MATH 420:   Mathematical Modeling

MWF 12pm,    MTH0303                                                          Spring 2011

For slides and exhibits from lectures, click here.

For R and MATLAB scripts, click here.

For information on getting started with R, click here.

For Current HW/Project Assignment, click here.

HW1 Assignment, due Friday, Feb. 11. HW2 due Feb. 16.
       HW3 due Mar. 4.

             Eric Slud, Stat. Program, Math. Dept., Office Rm. 2314, x5-5469,
             Wojtek Czaja, Math. Dept. Rm. 4406, x5-5106,
             Brian Hunt, Math. Dept. Rm. 1105, x5-5056,
             David Levermore, Math. Dept. Rm. 3313, x5-5127,

Office hours: initially M3, Th2 for Eric Slud

Prerequisites:    MATH 241, 246, and either 240 or 461;    STAT 400 is also desirable.

Recommended additional background: some computer proficiency either in MATLAB or other computing platform
(Mathematica, MAPLE, R , ...) where numerical analysis tasks and graphical outputs are easy to generate; and some
previous exposure to data (in STAT or MATH or an outside discipline like economics or biology or engineering).

Description: The main objective of this course is to learn from experience the various aspects of the
modeling process, including:

             Formulating and refining a mathematical model
             Mathematical and computational analysis of the model
             Evaluation and modification of model results and assumptions, and
             Oral and written communication of the results

Mathematical techniques discussed will be motivated by problems from areas such as physics, biology, economics, etc.

Brian Hunt's overview of the sample project problem he talked about
Wed. Feb.2 can be found here. You can also view the whole Sample Report.

Recommended Texts:

For basic techniques and approach to modeling:

(1) Guide to Mathematical Modeling, by D. Edwards and M. Hamson,   CRC Press, 1990, 268pp.,
       ISBN: 0849377005

(2) Concepts of Mathematical Modeling, by Walter J. Meyer,   Dover, 2004, 448pp.
       ISBN: 0486435156

For case studies and examples: web sources, the Meyer book and:

(3) Topics in Mathematical Modeling, by K. K. Tung,   Princeton Univ. Press, 2007, 300pp.,
       ISBN: 0691116423

Course syllabus  in pdf format can be found here.

Homework and Project Assignments

HW1, due Friday, Feb.11, in class: You are to choose one of the two HW/project problems
HW1A (with probability/statistics flavor) or HW1B (with calculus or ODE modeling flavor) to
work through individually. These are multi-part guided worksheets; after solving as many of
the parts as you can -- and some parts are free-form, with choices and assumptions for
you to make -- write the totality up in the form of a paper with words, explaining the
problem and results as coherently as you can. The writeup should not be more than about
4 or 5 pages: while you certainly can provide computations and pictures, especially in
HW1A, you should connect them to the whole mini-project and interpret them for the reader.

The raw data scatter-plot in HW1A can be seen here. You can get to the raw data
as an ASCII file by clicking on Births1978.txt.

An R Log of model-fitting steps in HW1A and associated pictures, which
includes a discussion of desirable elements in the narrative presentation,
can be found in the Scripts directory in HW1AScript.Rlog

HW2 problem assignment due Wed. February 16 in class.
Some software guidance for this HW can be found here.

Mini-project 3, due Friday, March 4, in class. You will have 2 choices
of Projects, which you are to work on in groups of 1, 2 or 3. You may choose
your own groups, but we reserve the right to prevent your groups from being
too unbalanced, e.g. with only one primary type of background. (You should
also make sure that your group contains at least one person capable of writing
and debugging code in MATLAB, R or Mathematica.) In these projects,
you are urged to combine analytical and computational approaches and tools
to get the best results you can. The project choices are   HW3A   or   HW3B.
As in HW1, after your group gets the best results you can for each of the
project parts, you should together write up the results to make as coherent
a report as possible.

Mini-project 4, due Wednesday, April 6, in class. There will be 3 choices
of Projects, which you are to work on in your own chosen groups of 1, 2 or 3.
The project choices are   HW4A   or   HW4B   or   HW4C. The guidelines for effective
report writeups are exactly as in the previous project, but in the middle of this
project (Thursday and Friday, March 17-18), each group will be asked to make
a brief (10 minute) presentation of results: you will get feedback and a grade on
these presentations, which will count 20% of your total grade for this mini-project.

Final Projects, due Friday May 13. There will be 3 or 4 choices
of projects: so far,
Project 5B (Machine Learning) or Project 5C
(Data Assimilation & Kalman Filter
, with additional information here)
or Project 5D (Queueing Simulation). Please try to form groups of
at least 3 for this project, and to vary your project choice
from the topics you have chosen previously.
Guidelines for
effective report writeups are as before --- shoot for 5 to 10
pages of text, with appropriately integrated pictures and tables,
but no undigested numerical outputs --- but now you will be
undertaking more of the foundational modeling choices than in the
mini-projects, and these choices should be motivated and justified.

