**MATH 420: Mathematical Modeling**

**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.**

**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.,

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

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.

The UMCP Math Department home page.

The University of Maryland home page.

Last updated April 15, 2011.