---
_id: '9821'
abstract:
- lang: eng
text: Heart rate variability (hrv) is a physiological phenomenon of the variation
in the length of the time interval between consecutive heartbeats. In many cases
it could be an indicator of the development of pathological states. The classical
approach to the analysis of hrv includes time domain methods and frequency domain
methods. However, attempts are still being made to define new and more effective
hrv assessment tools. Persistent homology is a novel data analysis tool developed
in the recent decades that is rooted at algebraic topology. The Topological Data
Analysis (TDA) approach focuses on examining the shape of the data in terms of
connectedness and holes, and has recently proved to be very effective in various
fields of research. In this paper we propose the use of persistent homology to
the hrv analysis. We recall selected topological descriptors used in the literature
and we introduce some new topological descriptors that reflect the specificity
of hrv, and we discuss their relation to the standard hrv measures. In particular,
we show that this novel approach provides a collection of indices that might be
at least as useful as the classical parameters in differentiating between series
of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering
from a stroke episode.
acknowledgement: We express our gratitude to the anonymous referees who provided constructive
comments that helped us improve the quality of the paper.
article_number: e0253851
article_processing_charge: Yes
article_type: original
author:
- first_name: Grzegorz
full_name: Graff, Grzegorz
last_name: Graff
- first_name: Beata
full_name: Graff, Beata
last_name: Graff
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: Dariusz
full_name: Gąsecki, Dariusz
last_name: Gąsecki
- first_name: Krzysztof
full_name: Narkiewicz, Krzysztof
last_name: Narkiewicz
citation:
ama: Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. Persistent
homology as a new method of the assessment of heart rate variability. PLoS
ONE. 2021;16(7). doi:10.1371/journal.pone.0253851
apa: Graff, G., Graff, B., Pilarczyk, P., Jablonski, G., Gąsecki, D., & Narkiewicz,
K. (2021). Persistent homology as a new method of the assessment of heart rate
variability. PLoS ONE. Public Library of Science. https://doi.org/10.1371/journal.pone.0253851
chicago: Graff, Grzegorz, Beata Graff, Pawel Pilarczyk, Grzegorz Jablonski, Dariusz
Gąsecki, and Krzysztof Narkiewicz. “Persistent Homology as a New Method of the
Assessment of Heart Rate Variability.” PLoS ONE. Public Library of Science,
2021. https://doi.org/10.1371/journal.pone.0253851.
ieee: G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, and K. Narkiewicz,
“Persistent homology as a new method of the assessment of heart rate variability,”
PLoS ONE, vol. 16, no. 7. Public Library of Science, 2021.
ista: Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. 2021.
Persistent homology as a new method of the assessment of heart rate variability.
PLoS ONE. 16(7), e0253851.
mla: Graff, Grzegorz, et al. “Persistent Homology as a New Method of the Assessment
of Heart Rate Variability.” PLoS ONE, vol. 16, no. 7, e0253851, Public
Library of Science, 2021, doi:10.1371/journal.pone.0253851.
short: G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, K. Narkiewicz,
PLoS ONE 16 (2021).
date_created: 2021-08-08T22:01:28Z
date_published: 2021-07-01T00:00:00Z
date_updated: 2026-04-02T13:56:42Z
day: '01'
ddc:
- '006'
department:
- _id: HeEd
doi: 10.1371/journal.pone.0253851
external_id:
isi:
- '000678124900050'
pmid:
- '34292957'
file:
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checksum: 0277aa155d5db1febd2cb384768bba5f
content_type: application/pdf
creator: asandaue
date_created: 2021-08-09T09:25:41Z
date_updated: 2021-08-09T09:25:41Z
file_id: '9832'
file_name: 2021_PLoSONE_Graff.pdf
file_size: 2706919
relation: main_file
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file_date_updated: 2021-08-09T09:25:41Z
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license: https://creativecommons.org/licenses/by/4.0/
month: '07'
oa: 1
oa_version: Published Version
pmid: 1
publication: PLoS ONE
publication_identifier:
eissn:
- 1932-6203
publication_status: published
publisher: Public Library of Science
quality_controlled: '1'
scopus_import: '1'
status: public
title: Persistent homology as a new method of the assessment of heart rate variability
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 16
year: '2021'
...
