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Bayesian topological signal processing
1. | University of Notre Dame, Department of Aerospace and Mechanical Engineering, Fitzpatrick Hall of Engineering and Cushing Hall, 112 N Notre Dame Ave, Notre Dame, IN 46556 |
2. | University of Tennessee, Department of Mathematics, 1403 Circle Drive, Knoxville, TN 37996-1320 |
3. | US Army Research Laboratory, 7101 Mulberry Point Road, Bldg. 459, Aberdeen Proving Ground, MD 21005-5425 |
Topological data analysis encompasses a broad set of techniques that investigate the shape of data. One of the predominant tools in topological data analysis is persistent homology, which is used to create topological summaries of data called persistence diagrams. Persistent homology offers a novel method for signal analysis. Herein, we aid interpretation of the sublevel set persistence diagrams of signals by 1) showing the effect of frequency and instantaneous amplitude on the persistence diagrams for a family of deterministic signals, and 2) providing a general equation for the probability density of persistence diagrams of random signals via a pushforward measure. We also provide a topologically-motivated, efficiently computable statistical descriptor analogous to the power spectral density for signals based on a generalized Bayesian framework for persistence diagrams. This Bayesian descriptor is shown to be competitive with power spectral densities and continuous wavelet transforms at distinguishing signals with different dynamics in a classification problem with autoregressive signals.
References:
[1] |
H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta and L. Ziegelmeier,
Persistence images: A stable vector representation of persistent homology, The Journal of Machine Learning Research, 18 (2017), 218-252.
|
[2] |
M. Bandarabadi, A. Dourado, C. A. Teixeira, T. I. Netoff and K. K. Parhi, Seizure prediction with bipolar spectral power features using adaboost and svm classifiers, Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), (2013), 6305–6308. |
[3] |
S. Barbarossa and S. Sardellitti,
Topological signal processing over simplicial complexes, IEEE Transactions on Signal Processing, 68 (2020), 2992-3007.
doi: 10.1109/TSP.2020.2981920. |
[4] |
R. J. Barry, A. R. Clarke, S. J. Johnstone, C. A. Magee and J. A. Rushby,
EEG differences between eyes-closed and eyes-open resting conditions, Clinical Neurophysiology, 118 (2007), 2765-2773.
doi: 10.1016/j.clinph.2007.07.028. |
[5] |
J. Berwald and M. Gidea,
Critical transitions in a model of a genetic regulatory system, Mathematical Biosciences and Engineering, 11 (2014), 723-740.
doi: 10.3934/mbe.2014.11.723. |
[6] |
P. Bromiley, Products and convolutions of gaussian probability density functions, Tina-Vision Memo, 3.4 (2003), 13 pp. |
[7] |
P. Bubenik,
Statistical topological data analysis using persistence landscapes, The Journal of Machine Learning Research, 16 (2015), 77-102.
|
[8] |
G. Carlsson,
Topology and data, Bulletin of the American Mathematical Society, 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[9] |
G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, in Symposium on Geometry Processing, (eds. R. Scopigno and D. Zorin), The Eurographics Association, (2004), 124–135.
doi: 10.1145/1057432.1057449. |
[10] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete & Computational Geometry, 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[11] |
W. Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, Journal of Algebra and Its Applications, 14 (2015), 1550066.
doi: 10.1142/S0219498815500668. |
[12] |
H. Edelsbrunner, D. Letscher and A. Zomorodian,
Topological persistence and simplification, Discrete & Computational Geometry, 28 (2002), 511-533.
doi: 10.1007/s00454-002-2885-2. |
[13] |
H. Edelsbrunner and J. Harer, Computational Topology, American Mathematical Society, 2010.
doi: 10.1090/mbk/069. |
[14] |
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan, A. Singh and et al.,
Confidence sets for persistence diagrams, The Annals of Statistics, 42 (2014), 2301-2339.
doi: 10.1214/14-AOS1252. |
[15] |
P. J. Franaszczuk and K. J. Blinowska,
Linear model of brain electrical activity? EEG as a superposition of damped oscillatory modes, Biological Cybernetics, 53 (1985), 19-25.
doi: 10.1007/BF00355687. |
[16] |
P. J. Franaszczuk, G. K. Bergey, P. J. Durka and H. M. Eisenberg,
Time-frequency analysis using the matching pursuit algorithm applied to seizures originating from the mesial temporal lobe, Electroencephalography and Clinical Neurophysiology, 106 (1998), 513-521.
doi: 10.1016/S0013-4694(98)00024-8. |
[17] |
S. Gholizadeh and W. Zadrozny, A short survey of topological data analysis in time series and systems analysis, (2018). |
[18] |
R. Ghrist,
Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[19] |
C. Ieracitano, N. Mammone, A. Bramanti, S. Marino, A. Hussain and F. C. Morabito, A time-frequency based machine learning system for brain states classification via eeg signal processing, in International Joint Conference on Neural Networks (IJCNN), (2019), 1–8.
doi: 10.1109/IJCNN.2019.8852240. |
[20] |
F. Khasawneh and E. Munch, Exploring Equilibria in Stochastic Delay Differential Equations Using Persistent Homology, 2014.
doi: 10.1115/DETC2014-35655. |
[21] |
J. F. C. Kingman, Poisson Processes, Oxford Studies in Probability, 3, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.
