ArticlePDF Available

Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering

Authors:

Abstract and Figures

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.
Content may be subject to copyright.
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) =
N1
t=0
x(t)a(t)ej2πf 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
N1
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) =
N1
t=0
x(t)vk(t)ej2πf t, (3)
obtained with KSlepian sequences, vk(t), that verify [48]:
N1
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) =
N1
t=0
d2
k(f)|Yk(f)|2
N1
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
aR
,
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=p1jcot α=exp{−j[πsgn(sin α)/4 α/2]}
p|sin α|. (8)
For
a=
2
k
,
kZ
,
απk
, 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(τt), (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 [5355]:
Wψx(t)(a,b) = 1
aZ+
x(t)ψtb
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 =SnhW
ψxi(t)i(a,b)oWψxj(t)(a,b)
2
SWψxi(t)(a,b)
2SWψxj(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
xiI
and
xjJ
, and the
dissimilarity between clusters,
d(I
,
J)
, is calculated by the maximum, minimum, or average linkage,
given by:
dmax (I,J)=max
xiI,xjJdxi,xj, (15)
dmin (I,J)=min
xiI,xjJdxi,xj, (16)
dave (I,J)=1
kIkk Jk
xiI,xjJ
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
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:
(p) = w(p)
pw(p), (21)
where the function (p)is interpreted as a probability distribution. Finally, we feed the HC with the
matrix
= [δij ]
,
i
,
j=
1, ..., 75, where
δij =JSD(i
,
j)
represents the JSD between the normalized
vectors (i,j).
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(i
,
j)
, and
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
ei2πfctet2
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
h1
and
f=
0.042
h1
, 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
(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 49reveal 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(i
,
j)
, and
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, f24 h1.
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]h1
(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]h1
, 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 → L3L4
. 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 t 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 t 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.
References
1.
Takens, F. Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence, Warwick 1980;
Springer: Berlin, Germany, 1981; pp. 366–381.
2.
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.
Entropy 2017,19, 390 16 of 18
3.
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.
4.
Stein, E.M.; Shakarchi, R. Fourier Analysis: An Introduction; Princeton University Press: Princeton, NJ, USA, 2003.
5. Dym, H.; McKean, H. Fourier Series and Integrals; Academic Press: San Diego, CA, USA, 1972.
6.
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.
7.
Vautard, R.; Yiou, P.; Ghil, M. Singular-spectrum analysis: A toolkit for short, noisy chaotic signals. Phys. D
Nonlinear Phenom. 1992,58, 95–126.
8.
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.
9.
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.
10.
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.
11. Cohen, L. Time-Frequency Analysis; Prentice-Hall: Upper Saddle River, NJ, USA, 1995.
12.
Almeida, L.B. The fractional Fourier transform and time–frequency representations. IEEE Trans. Signal Process.
1994,42, 3084–3091.
13.
Sejdi´c, E.; Djurovi´c, I.; Stankovi´c, L. Fractional Fourier transform as a signal processing tool: An overview of
recent developments. Signal Process. 2011,91, 1351–1369.
14.
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.
15. Qian, S.; Chen, D. Joint time-frequency analysis. IEEE Signal Process. Mag. 1999,16, 52–67.
16. Kemao, Q. Windowed Fourier transform for fringe pattern analysis. Appl. Opt. 2004,43, 2695–2702.
17.
Hlubina, P.; Lu ˇnáˇcek, 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.
18. Qian, S.; Chen, D. Discrete Gabor Transform. IEEE Trans. Signal Process. 1993,41, 2429–2438.
19. Yao, J.; Krolak, P.; Steele, C. The generalized Gabor transform. IEEE Trans. Image Process. 1995,4, 978–988.
20. Mallat, S. A Wavelet Tour of Signal Processing; Academic Press: Burlington, VT, USA, 1999.
21.
Yan, R.; Gao, R.X.; Chen, X. Wavelets for fault diagnosis of rotary machines: A review with applications.
Signal Process. 2014,96, 1–15.
