# Divergence-Based Segmentation Algorithm for Heavy-Tailed Acoustic Signals with Time-Varying Characteristics

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### Experimental Illustration of the Problem

## 3. Methodology

#### 3.1. Preliminary Study

#### 3.2. The Stable Distribution with Changing Parameters

#### 3.3. Signal Parameters Identification and Modelling

#### 3.4. Divergence Measure

#### 3.5. Segmentation Procedure

- Divide the signal into M segments of length equal to L.
- Estimate the pdf in each segment, ${\widehat{p}}_{1}\left(x\right),\dots ,{\widehat{p}}_{M}\left(x\right)$.
- Estimate the pdf corresponding to the last L samples in the signal, ${\widehat{p}}_{*}\left(x\right)$.
- Calculate the Jeffreys distance between the pdfs in subsequent segments and the pdf of the last L samples,$$\left(\widehat{J}({\widehat{p}}_{1}\left(x\right),{\widehat{p}}_{*}\left(x\right)),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\widehat{J}({\widehat{p}}_{M}\left(x\right),{\widehat{p}}_{*}\left(x\right))\right).$$
- Calculate the increments (differences) of Jeffreys distance,$$\begin{array}{c}\left({D}_{1},\dots ,{D}_{M-1}\right)=\left(\widehat{J}({\widehat{p}}_{2}\left(x\right),{\widehat{p}}_{*}\left(x\right))-\widehat{J}({\widehat{p}}_{1}\left(x\right),{\widehat{p}}_{*}\left(x\right)),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\right.\\ \left.\widehat{J}({\widehat{p}}_{M}\left(x\right),{\widehat{p}}_{*}\left(x\right))-\widehat{J}({\widehat{p}}_{M-1}\left(x\right),{\widehat{p}}_{*}\left(x\right))\right).\end{array}$$
- Use the LLR method to find the index ${i}^{*}$ in $1,\dots ,M-1$ corresponding to the change in scale of $\left({D}_{1},\dots ,{D}_{M-1}\right)$.
- Consider the sub-samples $\left({D}_{1},\dots ,{D}_{{i}^{*}-1}\right)$ and $\left({D}_{{i}^{*}},\dots ,{D}_{M-1}\right)$. Use the LLR method to find the indexes ${i}^{**}$ and ${i}^{***}$ in $1,\dots ,{i}^{*}-1$ and in ${i}^{*},\dots ,M-1$, respectively, corresponding to changes in scale.

Algorithm 1: Segmentation procedure based on Jeffreys distance |

Data: Input dataDivide data into M segments of length equal to L: Segment 1, Segment 2, …, Segment MEstimate the pdf in each segment:
_{1}**for**$k\leftarrow 1$**to**M**do**_{2}- $\lfloor $ Calculate ${\widehat{p}}_{k}\left(x\right)$ for Segment k
Estimate the pdf of the last L samples in the signal: ${\widehat{p}}_{*}\left(x\right)$Calculate the Jeffreys distance:
_{3}**for**$k\leftarrow 1$**to**M**do**_{4}- $\lfloor $ Calculate $\widehat{J}({\widehat{p}}_{k}\left(x\right),{\widehat{p}}_{*}\left(x\right))$
Calculate the increments of Jeffreys distance:
_{5}**for**$k\leftarrow 1$**to**$M-1$**do**_{6}- $\lfloor $ Calculate ${D}_{k}=\widehat{J}({\widehat{p}}_{k+1}\left(x\right),{\widehat{p}}_{*}\left(x\right))-\widehat{J}({\widehat{p}}_{k}\left(x\right),{\widehat{p}}_{*}\left(x\right))$
Use the LLR method to find change in scale in $\left({D}_{1},\dots ,{D}_{M-1}\right)$: ${i}^{*}$Use the LLR method to find change in scale in $\left({D}_{1},\dots ,{D}_{{i}^{*}-1}\right)$: ${i}^{**}$Use the LLR method to find change in scale in $\left({D}_{1},\dots ,{D}_{{i}^{*}-1}\right)$: ${i}^{***}$ |

