# Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Fundamentals

#### 2.1. Multitaper Method

#### 2.2. Fractional Fourier Transform

#### 2.3. Wavelet Transform

#### 2.4. Jensen–Shannon Divergence

#### 2.5. Hierarchichal Clustering

## 3. Dataset

## 4. Analysis and Visualization of Tidal Data

#### 4.1. HC Analysis in the Frequency Domain

#### 4.2. HC Analysis in the Time–Frequency Domain

#### 4.2.1. The FFrT-Based Approach

#### 4.2.2. The CWT-Based Approach

## 5. Long-Range Behavior of Tides

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Takens, F. Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence, Warwick 1980; Springer: Berlin, Germany, 1981; pp. 366–381. [Google Scholar]
- Dergachev, V.A.; Gorban, A.; Rossiev, A.; Karimova, L.; Kuandykov, E.; Makarenko, N.; Steier, P. The filling of gaps in geophysical time series by artificial neural networks. Radiocarbon
**2001**, 43, 365–371. [Google Scholar] - Ghil, M.; Allen, M.; Dettinger, M.; Ide, K.; Kondrashov, D.; Mann, M.; Robertson, A.W.; Saunders, A.; Tian, Y.; Varadi, F.; Yiou, P. Advanced spectral methods for climatic time series. Rev. Geophys.
**2002**, 40, 1–41. [Google Scholar] - Stein, E.M.; Shakarchi, R. Fourier Analysis: An Introduction; Princeton University Press: Princeton, NJ, USA, 2003. [Google Scholar]
- Dym, H.; McKean, H. Fourier Series and Integrals; Academic Press: San Diego, CA, USA, 1972. [Google Scholar]
- Wu, D.L.; Hays, P.B.; Skinner, W.R. A least squares method for spectral analysis of space-time series. J. Atmos. Sci.
**1995**, 52, 3501–3511. [Google Scholar] [CrossRef] - Vautard, R.; Yiou, P.; Ghil, M. Singular-spectrum analysis: A toolkit for short, noisy chaotic signals. Phys. D Nonlinear Phenom.
**1992**, 58, 95–126. [Google Scholar] [CrossRef] - Thomson, D.J. Multitaper analysis of nonstationary and nonlinear time series data. In Nonlinear and Nonstationary Signal Processing; Cambridge University Press: London, UK, 2000; pp. 317–394. [Google Scholar]
- Ding, H.; Chao, B.F. Detecting harmonic signals in a noisy time-series: The z-domain Autoregressive (AR-z) spectrum. Geophys. J. Int.
**2015**, 201, 1287–1296. [Google Scholar] [CrossRef] - Donelan, M.; Babanin, A.; Sanina, E.; Chalikov, D. A comparison of methods for estimating directional spectra of surface waves. J. Geophys. Res. Oceans
**2015**, 120, 5040–5053. [Google Scholar] [CrossRef] - Cohen, L. Time-Frequency Analysis; Prentice-Hall: Upper Saddle River, NJ, USA, 1995. [Google Scholar]
- Almeida, L.B. The fractional Fourier transform and time–frequency representations. IEEE Trans. Signal Process.
**1994**, 42, 3084–3091. [Google Scholar] [CrossRef] - Sejdić, E.; Djurović, I.; Stanković, L. Fractional Fourier transform as a signal processing tool: An overview of recent developments. Signal Process.
**2011**, 91, 1351–1369. [Google Scholar] - Portnoff, M. Time-frequency representation of digital signals and systems based on short-time Fourier analysis. IEEE Trans. Acoust. Speech Signal Process.
**1980**, 28, 55–69. [Google Scholar] - Qian, S.; Chen, D. Joint time-frequency analysis. IEEE Signal Process. Mag.
**1999**, 16, 52–67. [Google Scholar] - Kemao, Q. Windowed Fourier transform for fringe pattern analysis. Appl. Opt.
**2004**, 43, 2695–2702. [Google Scholar] [CrossRef] [PubMed] - Hlubina, P.; Luňáček, J.; Ciprian, D.; Chlebus, R. Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals. Opt. Commun.
**2008**, 281, 2349–2354. [Google Scholar] [CrossRef] [Green Version] - Qian, S.; Chen, D. Discrete Gabor Transform. IEEE Trans. Signal Process.
**1993**, 41, 2429–2438. [Google Scholar] [CrossRef] - Yao, J.; Krolak, P.; Steele, C. The generalized Gabor transform. IEEE Trans. Image Process.
**1995**, 4, 978–988. [Google Scholar] [PubMed] - Mallat, S. A Wavelet Tour of Signal Processing; Academic Press: Burlington, VT, USA, 1999. [Google Scholar]
- Yan, R.; Gao, R.X.; Chen, X. Wavelets for fault diagnosis of rotary machines: A review with applications. Signal Process.
