Spectral approach for time series analysis
Inventory Model With Gamma Distribution
Hadi Sumadibrata, Ismail
Bin Mohd
642
Accuracy Analysis Of Naive Bayesian
Ruslam, Armin Lawi, And
Sri Astuti Thamrin
649
A New Method For Generating Fuzzy Rules From Training Data
And Its Application In Financial Problems
Agus Maman Abadi,
Subanar, Widodo, Samsubar
Saleh
655
The Application Of Laws Of Large Numbers In Convergence
Concept In Probability And Distribution
Georgina M. Tinungki
662
An Empirical Bayes Approach for Binary Response Data in
Small Area Estimation
Dian Handayani, Noor
Akma Ibrahim, Khairil A.
Notodiputro, MOhd. Bakri
Adam
669
Statistical Models For Small Area Estimation
Khairil A Notodiputro,
Anang Kurnia, and Kusman
Sadik
677
Maximum Likelihood Estimation For The Non-Separable Spatial
Unilateral Autoregressive Model
Norhashidah Awang,
Mahendran Shitan
685
Small Area Estimation Using Natural Exponential Families With
Quadratic Variance Function (Nef-Qvf) For Binary Data
Kismiantini
691
Using An Extended And Ensemble Kalman Filter Algorithm For
The Training Of Feedforward Neural Network In Time Series
Forecasting
Zaqiatud Darojah, M. Isa
Irawan, And Erna Apriliani
696
Estimation Of Outstanding Claims Liability And Sensitivity
Analysis: Probabilistic Trend Family (PTF) Model
Arif Herlambang, Dumaria
R Tampubolon
704
Expected Value Of Shot Noise Processes
Suyono
711
Modelling Malaysian Wind Speed Data Via Two Paramaters
Weibull
Nur Arina Basilah Kamisan,
Yong Zulina Zubairi, Abdul
Ghapor Hussin, Mohd.
Sahar Yahya
718
Application Of Latin Hypercube Sampling And Monte Carlo
Simulation Methods: Case Study The Reliability Of Stress
Intensity Factor And Energy Release Rate Of Indonesian
Hardwoods
Yosafat Aji Pranata And
Pricillia Sofyan Tanuwijaya
726
The Development Of Markov Chain Monte Carlo (Mcmc)
Algorithm For Autologistic Regression Parameters Estimation
Suci Astutik, Rahma
Fitriani, Umu Sa’adah, And
Agustin Iskandar
734
A Note About Dh-Fever Estimation With ARIMAX Models
Elly Ana, Dwi Atmono
Agus W
741
Evaluation Of Additive-Innovational Outlier Identification
Procedure For Some Bilinear Models
I
745
Anti-Spam Filter
ix
smail, M.I., Mohamed, I.B.,
Yahya, M.S.
Model By Spectral Regression Methods
Iriawan, Suhartono
Application Of Cluster Analysis To Developing Core Collection
In Plant Genetic Resources
Sutoro
875
Small Area Estimation With Time And Area Effects Using A
Dynamic Linear Model
Kusman Sadik And Khairil
Anwar Notodiputro
880
Statistical Analysis Of Wind Direction Data
Ahmad Mahir Razali, Arfah
Ahmad, Azami Zaharim
And Kamaruzzaman Sopian
886
Generalized Additive Mixed Models in Small Area Estimation
Anang Kurnia, Khairil A.
