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entropy

Article

Tidal Analysis Using Time–Frequency Signal

Processing and Information Clustering

António M. Lopes 1,*ID and José A. Tenreiro Machado 2ID

1UISPA–LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias,

4200-465 Porto, Portugal

2Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering,

Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal; jtm@isep.ipp.pt

*Correspondence: aml@fe.up.pt; Tel.: +351-913-499-471

Received: 1 June 2017; Accepted: 26 July 2017; Published: 29 July 2017

Abstract:

Geophysical time series have a complex nature that poses challenges to reaching assertive

conclusions, and require advanced mathematical and computational tools to unravel embedded

information. In this paper, time–frequency methods and hierarchical clustering (HC) techniques

are combined for processing and visualizing tidal information. In a ﬁrst phase, the raw data are

pre-processed for estimating missing values and obtaining dimensionless reliable time series. In a

second phase, the Jensen–Shannon divergence is adopted for measuring dissimilarities between data

collected at several stations. The signals are compared in the frequency and time–frequency domains,

and the HC is applied to visualize hidden relationships. In a third phase, the long-range behavior of

tides is studied by means of power law functions. Numerical examples demonstrate the effectiveness

of the approach when dealing with a large volume of real-world data.

Keywords:

multitaper method; wavelet transform; Jensen–Shannon divergence; hierarchical clustering;

power law; tidal time series

1. Introduction

Geophysical time series (TS) can be interpreted as the output of multidimensional dynamical

systems inﬂuenced by many distinct factors at different scales in space and time. In light of

Takens’ embedding theorem, these TS can reveal—at least partially—the underlying dynamics of the

corresponding systems [1].

Some common properties of geophysical TS are their complex structure, non-linearity, and

non-stationarity [

2

,

3

]. These characteristics pose difﬁculties in processing the data that are not

easily addressed by means of tools such as Fourier analysis [

4

,

5

]. To overcome such limitations,

other techniques for spectral estimation are adopted, such as the least-squares [

6

] and singular

spectrum analysis [

7

], the multitaper method (MM) [

8

], and the autoregressive moving average [

9

] and

maximum entropy techniques [

10

]. Alternatively, time–frequency methods [

11

] have proven powerful

for processing non-linear and non-stationary data. We can mention not only the

fractional [12,13]

, short

time [

14

,

15

], and windowed Fourier [

16

,

17

] transforms, but also the

Gabor [18,19]

,

wavelet [20,21]

,

Hilbert–Huang [

22

,

23

], and

S

[

24

,

25

] transforms. Additionally, distinct complexity measures

(e.g., entropy, Lyapunov exponent, Komologrov estimates, and fractal dimension) [

26

], detrended

ﬂuctuation analysis [

27

], and recurrence plots [

28

], among others [

3

,

29

–

33

], are also adopted for

analyzing complex TS.

Jalón-Rojas et al. [

34

] compared different spectral methods for the analysis of high-frequency and

long TS collected at the Girond estuary. They considered speciﬁc evaluation criteria and concluded

that the combination of distinct methods could be a good strategy for dealing with data measured

Entropy 2017,19, 390; doi:10.3390/e19080390 www.mdpi.com/journal/entropy

Entropy 2017,19, 390 2 of 18

at coastal waters. Grinsted et al. [

35

] adopted the cross wavelet transform and wavelet coherence

for examining relationships in time and frequency between two TS. They applied these methods

to the Arctic Oscillation index and the Baltic maximum sea ice extent record. Vautard et al. [

7

]

used the singular-spectrum analysis, demonstrating the effectiveness of the technique when dealing

with short and noisy TS. Malamud and Turcotte [

36

] introduced the self-afﬁne TS, characterized

by a power spectral density (PSD) that is described by a power law (PL) function of the frequency.

They addressed a variety of techniques to quantify the strength of long-range persistence—namely,

the Fourier power spectral, semivariogram, rescaled-range, average extreme-event, and wavelet

variance analysis. Ding and Chao [

9

] adopted autoregressive methods for detecting harmonic signals

with exponential decay or growth contained in noisy TS. Donelan et al. [

10

] used the maximum

likelihood, maximum entropy, and wavelets for estimating the directional spectra of water waves.

Gong et al. [

37

] adopted the

S

-transform for analyzing seismic data. Huang et al. [

22

] proposed

empirical mode decomposition and the Hilbert–Huang transform. First, a TS is decomposed into a

ﬁnite and often small number of intrinsic mode functions, and then the Hilbert transform is applied to

the modes. Forootan and Kusche [

38

] used independent component analysis to separate unknown

mixtures of deterministic sinusoids with non-null trend. Doner et al. [

39

] explored recurrence networks,

interpreting the recurrence matrix of a TS as the adjacency matrix of an associated complex network

that links different points in time if the considered states are closely neighbored in the phase space.