## 4. Results

## 5. Simulations

## 6. Discussion and Validation of the Procedure

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cho, H.; Fryzlewicz, P. Multiple-change-point detection for high dimensional time series via sparsified binary segmentation. J. R. Stat. Soc. Ser. B Stat. Methodol.
**2015**, 77, 475–507. [Google Scholar] [CrossRef] [Green Version] - Raphael, C. Automatic segmentation of acoustic musical signals using Hidden Markov Models. IEEE Trans. Pattern Anal. Mach. Intell.
**1999**, 21, 360–370. [Google Scholar] [CrossRef] [Green Version] - Andre-Obrecht, R. A New Statistical Approach for the Automatic Segmentation of Continuous Speech Signals. IEEE Trans. Acoust. Speech, Signal Process.
**1988**, 36, 29–40. [Google Scholar] [CrossRef] - Zimroz, R.; Madziarz, M.; Żak, G.; Wyłomańska, A.; Obuchowski, J. Seismic signal segmentation procedure using time-frequency decomposition and statistical modelling. J. Vibroeng.
**2015**, 17, 3111–3120. [Google Scholar] - Popescu, T. Signal segmentation using changing regression models with application in seismic engineering. Digit. Signal Process. A Rev. J.
**2014**, 24, 14–26. [Google Scholar] [CrossRef] - Gaby, J.; Anderson, K. Hierarchical segmentation of seismic waveforms using affinity. Geoexploration
**1984**, 23, 1–16. [Google Scholar] [CrossRef] - Kucharczyk, D.; Wyłomanska, A.; Obuchowski, J.; Zimroz, R.; Madziarz, M. Stochastic Modelling as a Tool for Seismic Signals Segmentation. Shock Vib.
**2016**, 2016, 8453426. [Google Scholar] [CrossRef] [Green Version] - Chen, C.H. On a segmentation algorithm for seismic signal analysis. Geoexploration
**1984**, 23, 35–40. [Google Scholar] [CrossRef] - Chiaruttini, C.; Roberto, V. Automation of seismic network signal interpolation: An artificial intelligence approach. Geoexploration
**1988**, 11, 327–338. [Google Scholar] [CrossRef] - Theodor, P.; Cioboată, D. Performance Evaluation of Some Change Detection and Data Segmentation Algorithms. Int. J. Math. Comput. Methods
**2016**, 1, 236. [Google Scholar] - Pikoulis, E.V.; Psarakis, E. A New Automatic Method for Seismic Signals Segmentation. In Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, Japan, 25–30 March 2012. [Google Scholar]
- Vaezi, Y.; Van der Baan, M. Comparison of the STA/LTA and power spectral density methods for microseismic event detection. Geophys. J. Int.
**2015**, 203, 1896–1908. [Google Scholar] [CrossRef] [Green Version] - Lopatka, M.; Laplanche, C.; Adam, O.; Motsch, J.; Zarzycki, J. Non-stationary time-series segmentation based on the Schur prediction error analysis. In Proceedings of the IEEE/SP 13th Workshop on Statistical Signal Processing, Bordeaux, France, 17–20 July 2005; Volume 2005, pp. 251–255. [Google Scholar]
- Makowski, R.; Hossa, R. Automatic speech signal segmentation based on the innovation adaptive filter. Int. J. Appl. Math. Comput. Sci.
**2014**, 24, 259–270. [Google Scholar] [CrossRef] [Green Version] - Khanagha, V.; Daoudi, K.; Pont, O.; Yahia, H. Phonetic segmentation of speech signal using local singularity analysis. Digit. Signal Process. A Rev. J.
**2014**, 35, 86–94. [Google Scholar] [CrossRef] [Green Version] - Micó, P.; Mora, M.; Cuesta-Frau, D.; Aboy, M. Automatic segmentation of long-term ECG signals corrupted with broadband noise based on sample entropy. Comput. Methods Programs Biomed.
**2010**, 98, 118–129. [Google Scholar] [CrossRef] - Kucharczyk, D.; Wyłomańska, A.; Zimroz, R. Structural break detection method based on the Adaptive Regression Splines technique. Phys. A Stat. Mech. Its Appl.
**2017**, 471, 499–511. [Google Scholar] [CrossRef] [Green Version] - Montiel, D.; Cang, H.; Yang, H. Quantitative Characterization of Changes in Dynamical Behavior for Single-Particle Tracking Studies. J. Phys. Chem.
**2006**, 110, 19763–19770. [Google Scholar] [CrossRef] - Estarellas, C.; Serra, L. Resonant Anderson localization in segmented wires. Phys. Rev. E
**2016**, 93, 32–105. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mordant, N.; Crawford, A.; Bodenschatz, E. Three-Dimensional Structure of the Lagrangian Acceleration in Turbulent Flows. Phys. Rev. Lett.
**2004**, 93, 214–501. [Google Scholar] [CrossRef] [Green Version] - Han, Z.; Chen, H.; Yan, T.; Jiang, G. Time Series Segmentation to Discover Behavior Switching in Complex Physical Systems. In Proceedings of the IEEE International Conference on Data Mining, Atlantic City, NJ, USA, 14–17 November 2015; pp. 161–170. [Google Scholar]
- Gajda, J.; Sikora, G.; Wyłomańska, A. Regime variance testing - a quantile approach. Acta Phys. Pol. B
