Download Mathematical Modelling 2013

Mathematical Modelling 2013
Instructor: Prof. Ganser, Armstrong Hall 408K, ,[email protected]
PR: A background in differential equations, linear algebra and statistics/probability is
necessary for this course. The course in statistics should be at the calculus level such as
Stat 461 here at WVU. Students must have knowledge of a spread sheet program like
JMP or Excel as well as a program for doing math such as Mathematica or Matlab. The
first set assignments should help you decide if you have the proper background.
Grading: 50% (assignments, projects)
20% (Midterm)
30%(Final)
Assignments are to be done individually and typed. The project write-up should be
clear and concise with page numbers and include:
i) Statement of the problem
ii) Summary of the solution with reference (page numbers) to calculations and data
analysis in the back.
iii) Computer code and output in the back
Assignments will have a due date. Sometimes the date is relaxed for all students because
of unforeseen circumstances. However, once a date is set the assignments are due on that
date. Any projects that are turned in late will either not be accepted or the grade will be
reduced.
Goals: This course covers many models. A summary is discussed the first day of class.
It is more of a survey of mathematical models than a specialized course in particular
models. Topics will not be studied in complete detail in order to see more examples.
There is a danger that a student will feel that they have not learned the topic well enough
to use it in the future. However, it is hoped that the student will have sufficient
understanding of the models discussed so that he or she may know “where to start” when
faced with a new problem. Also, the course should help students better understand the
models developed by others.
Outline
Linear Models (such as y  0  1 x1  2 x2   . linear in the betas)
Basic Theory of Least Squares
Normally Distributed  ' s
Parameter Table
Focus on model selection using AIC, SIC and CP statistics
Prediction
Dimensional Analysis
How it works
Examples (simple ones plus drag on a sphere, period of a pendulum, etc)
Probability Models
Review of selected distributions
Maximum Likelihood
Approximate confidence intervals
Goodness of fit
Main Effects / Additive Models
Design of Experiments
Orthogonal Arrays
Time Series (if there is time)
Spectral Analysis
ARMA models
Assignments
1. Enter the data for the tape problem into a spread sheet. The physical problem will be
explained in class. Estimate the value of the angle A for c  24cm and w  15cm. in
any way.
10 10 10 10 10 20 20 20 20 20 30 30 30 30 30
Circum
c/cm
Tape
3
5
7
9
11 3
5
7
9
11 3
5
7
9
11
width
w/cm
Angle 17 30 44 64 --- 9
14 20 27 33 6
10 14 17 21
of
Pitch
A/deg
2.
x
1
1
2
4
5
6
6
7
4
y
3
6
4
3
5
9
10
8
6
Use linear regression to fit the line y  0  1 x to the data. Include the parameter table
for the fit. This should have at minimum the estimates and standard errors for the
parameters, t-statistic, p-values or probabilities and the standard error of regression. It is
not necessary to know what these numbers mean yet. It is important that you know how
to use some software to find these numbers.
3. The input –output model problem. The program is on e-campus. See if you can find a
formula for predicting the output from the input by doing different experiments.
Remember that the “system” has the properties discussed in class.
4. Suppose you experiment with two different shoes and two different balls to determine
the best combination for scoring the highest in bowling. Below is the data. Can you
come to some conclusion as to what may be the best combination? One possible answer
is there is no conclusion.

Shoes
Ball
176
A1
B1
176
A1
B2
186
A2
B1
184
A2
B2
180
A1
B1
182
A1
B2
182
A2
B1
188
A2
B2
5. The following data set gives the mass of a chunk of cement and the mass of the four
components used in the mix. The problem is to use these masses to predict the heat that
was produced given in the y column. What variables would you use in the model?
This is very basic, and the question has to do with the initial pencil and paper analyisis.
Mass(gm)
1005
990
985
1025
900
905
1005
890
955
1000
870
980
960
x1(gm)
70.35
9.9
108.35
112.75
63
99.55
30.15
8.9
19.1
210
8.7
107.8
96
x2(gm)
261.3
287.1
551.6
317.75
468
497.75
713.55
275.9
515.7
470
348
646.8
652.8
x3(gm)
60.3
148.5
78.8
82
54
81.4
170.85
195.8
171.9
40
200.1
88.2
76.8
x4(gm)
603
514.8
197
481.75
297
199.1
60.3
391.6
210.1
260
295.8
117.6
115.2
y(calories)
78892.5
73557
102735.5
89790
85950
98826
103213.5
64525
88910.5
115900
72906
111034
105024
6. Read the Dimensional Analysis Notes that are on the e-campus page for our class. Do
problems 3 and 4 on page 9.
7. Crater Ejecta Scaling Laws Article. (a) Do the dimensional analysis of Eq.(1)( the
authors never do this) (b) In your own words derive Eq(6) from Eq.(1). Pay
attention to how the authors do it and why they do it.
8. One production process yields items with a quality characteristic with mean
  1.5 and a standard deviation of   .1 , while another yields a mean of 1.2 and
standard deviation .4. If the target is m  1.2 , which approach is better using a
quadratic loss function?
9. A company sells a product for $500 that is quaranteed to be on target with a
variation of at most 2 . The company knows that the production process yields
items on target on average with a variation of   .5 with no further testing. An
additional test on each individual unit costs $15 to determine if it is within the
guaranteed specifications. Make a case for or against the cost of further testing
using a quadratic loss function.
10. Consider the experiment given in the Table with 3 Factors A ,B, and C. Each with
2 levels. What factor affects the loss function L[y] the most and what settings would
minimize L[y]? How would you approach this problem?
3.567
2.145
1.678
2.564
1
2
3
A1
A1
A2
A2
B1
B2
B1
B2
C1
C2
C2
C1
11. The problem is to predict the energy consumption for the winter of 1971(the first
quarter of 1971).
(a)Graph the data and determine a linear model. Model the slight increasing trend with a
straight line. Write out the linear model y  X    .
Year
1965
1966
1967
1968
1969
1
874
866
843
906
952
2
679
700
719
703
745
3
616
603
594
634
635
4
816
814
819
844
871
12. Turkey data. The following data corresponds to the weight of turkeys(y) at age (x)
raised in Georgia, Virginia, or Wisconsin.
y
13.3
8.9
15.1
10.4
13.1
12.4
13.2
11.8
11.5
14.2
15.4
13.1
13.8
x
28
20
32
22
29
27
28
26
21
27
29
23
25
Origin
G
G
G
G
V
V
V
V
W
W
W
W
W
(a) Write out the linear model y  X    for this problem using a dummy variable
to model the state of origin and assuming the weight grows linearly with age (
straight line). Use a1 , a2 , a3 (G,V,W) with a2  a1  a3 for the coefficients
corresponding to the states.
k
k
 .5cos 23.1
 .3cos 21 k k  1,2,...50. Use this data to calculate the
13. Plot y(k)  cos 22.4
Sample Spectrum.
14. Let wt be a sequence of independent and normally distributed random variables with
mean zero and variance one for every t . Use any program to graph a realization of wt as
a function of t for t  1,2,...100. Use this data to calculate the Sample Spectrum.
15. An experiment is done to see if a coin is fair(the probability of heads is .5). In one
experiment the coin is tossed 100 times and heads show up 57 times. Use the likelihood
ratio test and the result that 2 log  is approximately  2 (when is large) to test p0  .5 .
Do the same calculation for n  1000 and heads appeared 570 times. Why do you think
(use common sense) the conclusions are different?
Name:___________________________________
Major/Year:
Statistics Background:
Software for doing Mathematics/spreadsheets:
What do you expect from a Math Modelling course?