**2014**, 96, 1–15. [Google Scholar] [CrossRef] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Wu, Z.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal.
**2009**, 1, 1–41. [Google Scholar] [CrossRef] - Zayed, A.I. Hilbert transform associated with the fractional Fourier transform. IEEE Signal Process. Lett.
**1998**, 5, 206–208. [Google Scholar] [CrossRef] - Peng, Z.; Peter, W.T.; Chu, F. A comparison study of improved Hilbert–Huang transform and wavelet transform: Application to fault diagnosis for rolling bearing. Mech. Syst. Signal Process.
**2005**, 19, 974–988. [Google Scholar] [CrossRef] - Olsson, J.; Niemczynowicz, J.; Berndtsson, R. Fractal analysis of high-resolution rainfall time series. J. Geophys. Res. Atmos.
**1993**, 98, 23265–23274. [Google Scholar] [CrossRef] - Matsoukas, C.; Islam, S.; Rodriguez-Iturbe, I. Detrended fluctuation analysis of rainfall and streamflow time series. J. Geophys. Res.
**2000**, 105, 29165–29172. [Google Scholar] [CrossRef] - Marwan, N.; Donges, J.F.; Zou, Y.; Donner, R.V.; Kurths, J. Complex network approach for recurrence analysis of time series. Phys. Lett. A
**2009**, 373, 4246–4254. [Google Scholar] [CrossRef] - Donner, R.V.; Donges, J.F. Visibility graph analysis of geophysical time series: Potentials and possible pitfalls. Acta Geophys.
**2012**, 60, 589–623. [Google Scholar] [CrossRef] - Machado, J.T. Fractional order description of DNA. Appl. Math. Model.
**2015**, 39, 4095–4102. [Google Scholar] [CrossRef] - Lopes, A.M.; Machado, J.T. Integer and fractional-order entropy analysis of earthquake data series. Nonlinear Dyn.
**2016**, 84, 79–90. [Google Scholar] [CrossRef] - Machado, J.A.T.; Lopes, A.M. Analysis and visualization of seismic data using mutual information. Entropy
**2013**, 15, 3892–3909. [Google Scholar] [CrossRef] - Machado, J.A.; Mata, M.E.; Lopes, A.M. Fractional state space analysis of economic systems. Entropy
**2015**, 17, 5402–5421. [Google Scholar] [CrossRef] - Jalón-Rojas, I.; Schmidt, S.; Sottolichio, A. Evaluation of spectral methods for high-frequency multiannual time series in coastal transitional waters: Advantages of combined analyses. Limnol. Oceanogr. Methods
**2016**, 14, 381–396. [Google Scholar] [CrossRef] - Grinsted, A.; Moore, J.C.; Jevrejeva, S. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Process. Geophys.
**2004**, 11, 561–566. [Google Scholar] [CrossRef] - Malamud, B.D.; Turcotte, D.L. Self-affine time series: I. Generation and analyses. Adv. Geophys.
**1999**, 40, 1–90. [Google Scholar] - Gong, D.; Feng, L.; Li, X.T.; Zhao, J.-M.; Liu, H.-B.; Wang, X.; Ju, C.-H. The Application of S-transform Spectrum Decomposition Technique in Extraction of Weak Seismic Signals. Chin. J. Geophys.
**2016**, 59, 43–53. [Google Scholar] [CrossRef] - Forootan, E.; Kusche, J. Separation of deterministic signals using independent component analysis (ICA). Stud. Geophys. Geod.
**2013**, 57, 17–26. [Google Scholar] [CrossRef] - Donner, R.V.; Zou, Y.; Donges, J.F.; Marwan, N.; Kurths, J. Recurrence networks—A novel paradigm for nonlinear time series analysis. New J. Phys.
**2010**, 12, 033025. [Google Scholar] [CrossRef] - Lopes, A.M.; Machado, J.T. Analysis of temperature time-series: Embedding dynamics into the MDS method. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 851–871. [Google Scholar] [CrossRef] - Machado, J.T.; Lopes, A.M. The persistence of memory. Nonlinear Dyn.
**2015**, 79, 63–82. [Google Scholar] [CrossRef] - Pugh, D.; Woodworth, P. Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Shankar, D. Seasonal cycle of sea level and currents along the coast of India. Curr. Sci.
**2000**, 78, 279–288. [Google Scholar] - Erol, S. Time-frequency analyses of tide-gauge sensor data. Sensors
**2011**, 11, 3939–3961. [Google Scholar] [CrossRef] [PubMed] - Brigham, E.O. The Fast Fourier Transform and Its Applications; Number 517.443; Prentice Hall: Upper Sadlle River, NJ, USA, 1988. [Google Scholar]
- Prieto, G.; Parker, R.; Thomson, D.; Vernon, F.; Graham, R. Reducing the bias of multitaper spectrum estimates. Geophys. J. Int.