**2013**, 44, 1015–1035. [Google Scholar] [CrossRef] [Green Version] - Zhang, C.; Gao, F.; Zhang, R. Segmentation algorithm for DNA sequences. Phys. Rev. E
**2005**, 72, 041917. [Google Scholar] [CrossRef] - Sippl, M. Calculation of conformational ensembles from potentials of mean force. An approach to the knowledge-based prediction of local structures in globular proteins. J. Mol. Biol.
**1990**, 213, 859–883. [Google Scholar] [CrossRef] - Camargo, S.; Duarte Queirós, S.M.; Anteneodo, C. Nonparametric segmentation of nonstationary time series. Phys. Rev. E
**2011**, 84, 046702. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jamali, S.; Jönssonv, P.; Eklundh, L.; Ardö, J.; Seaquist, J. Detecting changes in vegetation trends using time series segmentation. Remote Sens. Environ.
**2015**, 156, 182–195. [Google Scholar] [CrossRef] - Omranian, N.; Mueller-Roeber, B.; Nikoloski, Z. Segmentation of biological multivariate time-series data. Sci. Rep.
**2015**, 5, 1–6. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Puchalski, A.; Komorska, I. Stable distributions, generalised entropy, and fractal diagnostic models of mechanical vibration signals. Diagnostyka
**2017**, 18, 103–110. [Google Scholar] - Wyłomańska, A.; Zimroz, R. Signal segmentation for operational regimes detection of heavy duty mining mobile machines—A statistical approach. Diagnostyka
**2014**, 15, 33–42. [Google Scholar] - Gąsior, K.; Urbańska, H.; Grzesiek, A.; Zimroz, R.; Wyłomańska, A. Identification, decomposition and segmentation of impulsive vibration signals with deterministic components—A sieving screen case study. Sensors
**2020**, 20, 5648. [Google Scholar] [CrossRef] - Michalak, A.; Wodecki, J.; Drozda, M.; Wyłomańska, A.; Zimroz, R. Model of the vibration signal of the vibrating sieving screen suspension for condition monitoring purposes. Sensors
**2021**, 21, 213. [Google Scholar] [CrossRef] - Crossman, J.; Guo, H.; Murphey, Y.; Cardillo, J. Automotive signal fault diagnostics—Part I: Signal fault analysis, signal segmentation, feature extraction and quasi-optimal feature selection. IEEE Trans. Veh. Technol.
**2003**, 52, 1063–1075. [Google Scholar] [CrossRef] - Douglas, H.; Zamba, K. Statistical Process Control for Shifts in Mean or Variance Using a Changepoint Formulation. Technometrics
**2005**, 47, 164–173. [Google Scholar] - Naruse, Y.; Takiyama, K.; Okada, M.; Umehara, H. Statistical method for detecting phase shifts in alpha rhythm from human electroencephalogram data. Phys. Rev. E
**2013**, 87, 042708. [Google Scholar] [CrossRef] [PubMed] - Niu, Y.S.; Zhang, H. The screening and ranking algorithm to detect DNA copy number variations. Ann. Appl. Stat.
**2012**, 6, 1306–1326. [Google Scholar] [CrossRef] [Green Version] - Fryzlewicz, P. Wild binary segmentation for multiple change-point detection. Ann. Appl. Stat.
**2014**, 42, 2243–2281. [Google Scholar] [CrossRef] - Grzesiek, A.; Zimroz, R.; Sliwinski, P.; Gomolla, N.; Wyłomańska, A. Long term belt conveyor gearbox temperature data analysis – Statistical tests for anomaly detection. Measurement
**2020**, 165, 108124. [Google Scholar] [CrossRef] - Zhou, C.; Zou, C.; Zhang, Y.; Wang, Z. Nonparametric control chart based on change-pointmodel. Stat. Pap.
**2009**, 50, 13–28. [Google Scholar] [CrossRef] - Jewell, S.; Fearnhead, P.; Witten, D. Testing for a Change in Mean After Changepoint Detection. arXiv
**2021**, arXiv:1910.04291. [Google Scholar] - Wenger, K.; Leschinski, C.; Sibbertsen, P. Change-in-mean tests in long-memory time series: A review of recent developments. AStA Adv. Stat. Anal.