![]() ![]() |
[22] |
S. G. Mallat and {Z hifeng Zhang},
Matching pursuits with time-frequency dictionaries, IEEE Transactions on Signal Processing, 41 (1993), 3397-3415.
|
[23] |
A. Marchese and V. Maroulas,
Signal classification with a point process distance on the space of persistence diagrams, Advances in Data Analysis and Classification, 12 (2018), 657-682.
doi: 10.1007/s11634-017-0294-x. |
[24] |
V. Maroulas, J. L. Mike and C. Oballe, Nonparametric estimation of probability density functions of random persistence diagrams, Journal of Machine Learning Research, 20 (2019), 1–49. Available from: http://jmlr.org/papers/v20/18-618.html. |
[25] |
V. Maroulas, F. Nasrin and C. Oballe,
A bayesian framework for persistent homology, SIAM Journal on Mathematics of Data Science, 2 (2020), 48-74.
doi: 10.1137/19M1268719. |
[26] |
Y. Mileyko, S. Mukherjee and J. Harer, Probability measures on the space of persistence diagrams, Inverse Problems, 27 (2011), 124007.
doi: 10.1088/0266-5611/27/12/124007. |
[27] |
A. Monod, S. Kalisnik, J. A. Patino-Galindo and L. Crawford, Tropical sufficient statistics for persistent homology, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 337–371.
doi: 10.1137/17M1148037. |
[28] |
F. Nasrin, C. Oballe, D. Boothe and V. Maroulas, Bayesian topological learning for brain state classification, in 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), (2019), 1247–1252. |
[29] |
A. V. Oppenheim, J. R. Buck and R. W. Schafer, Discrete-Time Signal Processing, 2nd edition, Prentice-Hall signal processing, Prentice-Hall, Upper Saddle River, NJ, 1999. Available from: https://cds.cern.ch/record/389969. |
[30] |
J. A. Perea and J. Harer,
Sliding windows and persistence: An application of topological methods to signal analysis, Found. Comput. Math., 15 (2015), 799-838.
doi: 10.1007/s10208-014-9206-z. |
[31] |
R. Pintelon and J. Schoukens,
Time series analysis in the frequency domain, IEEE Transactions on Signal Processing, 47 (1999), 206-210.
|
[32] |
M. Robinson, Topological Signal Processing, Springer, 2014.
doi: 10.1007/978-3-642-36104-3. |
[33] |
M. D. Sacchi, T. J. Ulrych and C. J. Walker,
Interpolation and extrapolation using a high-resolution discrete fourier transform, IEEE Transactions on Signal Processing, 46 (1998), 31-38.
doi: 10.1109/78.651165. |
[34] |
N. Sanderson, E. Shugerman, S. Molnar, J. D. Meiss and E. Bradley, Computational topology techniques for characterizing time-series data, in Advances in Intelligent Data Analysis XVI, Springer International Publishing, (2017), 284–296. |
[35] |
K. F. Swaiman, S. Ashwal and M. I. Shevell, Swaiman's Pediatric Neurology, Elsevier, 2018.
doi: 10.1016/c2013-1-00079-0. |
[36] |
T. Shiraishi, T. Le, H. Kashima and M. Yamada, Topological bayesian optimization with persistence diagrams, preprint, arXiv: 1902.09722. |
[37] |
B. W. Silverman, Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1986. |
[38] |
P. Skraba, V. de Silva and M. Vejdemo-Johansson, Topological analysis of recurrent systems, in NIPS 2012, 2012. |
[39] |
Y. Umeda,
Time series classification via topological data analysis, Transactions of The Japanese Society for Artificial Intelligence, 32 (2017), 1-12.
doi: 10.1527/tjsai.D-G72. |
[40] |
Y. Wang, H. Ombao and M. K. Chung,
Topological data analysis of single-trial electroencephalographic signals, Ann. Appl. Stat., 12 (2018), 1506-1534.
