# A Sequential Algorithm for Signal Segmentation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bayesian Model for Power Switch

## 3. The Sequential Segmentation Algorithm

**seg**($y\in {\Re}^{N}$,${n}_{min}$) as:

- 1
**If**$N<{n}_{min}$, stop, returning the empty vector $t=\left[\right]$;- 2
**Else**- (a)
- Obtain the MAP estimate $\overline{t}$;
- (b)
- Define ${y}^{1}={y}_{1,\dots ,\overline{t}}$ and ${y}^{2}={y}_{\overline{t}+1,\dots ,N}$;
- (c)
- (Stopping criterion)
**If**$var({y}^{1})=var({y}^{2})$, stop, returning the empty vector $t=\left[\right]$; - (d)
**Else**return the concatenated vector $[seg({y}^{1},{n}_{min})\phantom{\rule{1.em}{0ex}}\overline{t}\phantom{\rule{1.em}{0ex}}\overline{t}+seg({y}^{2},{n}_{min})]$.

**seg**appearing in the last line refers to the function itself; our algorithm, then, is of a recursive nature.

- Obtain ${s}_{0}=su{p}_{{\Theta}_{0}}P({\sigma}_{0},\delta |y)$;
- Obtain the evidence favoring ${H}_{0}$ : $ev({H}_{0},y)=1-{\int}_{T(y)}P({\sigma}_{0},\delta |y)d{\sigma}_{0}d\delta $;
**If**$ev({H}_{0},y)<{\alpha}_{min}$, return 1 ($var({y}^{1})\ne var({y}^{2})$);**else**return 0.

## 4. Results: Simulated Signal

^{®}with little concern for computational performance. In particular, no parallelization strategy was adopted, and both steps are strongly parallelizable. With this issue in mind, we are working on a new version of the algorithm implemented in Python and parallelized using the multiprocessing library; we expect this new version to sensibly improve the computational performance of our algorithm.