The recurrence matrix yields new quantitative characteristics (such as average path length, clustering

coefﬁcient, or centrality measures of the recurrence network) related to the dynamical complexity of

the TS.

Lopes at al. [32,40,41]

investigated geophysical data by means of multidimensional scaling and

fractional order techniques.

Tides are variations in the sea level mainly caused by astronomical components, such as

gravitational forces exerted by the Moon, the Sun, and the rotation of the Earth, but also reﬂect

non-astronomical sources such as the weather [

42

]. Understanding the sea-level variations is of great

importance for both safe navigation and for planning and promoting the sustainable development of

coastal areas. Moreover, sea-level observations provide valuable data to ocean sciences, geodynamics,

and geosciences [

43

,

44

]. Tides can be measured by means of gauges, with respect to a datum, and the

values are recorded over time. A large volume of tidal information is presently available for scientiﬁc

research. Tidal TS include harmonic constituents and other components with multiple time scales that

span from hours to decades. On such time scales, tidal data are often non-stationary, and as with most

geophysical TS, standard mathematical tools are insufﬁcient to satisfactorily assess the information

that they embed.

In this paper we combine time–frequency methods and hierarchical clustering (HC) techniques

to process and visualize tidal information. In a ﬁrst phase, we pre-process the raw data (i.e., we ﬁll

the gaps in the TS with values calculated with a suitable tidal model), and then we normalize the

data to obtain dimensionless TS. In a second phase, we use the Jensen–Shannon divergence (JSD) to

measure the dissimilarities between TS collected at several stations located worldwide. The TS are

compared in the frequency and time–frequency domains. The frequency domain information consists

of the PSD generated by the MM. The time–frequency information corresponds to the magnitudes of

the fractional Fourier transform (FrFT) and the continuous wavelet transform (CWT) of the TS. In the

three cases, HC generates maps that are interpreted based on the emerging clusters of the points that

represent tidal stations. In a third phase, the long-range behavior of tides is modeled by means of PL

functions using the TS spectra at low frequencies. Numerical examples demonstrate the effectiveness

of the approach when dealing with a large volume of real-world data.

In this line of thought, the structure of the paper is as follows. Section 2presents the main

mathematical tools used for processing the TS. Section 3introduces the data set and the pre-processing

used to generate well-formatted TS. Section 4applies the HC method and discusses the results.

Section 5studies the long-range behavior of tides by means of PL functions. Finally, Section 6draws

the main conclusions.

Entropy 2017,19, 390 3 of 18

2. Mathematical Fundamentals

This section introduces the main mathematical tools adopted for data processing; namely,

the MM, FrFt, CWT, JSD, and HC techniques. These tools are well-suited to TS generated by most

naturally-occurring phenomena, as is the case of biological, climatic, and geophysical processes.

2.1. Multitaper Method

The MM is a robust numerical algorithm for estimating the PSD of a signal. Given an

N

-length

sequence

x(t)

, its PSD can be estimated by the single-taper, or modiﬁed periodogram function,

T(f)

,

derived directly from the FT of x(t)[45]. Therefore, we have:

T(f) =

N−1

∑

t=0

x(t)a(t)e−j2πf t

2

, (1)

where

t

and

f

denote time and frequency, respectively, and

j=√−1

. The function

a(t)

is called a

taper, or window, and represents a series of weights that verify the condition

∑N−1

t=0|a(t)|=

1. If

a(t)

is

a rectangular (or boxcar) function, then (1) yields the standard periodogram of x(t)[46].

Expression

(1)

leads to a biased estimate of the PSD due to both spectral leakage (i.e., power

spreading from strong peaks at a given frequency towards neighboring frequencies) and variance of

T(f)

(i.e., noise affecting the spectra). To avoid these artifacts, the MM method was introduced by

Thomson [

47

]. In this method,

x(t)

is multiplied by a set of orthogonal sequences, or tapers, to obtain

a set of single-taper periodograms. The set is then averaged to yield an improved estimate of the PSD,

¯

S(f), given by:

¯

S(f) = 1

K

K

∑

k=1

Tk(f), (2)

where

Tk(f) = |Yk(f)|2

,

k=

1,

···

,

K

, are spectral estimates, or eigenspectra functions, and

Yk(f)

are

the eigencomponents:

Yk(f) =

N−1

∑

t=0

x(t)vk(t)e−j2πf t, (3)

obtained with KSlepian sequences, vk(t), that verify [48]:

N−1

∑

t=0

vj(t)vk(t) = δjk,i,j=1, ··· ,K. (4)

Instead of

(2)

, a weighted average is often adopted that minimizes some measure of discrepancy

of Yk, yielding the estimate:

ˆ

S(f) =

N−1

∑

t=0

d2

k(f)|Yk(f)|2

N−1

∑

t=0

d2

k(f)

, (5)

where dk(f)are weights [47].

