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Module description - Stochastic Processes and Time Series
(Stochastische Prozesse und Zeitreihen)

ECTS 3.0
Specification Modeling stochastic processes and understanding time series
Level Advanced
Content A stochastic process is the mathematical description of temporally (or spatially) ordered random processes. The theory of stochastic processes is an extension of probability theory.
Time-ordered data are analyzed, characterized, and modeled as a stochastic process. Extreme value analysis is used to describe and model the recurrence of extreme values within a process.
Finally, the modelling of a special category of stochastic processes is discussed: the Markov processes.
Learning outcomes Understanding time series analysis
Students recognize time series and can characterize them statistically (expected value, variance, trend, stationarity, ergodicity, correlation structure).
They can characterize the relationships between several time series (independence, correlation), and they can decompose time series into suitable components.

Model stochastic processes
Students will be able to model general stochastic processes using different approaches (moving average models, autoregressive models, and ARMA) and predict future outcomes. They can model components of time series using stochastic processes.
The studentts know specific stochastic processes (Bernoulli process, Gaussian process, Poisson process) and their areas of application. They will be able to apply them in practical modelling problems.

Extreme value analysis
Students understand the concept of extreme values in stochastic processes. They can determine the extreme values of a stochastic process using data, and model them using extreme value distributions.

Markov processes and chains.
Students understand Markov processes and their characterization by Markov chains. They can define Markov chains (transition matrix and initial state) for practical problems, and determine properties of the underlying processes (limiting distribution, absorption behavior). They can integrate a Markov chain into a simulation.

Hidden Markov Model
Students understand the extension of a Markov model to Hidden Markov models. They know their areas of application and can solve practical problems. They understand and apply common algorithms (Forward, Backward, Viterbi) in a target-oriented way.
Evaluation Mark
Built on the following competences Probability Modelling
Modultype Portfolio Module
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