## 5. Results: OceanPod Samples

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Etter, P.C. Underwater Acoustic Modeling and Simulation; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Brethorst, L. Bayesian Spectrum Analysis and Parameter Estimation; Springer: New York, NY, USA, 1988. [Google Scholar]
- Jaynes, E. Bayesian Spectrum and Chirp Analysis. In Maximum-Entropy and Bayesian Spectral Analysis and Estimation Problems; Smith, C.R., Erickson, G.J., Eds.; D. Reidel Publishing Co.: Dordrecht, Nederlanden, 1987. [Google Scholar]
- Ruanaidh, J.J.K.; Fitzgerald, W.J. Numerical Bayesian Methods Applied to Signal Processing; Springer: New York, NY, USA, 1996. [Google Scholar]
- Bretthorst, L. Bayesian Analysis. I. Parameter Estimation Using Quadrature NMR Models. J. Magn. Reson.
**1990**, 88, 533–551. [Google Scholar] [CrossRef] - Bretthorst, L. Bayesian Analysis. II. Signal Detection and Model Selection. J. Magn. Reson.
**1990**, 88, 552–570. [Google Scholar] [CrossRef] - Bretthorst, L. Bayesian Analysis. III. Applications to NMR Signal Detection, Model Selection and Parameter Estimation. J. Magn. Reson.
**1990**, 88, 571–595. [Google Scholar] [CrossRef] - Hubert, P.; Padovese, L.R.; Stern, J.M. Full bayesian approach for signal detection with an application to boat detection on underwater soundscape data. In Maximum Entropy Methods in Science and Engineering; Springer-Verlag: New York, NY, USA, 2017. [Google Scholar]
- Makowsky, 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] - Ukil, A.; Zivanovic, R. Automatic Signal Segmentation based on Abrupt Change Detection for Power Systems Applications. In Proceedings of the Power India Conference, New Delhi, India, 10–12 April 2006. [Google Scholar]
- Schwartzman, A.; Gavrilov, Y.; Adler, R.J. Multiple Testing of Local Maxima for Detection of Peaks in 1D. Ann. Stat.
**2011**, 39, 3290–3319. [Google Scholar] [CrossRef] [PubMed] - Kuntamalla, S.; Reddy, L.R.G. An Efficient and Automatic Systolic Peak Detection Algorithm for Photoplethysmographic Signals. Int. J. Comput. Appl.
**2014**, 97, 19. [Google Scholar] [CrossRef] - Thedorou, T.; Mporas, I.; Fakotakis, N. An Overview of Automatic Audio Segmentation. Int. J. Inf. Technol. Comput. Sci.
**2014**, 11, 1–9. [Google Scholar] [CrossRef] - Jaynes, E.T. On the Rationale of Maximum-Entropy Methods. IEEE Proc.
**1982**, 70, 939–952. [Google Scholar] [CrossRef] - Jaynes, E.T. Prior Probabilities. IEEE Trans. Syst. Sci. Cybern.
**1968**, 4, 227–241. [Google Scholar] [CrossRef] - Jeffreys, H. An Invariant Form for the Prior Probability in Estimation Problems. In Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences; The Royal Society: London, UK, 1946; Volume 186, pp. 453–461. [Google Scholar]
- Good, I.J. The Bayes/Non-bayes compromise: A Brief Review. J. Am. Stat. Assoc.
**1992**, 87, 597–606. [Google Scholar] [CrossRef] - Perez, M.; Pericchi, L.R. Changing Statistical Significance with the Amount of Information: The Adaptative α Significance Level. Stat. Probab. Lett.
**2014**, 85, 20–24. [Google Scholar] [CrossRef] [PubMed] - Pereira, C.A.B.; Wechsler, S. On the Concept of P-value. Revis. Bras. Probab. Estat.
**1993**, 7, 159–177. [Google Scholar] - Pereira, C.A.B.; Stern, J.M. Evidence and credibility: Full Bayesian significance test for precise hypotheses. Entropy
**1999**, 1, 99–110. [Google Scholar] [CrossRef] - Chakrabarty, D. A New Bayesian Test to Test for the Intractability-Countering Hypothesis. JASA
**2017**, 112, 561–577. [Google Scholar] [CrossRef] - Hubert, P.; Lauretto, M.; Stern, J.M. FBST for Generalized Poisson Distributions. AIP Conf. Procee.
**2009**, 1193, 210–217. [Google Scholar] - Lauretto, M.S.; Nakano, F.; Faria, S.R.; Pereira, C.A.B.; Stern, J.M. A Straightforward Multiallelic Significance Test for the Hardy-Weinberg Equilibrium Law. Genet. Mol. Biol.
**2009**, 32, 619–625. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stern, J.M.; Zacks, S. Testing the Independence of Poisson Variates Under the Holgate Bivariate Distribution: The Power of a New Evidence Test. Stat. Probab. Lett.
**2002**, 60, 313–320. [Google Scholar] [CrossRef] - Palshikar, G.K. Simple Algorithms for Peak Detection in Time Series. In Proceedings of the 1st International Conference Advanced Data Analysis, Business Analytics and Intelligence, Ahmedabad, India, 6–7 June 2009. [Google Scholar]
- Chiachio, M.; Beck, J.M.; Chiachio, J.; Rus, G. Approximate Bayesian Computation by Subset Simulation. SIAM J. Sci. Comput.
**2014**, 36, A1339–A1358. [Google Scholar] [CrossRef] - Beck, J.L.; Zuev, K.M. Asymptotically independent Markov Sampling: A new MCMC scheme for Bayesian inference. In Proceedings of the Second International Conference on Vulnerability and Risk Analysis and Management (ICVRAM) and the Sixth International Symposium on Uncertainty Modeling and Analysis (ISUMA), Liverpool, UK, 13–16 July 2014. [Google Scholar]
- Beck, J.L.; Taflanidis, A. Prior and Posterior Robust Stochastic Predictions for Dynamical Systems Using Probability Logic. Int. J. Uncertain. Quantif.
**2014**, 3, 271–288. [Google Scholar] [CrossRef] - Chiachio, J.; Bochud, N.; Chiachio, M.; Cantero, S.; Rus, G. A Multilevel Bayesian Method for Ultrasound-Based Damage Identification in Composites Laminates. Mech. Syst. Signal Process.
**2017**, 88, 462–477. [Google Scholar] [CrossRef]

Algorithm | Parameters | Elapsed Time (s) | # of Segments | First Cutting Point | Last Cutting Point |
---|---|---|---|---|---|