**2007**, 171, 1269–1281. [Google Scholar] [CrossRef] - Thomson, D.J. Spectrum estimation and harmonic analysis. Proc. IEEE
**1982**, 70, 1055–1096. [Google Scholar] [CrossRef] - Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and uncertainty–V: The discrete case. Bell Labs Tech. J.
**1978**, 57, 1371–1430. [Google Scholar] [CrossRef] - Fodor, I.K.; Stark, P.B. Multitaper spectrum estimation for time series with gaps. IEEE Trans. Signal Process.
**2000**, 48, 3472–3483. [Google Scholar] - Smith-Boughner, L.; Constable, C. Spectral estimation for geophysical time-series with inconvenient gaps. Geophys. J. Int.
**2012**, 190, 1404–1422. [Google Scholar] [CrossRef] - Bultheel, A.; Sulbaran, H.E.M. Computation of the fractional Fourier transform. Appl. Comput. Harmon. Anal.
**2004**, 16, 182–202. [Google Scholar] [CrossRef] [Green Version] - Ozaktas, H.M.; Zalevsky, Z.; Kutay, M.A. The Fractional Fourier Transform; Wiley: Chichester, UK, 2001. [Google Scholar]
- Machado, J.T.; Costa, A.C.; Quelhas, M.D. Wavelet analysis of human DNA. Genomics
**2011**, 98, 155–163. [Google Scholar] [CrossRef] [PubMed] - Cattani, C. Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure; World Scientific: Singapore, 2007; Volume 74. [Google Scholar]
- Stark, H.G. Wavelets and Signal Processing: An Application-Based Introduction; Springer Science & Business Media: New York, NY, USA, 2005. [Google Scholar]
- Ngui, W.K.; Leong, M.S.; Hee, L.M.; Abdelrhman, A.M. Wavelet analysis: Mother wavelet selection methods. Appl. Mech. Mater.
**2013**, 393, 953–958. [Google Scholar] [CrossRef] - Cui, X.; Bryant, D.M.; Reiss, A.L. NIRS-based hyperscanning reveals increased interpersonal coherence in superior frontal cortex during cooperation. Neuroimage
**2012**, 59, 2430–2437. [Google Scholar] [CrossRef] [PubMed] - Jeong, D.H.; Kim, Y.D.; Song, I.U.; Chung, Y.A.; Jeong, J. Wavelet Energy and Wavelet Coherence as EEG Biomarkers for the Diagnosis of Parkinson’s Disease-Related Dementia and Alzheimer’s Disease. Entropy
**2015**, 18, 8. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
- Sato, A.H. Frequency analysis of tick quotes on the foreign exchange market and agent-based modeling: A spectral distance approach. Phys. A Stat. Mech. Appl.
**2007**, 382, 258–270. [Google Scholar] [CrossRef] - Felsenstein, J. PHYLIP (Phylogeny Inference Package) version 3.6. Distributed by the author. Department of Genome Sciences, University of Washington, Seattle. Available online: http://evolution.genetics.washington.edu/phylip.html (accessed on 29 July 2017).
- Tenreiro Machado, J.; Duarte, F.B.; Duarte, G.M. Analysis of stock market indices with multidimensional scaling and wavelets. Math. Probl. Eng.
**2012**, 2012, 819503. [Google Scholar] [CrossRef] - Lopes, A.M.; Machado, J.A.T. Fractional order models of leaves. J. Vib. Control
**2014**, 20, 998–1008. [Google Scholar] [CrossRef] - Baleanu, D. Fractional Calculus: Models and Numerical Methods; World Scientific: Singapore, 2012; Volume 3. [Google Scholar]
- Mandelbrot, B.B.; Van Ness, J.W. Fractional Brownian motions, fractional noises and applications. SIAM Rev.
**1968**, 10, 422–437. [Google Scholar] [CrossRef] - Keshner, M.S. 1/f noise. Proc. IEEE
**1982**, 70, 212–218. [Google Scholar] [CrossRef] - Mandelbrot, B.B. The Fractal Geometry of Nature; Macmillan: London, UK, 1983; Volume 173. [Google Scholar]