Notodiputro, Asep
Saefuddin, I Wayan
Mangku
891
Kernel Principal Component Analysis In Data Visualization
Ismail Djakaria, Suryo
Guritno, Sri Haryatmi
898
GARCH Models And The Simulations
Nelson Nainggolan, Budi
Nurani Ruchjana And
Sutawanir Darwis
906
Rainfall Prediction Using Bayesian Network
Hera Faizal Rachmat, Aji
Hamim Wigena, and Erfiani
911
Identifying Item Bias Using The Simple Volume Indices And
Multidimensional Item Response Theory Likelihood Ratio
(Irt-Lr) Test
Heri Retnawati
916
Ordinary Kriging And Inverse Distance Weighting For Mapping
Soil Phosphorus In Paddy Field
Mohammad Masjkur,
Muhammad Nuraidi and
Chichi Noviant
924
K-Means Clustering Visualization
On Agriculture Potential Data
For Villages In Bogor Using Mapserver
Imas S. Sitanggang, Henri
Harianja, and Lailan
Syaufina
932
Some Methods To Estimate The Number Of Components In A
Mixture
M. A. Satyawan, A. H.
Wigena, Erfiani
941
A Probabilistic Model For Finding A Repeat
Triplet Region In DNA Sequence
Application Of Spherical Harmonics In Determination Of Tec
Using Gps Observable
Tigor Nauli
947
Mardina Abdullah, Siti
Aminah Bahari, Baharudin
Yatim, Azami Zaharim,
Ahmad Mahir Razali
954
Testing Structure Correlation Of Global Market By Statistic Vvsv
Erna Tri Herdiani, and
Maman A. Djauhari
961
Exploring the MAUP from a spatial perspective
Gandhi Pawitan
967
Estimation of RCA(1) Model using EF:
A new procedure and its robustness
1Norli Anida Abdullah,
2Ibrahim Mohamed,
3Shelton Peiris
996
Second Order Linear Elliptic Operators
In The Unit Square
Abdul Rouf Alghofari
xi
1008
POSTER
Study Of Fractional Factorial Split-Plot Experiment
Sri Winarni, Budi Susetyo,
and Bagus Sartono
1012
Improving Model Performance For Predicting Poverty Village
Category Using Neighborhood Information In Bogor
Bagus Sartono, Utami Dyah
S, and Zulhelmi Thaib
1019
Ammi Models On Count Data: Log-Bilinear Models
Alfian Futuhul Hadi
H. Ahmad Ansori Mattjik
I Made Sumertajaya
Halimatus Sa’diyah
Budi Nurani R , and
Kartlos J. Kachiashvili
1026
Atje Setiawan A. ,
Retantyo Wardoyo , Sri
Hartati , and Agus Harjoko
1045
Validation Of Training Model
For Robust Tests Of Spread
Teh Sin Yin, and Abdul
Rahman Othman
1056
Spectral Approach For Time Series Analysis
Kusman Sadik
1063
The ACE Algorithm for Optimal Transformations in Multiple
Regression
The Relation Between The Students’ Interaction And The
Construction Of Mathematical Knowledge
Application of Auto Logistic Regression Spatial Model using
Variogram Based Weighting Matrix to Predict Poverty Village
Category
Kusman Sadik
1066
Rini Setianingsih
1069
Utami Dyah Syafitri, Bagus
Sartono, Vinda Pratama
1075
Prediction Of Oil Production Using Non Linear Regression By
Sdpro Software
(Special Program Package)*)
An Implementation Of Spatial Data Mining
Using Spatial Autoregressive (Sar) Model
For Education Quality Mapping At West Java*)
xii
1038
The 3"* International Conference on Mathematics and Statistics (ICoMS-3)
Institut Pertanian Bogor, Indonesia, 5-6 August 2008
SPECTRAL APPROACH FOR TIME SERIES ANALYSIS
Kusman Sadik
Department of Statistics, Institut Pertanian Bogor Jl. Meranti, Wing 22 Level 4,
Kampus IPB Darmaga, Bogor 16680 – Indonesia
e-mail : kusmans(a),ipb.ac.id
Key Words: Spectral analysis, seasonal adjustment procedures, cross spectral, fuzzy logic systems, input signals,
fourier transform, fractal space-time.
1. Introduction
The first appearance of spectral analysis in the study of macroeconomic time series dates motivated by the
requirement of a more insightful knowledge of the series structure and supported by the contemporaneous
progress in spectral estimation and computation. The first works focused on the problem of seasonal
adjustment procedures and on the general spectral structure of economic data. Cross spectral methods were
pointed out from the outset as being important in discovering and interpreting the relationships between
economic variables. After the early years, the range of application of such analysis was extended to the study of
other econometric issues, among which the controversial trend-cycle separation, the related problem of
business cycles extraction and the analysis of co-movements among series, usefiil in the study of international
business cycles. In particular, cross spectral analysis allows a detailed study of the correlation among series. An
empirical investigation about the possibility that the market is in a self-organized critical state (SOC) show a
power law behaviour in the avalanche size, duration and laminar times during high activity period (Bartolozzi,
Leinweber and Thomas, 2005).