Some variants of the MM can process TS with gaps [

49

,

50

], but we consider herein the “standard”

MM implementation, which requires evenly sampled TS without gaps.

2.2. Fractional Fourier Transform

The FrFT of order

a∈R

,

Fa

, is a linear integral operator that maps a given function (or signal)

x(t)onto xa(τ),{t,τ} ∈ R, by the expression [51]:

Entropy 2017,19, 390 4 of 18

xa(τ) = Fa(τ) = Z∞

−∞Ka(τ,t)x(t)dt, (6)

where, setting α=aπ/2, the kernel Ka(τ,t)is deﬁned as:

Ka(τ,t) = Cαexp −jπ2tτ

sin α−(t2+τ2)cot α, (7)

with

Cα=p1−jcot α=exp{−j[πsgn(sin α)/4 −α/2]}

p|sin α|. (8)

For

a=

2

k

,

k∈Z

,

α∈πk

, we should take limiting values. Furthermore, when

a=

4

k

and

a=

2

+

4

k

, the FrFT becomes

f4k(τ) = f(τ)

and

f2+4k(τ) = f(−τ)

, respectively, and the kernels are:

K4k(t,τ) = δ(τ−t), (9a)

K2+4k(t,τ) = δ(τ+t). (9b)

When

a=

1

+

4

k

, we have

Fa=F1

that corresponds to the ordinary Fourier transform (FT),

and when

a=

3

+

4

k

, we have

Fa=F3=F2F1

. Therefore, the operator

Fa

can be interpreted as the

a

th power of the ordinary FT, that may be considered modulo 4 [

51

,

52

]. For the digital computation

of

Fa

, different algorithms were proposed [

51

]. Here we adopt the Fast Approximate FrFT [

51

]

(https://nalag.cs.kuleuven.be/research/software/FRFT/). The signal

x(t)

must be evenly sampled

and without gaps.

2.3. Wavelet Transform

The wavelet transform converts a given function,

x(t)

, from standard time into the generalized

time–frequency domain, and represents a powerful tool for identifying intermittent periodicities in the

data. The discrete wavelet transform is particularly useful for noise reduction and data compression,

while the CWT is better for feature extraction [35].

The CWT of x(t)is given by [53–55]:

Wψx(t)(a,b) = 1

√aZ+∞

−∞x(t)ψ∗t−b

adt,a>0, (10)

where

ψ

denotes the mother wavelet function,

(·)∗

represents the complex conjugate of the argument,

and the parameters (a,b)represent the dyadic dilation and translation of ψ, respectively.

The CWT processes data at different scales. The temporal analysis is performed with a contracted

version of the prototype wavelet, while frequency analysis is derived with a dilated version of

ψ

.

The parameter ais related to frequency, and boften represents time or space.

The choice of an appropriate mother wavelet represents a key issue in the analysis [

53

,

56

].

Some initial knowledge about the signal characteristics is important, but we often choose based on

several trials and the results obtained. Therefore, the best

ψ

is the one that more assertively highlights

the features that we are looking for.

Two TS can be compared directly by computing their wavelet coherence as a function of time

and frequency. In other words, wavelet coherence measures time-varying correlations as a function of

frequency [35,44,57].

Given two TS, xi(t)and xj(t), their wavelet coherence is given by [35,44,58]:

Cij =SnhW∗

ψxi(t)i(a,b)oWψxj(t)(a,b)

2

SWψxi(t)(a,b)

2SWψxj(t)(a,b)

2, (11)

Entropy 2017,19, 390 5 of 18

where S(·)is a smoothing function in time and frequency.

Similarly to the MM and FrFT, the CWT can be applied to TS evenly sampled and without

missing data.

2.4. Jensen–Shannon Divergence

The JSD measures the dissimilarity between two probability distributions,

P

and

Q

[

59

], and is the

smoothed and symmetrical version of the Kullback–Leibler divergence, or relative entropy, given by:

KLD (P,Q)=∑

k

p(k)ln p(k)

q(k). (12)

The JSD is formulated as:

JSD (P,Q)=1

2[KLD (P,M)+KLD (Q,M)] , (13)

where M=P+Q

2is a mixture distribution.