**Figure 2.**Original, $x\left(t\right)$, and the reconstructed, $\tilde{x}\left(t\right)$, time series (TS) of Boston.

**Figure 3.**The power spectral density (PSD) for Boston tidal TS calculated through the classical periodogram, $T\left(f\right)$, and multitaper method (MM) $\widehat{S}\left(f\right)$.

**Figure 4.**The hierarchical tree resulting from $\left[{\delta}_{ij}\right]$, $i,j=1,\dots ,75$, with ${\delta}_{ij}=JSD({\mathsf{\Phi}}_{i},{\mathsf{\Phi}}_{j})$, and $\mathsf{\Phi}$ calculated based on the MM. JSD: Jensen–Shannon divergence.

**Figure 5.**Locus of magnitude of the fractional Fourier transform (FrFT) (in log scale) versus ($a,\tau $) for Boston (lat: ${42.35}^{\circ}$, lon: $-{71.05}^{\circ}$) and Christmas Is. (lat: ${1.983}^{\circ}$, lon: $-{157.467}^{\circ}$) tidal stations.

**Figure 6.**The hierarchical tree resulting from $\left[{\delta}_{ij}\right]$, $i,j=1,\dots ,75$, with ${\delta}_{ij}=JSD({\mathsf{\Omega}}_{i},{\mathsf{\Omega}}_{j})$, and $\mathsf{\Omega}$ calculated based on the FrFT.

**Figure 7.**The continuous wavelet transform (CWT) for Boston (lat: ${42.35}^{\circ}$, lon: $-{71.05}^{\circ}$) and Christmas Is. (lat: ${1.983}^{\circ}$, lon: $-{157.467}^{\circ}$) tidal stations. The dashed white lines represent the cones on influence.

**Figure 8.**The wavelet coherence between Boston (lat: ${42.35}^{\circ}$, lon: $-{71.05}^{\circ}$) vs. Christmas Is. (lat: ${1.983}^{\circ}$, lon: $-{157.467}^{\circ}$) and Boston (lat: ${42.35}^{\circ}$, lon: $-{71.05}^{\circ}$) vs. New York (lat: ${40.7}^{\circ}$, lon: $-{74.02}^{\circ}$) tidal stations. The dashed white lines represent the cones on influence.

**Figure 9.**The hierarchical tree resulting from $\left[{\delta}_{ij}\right]$, $i,j=1,\dots ,75$, with ${\delta}_{ij}=JSD({\mathsf{\Omega}}_{i},{\mathsf{\Omega}}_{j})$, and $\mathsf{\Omega}$ calculated based on the CWT.