22.
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.
23.
Wu, Z.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method.
Adv. Adapt. Data Anal. 2009,1, 1–41.
24.
Zayed, A.I. Hilbert transform associated with the fractional Fourier transform. IEEE Signal Process. Lett.
1998,5, 206–208.
25.
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.
26.
Olsson, J.; Niemczynowicz, J.; Berndtsson, R. Fractal analysis of high-resolution rainfall time series. J. Geophys.
Res. Atmos. 1993,98, 23265–23274.
27.
Matsoukas, C.; Islam, S.; Rodriguez-Iturbe, I. Detrended fluctuation analysis of rainfall and streamflow time
series. J. Geophys. Res. 2000,105, 29165–29172.
28.
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.
29.
Donner, R.V.; Donges, J.F. Visibility graph analysis of geophysical time series: Potentials and possible pitfalls.
Acta Geophys. 2012,60, 589–623.
30. Machado, J.T. Fractional order description of DNA. Appl. Math. Model. 2015,39, 4095–4102.
31.
Lopes, A.M.; Machado, J.T. Integer and fractional-order entropy analysis of earthquake data series.
Nonlinear Dyn. 2016,84, 79–90.
32.
Machado, J.A.T.; Lopes, A.M. Analysis and visualization of seismic data using mutual information. Entropy
2013,15, 3892–3909.
Entropy 2017,19, 390 17 of 18
33.
Machado, J.A.; Mata, M.E.; Lopes, A.M. Fractional state space analysis of economic systems. Entropy
2015
,
17, 5402–5421.
34.
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.
35.
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.
36.
Malamud, B.D.; Turcotte, D.L. Self-affine time series: I. Generation and analyses. Adv. Geophys.
1999
,40,
1–90.
37.
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.
38.
Forootan, E.; Kusche, J. Separation of deterministic signals using independent component analysis (ICA).
Stud. Geophys. Geod. 2013,57, 17–26.
39.
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.
40.
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.
41. Machado, J.T.; Lopes, A.M. The persistence of memory. Nonlinear Dyn. 2015,79, 63–82.
42.
Pugh, D.; Woodworth, P. Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes;
Cambridge University Press: Cambridge, UK, 2014.
43. Shankar, D. Seasonal cycle of sea level and currents along the coast of India. Curr. Sci. 2000,78, 279–288.
44. Erol, S. Time-frequency analyses of tide-gauge sensor data. Sensors 2011,11, 3939–3961.
45.
Brigham, E.O. The Fast Fourier Transform and Its Applications; Number 517.443; Prentice Hall: Upper Sadlle
River, NJ, USA, 1988.
46.
Prieto, G.; Parker, R.; Thomson, D.; Vernon, F.; Graham, R. Reducing the bias of multitaper spectrum
estimates. Geophys. J. Int. 2007,171, 1269–1281.
47. Thomson, D.J. Spectrum estimation and harmonic analysis. Proc. IEEE 1982,70, 1055–1096.
48.
Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and uncertainty–V: The discrete case.
Bell Labs Tech. J. 1978,57, 1371–1430.
49.
Fodor, I.K.; Stark, P.B. Multitaper spectrum estimation for time series with gaps. IEEE Trans. Signal Process.
2000,48, 3472–3483.
50.
Smith-Boughner, L.; Constable, C. Spectral estimation for geophysical time-series with inconvenient gaps.
Geophys. J. Int. 2012,190, 1404–1422.
51.
Bultheel, A.; Sulbaran, H.E.M. Computation of the fractional Fourier transform. Appl. Comput. Harmon. Anal.
2004,16, 182–202.
52. Ozaktas, H.M.; Zalevsky, Z.; Kutay, M.A. The Fractional Fourier Transform; Wiley: Chichester, UK, 2001.
53. Machado, J.T.; Costa, A.C.; Quelhas, M.D. Wavelet analysis of human DNA. Genomics 2011,98, 155–163.
54.
