Random Processes (隨機過程)



Professor Ke-Sheng Cheng
E-mail: rslab@ntu.edu.tw
(This course will be taught in English.)



Almost all phenomena occurred to or experienced by us
involve some degree of randomness. Examples of such
phenomena
include occurrences of natural disasters (for examples, earth
quakes, typhoons, etc.), outbreaks of epidemic diseases,
spatial and temporal variations of storm rainfall, number of
transportation accidents occurred every year, just to name a
few. Understanding how such random phenomena occur and
modeling these phenomena have many practical applications.
The objective of this course is to introduce (1) fundamental
properties of random processes, (2) several important models
of random processes, especially those related to hydrology
and
water resources engineering, and (3) methods of stochastic
simulation.
Prerequisites of this course include (1) an entry level of
statistics course or familiarity of probability distributions and
(2) capability of programing with one computer language.
Fundamental probability theory and major probability
distributions will not be covered in this course.


Syllabus

1.        Introduction  (updated on 10/10/2005)
  • Definition of a random process
  • Characterizing a random process
  • Stationarity
Homework 1  (Updated on 10/10/2005. Problem 3 corrected.)
  • Random walk process
  • Gaussian random processes
  • Gaussian-Markov random process
  • ARMA random processes
2.        Markov Processes and Markov Chains   (Updated on 12/09)
  • Definition of a Markov process
  • Transition probability matrices of a Markov chain
  • Markov chain models
  • First step analysis
  • Classification of states  (uploaded on 12/09)
  • Long rum behavior of Markov processes
          (Homework 2)   (12/24)
3.        Poisson Processes  (updated 01/13/2006)
  • The law of rare events
  • Distributions associated with the Poisson processes
  • Spatial Poisson processes
4.        Renewal Processes  (01/13/2006)

5.        Linear Systems
  • The structure of state-space models
  • Linear models
  • System properties and model properties
  • Solution of state-space equations
6.        Optimal Linear Systems – the Kalman Filtering Approach
  • Mean square estimation
  • Filtering and prediction
  • Kalman filters
7.        Stochastic Simulation  (uploaded on Jan. 1, 2006)
  • Random number generation
  • Random process simulation
  • Random field simulation
  • Conditional random field simulation
8.        Examples of random processes in natural environment
  • Rainfall processes – storm hyetographs
  • River channel network
  • Percolation processes

Grade:
Midterm exam – 25%
Final exam – 25%
Homeworks – 50%

Reference books:
  1. Hoel, P.G., Port, S.C., and Stone, C.J., 1972. Introduction to Stochastic
    Processes. Houghton Mifflin, Boston, Mass. 203p.
  2. Taylor, H.M. and Karlin, S., 1994. An Introduction to Stochastic Modeling.
    Academic Press, Inc., San Diego, CA, 566p.


Weekly Power Point files
Week 1 (09/23) - Random Processes (1)
Week 2 (09/30) - Random Processes (2)
Week 3 (10/07) - Markov Chains (1)
Week 4 (10/14)
Week 5 (10/21)
Week 6 (10/28)
Week 7 (11/04) - Midterm Exam
Week 8 (11/11)
Week 9 (11/18)
Week 10 (11/25)
Week 11 (12/02)
Week 12 (12/09)
Week 13 (12/16)
Week 14 (12/23)
Week 15 (12/30)
Week 16 (01/06)
Week 17 (01/13) - Final Exam