Alternatively, we can write:

JSD (P,Q)=1

2"∑

k

p(k)ln p(k) + ∑

k

q(k)ln q(k)#−∑

k

m(k)ln m(k). (14)

2.5. Hierarchichal Clustering

Clustering analysis groups objects that are similar to each other in some sense. In HC, a hierarchy

of object clusters is built based on one of two alternative algorithms. In agglomerative clustering,

each object starts in its own singleton cluster, and at each step the two most similar clusters are greedily

merged. The algorithm stops when all objects are in the same cluster. In divisive clustering, all objects

start in one cluster, and at each step the algorithm removes the outsiders from the least cohesive cluster.

The iterations stop when each object is in its own singleton cluster. The clusters are combined (split) for

agglomerative (divisive) clustering based on their dissimilarity. Therefore, given two clusters,

I

and

J

,

a metric is speciﬁed to measure the distance,

δ(xi

,

xj)

, between objects

xi∈I

and

xj∈J

, and the

dissimilarity between clusters,

d(I

,

J)

, is calculated by the maximum, minimum, or average linkage,

given by:

dmax (I,J)=max

xi∈I,xj∈Jdxi,xj, (15)

dmin (I,J)=min

xi∈I,xj∈Jdxi,xj, (16)

dave (I,J)=1

kIkk Jk∑

xi∈I,xj∈J

dxi,xj. (17)

The results of HC are usually presented in a dendrogram or a tree diagram.

3. Dataset

The tidal information are available at the University of Hawaii Sea Level Center (http://uhslc.

soest.hawaii.edu/). Worldwide stations have records covering different time periods. We consider

hourly data collected between January, 1 1994 and December, 31 2014 at

s=

75 stations. Their labels,

names, and percentage of missing data are shown in Table 1. The stations’ geographical location is

depicted in Figure 1.

Entropy 2017,19, 390 6 of 18

Table 1. Stations’ labels, names, and percentage of missing data.

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

Antofagasta

Atlantic City

Balboa

Boston

Broome

Buenaventura

Callao

Charlotte Amalie

Chichijima

Christmas Is.

Cocos Is.

Cuxhaven

Darwin

Durban

Dzaoudzi

East London

Eastport

Esperance

Fort Denison

Fort-de-France

Fremantle

Funafuti

Galveston

Gan

Grand Isle

Granger Bay

Guam

Kahului Harbor Kaohsiung

Keelung

Knysna

Ko Lak

Langkawi

Legaspi

Lime Tree Bay

Lobos de Afuera

Luderitz

Maisaka

Malakal

Marseille

Mera

Mombasa

Nain

Napier

New York

Newport

Ny-Alesund

Pago Pago

Paita

Papeete

Pensacola

Petersburg

Ponta Delgada

Port Isabel

Portland

Prince Rupert

Pte Des Galets

Puerto Montt

Richard's Bay

Rockport

Rorvik

Saipan

Salalah

San Juan Puerto Rico

Santa Monica

Simon's Bay

Spring Bay

Tofino

Toyama

Vardoe

Victoria Wakkanai

Walvis Bay

Yap

Zanzibar

Figure 1. Geographic location of the s=75 stations considered in the study.

Occasional gaps in the TS,

x(t)

, must be ﬁlled before applying the MM and CWT processing tools.

The missing values are replaced by artiﬁcial data generated by a tidal model, ˆ

x(t), given by:

ˆ

x(t) = U0+

T

∑

k=1

Ukcos(2πfkt+φk), (18)

where

U0=hx(t)i

denotes the average value of

x(t)

, and the sinusoidal terms represent standard

tidal constituents of known frequency,

fk

,

k=

1,

···

,

T

, according to the International Hydrographic

Organization (https://www.iho.int/srv1/index.php?lang=en). The amplitude and phase shift,

Uk

and

φk, are computed by the least-squares method.

Entropy 2017,19, 390 7 of 18

Herein, we adopt

T=

37 and the components listed in Table 2. For example, Figure 2depicts

the original,

x(t)

, and the reconstructed,

˜

x(t)

, TS of Boston, illustrating the effectiveness of the model.

Identical results are obtained for other tidal stations.

Table 2. Standard tidal constituents.

