This course deals with mathematical basics of stochastic process. After the introduction of probability theory, concept of stochastic processes are introduced. It includes definition, mathematical representation, and spectral representation. To understand the theoretical background, a project is developed. The project requests that students generate numerically a stochastic process with a certain shape of power spectrum and confirm the stochastic properties of the power spectrum. The results of project will be presented and discussed at the final class. Furthermore, students submit a resume, which includes the summary of the results.
To consider the uncertainties of the natural phenomena, stochastic techniques are very important in the field of civil engineering and disaster mitigation. This course deals with time-varying phenomena such as earthquake ground motions and discuss their stochastic model. The mathematical basics of stochastic process is introduced: (1) understanding probabilistic space, (2) understanding calculation of probability, (3) understanding the definition of stochastic process, (4) understanding representation of stochastic process in frequency domain using Fourier transform, (5) understanding physical meaning of power spectrum, (6) students can generate numerically stochastic process with any shapes of power spectrum using random number.
stochastic process, Fourier transform, Fourier Stieltjes integral, auto-correlation function, power spectrum, cross-correlation function, cross spectrum, Rice's representation, numerical calculation, random number
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
Important points will be written on blackboard. Details will be understood through the project.
|Course schedule||Required learning|
|Class 1||Introduction and about Project||introduction and about Project|
|Class 2||Basics of probability theory (probability space)||definitions of set theory, probability space, and probability|
|Class 3||Basics of probability theory (probability density function)||definition of probability variable and probability density function, calculation of probability|
|Class 4||Basics of stochastic process||definition of stochastic process|
|Class 5||Spectral representation of stochastic process||spectral representation, Fourier Stieltjes integral|
|Class 6||Power and phase spectra||definition of power and phase spectra|
|Class 7||Rice's representation and numerical generation of stochastic process||Rice's representation and numerical method to generate stochastic process|
|Class 8||Presentation and discussion on results of the project||Presentation and discussion on results of the project|
No textbook is assigned.
Leon Cohen, "Time Frequency Analysis: Theory and Applications," Prentice Hall, 1994.
Students' knowledge of the topics on this course will be assessed through presentation and resume of the project.
Programming skills and environment for numerical calculation are required.
The contents and schedule will be changed to adjust the students.