**2019**, 103, 237–256. [Google Scholar] [CrossRef] [Green Version] - Inclan, C.; Tiao, G. Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. J. Am. Stat. Assoc.
**1994**, 89, 913–923. [Google Scholar] - Chen, J.; Gupta, A. Change point analysis of a Gaussian model. Stat. Pap.
**1999**, 40, 323–333. [Google Scholar] [CrossRef] - Killick, R.; Fearnhead, P.; Eckley, I.A. Optimal Detection of Changepoints With a Linear Computational Cost. J. Am. Stat. Assoc.
**2012**, 107, 1590–1598. [Google Scholar] [CrossRef] - Gabbanini, F.; Vannucci, M.; Bartoli, G.; Moro, A. Wavelet Packet Methods for the Analysis of Variance of Time Series With Application to Crack Widths on the Brunelleschi Dome. J. Comput. Graph. Stat.
**2004**, 13, 639–658. [Google Scholar] [CrossRef] [Green Version] - Whitcher, B.; Byers, S.; Guttorp, P.; Percival, D. Testing for homogeneity of variance in time series: Long memory, wavelets, and the Nile River. Water Resour. Res.
**2002**, 38, 12–16. [Google Scholar] [CrossRef] [Green Version] - Hawkins, D.M.; Zamba, K.D. A Change-Point Model for a Shift in Variance. J. Qual. Technol.
**2005**, 37, 21–31. [Google Scholar] [CrossRef] - Jong-Min, K.; Jaiwook, B.; Mitch, R. Detecting the Change of Variance by Using Conditional Distribution with Diverse Copula Functions; Springer: Singapore, 2018; pp. 145–154. [Google Scholar]
- Qin, R.; Ma, J. An efficient algorithm to estimate the change in variance. Econ. Lett.
**2018**, 168, 15–17. [Google Scholar] [CrossRef] - Sánchez-Pérez, L.; Sánchez-Fernández, L.; Suárez-Guerra, S.; Carbajal-Hernández, J. Aircraft class identification based on take-off noise signal segmentation in time. Expert Syst. Appl.
**2013**, 40, 5148–5159. [Google Scholar] [CrossRef] - Tóth, B.; Lillo, F.; Farmer, J. Segmentation algorithm for non-stationary compound Poisson processes: With an application to inventory time series of market members in a financial market. Eur. Phys. J. B
**2010**, 78, 235–243. [Google Scholar] [CrossRef] - Lee, S.X.; McLachlan, G.J. Scale Mixture Distribution. In Wiley StatsRef: Statistics Reference Online; American Cancer Society: Atlanta, GA, USA, 2019; pp. 1–16. [Google Scholar]
- Charytoniuk, W.; Nazarko, J. An application of compound probability distributions to electric load modeling. Stoch. Anal. Appl.
**1994**, 12, 31–40. [Google Scholar] [CrossRef] - Willmot, G.E.; Lin, X.S. Compound distributions. In Lundberg Approximations for Compound Distributions with Insurance Applications; Springer: New York, NY, USA, 2001; pp. 51–80. [Google Scholar]
- Forbes, F.; Wraith, D. A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: Application to robust clustering. Stat. Comput.
**2014**, 24, 971–984. [Google Scholar] [CrossRef] - Kim, J.; Mahmassani, H.S. Compound Gamma representation for modeling travel time variability in a traffic network. Transp. Res. Part B Methodol.
**2015**, 80, 40–63. [Google Scholar] [CrossRef] [Green Version] - Maxwell, O.; Oyamakin, S.; Chukwu, A. On making an informed decision between four exponential-based continuous compound distributions. J. Adv. Appl. Math.
**2019**, 4, 75–81. [Google Scholar] [CrossRef] - Afuecheta, E.; Semeyutin, A.; Chan, S.; Nadarajah, S.; Andrés Pérez Ruiz, D. Compound distributions for financial returns. PLoS ONE
**2020**, 15, 1–25. [Google Scholar] [CrossRef] - Andel, J. Autoregressive series with random parameters. Math. Oper. Und Stat.
**1976**, 7, 735–741. [Google Scholar] [CrossRef] - Liu, L.; Tiao, G.C. Random coefficient first-order autoregressive models. J. Econom.
**1980**, 13, 305–325. [Google Scholar] [CrossRef] - Nicholls, D.F.; Quinn, B.G. The estimation of random coefficient autoregressive models. I. J. Time Ser. Anal.
**1980**, 1, 37–46. [Google Scholar] [CrossRef] - Nicholls, D.; Quinn, B. Random Coefficient Autoregressive Models: An Introduction; Lecture Notes in Statistics; Springer: New York, NY, USA, 1982. [Google Scholar]
- Regis, M.; Serra, P.; van den Heuvel, E.R. Random autoregressive models: A structured overview. Econom. Rev.