Cattani, C. Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure; World Scientific:
Singapore, 2007; Volume 74,
55.
Stark, H.G. Wavelets and Signal Processing: An Application-Based Introduction; Springer Science & Business
Media: New York, NY, USA, 2005.
56.
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.
57.
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.
58.
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.
59. Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: Hoboken, NJ, USA, 2012.
60.
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.
Entropy 2017,19, 390 18 of 18
61.
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).
62.
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.
63. Lopes, A.M.; Machado, J.A.T. Fractional order models of leaves. J. Vib. Control 2014,20, 998–1008.
64. Baleanu, D. Fractional Calculus: Models and Numerical Methods; World Scientific: Singapore, 2012; Volume 3.
65.
Mandelbrot, B.B.; Van Ness, J.W. Fractional Brownian motions, fractional noises and applications. SIAM Rev.
1968,10, 422–437.
66. Keshner, M.S. 1/f noise. Proc. IEEE 1982,70, 212–218.
67. Mandelbrot, B.B. The Fractal Geometry of Nature; Macmillan: London, UK, 1983; Volume 173.
c
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/).
... 63 Hz was chosen to reflect frequencies affected by flow noise while 1500 Hz was chosen since it is outside of both the flow noise band, as well as the observer electrical (system) noise band. At 63 Hz, a pattern that follows the frequencies of tidal components is apparent, with peaks at the O1, K1, M2, and MK3 tidal frequencies (Lopes and Tenreiro Machado, 2017). At frequencies above 100 Hz, the periodogram in Fig. 7 shows an increase in spectral density at 24 h and 26 h. Figure 8 shows that the acoustic power at 1500 Hz follows tidal cycles, with peaks lining up with the O1, K1, and M2 frequencies. ...
Article
Full-text available
The main sources of noise in the Arctic Ocean are naturally occurring, rather than related to human activities. Sustained acoustic monitoring at high latitudes provides quantitative measures of changes in the sound field attributable to evolving human activity or shifting environmental conditions. A 12-month ambient sound time series (September 2018 to August 2019) recorded and transmitted from a real-time monitoring station near Gascoyne Inlet, Nunavut is presented. During this time, sound levels in the band 16-6400 Hz ranged between 10 and 135 dB re 1 lPa 2 /Hz. The average monthly sound levels follow seasonal ice variations with a dependence on the timing of ice melt and freeze-up and with higher frequencies varying more strongly than the lower frequencies. Ambient sound levels are higher in the summer during open water and quietest in the winter during periods of pack ice and shore fast ice. An autocorrelation of monthly noise levels over the ice freeze-up and complete cover periods reveal a $24 h periodic trend in noise power at high frequencies (>1000 Hz) caused by tidally driven surface currents in combination with increased ice block collisions or increased stress in the shore fast sea ice.
... There are also spectral peaks at 4.39 h (M6 in Figure 5) and 6.28 h (M4 in Figure 5) on OBS-1. M2, M4 and M6 correspond to the principal lunar semi-diurnal constituent and the first and the second overtide of principal lunar semi-diurnal constituent, respectively (Lopes and Tenreiro Machado, 2017). It can be argued that there are three small peaks related to M2 and other tidal constituents present on OBS-5 (M2? N2 and MU2 in Figure 5). ...
Article
Full-text available
Short duration events (SDEs) are reported worldwide from ocean-bottom seismometers (OBSs). Due to their high frequency (4–30 Hz) and short duration, they are commonly attributed to aseismic sources, such as fluid migration related processes from cold seeps, biological signals, or noise. We present the results of a passive seismic experiment that deployed an OBS network for 10-month (October 2015–July 2016) at an active seepage site on Vestnesa Ridge, West Svalbard continental margin. We characterize SDEs and their temporal occurrence using the conventional short-time-average over long-time-average approach. Signal periodograms show that SDEs have periodic patterns related to solar and lunar cycles. A monthly correlation between SDE occurrences and modelled tides for the area indicates that tides have a partial control on SDEs recorded over 10 months. The numbers of SDEs increase close to the tidal minima and maxima, although a correlation with tidal highs appears more robust. Large bursts of SDEs are separated by interim quiet cycles. In contrast, the periodicity analysis of tremors shows a different pattern, likely caused by the effect of tidally controlled underwater currents on the instrumentation. We suggest that SDEs at Vestnesa Ridge may be related to the dynamics of the methane seepage system which is characterized by a complex interaction between migration of deep sourced fluids, gas hydrate formation and seafloor gas advection through cracks. Our observation from this investigated area offshore west-Svalbard, is in line with the documentation of SDEs from other continental margins, where micro-seismicity and gas release into the water column are seemingly connected.