2. Mathematical Models for Fractal Fluctuations and Deterministic
The larger scale fluctuations incorporate the smaller scale fluctuations as intemal fine scale structure and
may be visualized as a continuum of eddies (waves). Dynamical systems in nature are basically fiizzy logic
systems integrating a unified v^ole communicating network of input signals (perturbations) with
self-organized ordered two-way information transpot between the larger and smaller scales. Self-similar
structures are generated by iteration (repetition) of simple rules for growth processes on all scales of space and
time. Such iterative processes are simulated mathematically by numerical computations such as X„+i = F(^«)
where X„+i the value of the variable X at (« + i f " computational step is a fiinction F of its earlier value X„. The
spectrum of { x } is defined to be the Fourier transform of Yx(/^),
3. Model Concepts of Universal Spectrum of Fluctuations
A general systems theory was first developed to quantify the observed fractal space-time fluctuations in
turbulent fluid flows. The model concepts are independent of the exact details of the physical, chemical or
other properties and therefore applicable to all dynamical systems. The observed inverse power law form for
power spectra implies that the fi-actal fluctuations can be visualized to result from a hierarchical eddy
continuum structure for the overall pattern. Starting from this simple basic concept that large scale eddies
form as envelopes of intemal small scale eddy circulations the following important model predictions are
derived.
Time-fi-equency approaches which represent the fi*equency content of a series, while keeping the time
description parameter to give a three-dimensional time-dependent spectrum will not be tackled in this paper.
This is for essentially two reasons: first, they would require more than a simple section; second, and more
importantly, because evolutionary spectral methods and wavelets are suitable when dealing wdth very long
time series, like those found in geophysics, astrophysics, neurosciences or finance. But their application to short
series the norm in macroeconomics is difficult and may give unstable parameter-dependent results. For such
series, traditional spectral analysis is probably more suitable.
1063
Let Zt is time series stationer process with autocorrelation γk. Its Fourier
transformation is (Yujima, 1998) :
f ( w) =
1
2π
∞
∑ γ k e −iwk
k = −∞
Function f(w) is continu function non-negative that representative of spectrum for
autocorrelation function γk. Spectrum of f(w) and autocorrelation function γk are unique
Fourier transformation. Generally, spectrum of ARMA(p,q) process can be written
(West, 1999):
f ( w) =
σ a2θ q (e − iw )θ p ( eiw )
2πφ p (e − iw )φ p (e iw )
Figure 1: Plots of the spectrum of MA(1) processes (0 = 0.5 for the left figure and 0 = - 0.5
right figure) – Simulated Data
for the
Figure 1 plots the spectrum of MA(1) processes with positive and negative coefficients. When 0 > 0,
we see that the spectrum is high for low frequencies and low for high frequencies. When 0 < 0, we observe
the opposite. This is because when 0 is positive, we have positive one lag correlation which makes the series
smooth with only small contribution from high frequency (say, day to day) components. When 0 is negative,
we have negative one lag correlation, therefore the series fluctuates rapidly about its mean value.
Figure 3: Window Functions and Their Frequency Response (Simulated Data). The rectangular
(dashedline), Hanning (dotted line) and Hamming (full line) time windows (left panel) and their
1064
respective Fourier transforms (right panel). For the latter, the number of points N = 15 has been
chosen rather small to emphasize the differences. Note in the zoom (right panel, inset) the reduced side
lobe amplitude and leakage of the Hanning and Hamming windows with respect to the rectangular one,
the Hamming window performing better in the first side lobe.
The only way to prevent this effect, would be to choose T (or equivalently N) as a multiple of
the largest period that is likely to occur. Unfortunately, this is feasible only if we have some idea of
the frequencies involved in the process and would in any case entail some loss of data at one or both
sample ends. As for the cutoff fr equencies ν l = k l /(N∆t) and ν h = k h /(N∆t), given the value of N,
they must be chosen to be multiples of T −1 , otherwise the filter does not completely remove the zero
frequency component (i.e. the signal mean) and cannot help in eliminating unit roots.
4.
Conclusion
This paper highlighted the main features of spectral analysis and their practical application. After a
general theoretical introduction, we approached the issue of filtering for the extraction of particular
components, mostly those related to the business cycle. In fact one of the advantages of the method is that
it allows a quantitative definition of the cycle, and the extraction of long, medium or short term
components, according to the researcher’s wish. Then, we sketched the theory and practice of cross
spectral analysis introducing some typical concepts, like coherency and phase spectrum, which may provide
some essential information, complementary to that given by time domain methods.