**2021**, 1–24. [Google Scholar] [CrossRef] - Cox, D.R. Some Statistical Methods Connected with Series of Events. J. R. Stat. Soc. Ser. B (Methodological)
**1955**, 17, 129–157. [Google Scholar] [CrossRef] - Lando, D. On Cox processes and credit risky securities. Rev. Deriv. Res.
**1998**, 2, 99–120. [Google Scholar] [CrossRef] - Krumin, M.; Shoham, S. Generation of Spike Trains with Controlled Auto- and Cross-Correlation Functions. Neural Comput.
**2009**, 21, 1642–1664. [Google Scholar] [CrossRef] - Chubynsky, M.V.; Slater, G.W. Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion. Phys. Rev. Lett.
**2014**, 113, 098302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jain, R.; Sebastian, K. Diffusing diffusivity: A new derivation and comparison with simulations. J. Chem. Sci.
**2017**, 129, 929–937. [Google Scholar] [CrossRef] - Wang, W.; Cherstvy, A.G.; Chechkin, A.V.; Thapa, S.; Seno, F.; Liu, X.; Metzler, R. Fractional Brownian motion with random diffusivity: Emerging residual nonergodicity below the correlation time. J. Phys. A Math. Theor.
**2020**, 53, 474001. [Google Scholar] [CrossRef] - Lévy, P. Calcul des Probabilites; Gauthier-Villars: Paris, France, 1925. [Google Scholar]
- Khinchine, A.Y.; Lévy, P. Sur les lois stables. CR Acad. Sci. Paris
**1936**, 202, 374–376. [Google Scholar] - Weron, A. Stable processes and measures: A survey. In Probability Theory on Vector Spaces III; Springer: Berlin/Heidelberg, Germany, 1984; pp. 306–364. [Google Scholar]
- Zolotarev, V. One-dimensional Stable Distributions; Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 1986. [Google Scholar]
- Janicki, A.; Weron, A. Simulation and Chaotic Behavior of α-Stable Stochastic Processes; Chapman & Hall/CRC Pure and Applied Mathematics; Taylor & Francis: Abingdon, UK, 1993. [Google Scholar]
- Samorodnitsky, G.; Taqqu, M. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance; Chapman and Hall: London, UK, 1994. [Google Scholar]
- Nikias, C.; Shao, M. Signal Processing with Alpha-Stable Distributions and Applications; Adaptive and Cognitive Dynamic Systems: Signal Processing, Learning, Communications and Control; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Nolan, J.P. Stable Distributions - Models for Heavy Tailed Data; Birkhauser: Boston, MA, USA, 2018. [Google Scholar]
- Burnecki, K.; Wyłomańska, A.; Beletskii, A.; Gonchar, V.; Chechkin, A. Recognition of stable distribution with Lévy index α close to 2. Phys. Rev. E
**2012**, 85, 056711. [Google Scholar] [CrossRef] [Green Version] - Koutrouvelis, I.A. Regression-type estimation of the parameters of stable laws. J. Am. Stat. Assoc.
**1980**, 75, 918–928. [Google Scholar] [CrossRef] - Csiszár, I. Information-Type Measures of Difference of Probability Distributions and Indirect Observations. Stud. Sci. Math. Hung.
**1967**, 2, 299–318. [Google Scholar] - Csiszár, I. I-Divergence Geometry of Probability Distributions and Minimization Problem. Ann. Probab.
**1975**, 3, 146–158. [Google Scholar] [CrossRef] - Basseville, M. Distance measures for signal processing and pattern recognition. Signal Process.
**1989**, 18, 349–369. [Google Scholar] [CrossRef] [Green Version] - Basseville, M. Divergence measures for statistical data processing—An annotated bibliography. Signal Process.
**2013**, 93, 621–633. [Google Scholar] [CrossRef] - Chung, J.; Kannappan, P.; Ng, C.; Sahoo, P. Measures of distance between probability distributions. J. Math. Anal. Appl.
**1989**, 138, 280–292. [Google Scholar] [CrossRef] [Green Version] - Hill, P.D. Kernel estimation of a distribution function. Commun. Stat. Theory Methods
**1985**, 14, 605–620. [Google Scholar] - Bowman, A.W.; Azzalini, A. Applied Smoothing Techniques for Data Analysis; Oxford University Press Inc.: New York, NY, USA, 1997. [Google Scholar]
- Silverman, B. Density Estimation: For Statistics and Data Analysis; Chapman & Hall: London, UK, 1986. [Google Scholar]
- Horvath, L. The Maximum Likelihood Method for Testing Changes in the Parameters of Normal Observations. Ann. Stat.
**1993**, 21, 671–680. [Google Scholar] [CrossRef]