doi: 10.1214/17-AOAS1119. |
show all references
References:
[1] |
H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta and L. Ziegelmeier,
Persistence images: A stable vector representation of persistent homology, The Journal of Machine Learning Research, 18 (2017), 218-252.
|
[2] |
M. Bandarabadi, A. Dourado, C. A. Teixeira, T. I. Netoff and K. K. Parhi, Seizure prediction with bipolar spectral power features using adaboost and svm classifiers, Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), (2013), 6305–6308. |
[3] |
S. Barbarossa and S. Sardellitti,
Topological signal processing over simplicial complexes, IEEE Transactions on Signal Processing, 68 (2020), 2992-3007.
doi: 10.1109/TSP.2020.2981920. |
[4] |
R. J. Barry, A. R. Clarke, S. J. Johnstone, C. A. Magee and J. A. Rushby,
EEG differences between eyes-closed and eyes-open resting conditions, Clinical Neurophysiology, 118 (2007), 2765-2773.
doi: 10.1016/j.clinph.2007.07.028. |
[5] |
J. Berwald and M. Gidea,
Critical transitions in a model of a genetic regulatory system, Mathematical Biosciences and Engineering, 11 (2014), 723-740.
doi: 10.3934/mbe.2014.11.723. |
[6] |
P. Bromiley, Products and convolutions of gaussian probability density functions, Tina-Vision Memo, 3.4 (2003), 13 pp. |
[7] |
P. Bubenik,
Statistical topological data analysis using persistence landscapes, The Journal of Machine Learning Research, 16 (2015), 77-102.
|
[8] |
G. Carlsson,
Topology and data, Bulletin of the American Mathematical Society, 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[9] |
G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, in Symposium on Geometry Processing, (eds. R. Scopigno and D. Zorin), The Eurographics Association, (2004), 124–135.
doi: 10.1145/1057432.1057449. |
[10] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete & Computational Geometry, 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[11] |
W. Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, Journal of Algebra and Its Applications, 14 (2015), 1550066.
doi: 10.1142/S0219498815500668. |
[12] |
H. Edelsbrunner, D. Letscher and A. Zomorodian,
Topological persistence and simplification, Discrete & Computational Geometry, 28 (2002), 511-533.
doi: 10.1007/s00454-002-2885-2. |
[13] |
H. Edelsbrunner and J. Harer, Computational Topology, American Mathematical Society, 2010.
doi: 10.1090/mbk/069. |
[14] |
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan, A. Singh and et al.,
Confidence sets for persistence diagrams, The Annals of Statistics, 42 (2014), 2301-2339.
doi: 10.1214/14-AOS1252. |
[15] |
P. J. Franaszczuk and K. J. Blinowska,
Linear model of brain electrical activity? EEG as a superposition of damped oscillatory modes, Biological Cybernetics, 53 (1985), 19-25.
doi: 10.1007/BF00355687. |
[16] |
P. J. Franaszczuk, G. K. Bergey, P. J. Durka and H. M. Eisenberg,
Time-frequency analysis using the matching pursuit algorithm applied to seizures originating from the mesial temporal lobe, Electroencephalography and Clinical Neurophysiology, 106 (1998), 513-521.
doi: 10.1016/S0013-4694(98)00024-8. |
[17] |
S. Gholizadeh and W. Zadrozny, A short survey of topological data analysis in time series and systems analysis, (2018). |
[18] |
R. Ghrist,
Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[19] |
C. Ieracitano, N. Mammone, A. Bramanti, S. Marino, A. Hussain and F. C. Morabito, A time-frequency based machine learning system for brain states classification via eeg signal processing, in International Joint Conference on Neural Networks (IJCNN), (2019), 1–8.
doi: 10.1109/IJCNN.2019.8852240. |
[20] |
F. Khasawneh and E. Munch, Exploring Equilibria in Stochastic Delay Differential Equations Using Persistent Homology, 2014.
doi: 10.1115/DETC2014-35655. |
[21] |
J. F. C. Kingman, Poisson Processes, Oxford Studies in Probability, 3, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.