... The results obtained for each theory, that is, the values assessing the virus genetic code under the light of the Kolmogorov's complexity and the Shannon's information, are further processed by means of advanced computational representation techniques. The final visualization is obtained using the hierarchical clustering (HC) [15][16][17][18][19][20] and multidimensional scaling (MDS) techniques [21][22][23][24][25][26][27]. Three alternative representations, namely dendrograms, hierarchical trees and three-dimensional loci, are considered. ...
Article
This paper tackles the information of 133 RNA viruses available in public databases under the light of several mathematical and computational tools. First, the formal concepts of distance metrics, Kol-mogorov complexity and Shannon information are recalled. Second, the computational tools available presently for tackling and visualizing patterns embedded in datasets, such as the hierarchical clustering and the multidimensional scaling, are discussed. The synergies of the common application of the mathematical and computational resources are then used for exploring the RNA data, cross-evaluating the normalized compression distance, entropy and Jensen-Shannon divergence, versus representations in two and three dimensions. The results of these different perspectives give extra light in what concerns the relations between the distinct RNA viruses.
... The results obtained for each theory, that is, the values assessing the virus genetic code under the light of the Kolmogorov's complexity and the Shannon's information, are further processed by means of advanced computational representation techniques. The final visualization is obtained using the hierarchical clustering (HC) [15][16][17][18][19][20] and multidimensional scaling (MDS) techniques [21][22][23][24][25][26][27]. Three alternative representations, namely dendrograms, hierarchical trees and three-dimensional loci, are considered. ...
Article
Full-text available
This paper tackles the information of 133 RNA viruses available in public databases under the light of several mathematical and computational tools. First, the formal concepts of distance metrics, Kolmogorov complexity and Shannon information are recalled. Second, the computational tools available presently for tackling and visualizing patterns embedded in datasets, such as the hierarchical clustering and the multidimensional scaling, are discussed. The synergies of the common application of the mathematical and computational resources are then used for exploring the RNA data, cross-evaluating the normalized compression distance, entropy and Jensen–Shannon divergence, versus representations in two and three dimensions. The results of these different perspectives give extra light in what concerns the relations between the distinct RNA viruses.
... It should be noted that other indices can be used for quantifying complexity and different techniques can be adopted for dimensionality reduction, clustering and visualization. We can mention, for example, the use of time-frequency signal processing and hierarchical clustering for studying tidal data [35], the Lempel-Ziv complexity, sample entropy, signal harmonics power ratio, and fractal dimension for analyzing temperature time series [26], and information theory, fractional calculus and hierarchical clustering for studying art [20]. ...
Article
Full-text available
Art is the output of a complex system based on the human spirit and driven by several inputs that embed social, cultural, economic and technological aspects of a given epoch. A solid quantitative analysis of art poses considerable difficulties and reaching assertive conclusions is a formidable challenge. In this paper, we adopt complexity indices, dimensionality-reduction and visualization techniques for studying the evolution of Escher’s art. Grayscale versions of 457 artworks are analyzed by means of complexity indices and represented using the multidimensional scaling technique. The results are correlated with the distinct periods of Escher’s artistic production. The time evolution of the complexity and the emergent patterns demonstrate the effectiveness of the approach for a quantitative characterization of art.
