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 . linear in the betas) Basic Theory of Least Squares Normally Distributed Parameter Table Focus on model selection using AIC, SIC and 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 for and in any way. 10 10 10 10 10 20 20 20 20 20 30 30 30 30 30 Circu m 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 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 y(calor ) x1(gm) x2(gm) x3(gm) x4(gm) ies) 78892 1005 70.35 261.3 60.3 603 .5 990 9.9 287.1 148.5 514.8 73557 10273 985 108.35 551.6 78.8 197 5.5 1025 112.75 317.75 82 481.75 89790 900 63 468 54 297 85950 905 99.55 497.75 81.4 199.1 98826 10321 1005 30.15 713.55 170.85 60.3 3.5 890 8.9 275.9 195.8 391.6 64525 88910 955 19.1 515.7 171.9 210.1 .5 11590 1000 210 470 40 260 0 870 8.7 348 200.1 295.8 72906 11103 980 107.8 646.8 88.2 117.6 4 10502 960 96 652.8 76.8 115.2 4

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 and

a standard deviation of , while another yields a mean of 1.2 and standard deviation .

4. If the target is , 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 . The company knows that the production process yields items

on target on average with a variation of 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? 1 2 3

3.567 A1 B1 C1 2.145 A1 B2 C2 1.678 A2 B1 C2 2.564 A2 B2 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. 12. Turkey data. The following data corresponds to the weight of turkeys(y) at age (x) raised in Georgia, Virginia, or Wisconsin. y x Origin 13.3 28 G 8.9 20 G 15.1 32 G 10.4 22 G 13.1 29 V 12.4 27 V 13.2 28 V 11.8 26 V 11.5 21 W 14.2 27 W 15.4 29 W 13.1 23 W 13.8 25 W

(a) Write out the linear model for this problem using a dummy variable to model the state of origin and assuming the weight grows linearly with age ( straight line). Use (G,V,W) with for the coefficients corresponding to the states.

13. Plot Use this data to calculate the Sample Spectrum.

14. Let be a sequence of independent and normally distributed random variables with mean zero and variance one for every. Use any program to graph a realization of as a function of for 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 is approximately (when is large) to test . Do the same calculation for 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?