CSI 709 / STAT 789
Monte Carlo Methods in Science
Spring, 2009
Instructor:
James Gentle
Lectures: Wednesday, 4:30-7:10pm, Research I, room 302
This course covers applications of Monte Carlo methods in science.
The course begins with a review of relevant background material
in probability and statistics, and then covers algorithms for computer
generation of random numbers, first from a uniform distribution,
and then from various distributions of interest.
Examples of applications from the physical, biological, and
social sciences will be studied. Individual class projects will
focus on topics chosen by the students.
Text
The text for the course is
Gentle, James E. (2003), Random Number Generation and Monte Carlo
Methods, second edition, Springer, New York.
This will be supplemented by various handouts.
Topics
- Some basics of probability
- Random variables and their relationships to each other
- Some basics of statistical estimation
- Generating random numbers on the computer
- First simple examples: integration by Monte Carlo
- Principles and examples of mathematical modeling
- Discrete simulation models
- Stochastic diffusion models
- Markov chain Monte Carlo
- Stochastic optimization
We will not cover these topics sequentially.
Background and interests of class will affect the relative weighting
of these topics.
Software
No particular software system will be required, but examples will be given in R.
If you have not used the open-source, freely-distributed R system, I recommend
that you look into it. It is available in precompiled binary distributions for
Linux, MacOS X, and MS Windows.
It can be downloaded from
www.r-project.org
From that web page, go to CRAN, select a mirror, then select the appropriate distribution.
I will introduce the system in the first class.
Assignments will require the use of
computer software that has facilities for generating random numbers and for
general programming. Instead of
R, this could
be Matlab, Mathematica, Maple, C, and/or Fortran.
Prerequisites
The mathematical and statistical background required is relatively elementary;
basically mathematics through multivariate calculus, and some introduction
to probability and statistics.
Grading
Performance in the class will be evaluated based on
an in-class midterm (25%)
a final exam consisting of a take-home portion and an in-class
portion (35%)
a project (30%)
a number of smaller assignments (10%)
Ethical Behavior
Each student enrolled in this course must assume the
responsibilities of an active participant in GMU's scholarly
community in which everyone's academic work and behavior are
held to the highest standards of honesty. The GMU policy on
academic conduct will be followed in this course.
The GMU Honor Code will be strictly observed.
Week 1, January 21
- Some basics of probability theory
- Distributions of transformed random variables
- The PDF decomposition
Notes on probability distributions
Assignment 1
Due January 28.
Week 2, January 28
- Statistical estimation
Notes
- Monte Carlo estimation
- Computer generation of uniform random numbers
Assignment 2: Page 271: Exercises 7.1, 7.2, 7.3, 7.5;
Page 57: Exercises 1.4, 1.5.
Due February 11.
Week 3, February 4
Week 4, February 11
- Term project
- More on computer generation of uniform random numbers
- Generation of nonuniform random numbers
Assignment 3: Page 159: Exercises 4.1, 4.2, 4.3, 4.4, 4.6, 4.7, 4.9 (Instead of Fortran
or C, you can use R or some other package.)
Due February 25.
Comments, solutions. Also, inverse CDF for bivariate
distribution.
Week 5, February 18
- Term project (brief written plan)
- Generation of nonuniform random numbers
Slides from lecture (to be continued next week).
Week 6, February 25
- Generation of nonuniform random numbers
Week 7, March 4
Midterm exam (open book)
This will cover basic material on probability distributions, Section 7.1,
Chapter 1 (but not Exercise 1.4), and Sections 4.1-4.7.
The exam questions will be similar to the (easier) homework exercises,
except you will not be asked to use any computer software.
(No class March 11)
Week 8, March 18
Assignment 4: Page 161: Exercises 4.10, 4.11, 4.15
Due April 8. Notice: a lot of homework is due on that day!
Week 9, March 25
- More on general methods of random number generation
- Applications in Bayesian statistical methods
Slides from lecture.
Assignment 5: Page 162: Exercises 4.12, 4.14, 4.17
Due April 8. Notice: a lot of homework is due on that day!
Week 10, April 1
Week 11, April 8
Assignment 6: Exercises 2.1, 2.8, and 3.6
Due April 12.
Week 12, April 15
Week 13, April 22
Week 14, April 29
- Term project
- Simulation models
May 6
4:30pm - 7:15pm Final Exam.
James Gentle, jgentle at gmu dot edu