NOTES for Final Projects.
(I). (Project 5C) Brian Hunt has provided some initial instructions here,
including a URL for scalar Kalman-filter derivation and equations.

(II). (Simulation Topics, all projects.) See two general handouts and notes
on transformations of random variables, from my STAT 400 class web-pages,
Transformation of Random Variables
and Random-Number Generation and Simulation.

(III). (Simulation Topics, all projects.) See this directory for a series of
lectures explaining miscellaneous statistical computing devices in R, from
the course STAT 705. However, none of these are specific to queueing applications.

(IV). (Project 5D) For some background material that may give you ideas on
optimizing a criterion function whose values you can see only with noise,
look up in Wikipedia or elsewhere the keywords stochastic approximation
or response surface methodology.

Organization of the course:

The course will be team-taught by Professors Wojtek Czaja, Brian Hunt, David Levermore, and Eric Slud.

The course will begin with three 3-week thematic segments, introducing progressively more sophisticated
notions and tools of mathematical models. Each of these segments will be accompanied by a written assignment:
the first one a worksheet-style HW, the next two as Mini-projects which you can choose from a list or negotiate
with the instructor(s).

The other work for the course consists of attending (almost) all class sessions, which will be part lecture and
part discussion, and preparing and eventually completing one longer project to finish off the term,
on datasets or conceptual modeling projects you can choose in consultation with the instructors.

The Mini-projects and longer Term Project will be done in groups.

Some smaller projects previously used in Math 420 by Brian Hunt are shown here.
These can be viewed as examples of the types of worksheet mini-projects which
will be assigned in the first 8--9 weeks of this course.

Several slightly more open-ended projects previously used in Math 420 by
Brian Hunt can be seen here. These are examples of the types of projects from
which you can choose the larger 5-week term project at the end of the course.

Examples of thematic segments:

Unit 1. Data Display, Representation, and Parameterization.

       Exploratory techniques involving:
             -- plotting, data change-of-variables;
             -- search for pattern through differencing, basis representation, Fourier transform;
             -- unit-level versus aggregated models; averaging;
             -- cross-classification and disaggregation.

       Possible examples/case studies: chosen from among
             -- representation of signals, "signatures",
             -- relationships between variables in economics,
             -- representation of "time-between-failure" probability distributions.

Unit 2. Recursion and Causal Representation of Change.

             -- difference vs differential equation models;
             -- linear and nonlinear models, notion of "interaction";
             -- linear and nonlinear least-squares to choose parametric representations;
             -- Markov chains as probabilisitic recursion relations.

       Possible examples/case studies: chosen from among

             -- physical science examples;
             -- Fibonacci sequences in biology,
             -- compound interest, population growth, epidemics, traffic, as examples where
                 either deterministic (macro) or unit-level stochastic (micro) models make sense.

Unit 3. Qualitative Properties of Models, and Model Assessment.

             -- model predictions, assessment via metrics like sum of squared errors or average
                 one-step-ahead squared prediction error;
             -- residuals plotting as a way to refine models;
             -- qualitative properties, eg as defined by phase planes;
             -- dependence of model predictions on model parameters.

       Possible examples/case studies: chosen from among

             -- datasets on one-step-ahead prediction, e.g. "London Mortality Data";
             -- `compartmental' ODE-system models describing drug uptake, disease propagation, etc.
             -- predator-prey models;
             -- simplified climate models as in Tung (2007) book.

Extra topic to be introduced in connection with probabilistic/statistical models: simulation as
a device to supplement difficult or intractable probability calculations or to perform experiments
on model behavior.

COMPUTING in this Course

Many if not all of the modeling projects in this course will involve computing, for experimentation
with numerical solutions to dynamical equations, for optimization of parameter choices, for fitting
to data, etc. You may use any computing platform you choose, but the most likely choices are
MATLAB and R, especially if you want to discuss computing details with one of the instructors.

(1) You probably have some experience with MATLAB in previous MATH or engineering courses.
Some additional (free) text and tutorial materials to help you with numerical computing are linked here.

(2) If your projects involve probability, statistics, or data analysis, a good software choice is R. This
is a highly functional and freely downloadable package, containing numerical analysis modules as well
(which are good and serviceable but not quite as powerful or fast as the ones in MATLAB). Details on
downloading software and manual can be found by visiting the
R web-site. To get started with the
software, you can find many helpful free tutorials online (here is one). Or try another small
basic useful handout that I provided to one of my classes. There are also several authoritative books
(of which the one by Venables and Ripley is highly recommended), and many locations where you can
find working scripts and descriptions, such as the course web-pages for STAT 401 and STAT 705.

Important Dates

  • First Class: Monday, January 24, 2011.
  • Spring break: March 21--25, 2011.
  • Last class: May 9, 2011.
  • Final Project writeups due: May 13, 2011.

  • The UMCP Math Department home page.

    The University of Maryland home page.

    Last updated April 15, 2011.