1065
Hadi Sumadibrata, Ismail
Bin Mohd
642
Accuracy Analysis Of Naive Bayesian
Ruslam, Armin Lawi, And
Sri Astuti Thamrin
649
A New Method For Generating Fuzzy Rules From Training Data
And Its Application In Financial Problems
Agus Maman Abadi,
Subanar, Widodo, Samsubar
Saleh
655
The Application Of Laws Of Large Numbers In Convergence
Concept In Probability And Distribution
Georgina M. Tinungki
662
An Empirical Bayes Approach for Binary Response Data in
Small Area Estimation
Dian Handayani, Noor
Akma Ibrahim, Khairil A.
Notodiputro, MOhd. Bakri
Adam
669
Statistical Models For Small Area Estimation
Khairil A Notodiputro,
Anang Kurnia, and Kusman
Sadik
677
Maximum Likelihood Estimation For The Non-Separable Spatial
Unilateral Autoregressive Model
Norhashidah Awang,
Mahendran Shitan
685
Small Area Estimation Using Natural Exponential Families With
Quadratic Variance Function (Nef-Qvf) For Binary Data
Kismiantini
691
Using An Extended And Ensemble Kalman Filter Algorithm For
The Training Of Feedforward Neural Network In Time Series
Forecasting
Zaqiatud Darojah, M. Isa
Irawan, And Erna Apriliani
696
Estimation Of Outstanding Claims Liability And Sensitivity
Analysis: Probabilistic Trend Family (PTF) Model
Arif Herlambang, Dumaria
R Tampubolon
704
Expected Value Of Shot Noise Processes
Suyono
711
Modelling Malaysian Wind Speed Data Via Two Paramaters
Weibull
Nur Arina Basilah Kamisan,
Yong Zulina Zubairi, Abdul
Ghapor Hussin, Mohd.
Sahar Yahya
718
Application Of Latin Hypercube Sampling And Monte Carlo
Simulation Methods: Case Study The Reliability Of Stress
Intensity Factor And Energy Release Rate Of Indonesian
Hardwoods
Yosafat Aji Pranata And
Pricillia Sofyan Tanuwijaya
726
The Development Of Markov Chain Monte Carlo (Mcmc)
Algorithm For Autologistic Regression Parameters Estimation
Suci Astutik, Rahma
Fitriani, Umu Sa’adah, And
Agustin Iskandar
734
A Note About Dh-Fever Estimation With ARIMAX Models
Elly Ana, Dwi Atmono
Agus W
741
Evaluation Of Additive-Innovational Outlier Identification
Procedure For Some Bilinear Models
I
745
Anti-Spam Filter
ix
smail, M.I., Mohamed, I.B.,
Yahya, M.S.
Model By Spectral Regression Methods
Iriawan, Suhartono
Application Of Cluster Analysis To Developing Core Collection
In Plant Genetic Resources
Sutoro
875
Small Area Estimation With Time And Area Effects Using A
Dynamic Linear Model
Kusman Sadik And Khairil
Anwar Notodiputro
880
Statistical Analysis Of Wind Direction Data
Ahmad Mahir Razali, Arfah
Ahmad, Azami Zaharim
And Kamaruzzaman Sopian
886
Generalized Additive Mixed Models in Small Area Estimation
Anang Kurnia, Khairil A.