**Figure 3.**Examples of signals after the pre-processing step. Panel (

**a**) corresponds to Signal 1 and panel (

**b**) corresponds to Signal 2.

**Figure 7.**Jeffreys distance (panel (

**a**)) and differences of Jeffreys distance (panel (

**b**)) comparing the pdfs in the moving window of length 2500 with step equal to 250 to the pdf corresponding to the last window for Signal 1. Detected regime change points are marked in purple (dotted line), red (solid line), and yellow (dashed line).

**Figure 8.**Jeffreys distance (panel (

**a**)) and differences of Jeffreys distance (panel (

**b**)) comparing the pdfs in the moving window of length 2500 with step equal to 250 to the the pdf corresponding to the last window for Signal 2. Detected regime change points are marked in purple (dotted line), red (solid line) and yellow (dashed line).

**Figure 9.**Signals with marked regime changes. The first change is visible as a dotted purple line, second: a solid red line, and the last one is a dashed yellow line. Panel (

**a**) corresponds to Signal 1 and panel (

**b**) corresponds to Signal 2.

**Figure 10.**Estimated values of $\alpha $ on panel (

**a**) and $\sigma $ on panel (

**b**) in the subsequent segments of length 2500 for Signal 1. Fitted deterministic functions are marked in red.

**Figure 11.**Values of $\alpha $ (panel (

**a**)) and $\sigma $ (panel (

**b**)) in the subsequent segments of length 2500, the obtained simulated signal (panel (

**c**)), and the corresponding probability density map (panel (

**d**)).

**Figure 12.**Estimated parameters of the stable distribution for sample simulated signal presented in panel (

**c**) of Figure 11. Panels (

**a**), (

**b**), (

**c**) and (

**d**) correspond to $\alpha $, $\sigma $, $\beta $ and $\mu $, respectively.

**Figure 13.**Jeffreys distance (panel (

**a**)) and differences of Jeffreys distance (panel (

**b**)) comparing the pdfs in the moving window of length 2500 with step equal to 250 to the the pdf corresponding to the last window. Detected regime change points are marked in purple (dotted line), red (solid line), and yellow (dashed line) and the theoretical moments of $\sigma $ and $\alpha $ stabilization are marked with black dots.

**Figure 14.**Simulated signal with regime change points marked in purple (dotted line), red (solid line), and yellow (dashed line) and the theoretical moments of $\sigma $ and $\alpha $ stabilization marked with black dots.

**Figure 15.**Boxplots presenting the results of the Monte Carlo simulation study, i.e., the identified moments of the first (purple), the second (red), and the third (yellow) regime changes for 100 simulated signals. Panel (

**a**) corresponds to the procedure applied to non-overlapping windows of length 2500 (with step equal to 2500) and panels (

**b**) and (

**c**) correspond to the overlapping windows of length 2500 with step equal to 500 and 250, respectively. Black dash lines for the second and the third boxplot indicate the theoretical moments of $\sigma $ and $\alpha $ stabilization, respectively.

**Table 1.**Detected regime change points (in seconds) for all eight signals for different values of step at which the window of length 2500 is moved.