Name Symbol Period (h) Speed (◦/h)

Higher Harmonics

Shallow water overtides of principal lunar M46.210300601 57.9682084

Shallow water overtides of principal lunar M64.140200401 86.9523127

Shallow water terdiurnal MK38.177140247 44.0251729

Shallow water overtides of principal solar S46 60

Shallow water quarter diurnal MN46.269173724 57.4238337

Shallow water overtides of principal solar S64 90

Lunar terdiurnal M38.280400802 43.4761563

Shallow water terdiurnal 2”MK38.38630265 42.9271398

Shallow water eighth diurnal M83.105150301 115.9364166

Shallow water quarter diurnal MS46.103339275 58.9841042

Semi-Diurnal

Principal lunar semidiurnal M212.4206012 28.9841042

Principal solar semidiurnal S212 30

Larger lunar elliptic semidiurnal N212.65834751 28.4397295

Larger lunar evectional ν212.62600509 28.5125831

Variational MU212.8717576 27.9682084

Lunar elliptical semidiurnal second-order 2”N212.90537297 27.8953548

Smaller lunar evectional λ212.22177348 29.4556253

Larger solar elliptic T212.01644934 29.9589333

Smaller solar elliptic R211.98359564 30.0410667

Shallow water semidiurnal 2SM211.60695157 31.0158958

Smaller lunar elliptic semidiurnal L212.19162085 29.5284789

Lunisolar semidiurnal K211.96723606 30.0821373

Diurnal

Lunar diurnal K123.93447213 15.0410686

Lunar diurnal O125.81933871 13.9430356

Lunar diurnal OO122.30608083 16.1391017

Solar diurnal S124 15

Smaller lunar elliptic diurnal M124.84120241 14.4920521

Smaller lunar elliptic diurnal J123.09848146 15.5854433

Larger lunar evectional diurnal ρ26.72305326 13.4715145

Larger lunar elliptic diurnal Q126.86835 13.3986609

Larger elliptic diurnal 2Q128.00621204 12.8542862

Solar diurnal P124.06588766 14.9589314

Long Period

Lunar monthly Mm661.3111655 0.5443747

Solar semiannual Ssa 4383.076325 0.0821373

Solar annual Sa8766.15265 0.0410686

Lunisolar synodic fortnightly Msf 354.3670666 1.0158958

Lunisolar fortnightly Mf327.8599387 1.0980331

Entropy 2017,19, 390 8 of 18

Time (h) ×104

0 2 4 6 8 10 12 14 16 18

x(t), ˜x(t)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

×104

8.79 8.8 8.81 8.82 8.83 8.84 8.85

Original time series

Fitting data

Figure 2. Original, x(t), and the reconstructed, ˜

x(t), time series (TS) of Boston.

4. Analysis and Visualization of Tidal Data

In this section, we use HC for visualizing the relationships between

s=

75 tidal TS. The signals,

xi(t)

,

i=

1,

···

,

s

, are first normalized to zero mean and unit variance in order to get a dimensionless TS:

˜

xi(t) = xi(t)−µi

σi

, (19)

where µiand σirepresent the mean and standard deviation values of xi(t), respectively.

In Subsections 4.1 and 4.2, we use the JSD to measure the dissimilarities between the tidal data in

the frequency and time–frequency domains, respectively, and we apply the HC algorithm to visualize

relationships. It should be noted that other dissimilarity measures are possible [

32

,

33

], but several

numerical experiments led to the conclusion that the JSD yields reliable results.

4.1. HC Analysis in the Frequency Domain

Data in the frequency domain corresponds to the TS PSD estimates,

ˆ

S(f)

, calculated with the MM

as deﬁned in

(5)

. The superiority of the MM over the standard periodogram is illustrated in Figure 3

for data collected at the Boston tidal station (lat: 42.35

◦

, lon:

−

71.05

◦

). We observe that the variance

(spectral noise) of

ˆ

S(f)

is considerably smaller than the one obtained for the classical periodogram,

T(f). We obtain similar results for other tidal stations.

We normalize the PSD estimates, ˆ

S(f), by calculating the ratio:

Φ(f) = ˆ

S(f)

∑fˆ

S(f), (20)

where

Φ(f)

is interpreted as a probability distribution [

60

], and we feed the HC with the matrix

∆= [δij ]

,

i

,

j=

1, ..., 75, where

δij =JSD(Φi

,

Φj)

represents the JDS between the normalized PSD

estimates

(Φi

,

Φj)

. Figure 4depicts the tree generated by applying the successive (agglomerative) and

average-linkage methods [32,40]. The software PHYLIP was used for generating the graph [61].

Entropy 2017,19, 390 9 of 18

Figure 3.

The power spectral density (PSD) for Boston tidal TS calculated through the classical

periodogram, T(f), and multitaper method (MM) ˆ

S(f).

Figure 4.

The hierarchical tree resulting from

[δij ]

,

i

,

j=

1, ..., 75, with

δij =JSD(Φi

,

Φj)

, and

Φ

calculated based on the MM. JSD: Jensen–Shannon divergence.