---
_id: '15064'
abstract:
- lang: eng
text: We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available information
with chain maps on Delaunay complexes, we use persistent homology to quantify
the evidence of recurrent behavior. We establish a sampling theorem to recover
the eigenspaces of the endomorphism on homology induced by the self-map. Using
a combinatorial gradient flow arising from the discrete Morse theory for Čech
and Delaunay complexes, we construct a chain map to transform the problem from
the natural but expensive Čech complexes to the computationally efficient Delaunay
triangulations. The fast chain map algorithm has applications beyond dynamical
systems.
acknowledgement: This research has been supported by the DFG Collaborative Research
Center SFB/TRR 109 “Discretization in Geometry and Dynamics”, by Polish MNiSzW Grant
No. 2621/7.PR/12/2013/2, by the Polish National Science Center under Maestro Grant
No. 2014/14/A/ST1/00453 and Grant No. DEC-2013/09/N/ST6/02995. Open Access funding
provided by Projekt DEAL.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: U.
full_name: Bauer, U.
last_name: Bauer
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: M.
full_name: Mrozek, M.
last_name: Mrozek
citation:
ama: Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow
and homology inference for self-maps. Journal of Applied and Computational
Topology. 2020;4(4):455-480. doi:10.1007/s41468-020-00058-8
apa: Bauer, U., Edelsbrunner, H., Jablonski, G., & Mrozek, M. (2020). Čech-Delaunay
gradient flow and homology inference for self-maps. Journal of Applied and
Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00058-8
chicago: Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay
Gradient Flow and Homology Inference for Self-Maps.” Journal of Applied and
Computational Topology. Springer Nature, 2020. https://doi.org/10.1007/s41468-020-00058-8.
ieee: U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient
flow and homology inference for self-maps,” Journal of Applied and Computational
Topology, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020.
ista: Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient
flow and homology inference for self-maps. Journal of Applied and Computational
Topology. 4(4), 455–480.
mla: Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.”
Journal of Applied and Computational Topology, vol. 4, no. 4, Springer
Nature, 2020, pp. 455–80, doi:10.1007/s41468-020-00058-8.
short: U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and
Computational Topology 4 (2020) 455–480.
date_created: 2024-03-04T10:47:49Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2024-03-04T10:54:04Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.1007/s41468-020-00058-8
file:
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checksum: eed1168b6e66cd55272c19bb7fca8a1c
content_type: application/pdf
creator: dernst
date_created: 2024-03-04T10:52:42Z
date_updated: 2024-03-04T10:52:42Z
file_id: '15065'
file_name: 2020_JourApplCompTopology_Bauer.pdf
file_size: 851190
relation: main_file
success: 1
file_date_updated: 2024-03-04T10:52:42Z
has_accepted_license: '1'
intvolume: ' 4'
issue: '4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 455-480
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Čech-Delaunay gradient flow and homology inference for self-maps
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 4
year: '2020'
...
---
_id: '8580'
abstract:
- lang: eng
text: We evaluate the usefulness of persistent homology in the analysis of heart
rate variability. In our approach we extract several topological descriptors characterising
datasets of RR-intervals, which are later used in classical machine learning algorithms.
By this method we are able to differentiate the group of patients with the history
of transient ischemic attack and the group of hypertensive patients.
article_number: '9158054'
article_processing_charge: No
author:
- first_name: Grzegorz
full_name: Graff, Grzegorz
last_name: Graff
- first_name: Beata
full_name: Graff, Beata
last_name: Graff
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: Krzysztof
full_name: Narkiewicz, Krzysztof
last_name: Narkiewicz
citation:
ama: 'Graff G, Graff B, Jablonski G, Narkiewicz K. The application of persistent
homology in the analysis of heart rate variability. In: 11th Conference of
the European Study Group on Cardiovascular Oscillations: Computation and Modelling
in Physiology: New Challenges and Opportunities, . IEEE; 2020. doi:10.1109/ESGCO49734.2020.9158054'
apa: 'Graff, G., Graff, B., Jablonski, G., & Narkiewicz, K. (2020). The application
of persistent homology in the analysis of heart rate variability. In 11th Conference
of the European Study Group on Cardiovascular Oscillations: Computation and Modelling
in Physiology: New Challenges and Opportunities, . Pisa, Italy: IEEE. https://doi.org/10.1109/ESGCO49734.2020.9158054'
chicago: 'Graff, Grzegorz, Beata Graff, Grzegorz Jablonski, and Krzysztof Narkiewicz.
“The Application of Persistent Homology in the Analysis of Heart Rate Variability.”
In 11th Conference of the European Study Group on Cardiovascular Oscillations:
Computation and Modelling in Physiology: New Challenges and Opportunities, .