SeqSeg | $\beta =1,\alpha =0.01$ | 9.8912 | 5 | 4990 | 15,001 |

SeqSeg | $\beta =0.01,\alpha =0.01$ | 11.8567 | 5 | 4990 | 15,001 |

SeqSeg | $\beta =1,\alpha =0.1$ | 12.7605 | 5 | 4990 | 15,001 |

SeqSeg | $\beta =0.01,\alpha =0.1$ | 11.5746 | 5 | 4990 | 15,001 |

Palshikar ${S}_{1}$ | $h=3,k=100$ | 0.3116 | 31 | 4778 | 14,852 |

Palshikar ${S}_{2}$ | $h=3,k=100$ | 0.3303 | 13 | 5039 | 14,470 |

Palshikar ${S}_{3}$ | $h=3,k=100$ | 0.3443 | 9 | 1608 | 14,249 |

Palshikar ${S}_{1}$ | $h=3,k=500$ | 0.4264 | 5 | 5039 | 14,179 |

Palshikar ${S}_{2}$ | $h=3,k=500$ | 1.4768 | 55 | 1608 | 19,892 |

Palshikar ${S}_{3}$ | $h=3,k=500$ | 1.3090 | 25 | 964 | 19,892 |

Palshikar ${S}_{1}$ | $h=3,k=1000$ | 1.0846 | 16 | 964 | 19,892 |

Palshikar ${S}_{2}$ | $h=3,k=1000$ | 1.2406 | 10 | 964 | 19,892 |

Palshikar ${S}_{3}$ | $h=3,k=1000$ | 1.4050 | 55 | 1608 | 19,892 |

Palshikar ${S}_{1}$ | $h=3,k=2000$ | 0.9451 | 25 | 964 | 19,892 |

Palshikar ${S}_{2}$ | $h=3,k=2000$ | 0.9889 | 16 | 964 | 19,892 |

Palshikar ${S}_{3}$ | $h=3,k=2000$ | 1.1212 | 10 | 964 | 19,892 |

Palshikar ${S}_{1}$ | $h=4,k=100$ | 0.1718 | 5 | 13,171 | 14,311 |

Palshikar ${S}_{2}$ | $h=4,k=100$ | 0.2303 | 3 | 13,171 | 14,311 |

Palshikar ${S}_{3}$ | $h=4,k=100$ | 0.3023 | 4 | 12,606 | 14,727 |

Palshikar ${S}_{1}$ | $h=4,k=500$ | 0.4439 | 3 | 12,606 | 14,727 |

Palshikar ${S}_{2}$ | $h=4,k=500$ | 0.8924 | 16 | 1608 | 14,923 |

Palshikar ${S}_{3}$ | $h=4,k=500$ | 0.9922 | 10 | 1608 | 14,923 |

Palshikar ${S}_{1}$ | $h=4,k=1000$ | 1.0793 | 9 | 1608 | 14,727 |

Palshikar ${S}_{2}$ | $h=4,k=1000$ | 1.2486 | 6 | 1608 | 14,311 |

Palshikar ${S}_{3}$ | $h=4,k=1000$ | 0.8439 | 16 | 1608 | 14,923 |

Palshikar ${S}_{1}$ | $h=4,k=2000$ | 0.9167 | 10 | 1608 | 14,923 |

Palshikar ${S}_{2}$ | $h=4,k=2000$ | 1.0030 | 9 | 1608 | 14,727 |

Palshikar ${S}_{3}$ | $h=4,k=2000$ | 1.1367 | 6 | 1608 | 14,311 |

Algorithm | Sample | Elapsed Time (s) | Segments | First Segment (min) | Last Segment (min) |
---|---|---|---|---|---|

SeqSeg, $\beta =1{e}^{-5}$ | 8 February 2015 | 245.41 | 28 | 1.80 | 13.32 |

SeqSeg, $\beta =3{e}^{-5}$ | 8 February 2015 | 174.13 | 13 | 1.80 | 13.32 |

Palshikar $h=3$ | 8 February 2015 | 113.33 | 56 | 0.51 | 13.30 |

Palshikar $h=5$ | 8 February 2015 | 112.23 | 29 | 6.65 | 13.28 |

SeqSeg, $\beta =1{e}^{-5}$ | 30 January 2015 | 149.61 | 3 | 2.02 | 10.63 |

SeqSeg, $\beta =3{e}^{-5}$ | 30 January 2015 | 44.83 | 0 | - | - |

Palshikar $h=3$ | 30 January 2015 | 88.27 | 130 | 0.00 | 14.91 |

Palshikar $h=5$ | 30 January 2015 | 87.30 | 27 | 0.30 | 14.21 |

SeqSeg, $\beta =1{e}^{-5}$ | 2 February 2015 | 101.45 | 7 | 1.77 | 10.90 |

SeqSeg, $\beta =3{e}^{-5}$ | 2 February 2015 | 94.24 | 3 | 3.75 | 10.32 |

Palshikar $h=3$ | 2 February 2015 | 89.07 | 114 | 0.00 | 14.83 |

Palshikar $h=5$ | 2 February 2015 | 91.15 | 22 | 0.69 | 14.20 |

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Hubert, P.; Padovese, L.; Stern, J.M.
A Sequential Algorithm for Signal Segmentation. *Entropy* **2018**, *20*, 55.
https://doi.org/10.3390/e20010055

**AMA Style**

Hubert P, Padovese L, Stern JM.
A Sequential Algorithm for Signal Segmentation. *Entropy*. 2018; 20(1):55.
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**Chicago/Turabian Style**

Hubert, Paulo, Linilson Padovese, and Julio Michael Stern.
2018. "A Sequential Algorithm for Signal Segmentation" *Entropy* 20, no. 1: 55.
https://doi.org/10.3390/e20010055