We observe not only the emergence of two main (level 1) clusters,

C1

and

C2

, but also the presence

of various sub-clusters at different lower levels. For example, cluster

C1

is composed of level 2

sub-clusters

C11

and

C12

, while

C2

comprises

C21

,

C22

, and the “outlier” station 50. Nevertheless,

at lower levels of the hierarchical tree, the elements of certain sub-clusters emerge very close to each

other, making visualization more difﬁcult.

Entropy 2017,19, 390 10 of 18

4.2. HC Analysis in the Time–Frequency Domain

4.2.1. The FFrT-Based Approach

The FrFT converts a function to a continuum of intermediate domains between the orthogonal

time (or space) and frequency domains. Therefore, it can be thought of as an operator that rotates a

signal by any angle, instead of just π/2 radians as performed by the ordinary FT.

Figure 5depicts the log magnitude of the FrFT versus parameter

a∈[

0, 1

]

and

τ∈[

1, 184057

]

h

for Boston (lat: 42.35

◦

, lon:

−

71.05

◦

) and Christmas Is. (lat: 1.983

◦

, lon:

−

157.467

◦

) tidal stations.

For

a=

0, the FrFT corresponds to the time domain signal. For

a=

1, the FrFT yields the ordinary FT.

The main peaks observed in the time domain propagate along the continuum of pseudofrequency

(or time–frequency) domains (as

a

increases), originating high-energy paths that determine the shape

of the FrFT charts. Close to

τ=

92028 (i.e., to half of the total number of samples of the TS), we observe

a high-energy component that corresponds to the DC frequency, but other details are difﬁcult to

perceive. We obtain similar patterns for other tidal stations.

Figure 5.

Locus of magnitude of the fractional Fourier transform (FrFT) (in log scale) versus (

a

,

τ

) for

Boston (lat: 42.35◦, lon: −71.05◦) and Christmas Is. (lat: 1.983◦, lon: −157.467◦) tidal stations.

The structure of the FrFT plots reﬂect the characteristics of the TS. Nevertheless, to the authors

best knowledge, there are not yet assertive tools to explore this three-dimensional information.

For each TS, we calculate the corresponding FrFT, and we generate an

L×N

dimensional

complex-valued matrix,

W

, where

L

and

N

denote the number of points in frequency and time,

respectively. We then compute the

P=LN

dimensional vector

w(p)

,

p=

1,

···

,

P

, composed of the

columns of |W|, and we perform the normalization:

Ω(p) = w(p)

∑pw(p), (21)

where the function Ω(p)is interpreted as a probability distribution. Finally, we feed the HC with the

matrix

∆= [δij ]

,

i

,

j=

1, ..., 75, where

δij =JSD(Ωi

,

Ωj)

represents the JSD between the normalized

vectors (Ωi,Ωj).

Figure 6depicts the tree generated by the HC. As before, the successive (agglomerative) and

average-linkage methods were used [

32

,

40

]. We observe two main clusters,

U1

and

U2

, that are similar

to the ones identiﬁed by the MM-based approach,

C1

and

C2

, respectively, revealing good consistency

between the two processing alternatives.

Entropy 2017,19, 390 11 of 18

Figure 6.

The hierarchical tree resulting from

[δij ]

,

i

,

j=

1, ..., 75, with

δij =JSD(Ωi

,

Ωj)

, and

Ω

calculated based on the FrFT.

4.2.2. The CWT-Based Approach

The CWT is well suited to non-stationary signals and establishes a compromise between precision

analysis in the time and frequency domains [

62

]. We adopt here the complex Morlet wavelet, since several

numerical experiments were revealed to be a good choice in the context of continuous analysis and

feature extraction [35,44]. The complex Morlet wavelet is deﬁned as:

ψ(t) = 1

pπfb

ei2πfcte−t2

fb, (22)

where

fb

is related to the wavelet bandwidth and

fc

is its center frequency. These constants can be

interpreted as the parameters of a time-localized ﬁltering, or correlation, operator.

Figure 7depicts the CWT for Boston (lat: 42.35

◦

, lon:

−

71.05

◦

) and Christmas Is. (lat: 1.983

◦

,

lon:

−

157.467

◦

) tidal stations. We observe two main patterns at frequencies around

f=

0.08

h−1

and

f=

0.042

h−1

, corresponding to the semi-diurnal and diurnal tidal components, but other objects are

difﬁcult to identify. For other tidal stations we obtain similar patterns.