**Figure 10.**The $\widehat{S}\left(f\right)$, $f\in [{10}^{-5},{10}^{-2}]\text{}{\mathrm{h}}^{-1}$, and PL approximation for Boston tidal station, yielding $(a,b)=(58.86,0.32)$.

**Figure 11.**Locus of the (a, b) parameters and the polynomial (degree $n=3$) fit to ${\mathcal{L}}_{i}$, $i=1,\cdots ,4$. The size and color of the markers are proportional to the value of the root mean squared error (RMSE) of the PL fit to the MM estimates, ${\widehat{S}}_{i}\left(f\right)$, $i=1,\cdots ,75$.

Label | Name | Missing Data (%) | Label | Name | Missing Data (%) | Label | Name | Missing Data (%) |
---|---|---|---|---|---|---|---|---|

1 | Antofagasta | 5.4 | 26 | Granger Bay | 47.1 | 51 | Pensacola | 4.3 |

2 | Atlantic City | 3.5 | 27 | Guam | 11.4 | 52 | Petersburg | 0.6 |

3 | Balboa | 1.8 | 28 | Kahului Harbor | 0.3 | 53 | Ponta Delgada | 16.7 |

4 | Boston | 0.5 | 29 | Kaohsiung | 4.8 | 54 | Port Isabel | 0.4 |

5 | Broome | 1.7 | 30 | Keelung | 23.2 | 55 | Portland | 0.9 |

6 | Buenaventura | 12.9 | 31 | Knysna | 40.4 | 56 | Prince Rupert | 0.2 |

7 | Callao | 4.4 | 32 | Ko Lak | 5.3 | 57 | Pte Des Galets | 23.1 |

8 | Charlotte Amalie | 3.9 | 33 | Langkawi | 1.4 | 58 | Puerto Montt | 5.3 |

9 | Chichijima | 0 | 34 | Legaspi | 16.9 | 59 | Richard’s Bay | 36.4 |

10 | Christmas Is | 6.9 | 35 | Lime Tree Bay | 0.4 | 60 | Rockport | 0.1 |

11 | Cocos Is. | 0.9 | 36 | Lobos de Afuera | 14.8 | 61 | Rorvik | 15.2 |

12 | Cuxhaven | 0 | 37 | Luderitz | 63.8 | 62 | Saipan | 13.5 |

13 | Darwin | 0.2 | 38 | Maisaka | 0.1 | 63 | Salalah | 14.7 |

14 | Durban | 39.6 | 39 | Malakal | 1 | 64 | San Juan Puerto Rico | 1 |

15 | Dzaoudzi | 65.6 | 40 | Marseille | 31.5 | 65 | Santa Monica | 1.5 |

16 | East London | 37.6 | 41 | Mera | 0 | 66 | Simon’s Bay | 41.3 |

17 | Eastport | 2.4 | 42 | Mombasa | 30.5 | 67 | Spring Bay | 0.6 |

18 | Esperance | 2.5 | 43 | Nain | 50.8 | 68 | Tofino | 2.8 |

19 | Fort Denison | 1 | 44 | Napier | 19.4 | 69 | Toyama | 0 |

20 | Fort-de-France | 57.5 | 45 | New York | 13.1 | 70 | Vardoe | 1.3 |

21 | Fremantle | 0 | 46 | Newport | 0.3 | 71 | Victoria | 0.4 |

22 | Funafuti | 1.9 | 47 | Ny-Alesund | 0.3 | 72 | Wakkanai | 0 |

23 | Galveston | 2.2 | 48 | Pago Pago | 3.3 | 73 | Walvis Bay | 59 |

24 | Gan | 0.2 | 49 | Paita | 10.9 | 74 | Yap | 9 |

25 | Grand Isle | 3.1 | 50 | Papeete | 3.3 | 75 | Zanzibar | 5.3 |

Name | Symbol | Period (h) | Speed (${}^{\circ}$/h) |
---|---|---|---|

Higher Harmonics | |||

Shallow water overtides of principal lunar | ${M}_{4}$ | 6.210300601 | 57.9682084 |

Shallow water overtides of principal lunar | ${M}_{6}$ | 4.140200401 | 86.9523127 |

