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Persistent Homology of Delay Embeddings
Saba Emrani, Thanos Gentimis, Hamid Krim
TL;DR
The paper examines how to detect almost-periodic structure in signals with little prior information. It analyzes delay-coordinate embeddings with persistent homology and selects delay using an autocorrelation-like function; for wheezes, first persistent Betti numbers distinguish the signals from non-wheeze sounds in the reported analysis.
Problem
The paper asks how to detect almost-periodic structure in signals with little prior associated information.
Source: §1 · p. 1Method
The method analyzes delay-coordinate embeddings with persistent homology, uses an autocorrelation-like function to select delay, and applies the approach to wheezes using a harmonic signal model.
Source: §1 · p. 1, §6.2 · p. 12Results
The first Betti number of a wheeze-signal delay embedding is at least 1, while experiments show the first persistent Betti number of a non-wheeze signal is zero.
Source: §3.2 · p. 7Takeaways & Limitations
Subsampling a small number of embedding points reduces computational complexity and is presented as making the algorithm implementable in low-power systems.
Source: §5 · p. 8
Abstract
from arXiv · showhide
The objective of this study is to detect and quantify the periodic behavior of the signals using topological methods. We propose to use delay-coordinate embeddings as a tool to measure the periodicity of signals. Moreover, we use persistent homology for analyzing the structure of point clouds of delay-coordinate embeddings. A method for finding the appropriate value of delay is proposed based on the autocorrelation function of the signals. We apply this topological approach to wheeze signals by introducing a model based on their harmonic characteristics. Wheeze detection is performed using the first Betti numbers of a few number of landmarks chosen from embeddings of the signals.
1 Introduction
The paper proposes persistent-homology analysis of delay embeddings to find almost-periodic structure with little prior information. It applies the approach to wheeze signals, whose harmonic behavior is represented by a time-varying piecewise sinusoidal model.
1 Introduction
- The proposed approach uses an autocorrelation-like function to select delay and persistent homology to analyze delay embeddings for almost-periodic structure.The authors state that it describes global point-cloud structure rather than local geometric behavior and is robust to missing data points.Source: §1 · p. 1
- Using only a small number of points from the point cloud is presented as making the method computationally fast.The paper outlines sections on delay selection, persistent-homology analysis, wheeze-model validation, experiments, and conclusions.Source: §1 · p. 2
- Wheeze signals are modeled as piecewise sinusoids with time-varying frequencies and amplitudes, and are described as lying at ε distance from the model.The authors present wheezes as an illustration of topological analysis of delay embeddings.Source: §1 · p. 2
2 Proposed Framework
The framework embeds scalar signals in two dimensions, selects delay using an autocorrelation-like function, and applies persistent homology to subsampled point clouds.
2.1 Time Delay Embeddings
- A scalar time series is embedded in m dimensions; this study uses m = 2 for detecting almost periodicity and leaves m = 3 frequency estimation to future work.The delay time τ equals the index delay j multiplied by the sampling time Ts.Source: §2.1 · p. 2
2.2 Selecting Time Delay
- The ACL function selects delay between its first and second critical points, avoiding delays that compress the embedding or make adjacent components irrelevant.For non-stationary signals, the function is used strictly for delay selection, not to establish periodicity or detect frequencies.Source: §2.2 · pp. 2–3
2.3 Persistent Homology and Point Cloud Subsampling
- Persistent homology summarizes point-cloud topology with Betti numbers and persistence intervals; the study focuses on the first persistent Betti number for distinguishing wheeze from non-wheeze signals.The pipeline subsamples points and constructs a Rips complex; maxmin gives more evenly spaced subsamples than random selection but has higher computational complexity, and produces a smaller hole in the described point cloud.Source: §2.3 · p. 3
3 Application to Wheeze Detection
The section models monophonic wheezes as continuous piecewise sinusoids and characterizes their delay embeddings. With the delay selected between the first and second critical points, the embeddings have a persistent 1-dimensional hole, and experiments report different first Betti numbers for wheeze and non-wheeze signals.