IEEE, 2020. https://doi.org/10.1109/ESGCO49734.2020.9158054.'
ieee: 'G. Graff, B. Graff, G. Jablonski, and K. Narkiewicz, “The application of
persistent homology in the analysis of heart rate variability,” in 11th Conference
of the European Study Group on Cardiovascular Oscillations: Computation and Modelling
in Physiology: New Challenges and Opportunities, , Pisa, Italy, 2020.'
ista: 'Graff G, Graff B, Jablonski G, Narkiewicz K. 2020. The application of persistent
homology in the analysis of heart rate variability. 11th Conference of the European
Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology:
New Challenges and Opportunities, . ESGCO: European Study Group on Cardiovascular
Oscillations, 9158054.'
mla: 'Graff, Grzegorz, et al. “The Application of Persistent Homology in the Analysis
of Heart Rate Variability.” 11th Conference of the European Study Group on
Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges
and Opportunities, , 9158054, IEEE, 2020, doi:10.1109/ESGCO49734.2020.9158054.'
short: 'G. Graff, B. Graff, G. Jablonski, K. Narkiewicz, in:, 11th Conference of
the European Study Group on Cardiovascular Oscillations: Computation and Modelling
in Physiology: New Challenges and Opportunities, , IEEE, 2020.'
conference:
end_date: 2020-07-15
location: Pisa, Italy
name: 'ESGCO: European Study Group on Cardiovascular Oscillations'
start_date: 2020-07-15
date_created: 2020-09-28T08:59:27Z
date_published: 2020-08-01T00:00:00Z
date_updated: 2023-08-22T09:33:34Z
day: '01'
department:
- _id: HeEd
doi: 10.1109/ESGCO49734.2020.9158054
external_id:
isi:
- '000621172600045'
isi: 1
language:
- iso: eng
month: '08'
oa_version: None
publication: '11th Conference of the European Study Group on Cardiovascular Oscillations:
Computation and Modelling in Physiology: New Challenges and Opportunities, '
publication_identifier:
isbn:
- '9781728157511'
publication_status: published
publisher: IEEE
quality_controlled: '1'
scopus_import: '1'
status: public
title: The application of persistent homology in the analysis of heart rate variability
type: conference
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
year: '2020'
...
---
_id: '836'
abstract:
- lang: eng
text: Recent research has examined how to study the topological features of a continuous
self-map by means of the persistence of the eigenspaces, for given eigenvalues,
of the endomorphism induced in homology over a field. This raised the question
of how to select dynamically significant eigenvalues. The present paper aims to
answer this question, giving an algorithm that computes the persistence of eigenspaces
for every eigenvalue simultaneously, also expressing said eigenspaces as direct
sums of “finite” and “singular” subspaces.
alternative_title:
- PROMS
article_processing_charge: No
author:
- first_name: Marc
full_name: Ethier, Marc
last_name: Ethier
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: Marian
full_name: Mrozek, Marian
last_name: Mrozek
citation:
ama: 'Ethier M, Jablonski G, Mrozek M. Finding eigenvalues of self-maps with the
Kronecker canonical form. In: Special Sessions in Applications of Computer
Algebra. Vol 198. Springer; 2017:119-136. doi:10.1007/978-3-319-56932-1_8'
apa: 'Ethier, M., Jablonski, G., & Mrozek, M. (2017). Finding eigenvalues of
self-maps with the Kronecker canonical form. In Special Sessions in Applications
of Computer Algebra (Vol. 198, pp. 119–136). Kalamata, Greece: Springer. https://doi.org/10.1007/978-3-319-56932-1_8'
chicago: Ethier, Marc, Grzegorz Jablonski, and Marian Mrozek. “Finding Eigenvalues
of Self-Maps with the Kronecker Canonical Form.” In Special Sessions in Applications
of Computer Algebra, 198:119–36. Springer, 2017. https://doi.org/10.1007/978-3-319-56932-1_8.
ieee: M. Ethier, G. Jablonski, and M. Mrozek, “Finding eigenvalues of self-maps
with the Kronecker canonical form,” in Special Sessions in Applications of
Computer Algebra, Kalamata, Greece, 2017, vol. 198, pp. 119–136.
ista: 'Ethier M, Jablonski G, Mrozek M. 2017. Finding eigenvalues of self-maps with
the Kronecker canonical form. Special Sessions in Applications of Computer Algebra.