Figure 8shows the similarities between the two station pairs Boston (lat: 42.35

◦

, lon:

−

71.05

◦

)

vs. Christmas Is. (lat: 1.983

◦

, lon:

−

157.467

◦

) and Boston (lat: 42.35

◦

, lon:

−

71.05

◦

) vs. New York

(lat: 40.7

◦

, lon:

−

74.02

◦

). That is, we present one pair of distant and one pair of neighbor stations.

We verify that coherence between neighbors is higher and—as expected—we observe regions of strong

coherence at the frequencies of the main tidal components (Table 2). However, other strong coherence

regions emerge throughout the data which are difﬁcult to infer from the bare CWT charts. Therefore,

from Figure 8we conclude that wavelet coherence is a powerful tool for unveiling hidden similarities

between data. Yet, since it produces one chart per TS pair, a large amount of data is generated for all

combinations of pairs, and the global perspective is difﬁcult to obtain. To overcome these problems,

in the follow up, we combine CWT and HC tools.

Entropy 2017,19, 390 12 of 18

Figure 7.

The continuous wavelet transform (CWT) for Boston (lat: 42.35

◦

, lon:

−

71.05

◦

) and Christmas

Is. (lat: 1.983

◦

, lon:

−

157.467

◦

) tidal stations. The dashed white lines represent the cones on inﬂuence.

Figure 8.

The wavelet coherence between Boston (lat: 42.35

◦

, lon:

−

71.05

◦

) vs. Christmas Is. (lat: 1.983

◦

,

lon:

−

157.467

◦

) and Boston (lat: 42.35

◦

, lon:

−

71.05

◦

) vs. New York (lat: 40.7

◦

, lon:

−

74.02

◦

) tidal

stations. The dashed white lines represent the cones on inﬂuence.

For all TS, we determine the corresponding CWT, and as described in Subsection 4.2.1, we calculate

the function

Ω(p)

, where

w(p)

now denotes a vector obtained from

|W|

, with

W

generated by the

CWT. Finally, we feed the HC with the matrix ∆= [δij ],i,j=1, ..., 75.

Figure 9depicts the tree generated for matrix

∆

. We observe two main clusters,

V1

and

V2

, that are

similar to those already identiﬁed in the MM- and FrFT-based trees. For example, relative to

C1

and

C2

,

the main differences are for stations 30 (Keelung) and 50 (Papeete), which swapped places. Sub-clusters

at lower levels are now well separated, demonstrating the superiority of the time–frequency analysis

in discriminating differences between the data.

In conclusion, the trees from Figures 4–9reveal the same type of clusters, with slightly distinct

levels of discrimination of the sub-clusters, Figure 9apparently being slightly superior to the others.

This global comparison shows that geographically close stations can behave differently from each

other due to local factors. However, this may be not perceived when using standard processing tools.

Entropy 2017,19, 390 13 of 18

Figure 9.

The hierarchical tree resulting from

[δij ]

,

i

,

j=

1, ..., 75, with

δij =JSD(Ωi

,

Ωj)

, and

Ω

calculated based on the CWT.

5. Long-Range Behavior of Tides

The previous analysis revealed similarities embedded into distinct TS, but does not focus on

long memory effects that often occur in complex systems. Having this fact in mind, in this section we

study the long-range behavior of tides based on the characteristics of the TS PSD at low frequencies.

Therefore, we model the MM estimates,

ˆ

Si(f)

,

i=

1,

···

, 75, within the bandwidth

f∈[fL

,

fH]

, where

fLand fHdenote the lower and upper frequency limits by means of PL functions:

ˆ

Si(f)'a f −b,a,b∈ R+. (23)

In this perspective, “low frequencies” means the bandwidth bellow the ﬁrst harmonics with

signiﬁcant amplitude; that is, f≈24 h−1.

The values obtained for parameter

b

reveal underlying characteristics of the tidal dynamics;

namely, a fractional value of

b

may be indicative of dynamical properties similar to those usually

found in fractional-order systems [

41

,

63

,

64

]. Moreover, Equation

(23)

implies a relationship between

PL behavior and fractional Brownian motion (fBm) [

30

,

65

] (1

/f

noise [

66

]), since for many systems

fBm represents a signature of complexity [67].

Figure 10 illustrates the procedure for data from the Boston tidal station (lat: 42.35

◦

, lon:

−

71.05

◦

),

f∈[

10

−5

, 10

−2]h−1

(i.e., 4 days to 11.5 years), and the PL parameters determined by means of least

squares ﬁtting, yielding (a,b) = (58.86, 0.32).

Entropy 2017,19, 390 14 of 18

Figure 10.

