Module also offered within study programmes:
General information:
Name:
Analysis of nonstationary signals
Course of study:
2019/2020
Code:
ZSDA-3-0250-s
Faculty of:
Szkoła Doktorska AGH
Study level:
Third-cycle studies
Specialty:
-
Field of study:
Szkoła Doktorska AGH
Semester:
0
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
dr hab. Dudek Anna (aedudek@agh.edu.pl)
Dyscypliny:
informatyka techniczna i telekomunikacja, inżynieria mechaniczna, matematyka
Module summary

The student will acquire knowledge and skills in the analysis of cyclostationary and almost cyclostationary signals, having numerous applications in telecommunications, mechanics, vibroacoustics and economics. In addition, the student will acquire skills in using various bootstrap methods and applying them to construct confidence intervals for the characteristics of these processes.

Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Skills: he can
M_U001 Student is able to analyze cyclostationary and almost cyclostationary series in the time and frequency domain SDA3A_U01, SDA3A_U02 Completion of laboratory classes
M_U002 Student is able to judge his/hers degree of understanding of the problem and to point out the missing elements of reasoning, he/she is able to look for solution in the recent scientific publications SDA3A_U01 Execution of laboratory classes
M_U003 Student is able to choose the appropriate method of analysis for considered real data of a nonstationary nature, he/ she knows how to look for solutions in scientific publications on the problem under consideration SDA3A_U01 Completion of laboratory classes
Knowledge: he knows and understands
M_W001 Student knows the basic concepts and theorems of the theory of nonstationary time series with a periodic and almost periodic structure SDA3A_W01 Examination
M_W002 Student knows the basic resampling methods for nonstationary data SDA3A_W02 Execution of laboratory classes
M_W003 Student knows the most important consistency results for resampling methods for nonstationary data SDA3A_W02 Examination
Number of hours for each form of classes:
Sum (hours)
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Prace kontr. przejść.
Lektorat
60 30 0 30 0 0 0 0 0 0 0 0
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Prace kontr. przejść.
Lektorat
Skills
M_U001 Student is able to analyze cyclostationary and almost cyclostationary series in the time and frequency domain - - + - - - - - - - -
M_U002 Student is able to judge his/hers degree of understanding of the problem and to point out the missing elements of reasoning, he/she is able to look for solution in the recent scientific publications - - + - - - - - - - -
M_U003 Student is able to choose the appropriate method of analysis for considered real data of a nonstationary nature, he/ she knows how to look for solutions in scientific publications on the problem under consideration - - - - - - - - - - -
Knowledge
M_W001 Student knows the basic concepts and theorems of the theory of nonstationary time series with a periodic and almost periodic structure + - + - - - - - - - -
M_W002 Student knows the basic resampling methods for nonstationary data + - - - - - - - - - -
M_W003 Student knows the most important consistency results for resampling methods for nonstationary data + - - - - - - - - - -
Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 150 h
Module ECTS credits 6 ECTS
Udział w zajęciach dydaktycznych/praktyka 60 h
Preparation for classes 35 h
przygotowanie projektu, prezentacji, pracy pisemnej, sprawozdania 20 h
Realization of independently performed tasks 30 h
Examination or Final test 2 h
Contact hours 3 h
Module content
Lectures (30h):
  1. A reminder of the basic concepts of stationary series analysis.

    strong and weak stationarity, trend, seasonality, autocovariance, ARMA models

  2. Spectral analysis of stationary time series.

    Herglotz’s theorem, spectral density function – examples, interpretation, estimation

  3. Cyclostationary series – introduction. Spectral analysis of cyclostationary series.

    definition of the periodic process, examples, applications in telecommunications and mechanics, basic characteristics, Fourier representation of the autocovariance function, the impact of various operations on periodic series, spectral density.

  4. PARMA time series

    definition of periodic ARMA model, discussion of all stages of the PARMA series analysis

  5. Generalizations of cyclostationary series.

    almost periodically correlated time series, generalized almost periodically correlated time series, their characteristics and basic definitions

  6. Introduction to resampling methods. Bootstrap for independent data.

    idea of bootstrap, consistency for the overall mean, examples of inconsistency, second order correctness

  7. Bootstrap confidence intervals.

    construction of bootstrap confidence intervals, different types of confidence intervals

  8. Block bootstrap methods for nonstationary time series.

    Moving Block Bootstrap, Nonoverlapping Block Bootstrap, Circular Block Bootstrap, Stationary Bootstrap, consistency results for basic characteristics of stationary time series

  9. Block length choice.

    asymptotic efficiency, comparison of asymptotic variances for different block bootstrap methods, adaptive methods of block length choice

  10. Block bootstrap methods for cyclostationary and almost cyclostationary time series.

    Generalized Seasonal Block Bootstrap, Extension of the Moving Block Bootstrap, consistency results for basic characteristics

  11. Bootstrap for ARMA models

    basic bootstrap algorithms, discussion of possible generalizations into periodic series

  12. Bootstrap in the frequency domain.

    Frequency Domain Bootstrap, bootstrap algorithm for spectral density, consistency results

  13. Subsampling method.

    subsampling method, consistency results for stationary time series, subsampling consistency results for different characteristics of periodic and almost periodic processes

  14. Subsampling – methods of subsample length choice.

    Minimum Volatility Method, method based on logarithm of quantile

  15. A new method for estimating spectral density of nonstationary series.

    Method for the spectral density estimation of almost cyclostationary time series with non-zero mean function

Laboratory classes (30h):
  1. Analisis of stationary sequences.