Signal Number | Step | Purple Point | Red Point | Yellow Point |
---|---|---|---|---|

1 | 2500 | $11.8481$ | $18.9909$ | $25.5669$ |

500 | $12.4376$ | $18.7755$ | $25.4989$ | |

250 | $12.437$ | $18.9850$ | $25.4889$ | |

2 | 2500 | $12.6984$ | $19.1043$ | $24.7732$ |

500 | $14.2857$ | $19.1156$ | $24.7392$ | |

250 | $14.2914$ | $19.1156$ | $24.7449$ | |

3 | 2500 | $15.8163$ | $18.7642$ | $22.1088$ |

500 | $11.0544$ | $16.9841$ | $22.0295$ | |

250 | $11.0658$ | $17.0465$ | $21.9728$ | |

4 | 2500 | $12.8685$ | $19.7279$ | $25.6236$ |

500 | $12.8458$ | $18.2880$ | $25.5556$ | |

250 | $12.8231$ | $18.4751$ | $25.5612$ | |

5 | 2500 | $9.3537$ | $15.7029$ | $21.9388$ |

500 | $9.3651$ | $15.9864$ | $21.9728$ | |

250 | $9.3594$ | $16.0431$ | $22.0125$ | |

6 | 2500 | $15.0794$ | $18.7642$ | $24.8299$ |

500 | $15.1020$ | $18.7642$ | $24.5896$ | |

250 | $15.0794$ | $18.7642$ | $25.7959$ | |

7 | 2500 | $10.9410$ | $21.5986$ | $29.3084$ |

500 | $14.6825$ | $21.3605$ | $24.6145$ | |

250 | $14.6825$ | $21.3719$ | $25.1304$ | |

8 | 2500 | $11.2245$ | $15.8163$ | $21.3152$ |

500 | $12.3243$ | $15.8050$ | $20.8617$ | |

250 | $12.3186$ | $15.7937$ | $20.8560$ |

**Table 2.**Medians, intequartile ranges (IQR), and $80\%$ quantile intervals calculated based on the identified moments of the first (purple), the second (red), and the third (yellow) regime changes for $M=100$ simulated signals for the procedure applied to non-overlapping windows of length 2500 with step equal to 2500 and to the overlapping windows of length 2500 with step equal to 500 and 250.

Purple Point | Red Point | Yellow Point | |
---|---|---|---|

theoretical value | - | 875,000 | 1,125,000 |

step = 2500 | |||

${Q}_{0.50}$ | 500,000 | 875,000 | 1,120,000 |

$IQR$ | 126,250 | 2500 | 15,000 |

$[{Q}_{0.10};{Q}_{0.90}]$ | [395,000; 610,000] | [872,500; 883,750] | [1,093,750; 1,131,250] |

step = 500 | |||

${Q}_{0.50}$ | 450,000 | 874,500 | 1,118,500 |

$IQR$ | 71,250 | 1500 | 21,250 |

$[{Q}_{0.10};{Q}_{0.90}]$ | [390,250; 523,250] | [870,750; 875,500] | [1,079,250;1,140,250] |

step = 250 | |||

${Q}_{0.50}$ | 425,750 | 874,250 | 1,120,750 |

$IQR$ | 54,750 | 2000 | 15,625 |

$[{Q}_{0.10};{Q}_{0.90}]$ | [378,125; 478,000] | [869,000; 875,000] | [1,090,125; 1,128,625] |

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**MDPI and ACS Style**

Grzesiek, A.; Gąsior, K.; Wyłomańska, A.; Zimroz, R.
Divergence-Based Segmentation Algorithm for Heavy-Tailed Acoustic Signals with Time-Varying Characteristics. *Sensors* **2021**, *21*, 8487.
https://doi.org/10.3390/s21248487

**AMA Style**

Grzesiek A, Gąsior K, Wyłomańska A, Zimroz R.
Divergence-Based Segmentation Algorithm for Heavy-Tailed Acoustic Signals with Time-Varying Characteristics. *Sensors*. 2021; 21(24):8487.
https://doi.org/10.3390/s21248487

**Chicago/Turabian Style**

Grzesiek, Aleksandra, Karolina Gąsior, Agnieszka Wyłomańska, and Radosław Zimroz.
2021. "Divergence-Based Segmentation Algorithm for Heavy-Tailed Acoustic Signals with Time-Varying Characteristics" *Sensors* 21, no. 24: 8487.
https://doi.org/10.3390/s21248487