Article
Full-text available
Parkinson's disease (PD) and Alzheimer's disease (AD) can coexist in severely affected; elderly patients. Since they have different pathological causes and lesions and consequently require different treatments; it is critical to distinguish PD-related dementia (PD-D) from AD. Conventional electroencephalograph (EEG) analysis has produced poor results. This study investigated the possibility of using relative wavelet energy (RWE) and wavelet coherence (WC) analysis to distinguish between PD-D patients; AD patients and healthy elderly subjects. In EEG signals; we found that low-frequency wavelet energy increased and high-frequency wavelet energy decreased in PD-D patients and AD patients relative to healthy subjects. This result suggests that cognitive decline in both diseases is potentially related to slow EEG activity; which is consistent with previous studies. More importantly; WC values were lower in AD patients and higher in PD-D patients compared with healthy subjects. In particular; AD patients exhibited decreased WC primarily in the γ band and in links related to frontal regions; while PD-D patients exhibited increased WC primarily in the α andβ bands and in temporo-parietal links. Linear discriminant analysis (LDA) of RWE produced a maximum accuracy of 79.18% for diagnosing PD-D and 81.25% for diagnosing AD. The discriminant accuracy was 73.40% with 78.78% sensitivity and 69.47% specificity. In distinguishing between the two diseases; the maximum performance of LDA using WC was 80.19%. We suggest that using a wavelet approach to evaluate EEG results may facilitate discrimination between PD-D and AD. In particular; RWE is useful for differentiating individuals with and without dementia and WC is useful for differentiating between PD-D and AD.
Article
Full-text available
This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners' modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis.
Article
High-frequency monitoring is currently a major component in the management and research of the coastal system responses to ongoing global changes. This monitoring is essential in tidal systems to address the multiscale variability of physico-chemical parameters. The analysis of the resulting multiscale, nonlinear, non-stationary and noisy time series requires adequate techniques; however, to date, there are no standardized methods. Spectral methods might be useful tools to reveal the main variability time scales, and thus their associated forcings. The most widely used methods in coastal systems are Lomb-Scargle Periodogram (LSP), Singular Spectral Analysis (SSA), Continuous Wavelet Transform (CWT), and Empirical Mode Decomposition (EMD), but their relevance for high-frequency, long-term records is still largely unexplored. In this article, these spectral methods are tested and compared using a high-frequency 10-yr turbidity dataset in the Gironde estuary. Advantages and limitations of each method are evaluated on the basis of five criteria: (1) efficiency for incomplete time series, (2) appropriateness for time-varying analysis, (3) ability to recognize processes without complementary environmental variables, (4) capacity to calculate the relative importance of forcings, and (5) capacity to identify long-term trends. SSA is the only analysis method to satisfy all the criteria, even with 70% missing data. Combining methods is also a promising strategy; i.e., SSA + LSP for better recognition of processes; CWT + SSA and EMD + CWT for short-term (seasonal) and long-term (>1 yr) analysis, respectively. The purpose of this methodological framework is to serve as a reference for future post-processing of data from monitoring programs in coastal waters.