The

ˆ

S(f)

,

f∈[

10

−5

, 10

−2]h−1

, and PL approximation for Boston tidal station, yielding

(a,b) = (58.86, 0.32).

The parameters

(a

,

b)

are computed for the whole set of time-series (

s=

75 in total), and the

corresponding locus is depicted in Figure 11. The size and color of the markers are proportional to

the value of the root mean squared error (RMSE) of the PL ﬁt. We verify that

b

has values between

0.2 and 0.8, corresponding to TS including long memory effects typical of fBm. Values of

b

close to

zero mean that tidal TS are close to white noise; that is, to a random signal having equal intensity at

different frequencies. On the other hand, values of

b

close to 1 follow the so-called pink or 1

/f

noise,

which occurs in many physical and biological systems. In general, for non-integer values of

b

, signals

are related to the ubiquitous fractional Brownian noise. So, we can say loosely that the smaller/higher

the values of b, the less/more correlated are consecutive signal samples and the smaller/larger is the

content of long-range memory effects.

In Figure 11, we group points in the locus

(a

,

b)

into four clusters

Li

,

i=

1,

···

, 4, loosely having

the correspondence

V11 ∪V12 → L1

,

V21 → L2

, and

V22 → L3∪L4

. Therefore, we ﬁnd that the clusters

previously identiﬁed with the tree diagrams for the global time scale have a distinct arrangement

in the long-range perspective. The chart also includes the approximation curves to

Li

,

i=

1,

···

, 4,

yielding lines resembling isoclines in a vector ﬁeld. Third-order polynomials (i.e., degree

n=

3)

were interpolated since they lead to a good compromise between reducing the RMSE of the ﬁt and

avoiding overﬁtting. From the gradient generated by the isoclines approximation, we observe not

only a gradual and smooth evolution between the four isoclines, but also a clear separation between

them, with particular emphasis on

L3

and

L4

. This property was not clear in the previous diagram

trees. Long-range memory effects are diluted when handling TS simultaneously with long and short

time scales, but the

(a

,

b)

locus unveils properties that reﬂect distinct classes of phenomena, and their

identiﬁcation needs further study.

We should also note that the low-frequency range covers time scales between 1 year and several

decades. So, the results demonstrate the presence of phenomena inﬂuencing tides during long periods

of time. The limits of such time scales remain to be explored, since present-day TS do not include

sufﬁciently long records. In other words, the results point toward obtaining longer TS, since relevant

phenomena may be not completely captured with the available data.

Entropy 2017,19, 390 15 of 18

log(a)

0.5 1 1.5 2 2.5 3

b

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17 18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58 59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

RMSE for PL ﬁt to ˆ

Si(f)

242

1408

2574

3741

4907

6073

7239

V11

V12

V21

V22

L1

L2

L3

L4

n

1234567

RMSE for polynom ial ﬁt to Li

0.02

0.03

0.04

0.05

Figure 11.

Locus of the (

a

,

b

) parameters and the polynomial (degree

n=

3) ﬁt to

Li

,

i=

1,

···

, 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, ˆ

Si(f),i=1, ··· , 75.

6. Conclusions

Tidal TS embed rich information contributed by a plethora of factors at different scales. Local

features—namely, geography (e.g., shape of the shoreline, bays, estuaries, and inlets, presence of

shallow waters) and weather (e.g., wind, atmospheric pressure, and rainfall/river discharge)—may

have a non-negligible effect on tides. This means that, for example, geographically close stations may

register quite different tidal behavior. Knowing the relationships between worldwide distributed

stations may be important to better understanding tides. However, disclosing such relationships

requires powerful tools for TS analysis that are able to unveil all details embedded in the data.

A method for analyzing tidal TS that combines time–frequency signal processing and HC was

proposed. Real world information from worldwide tidal stations was pre-processed to obtain TS

with reliable quality. Frequency and time–frequency data were generated by means of the MM, FrFT,

and CWT. The JSD was used to measure dissimilarities, and the HC was applied for visualizing

information. PL functions were adopted for investigating the long-range dynamics of tides. Numerical

analysis showed that the combination of CWT and HC leads to a good graphical representation of

the relationships between tidal TS. The two distinct perspectives of study reveal similar regularities

embedded into the raw TS and motivate their adoption with other geophysical information.

Acknowledgments:

The authors acknowledge the University of Hawaii Sea Level Center (http://uhslc.soest.hawaii.

edu/) for the data used in this paper. This work was partially supported by FCT (“Fundção para a Ciência e Tecnologia,

Portugal”) funding agency, under the reference Projeto LAETA–UID/EMS/50022/2013.

Author Contributions: These authors contributed equally to this work.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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