Notodiputro, Asep
Saefuddin, I Wayan
Mangku
891
Kernel Principal Component Analysis In Data Visualization
Ismail Djakaria, Suryo
Guritno, Sri Haryatmi
898
GARCH Models And The Simulations
Nelson Nainggolan, Budi
Nurani Ruchjana And
Sutawanir Darwis
906
Rainfall Prediction Using Bayesian Network
Hera Faizal Rachmat, Aji
Hamim Wigena, and Erfiani
911
Identifying Item Bias Using The Simple Volume Indices And
Multidimensional Item Response Theory Likelihood Ratio
(Irt-Lr) Test
Heri Retnawati
916
Ordinary Kriging And Inverse Distance Weighting For Mapping
Soil Phosphorus In Paddy Field
Mohammad Masjkur,
Muhammad Nuraidi and
Chichi Noviant
924
K-Means Clustering Visualization
On Agriculture Potential Data
For Villages In Bogor Using Mapserver
Imas S. Sitanggang, Henri
Harianja, and Lailan
Syaufina
932
Some Methods To Estimate The Number Of Components In A
Mixture
M. A. Satyawan, A. H.
Wigena, Erfiani
941
A Probabilistic Model For Finding A Repeat
Triplet Region In DNA Sequence
Application Of Spherical Harmonics In Determination Of Tec
Using Gps Observable
Tigor Nauli
947
Mardina Abdullah, Siti
Aminah Bahari, Baharudin
Yatim, Azami Zaharim,
Ahmad Mahir Razali
954
Testing Structure Correlation Of Global Market By Statistic Vvsv
Erna Tri Herdiani, and
Maman A. Djauhari
961
Exploring the MAUP from a spatial perspective
Gandhi Pawitan
967
Estimation of RCA(1) Model using EF:
A new procedure and its robustness
1Norli Anida Abdullah,
2Ibrahim Mohamed,
3Shelton Peiris
996
Second Order Linear Elliptic Operators
In The Unit Square
Abdul Rouf Alghofari
xi
1008
POSTER
Study Of Fractional Factorial Split-Plot Experiment
Sri Winarni, Budi Susetyo,
and Bagus Sartono
1012
Improving Model Performance For Predicting Poverty Village
Category Using Neighborhood Information In Bogor
Bagus Sartono, Utami Dyah
S, and Zulhelmi Thaib
1019
Ammi Models On Count Data: Log-Bilinear Models
Alfian Futuhul Hadi
H. Ahmad Ansori Mattjik
I Made Sumertajaya
Halimatus Sa’diyah
Budi Nurani R , and
Kartlos J. Kachiashvili
1026
Atje Setiawan A. ,
Retantyo Wardoyo , Sri
Hartati , and Agus Harjoko
1045
Validation Of Training Model
For Robust Tests Of Spread
Teh Sin Yin, and Abdul
Rahman Othman
1056
Spectral Approach For Time Series Analysis
Kusman Sadik
1063
The ACE Algorithm for Optimal Transformations in Multiple
Regression
The Relation Between The Students’ Interaction And The
Construction Of Mathematical Knowledge
Application of Auto Logistic Regression Spatial Model using
Variogram Based Weighting Matrix to Predict Poverty Village
Category
Kusman Sadik
1066
Rini Setianingsih
1069
Utami Dyah Syafitri, Bagus
Sartono, Vinda Pratama
1075
Prediction Of Oil Production Using Non Linear Regression By
Sdpro Software
(Special Program Package)*)
An Implementation Of Spatial Data Mining
Using Spatial Autoregressive (Sar) Model
For Education Quality Mapping At West Java*)
xii
1038
The 3"* International Conference on Mathematics and Statistics (ICoMS-3)
Institut Pertanian Bogor, Indonesia, 5-6 August 2008
SPECTRAL APPROACH FOR TIME SERIES ANALYSIS
Kusman Sadik
Department of Statistics, Institut Pertanian Bogor Jl. Meranti, Wing 22 Level 4,
Kampus IPB Darmaga, Bogor 16680 – Indonesia
e-mail : kusmans(a),ipb.ac.id
Key Words: Spectral analysis, seasonal adjustment procedures, cross spectral, fuzzy logic systems, input signals,
fourier transform, fractal space-time.
1. Introduction
The first appearance of spectral analysis in the study of macroeconomic time series dates motivated by the
requirement of a more insightful knowledge of the series structure and supported by the contemporaneous
progress in spectral estimation and computation. The first works focused on the problem of seasonal
adjustment procedures and on the general spectral structure of economic data. Cross spectral methods were
pointed out from the outset as being important in discovering and interpreting the relationships between
economic variables. After the early years, the range of application of such analysis was extended to the study of
other econometric issues, among which the controversial trend-cycle separation, the related problem of
business cycles extraction and the analysis of co-movements among series, usefiil in the study of international
business cycles. In particular, cross spectral analysis allows a detailed study of the correlation among series. An
empirical investigation about the possibility that the market is in a self-organized critical state (SOC) show a
power law behaviour in the avalanche size, duration and laminar times during high activity period (Bartolozzi,
Leinweber and Thomas, 2005).
