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Магистратура 2020/2021

Анализ и прогнозирование временных рядов: методы и приложения

Статус: Курс по выбору (Науки о данных)
Направление: 01.04.02. Прикладная математика и информатика
Когда читается: 2-й курс, 1, 2 модуль
Формат изучения: без онлайн-курса
Прогр. обучения: Науки о данных
Язык: русский
Кредиты: 8
Контактные часы: 56

Программа дисциплины


How to forecast rates of a national currency? How to identify an oncoming heart attack in good time? – the answers to these questions are associated with the problems (1) to predict a chaotic time series and (2) to reveal typical sequences in an observed time series, respectively. All these problems along with many others comprise the field of time series prediction. The course combines real-world applications with a strong theoretical background: the authors selected mathematical topics required to solve complex problems of actual practice. On the other hand, several topics focus on “mathematics of future” that is theories that will have become the basis of applications in the decades to come. The course starts with simple concepts and gradually works in more advanced applications. To be specific, the course deals with main models to examine and predict regular time series (exemplified by ARIMA and GARCH models), chaotic time series (predictive clustering, constructive neural networks, deep learning models and others) as well as with state-of-the-art approaches used to distinguish regular and chaotic time series, using observations of the time series at hand only; particular topics deal with the concepts of forecasting (time) and stationarity horizons. Applications considered range from econometrics problem to mobile health.
Цель освоения дисциплины

Цель освоения дисциплины

  • To introduce the theoretical foundations of Prediction theory for regular and chaotic time series.
  • To provide the students with practical skills of modelling real-world system.
  • To overview the current applications of decision-making support systems in logistics.
Планируемые результаты обучения

Планируемые результаты обучения

  • Understand fundamental concepts, advantages and limitations of correlation analysis.
  • Understand fundamental concepts of Linear regression analysis.
  • Understand fundamental concepts of Non-linear regression analysis.
  • Understand fundamental concepts of Neural networks and time series predictions.
  • Understand fundamental concepts of Predictive regression models.
  • Design and develop real-world systems for forecasting tasks using Predictive regression models.
  • Understand fundamental concepts of Maximum likelihood estimation of regression parameters.
  • Understand fundamental concepts of regular time series.
  • Analyse regular time series.
  • Understand fundamental concepts, advantages and limitations of АRIМА-models.
  • Understand fundamental concepts of non-linear and chaotic time series.
  • Analyse non-linear and chaotic time series and forecast them.
  • Understand fundamental concepts of Attractor reconstruction for chaotic time series.
  • Understand fundamental concepts, advantages and limitations of Predictive clustering.
  • Understand applications of predictive clustering.
  • Understand fundamental concepts of Stationarity horizon.
  • Understand fundamental concepts of bifurcation precursors and time series.
  • Understand predicitve complexity.
Содержание учебной дисциплины

Содержание учебной дисциплины

  • Correlation analysis.
    Causal connection. The Pearson's product-moment coefficient. Its properties. Statistical tests for Pearson's coefficient. The partial correlation coefficient. The coefficient of multiple correlation. Statistical tests for it.
  • Linear regression analysis.
    Theoretical and applied regression definitions. Gauss-Markov conditions. The least squares method. Properties of least squares estimators. BLUE. Remainder variance. Tests for parameters and line of regression. How to compare regression models? How to compare estimated parameters corresponding to different samples? The weighted least squares.
  • Non-linear regression analysis.
    The correlation ration. Its properties. Non-linear regression analysis. Linearized regression and its properties.
  • Neural networks and time series predictions.
    Constructive neural networks. NEAT-model. Deep learning and time series prediction.
  • Predictive regression models.
    Unconditional prediction. Conditional prediction. Autoregression models prediction.
  • Maximum likelihood estimation of regression parameters.
    Properties of maximum likelihood estimation. Maximum likelihood estimation for linear models. Statistical tests. Non-linear constraints.
  • Regular time series.
    Time series. Unit roots and co-integration. Autocorrelation and partial autocorrelation functions. Properties of AR(1)- and AR(2)- processes. Properties of moving average processes.
  • АRIМА-models.
    Identification of АRІМА-models. Method to estimate model parameters. Forecasting with АRIМА. GARCH-models.
  • Non-linear and chaotic time series. Forecasting horizon.
    Chaos “fingerprints”. Forecasting horizon. How to calculate forecasting horizon for a given time series? Multiplicative ergodic (Oseledets) theorem. An invariant measure of a dynamic system. Kolmogorov-Sinai entropy: series that generate information.
  • Attractor reconstruction for chaotic time series.
    Takens theory. Optimal reconstruction parameters. Properties of correlation integral. Limitations of non-linear dynamics algorithms. Econophysics.
  • Predictive clustering.
    One and multi-step ahead prediction. Clustering algorithms employed for predictive clustering. Association with invariant measure. Non-predictable observations. Similar time series. Relation tensor of time series.
  • Applications of predictive clustering.
    Mobile health. A plant to produce technical indicators for stock markets. Prediction of hash-tag popularity. Weather forecasting. Energy consumption prediction. Text of literature pieces as chaotic time series.
  • Stationarity horizon.
    Non-stationary time series. Definition of stationarity horizon. Empirical probability density function. How to compare two PDFs? Estimated minimum sample size. Horizon series. Its probability density function.
  • Bifurcation precursors and time series.
    The concept of bifurcation precursors (early-warning signs). Various approaches to identify pre-bifurcation states with the employment of time series.
  • Predicitve complexity.
    Predictability and learning. Learning a parameterized process. Beyond finite parameterization: general considerations and model process.
Элементы контроля

Элементы контроля

  • неблокирующий Intermediate task 1
    A practical tasks associated with regular time series.
  • неблокирующий Intermediate task 2
    A practical tasks associated with chaotic time series.
  • неблокирующий Exam
    The final exam is oral. The prerequisite for the course is a course in basic statistics.
  • неблокирующий Colloquium 1
  • неблокирующий Colloquium 2
Промежуточная аттестация

Промежуточная аттестация

  • Промежуточная аттестация (2 модуль)
    0.1 * Colloquium 1 + 0.1 * Colloquium 2 + 0.4 * Exam + 0.2 * Intermediate task 1 + 0.2 * Intermediate task 2
Список литературы

Список литературы

Рекомендуемая основная литература

  • Unpingco, J. Python for Signal Processing. – Springer International Publishing, 2014. – 128 pp.

Рекомендуемая дополнительная литература

  • Nonlinear time series : nonparametric and parametric methods, Fan, J., 2003