare occurrence during the 20-th, the 21-st and the rising part of the 22-nd cycle. LDE-type flares (Long Duration Events in SXR) are associated with the interplanetary protons (SEP and STIP as well), energized coronal archs and radio type IV emission. Generally, in all the cycles considered, LDE-type flares mainly originated during a 6-year interval of the respective cycle (2 years before and 4 years after the sunspot cycle maximum). The following significant periodicities were found:• in the 20-th cycle: 1.4, 2.1, 2.9, 4.0, 10.7 and 54.2 of month,• in the 21-st cycle: 1.2, 1.6, 2.8, 4.9, 7.8 and 44.5 of month,• in the 22-nd cycle, till March 1992: 1.4, 1.8, 2.4, 7.2, 8.7, 11.8 and 29.1 of month,• in all interval (1969-1992):a)the longer periodicities: 232.1, 121.1 (the dominant at 10.1 of year), 80.7, 61.9 and 25.6 of month,b)the shorter periodicities: 4.7, 5.0, 6.8, 7.9, 9.1, 15.8 and 20.4 of month.Fourier analysis of the LDE-type flare index (FI) yields significant peaks at 2.3 - 2.9 months and 4.2 - 4.9 months. These short periodicities correspond remarkably in the all three last solar cycles. The larger periodicities are different in respective cycles.

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