    Fitting ARMA series to real data, removing of trend and seasonality, ACF and PACF functions

  2. Spectral analysis of stationary time series.

    Generation of various types of stationary series, estimation of their spectral density functions, discussion of the leakage effect

  3. Cyclostationary processes

    Generation of various types of processes that are periodically correlated, estimation of their characteristics

  4. PARMA series

    Fitting the PARMA model to the real data: identification, estimation, forecast

  5. Generalizations of cyclostationary series.

    Generation of various types of processes that are almost periodically correlated, estimation their characteristics

  6. IID bootstrap. Bootstrap confidence intervals.

    Coding of IID bootstrap, analysis of bootstrap statistics distribution. Construction of different types of confidence intervals for various of parameters, discussion and interpretation of results

  7. Block bootstrap methods for stationary time series.

    Coding algorithms of block bootstrap methods for stationary series. Estimation of mean, construction of confidence intervals

  8. Methods for block length choice.

    Comparison of performance of different block bootstrap algorithms for various types of block lengths, calculation of actual coverage probabilities

  9. Bootstrapping ARMA models

    Construction of bootstrap parameter estimators of the ARMA model

  10. Block bootstrap methods for cyclostationary and almost cyclostationary time series.

    Coding algorithms of block bootstrap methods for cyclostationary and almost cyclostationary series. Estimation of characteristics in time and frequency domain, construction of confidence intervals

  11. Bootstrap in the frequency domain.

    Frequency Domain Bootstrap algorithm, application for real data

  12. Subsampling.

    The use of subsampling algorithm to estimate the overall mean of stationary series, comparison of results with block bootstrap methods

  13. Subsampling for nonstationary data.

    Use of subsampling algorithm for estimation of characteristics of cyclostationary series, comparison of results with block bootstrap methods

  14. Subsampling – methods of subsample length choice.

    Finding optimal block lengths using various adaptive methods

  15. Presentations of projects

    Defense of projects

Additional information
Teaching methods and techniques:
  • Lectures: Nie określono
  • Laboratory classes: Nie określono
Warunki i sposób zaliczenia poszczególnych form zajęć, w tym zasady zaliczeń poprawkowych, a także warunki dopuszczenia do egzaminu:

The condition of admission to the exam is having the positive grade from laboratory classes. If the student did not pass the laboratory classes, the student has the second opportunity to defend the project. The grade from laboratory classes is calculated on the basis of activity during course and grade from the project. Two unjustified absences during laboratory classes are possible.
The oral exam covers the issues presented in the lecture.

Participation rules in classes:
  • Lectures:
    – Attendance is mandatory: No
    – Participation rules in classes: Presence is not obligatory.
  • Laboratory classes:
    – Attendance is mandatory: Yes
    – Participation rules in classes: Presence is obligatory, 2 unjustified absences allowed.
Method of calculating the final grade:

The final grade is calculated as a weighted average of the exam and laboratory grades. The grade from laboratory exercises depends on the preparation and presentation of the solution to the problem. The weight of the evaluation of the laboratory = 0.5. The weight of the oral exam grade = 0.5.

Sposób i tryb wyrównywania zaległości powstałych wskutek nieobecności studenta na zajęciach:

In the case of absence the students are obligated to prepare the material for the next classes on their own.

Prerequisites and additional requirements:

Basic knowledge of statistics and programming.

Recommended literature and teaching resources:

Brockwell P.J. and Davis, R.A. (1991) Time Series Theory and Methods. 2nd Edition, Springer-Verlag, New York.

Hurd, H.L., Miamee, A.G. (2007). Periodically Correlated Random Sequences: Spectral. Theory and Practice. Wiley.

Lahiri, S.N. (2003) Resampling Methods for Dependent Data, Springer, New York.

Napolitano, A. (2012). Generalizations of Cyclostationary Signal Processing: Spectral Analysis and applications. Wiley-IEEE Press.

Scientific publications of module course instructors related to the topic of the module:

A.E. Dudek, J. Le±kow, E. Paparoditis and D. Politis (2014). A generalized block bootstrap for seasonal time series, J. Time Ser. Anal., 35, 89-114.

D. Dehay, A. Dudek and J. Le±kow (2014). Subsampling for continuous time nonstationary stochastic processes, J. Stat. Plan. Inf., 150, 142-158.

A.E. Dudek, M. Maiz and M. Elbadaoui (2014). Generalized Seasonal Block Bootstrap in frequency analysis of cyclostationary signals, Signal Process., 104C, 358-368.

A.E. Dudek (2015). Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series, Metrika, 78(3), 313-335

A.E. Dudek, E. Paparoditis and D. Politis (2016). Generalized Seasonal Tapered Block Bootstrap, Statistics and Probability Letters, 115, 27-35.

A.E. Dudek, H. Hurd and W. Wójtowicz (2016). Periodic autoregressive moving average methods based on Fourier representation of periodic coefficients, Wiley Interdisciplinary Reviews: Computational Statistics,
8(3), 130-149.

A.E. Dudek (2018). Block bootstrap for periodic characteristics of periodically correlated time series. Journal of Nonparametric Statistics, 30(1), 87-124.

D. Dehay, A.E. Dudek and M. Elbadaoui (2018). Bootstrap for almost cyclostationary processes with jitter effect, Digital Signal Processing, 73, 93-105.

Additional information:

None