3.1 Wheeze Modeling
- 0.84% of peak-to-peak value is the average model-to-signal Hausdorff distance for wheezes, compared with 9.7% for non-wheeze signals.The paper says this indicates the proposed model is inadequate to represent normal signals.Source: §3.1 · p. 4
3.2 Topological Characterization of Delay Embeddings
- The analysis generalizes from constant to time-varying amplitude, and phase differences between sinusoids can be ignored in characterizing their delay embeddings.The paper states that phase-shifted sinusoidal delay embedding sets are eventually equal.Source: §3.2 · p. 5
- With constant amplitude, the segment embeddings form concentric ellipses inside one circumscribed square; with varying amplitude, they lie in squares of different side lengths.The ellipses have angles of rotation ±45°.Source: §3.2 · pp. 6–7
- For delays between tc1 and tc2, the embedding is close to concentric ellipses, and its Rips complex has at least one persistent 1-dimensional hole.The ellipses have angles of rotation ±45°, different radii, and different circumscribed-square side lengths.Source: §3.2 · p. 7
- Recorded wheeze embeddings have an average Hausdorff distance δ of 0.0119 from the model embeddings, while barcode distance is bounded by Hausdorff distance.The barcode comparison uses the Wasserstein distance bound stated in the paper.Source: §3.2 · p. 7
- At least 1 first Betti number is reported for wheeze embeddings, versus zero for non-wheeze embeddings.The paper attributes the wheeze result to the model’s close persistent barcodes and reports the non-wheeze result experimentally.Source: §3.2 · p. 7
4 Experimental Results
The approach achieved 98.39% wheeze-detection accuracy, exceeding the reported accuracies of the techniques in and. Experiments used nine recorded breathing sounds and selected delays from the signals’ ACL functions.
4 Experimental Results
- 98.39% accuracy exceeded the 86.2% and 95.5% accuracies reported for techniques in and, respectively.The comparison is reported for the proposed technique and the two cited techniques.Source: §4 · p. 8
- The experiment used nine recorded signals: three normal sounds from different body locations and six wheeze types.Signals were sampled at 44.1 kHz; wheeze intervals were selected by ear and confirmed by experts.Source: §4 · p. 8
- The zero of each signal’s ACL function was selected as the delay; Figure 3 shows ACL functions for non-wheeze and wheeze sounds.The paper notes that a strict autocorrelation function cannot be used for periodicity analysis of these time-varying, non-stationary sounds.Source: §4 · p. 8, Figure 3 · p. 8
5 Conclusion
The conclusion presents persistent homology on delay-coordinate embeddings as a way to capture almost harmonic behavior. It also emphasizes that subsampling reduces computational complexity and supports implementation in low-power systems.
5 Conclusion
- Persistent homology applied to delay-coordinate embeddings is proposed to capture almost harmonic behavior.The method analyzes a point cloud constructed from a signal’s delay-coordinate embedding.Source: §5 · p. 8
- The method selects a very small number of points from the full point cloud to represent the sound signal.This is the subsampling step highlighted in the conclusion.Source: §5 · p. 8
- Subsampling reduces computational complexity, making the algorithm implementable in low-power systems.Source: §5 · p. 8
- Figure 4 presents embeddings for three non-wheeze recording locations and several wheeze types, as well as inspiratory stridor.Source: Figure 4 · p. 9
6 Appendix
The appendix relates delay embeddings of sinusoids to ellipses and persistent bars, then describes a wheeze-signal model built from estimated frequencies and an amplitude envelope.
6.1 Mathematical Proofs
- Figure 6 highlights the significant barcode in red and uses it to distinguish wheeze from non-wheeze breathing sounds.Source: Figure 6 · p. 10
- When τ ≠ kπ, the delay embedding of sin(t) forms an origin-centered ellipse rotated by 45°.The passage also states that the same result holds for delay 2π.Source: §6.1 · p. 11
- For delays between tc1 and tc2, the smallest ellipse yields a significant 1-dimensional persistent bar near βs; experiments favor the second ACL zero.In the discrete case, the bar’s birth is near 0 and its death is at least βs.Source: §6.1 · p. 12
6.2 The model construction
- The wheeze model estimates piecewise-constant frequency intervals from zero-crossing gaps, then scales the resulting sinusoids by an interpolated amplitude envelope.The envelope is interpolated from positive-amplitude critical points.Source: §6.2 · p. 12