Article
In processing of deep seismic reflection data, when the frequency band difference between the weak useful signal and noise both from the deep subsurface is very small and hard to distinguish, the traditional method of filtering will be limited. To solve this problem, we apply different spectral decomposition methods respectively to experimental data and real data and compare the results from these methods. Our purpose is to find an effective way to protect weak signals during processing deep seismic reflection data. The spectral decomposition method is based on the discrete Fourier transform, which uses the signal frequency-amplitude spectrum and other information to generate a high-resolution seismic image. Typically, it is used to identify the lateral distribution of media properties, solve spectrum changes within complex media and local phase instability and other issues, such as locating faults and small-scale complex fractures. S transform as a new time-frequency analysis method, which is a generalization of STFT developed by Stockwell in 1994, has the ability to automatically adjust the resolution. This method has been widely applied to exploration seismic, MT and other geophysical datasets in recent years. It has become one of the effective methods in noise suppressing during geophysical data processing. Comparing deep seismic reflection data with conventional oil reflection seismic data, in order to probe deep structure, this approach employs a large number of explosives, long observing systems, leading to a phenomenon that valid signals from the deep and noise are mixed together both in the time domain and frequency domain. Considering these characteristics of deep reflection data, this paper combines spectral decomposition with S transform technology. First we design a simple pulse function experimental data to confirm the validity of the S transform method. Then we illustrate the effect of spectral decomposition which is influenced by choosing frequency analysis methods and the transform window function which determines the strength of the resolving power of the method. On this basis, S transform spectrum decomposition is applied to a single channel of deep reflection seismic data and the stacked profile, then the application results of traditional transform spectral decomposition and S transform spectral decomposition are compared. Comparison of single channel data shows that compared with traditional spectral decomposition, the S transform spectral decomposition method is able to automatically adjust the resolution, accurately calibrate frequency component of weak signals at different times in deep reflection seismic data. Application to stacked profile data shows that the stacked profile results obtained by the S transform spectral decomposition and those from other spectral decomposition method are largely consistent, while the results of S transform spectral decomposition clearly depict the characteristics of low-frequency details which are superimposed by noise in original stacked profile. At the same time, it improves the resolution and enhances the phase axis continuity on the stacked profile. Comparison also clearly indicates that the phase axis on the resultant profile obtained by Gabor transform spectral decomposition is more broken, which is caused by fixed-length window function used by Gabor transform decomposition, in which the window length does not change with the signal frequency. In Gabor transform decomposition, the length of the window function parameters can only be selected from the start of processing and is set to a certain value, while the S transform spectral decomposition method chooses the variable length of the window function according to signal change. It can automatically adjust the frequency characteristics of the signal by the local window length to better characterize the details of each frequency range. Such an effect is very obvious in deep reflection seismic imaging. Our results show that the key of the spectral decomposition technique is to select the transform window function. The S transform spectral decomposition technology used in real deep reflection seismic data processing can effectively protect the weak low-frequency signals. It can effectively improve the signal to noise ratio and resolution of weak reflection signals from the deep subsurface, while depicting the characteristics of low-frequency details on the stacked section and ultimately obtaining better imaging results.
Article
Three methods of estimating the directional spectra of water waves are intercompared. The Maximum Likelihood Method (MLM) and the Maximum Entropy Method (MEM) require stationarity of the time series and yield only the frequency-direction spectra. The Wavelet Directional Method (WDM) does not require stationarity and also yields the wave number-direction spectra and is suitable for event analysis. The comparison includes three cases of wind-generated waves on a large lake and two cases of model-generated waves with different directional spreading. The comparisons of the frequency-direction spectra show that the Wavelet Directional Method yields the best estimates of the directional spectra.
Article
This paper studies the statistical distributions of worldwide earthquakes from year 1963 up to year 2012. A Cartesian grid, dividing Earth into geographic regions, is considered. Entropy and the Jensen–Shannon divergence are used to analyze and compare real-world data. Hierarchical clustering and multi-dimensional scaling techniques are adopted for data visualization. Entropy-based indices have the advantage of leading to a single parameter expressing the relationships between the seismic data. Classical and generalized (fractional) entropy and Jensen–Shannon divergence are tested. The generalized measures lead to a clear identification of patterns embedded in the data and contribute to better understand earthquake distributions.
Article
Leaves are mainly responsible for food production in vascular plants. Studying individual leaves can reveal important characteristics of the whole plant, namely its health condition, nutrient status, the presence of viruses and rooting ability. One technique that has been used for this purpose is Electrical Impedance Spectroscopy, which consists of determining the electrical impedance spectrum of the leaf. In this paper we use EIS and apply the tools of Fractional Calculus to model and characterize six species. Two modeling approaches are proposed: firstly, Resistance, Inductance, Capacitance electrical networks are used to approximate the leaves’ impedance spectra; afterwards, fractional-order transfer functions are considered. In both cases the model parameters can be correlated with physical characteristics of the leaves.