MATH 420: Mathematical Modeling
MWF 12pm, MTH0303 Spring 2011For 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.
Instructors:
Eric Slud, Stat. Program, Math.
Dept., Office Rm. 2314, x5-5469, evs@math.umd.edu
Wojtek Czaja,
Math. Dept. Rm. 4406, x5-5106, wojtek@math.umd.edu
Brian Hunt,
Math. Dept. Rm. 1105, x5-5056, bhunt@umd.edu
David Levermore,
Math. Dept. Rm. 3313, x5-5127, lvrmr@math.umd.edu
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., (2) Concepts of Mathematical Modeling, by Walter J. Meyer,
Dover, 2004, 448pp.
ISBN: 0849377005
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 Final Projects, due Friday May 13. There will
be 3 or 4 choices NOTES for Final Projects. (II). (Simulation Topics, all projects.) See two
general handouts and notes (III). (Simulation Topics, all projects.) See this
directory for a series of (IV). (Project 5D) For some background material that may give you
ideas on Organization of the course: Some smaller projects
previously used in Math 420 by Brian Hunt are shown here. Several slightly more open-ended projects
previously used in Math 420 by Examples of thematic segments:
COMPUTING in this Course (1) You probably have some experience with MATLAB in previous
MATH or engineering courses. (2) If your projects involve probability, statistics, or data
analysis, a good software choice is R. This The UMCP
Math Department home page. The University of
Maryland home page. Last updated April 15, 2011.
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.
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.
(I). (Project 5C) Brian Hunt has provided some initial
instructions here,
including a
URL for scalar Kalman-filter derivation and equations.
on transformations of random variables,
from my STAT 400 class web-pages,
Transformation of Random
Variables
and Random-Number
Generation and Simulation.
lectures explaining miscellaneous
statistical computing devices in R, from
the course STAT 705.
However, none of these are specific to queueing applications.
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.
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.
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.
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.
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.
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.
Some additional (free) text and tutorial
materials to help you with numerical computing are linked
here.
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