![]() ![]() |
[22] |
S. G. Mallat and {Z hifeng Zhang},
Matching pursuits with time-frequency dictionaries, IEEE Transactions on Signal Processing, 41 (1993), 3397-3415.
|
[23] |
A. Marchese and V. Maroulas,
Signal classification with a point process distance on the space of persistence diagrams, Advances in Data Analysis and Classification, 12 (2018), 657-682.
doi: 10.1007/s11634-017-0294-x. |
[24] |
V. Maroulas, J. L. Mike and C. Oballe, Nonparametric estimation of probability density functions of random persistence diagrams, Journal of Machine Learning Research, 20 (2019), 1–49. Available from: http://jmlr.org/papers/v20/18-618.html. |
[25] |
V. Maroulas, F. Nasrin and C. Oballe,
A bayesian framework for persistent homology, SIAM Journal on Mathematics of Data Science, 2 (2020), 48-74.
doi: 10.1137/19M1268719. |
[26] |
Y. Mileyko, S. Mukherjee and J. Harer, Probability measures on the space of persistence diagrams, Inverse Problems, 27 (2011), 124007.
doi: 10.1088/0266-5611/27/12/124007. |
[27] |
A. Monod, S. Kalisnik, J. A. Patino-Galindo and L. Crawford, Tropical sufficient statistics for persistent homology, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 337–371.
doi: 10.1137/17M1148037. |
[28] |
F. Nasrin, C. Oballe, D. Boothe and V. Maroulas, Bayesian topological learning for brain state classification, in 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), (2019), 1247–1252. |
[29] |
A. V. Oppenheim, J. R. Buck and R. W. Schafer, Discrete-Time Signal Processing, 2nd edition, Prentice-Hall signal processing, Prentice-Hall, Upper Saddle River, NJ, 1999. Available from: https://cds.cern.ch/record/389969. |
[30] |
J. A. Perea and J. Harer,
Sliding windows and persistence: An application of topological methods to signal analysis, Found. Comput. Math., 15 (2015), 799-838.
doi: 10.1007/s10208-014-9206-z. |
[31] |
R. Pintelon and J. Schoukens,
Time series analysis in the frequency domain, IEEE Transactions on Signal Processing, 47 (1999), 206-210.
|
[32] |
M. Robinson, Topological Signal Processing, Springer, 2014.
doi: 10.1007/978-3-642-36104-3. |
[33] |
M. D. Sacchi, T. J. Ulrych and C. J. Walker,
Interpolation and extrapolation using a high-resolution discrete fourier transform, IEEE Transactions on Signal Processing, 46 (1998), 31-38.
doi: 10.1109/78.651165. |
[34] |
N. Sanderson, E. Shugerman, S. Molnar, J. D. Meiss and E. Bradley, Computational topology techniques for characterizing time-series data, in Advances in Intelligent Data Analysis XVI, Springer International Publishing, (2017), 284–296. |
[35] |
K. F. Swaiman, S. Ashwal and M. I. Shevell, Swaiman's Pediatric Neurology, Elsevier, 2018.
doi: 10.1016/c2013-1-00079-0. |
[36] |
T. Shiraishi, T. Le, H. Kashima and M. Yamada, Topological bayesian optimization with persistence diagrams, preprint, arXiv: 1902.09722. |
[37] |
B. W. Silverman, Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1986. |
[38] |
P. Skraba, V. de Silva and M. Vejdemo-Johansson, Topological analysis of recurrent systems, in NIPS 2012, 2012. |
[39] |
Y. Umeda,
Time series classification via topological data analysis, Transactions of The Japanese Society for Artificial Intelligence, 32 (2017), 1-12.
doi: 10.1527/tjsai.D-G72. |
[40] |
Y. Wang, H. Ombao and M. K. Chung,
Topological data analysis of single-trial electroencephalographic signals, Ann. Appl. Stat., 12 (2018), 1506-1534.
doi: 10.1214/17-AOAS1119. |








Signal Length | 1 Second | 5 Seconds | ||||||||||||||
Signal 1 | 0 | 5.87 | 18.59 | - | 344.80 | 5.37 | 16.6 | - | 0 | 6.00 | 14.4 | 20.85 | 24.98 | 10.54 | 31.64 | 26.97 |
Signal 2 | 0 | 10.70 | - | - | 202.78 | 7.41 | - | - | 0 | 10.16 | 23.02 | - | 17.24 | 4.06 | 20.37 | - |
Signal Length | 1 Second | 5 Seconds | ||||||||||||||
Signal 1 | 0 | 5.87 | 18.59 | - | 344.80 | 5.37 | 16.6 | - | 0 | 6.00 | 14.4 | 20.85 | 24.98 | 10.54 | 31.64 | 26.97 |
Signal 2 | 0 | 10.70 | - | - | 202.78 | 7.41 | - | - | 0 | 10.16 | 23.02 | - | 17.24 | 4.06 | 20.37 | - |
Bayesian | PSD | CWT | ||||
Classifier | Precision | Recall | Precision | Recall | Precision | Recall |
LR | ||||||
SVM - Lin. | ||||||
MLP |
Bayesian | PSD | CWT | ||||
Classifier | Precision | Recall | Precision | Recall | Precision | Recall |
LR | ||||||
SVM - Lin. | ||||||
MLP |
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