ACA: Applications of Computer Algebra, PROMS, vol. 198, 119–136.'
mla: Ethier, Marc, et al. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical
Form.” Special Sessions in Applications of Computer Algebra, vol. 198,
Springer, 2017, pp. 119–36, doi:10.1007/978-3-319-56932-1_8.
short: M. Ethier, G. Jablonski, M. Mrozek, in:, Special Sessions in Applications
of Computer Algebra, Springer, 2017, pp. 119–136.
conference:
end_date: 2015-07-23
location: Kalamata, Greece
name: 'ACA: Applications of Computer Algebra'
start_date: 2015-07-20
date_created: 2018-12-11T11:48:46Z
date_published: 2017-07-27T00:00:00Z
date_updated: 2025-04-15T08:37:55Z
day: '27'
department:
- _id: HeEd
doi: 10.1007/978-3-319-56932-1_8
ec_funded: 1
external_id:
isi:
- '000434088200008'
intvolume: ' 198'
isi: 1
language:
- iso: eng
month: '07'
oa_version: None
page: 119 - 136
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Special Sessions in Applications of Computer Algebra
publication_identifier:
isbn:
- 978-331956930-7
publication_status: published
publisher: Springer
publist_id: '6812'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Finding eigenvalues of self-maps with the Kronecker canonical form
type: conference
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 198
year: '2017'
...
---
_id: '2035'
abstract:
- lang: eng
text: "Considering a continuous self-map and the induced endomorphism on homology,
we study the eigenvalues and eigenspaces of the latter. Taking a filtration of
representations, we define the persistence of the eigenspaces, effectively introducing
a hierarchical organization of the map. The algorithm that computes this information
for a finite sample is proved to be stable, and to give the correct answer for
a sufficiently dense sample. Results computed with an implementation of the algorithm
provide evidence of its practical utility.\r\n"
acknowledgement: This research is partially supported by the Toposys project FP7-ICT-318493-STREP,
by ESF under the ACAT Research Network Programme, by the Russian Government under
mega project 11.G34.31.0053, and by the Polish National Science Center under Grant
No. N201 419639.
article_processing_charge: No
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Grzegorz
full_name: Jablonski, Grzegorz
id: 4483EF78-F248-11E8-B48F-1D18A9856A87
last_name: Jablonski
orcid: 0000-0002-3536-9866
- first_name: Marian
full_name: Mrozek, Marian
last_name: Mrozek
citation:
ama: Edelsbrunner H, Jablonski G, Mrozek M. The persistent homology of a self-map.
Foundations of Computational Mathematics. 2015;15(5):1213-1244. doi:10.1007/s10208-014-9223-y
apa: Edelsbrunner, H., Jablonski, G., & Mrozek, M. (2015). The persistent homology
of a self-map. Foundations of Computational Mathematics. Springer. https://doi.org/10.1007/s10208-014-9223-y
chicago: Edelsbrunner, Herbert, Grzegorz Jablonski, and Marian Mrozek. “The Persistent
Homology of a Self-Map.” Foundations of Computational Mathematics. Springer,
2015. https://doi.org/10.1007/s10208-014-9223-y.
ieee: H. Edelsbrunner, G. Jablonski, and M. Mrozek, “The persistent homology of
a self-map,” Foundations of Computational Mathematics, vol. 15, no. 5.
Springer, pp. 1213–1244, 2015.
ista: Edelsbrunner H, Jablonski G, Mrozek M. 2015. The persistent homology of a
self-map. Foundations of Computational Mathematics. 15(5), 1213–1244.
mla: Edelsbrunner, Herbert, et al. “The Persistent Homology of a Self-Map.” Foundations
of Computational Mathematics, vol. 15, no. 5, Springer, 2015, pp. 1213–44,
doi:10.1007/s10208-014-9223-y.
short: H. Edelsbrunner, G. Jablonski, M. Mrozek, Foundations of Computational Mathematics
15 (2015) 1213–1244.
date_created: 2018-12-11T11:55:20Z
date_published: 2015-10-01T00:00:00Z
date_updated: 2025-09-23T14:08:54Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s10208-014-9223-y
ec_funded: 1
external_id:
isi:
- '000360862900004'
file:
- access_level: open_access
checksum: 3566f3a8b0c1bc550e62914a88c584ff
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creator: system
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file_name: IST-2016-486-v1+1_s10208-014-9223-y.pdf
file_size: 1317546
relation: main_file
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intvolume: ' 15'
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issue: '5'
language:
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month: '10'
oa: 1
oa_version: Published Version
page: 1213 - 1244
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Foundations of Computational Mathematics
publication_status: published
publisher: Springer
publist_id: '5022'
pubrep_id: '486'
quality_controlled: '1'
scopus_import: '1'
status: public
title: The persistent homology of a self-map
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...