2. Mathematical Models for Fractal Fluctuations and Deterministic
The larger scale fluctuations incorporate the smaller scale fluctuations as intemal fine scale structure and
may be visualized as a continuum of eddies (waves). Dynamical systems in nature are basically fiizzy logic
systems integrating a unified v^ole communicating network of input signals (perturbations) with
self-organized ordered two-way information transpot between the larger and smaller scales. Self-similar
structures are generated by iteration (repetition) of simple rules for growth processes on all scales of space and
time. Such iterative processes are simulated mathematically by numerical computations such as X„+i = F(^«)
where X„+i the value of the variable X at (« + i f " computational step is a fiinction F of its earlier value X„. The
spectrum of { x } is defined to be the Fourier transform of Yx(/^),
3. Model Concepts of Universal Spectrum of Fluctuations
A general systems theory was first developed to quantify the observed fractal space-time fluctuations in
turbulent fluid flows. The model concepts are independent of the exact details of the physical, chemical or
other properties and therefore applicable to all dynamical systems. The observed inverse power law form for
power spectra implies that the fi-actal fluctuations can be visualized to result from a hierarchical eddy
continuum structure for the overall pattern. Starting from this simple basic concept that large scale eddies
form as envelopes of intemal small scale eddy circulations the following important model predictions are
derived.
Time-fi-equency approaches which represent the fi*equency content of a series, while keeping the time
description parameter to give a three-dimensional time-dependent spectrum will not be tackled in this paper.
This is for essentially two reasons: first, they would require more than a simple section; second, and more
importantly, because evolutionary spectral methods and wavelets are suitable when dealing wdth very long
time series, like those found in geophysics, astrophysics, neurosciences or finance. But their application to short
series the norm in macroeconomics is difficult and may give unstable parameter-dependent results. For such
series, traditional spectral analysis is probably more suitable.
1063
Let Zt is time series stationer process with autocorrelation γk. Its Fourier
transformation is (Yujima, 1998) :
f ( w) =
1
2π
∞
∑ γ k e −iwk
k = −∞
Function f(w) is continu function non-negative that representative of spectrum for
autocorrelation function γk. Spectrum of f(w) and autocorrelation function γk are unique
Fourier transformation. Generally, spectrum of ARMA(p,q) process can be written
(West, 1999):
f ( w) =
σ a2θ q (e − iw )θ p ( eiw )
2πφ p (e − iw )φ p (e iw )
Figure 1: Plots of the spectrum of MA(1) processes (0 = 0.5 for the left figure and 0 = - 0.5
right figure) – Simulated Data
for the
Figure 1 plots the spectrum of MA(1) processes with positive and negative coefficients. When 0 > 0,
we see that the spectrum is high for low frequencies and low for high frequencies. When 0 < 0, we observe
the opposite. This is because when 0 is positive, we have positive one lag correlation which makes the series
smooth with only small contribution from high frequency (say, day to day) components. When 0 is negative,
we have negative one lag correlation, therefore the series fluctuates rapidly about its mean value.
Figure 3: Window Functions and Their Frequency Response (Simulated Data). The rectangular
(dashedline), Hanning (dotted line) and Hamming (full line) time windows (left panel) and their
1064
respective Fourier transforms (right panel). For the latter, the number of points N = 15 has been
chosen rather small to emphasize the differences. Note in the zoom (right panel, inset) the reduced side
lobe amplitude and leakage of the Hanning and Hamming windows with respect to the rectangular one,
the Hamming window performing better in the first side lobe.
The only way to prevent this effect, would be to choose T (or equivalently N) as a multiple of
the largest period that is likely to occur. Unfortunately, this is feasible only if we have some idea of
the frequencies involved in the process and would in any case entail some loss of data at one or both
sample ends. As for the cutoff fr equencies ν l = k l /(N∆t) and ν h = k h /(N∆t), given the value of N,
they must be chosen to be multiples of T −1 , otherwise the filter does not completely remove the zero
frequency component (i.e. the signal mean) and cannot help in eliminating unit roots.
4.
Conclusion
This paper highlighted the main features of spectral analysis and their practical application. After a
general theoretical introduction, we approached the issue of filtering for the extraction of particular
components, mostly those related to the business cycle. In fact one of the advantages of the method is that
it allows a quantitative definition of the cycle, and the extraction of long, medium or short term
components, according to the researcher’s wish. Then, we sketched the theory and practice of cross
spectral analysis introducing some typical concepts, like coherency and phase spectrum, which may provide
some essential information, complementary to that given by time domain methods.
1065