Shallow water terdiurnal | $M{K}_{3}$ | 8.177140247 | 44.0251729 |

Shallow water overtides of principal solar | ${S}_{4}$ | 6 | 60 |

Shallow water quarter diurnal | $M{N}_{4}$ | 6.269173724 | 57.4238337 |

Shallow water overtides of principal solar | ${S}_{6}$ | 4 | 90 |

Lunar terdiurnal | ${M}_{3}$ | 8.280400802 | 43.4761563 |

Shallow water terdiurnal | $2\u2033M{K}_{3}$ | 8.38630265 | 42.9271398 |

Shallow water eighth diurnal | ${M}_{8}$ | 3.105150301 | 115.9364166 |

Shallow water quarter diurnal | $M{S}_{4}$ | 6.103339275 | 58.9841042 |

Semi-Diurnal | |||

Principal lunar semidiurnal | ${M}_{2}$ | 12.4206012 | 28.9841042 |

Principal solar semidiurnal | ${S}_{2}$ | 12 | 30 |

Larger lunar elliptic semidiurnal | ${N}_{2}$ | 12.65834751 | 28.4397295 |

Larger lunar evectional | ${\nu}_{2}$ | 12.62600509 | 28.5125831 |

Variational | $M{U}_{2}$ | 12.8717576 | 27.9682084 |

Lunar elliptical semidiurnal second-order | $2\u2033{N}_{2}$ | 12.90537297 | 27.8953548 |

Smaller lunar evectional | ${\lambda}_{2}$ | 12.22177348 | 29.4556253 |

Larger solar elliptic | ${T}_{2}$ | 12.01644934 | 29.9589333 |

Smaller solar elliptic | ${R}_{2}$ | 11.98359564 | 30.0410667 |

Shallow water semidiurnal | $2S{M}_{2}$ | 11.60695157 | 31.0158958 |

Smaller lunar elliptic semidiurnal | ${L}_{2}$ | 12.19162085 | 29.5284789 |

Lunisolar semidiurnal | ${K}_{2}$ | 11.96723606 | 30.0821373 |

Diurnal | |||

Lunar diurnal | ${K}_{1}$ | 23.93447213 | 15.0410686 |

Lunar diurnal | ${O}_{1}$ | 25.81933871 | 13.9430356 |

Lunar diurnal | $O{O}_{1}$ | 22.30608083 | 16.1391017 |

Solar diurnal | ${S}_{1}$ | 24 | 15 |

Smaller lunar elliptic diurnal | ${M}_{1}$ | 24.84120241 | 14.4920521 |

Smaller lunar elliptic diurnal | ${J}_{1}$ | 23.09848146 | 15.5854433 |

Larger lunar evectional diurnal | $\rho $ | 26.72305326 | 13.4715145 |

Larger lunar elliptic diurnal | ${Q}_{1}$ | 26.86835 | 13.3986609 |

Larger elliptic diurnal | $2{Q}_{1}$ | 28.00621204 | 12.8542862 |

Solar diurnal | ${P}_{1}$ | 24.06588766 | 14.9589314 |

Long Period | |||

Lunar monthly | ${M}_{m}$ | 661.3111655 | 0.5443747 |

Solar semiannual | ${S}_{sa}$ | 4383.076325 | 0.0821373 |

Solar annual | ${S}_{a}$ | 8766.15265 | 0.0410686 |

Lunisolar synodic fortnightly | ${M}_{sf}$ | 354.3670666 | 1.0158958 |

Lunisolar fortnightly | ${M}_{f}$ | 327.8599387 | 1.0980331 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

M. Lopes, A.; Tenreiro Machado, J.A.
Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering. *Entropy* **2017**, *19*, 390.
https://doi.org/10.3390/e19080390

**AMA Style**

M. Lopes A, Tenreiro Machado JA.
Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering. *Entropy*. 2017; 19(8):390.
https://doi.org/10.3390/e19080390

**Chicago/Turabian Style**

M. Lopes, Antonio, and Jose A. Tenreiro Machado.
2017. "Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering" *Entropy* 19, no. 8: 390.
https://doi.org/10.3390/e19080390