Problem set #2

The following problems require you to compute least squares fits and linear interpolants. Many software packages will perform basic operations for you. Maple and Matlab are examples, but even simple plotting packages often have ``linear regression'' subroutines that will find the best fit for you.

1.

Download dataset A from the course web site. Find an interpolating polynomial through this dataset, and record the coefficients. Find the best fit of this data for linear, quadratic, cubic, fourth and fifth order polynomials and record the root mean squared errors. Assuming the data came from a systematic source, f(x), with random noise added, speculate the form of the underlying system. Justify your answer.

2.

Download dataset B from the course web site. This is a second set of measurements from the same source. Find an interpolating polynomial through these points and compare with the interpolating polynomial from dataset A. Are the interpolants consistent in any way?

3.

Download dataset C from the course web site. This is a third set of measurements from the same source. Find the best fit of this data for linear, quadratic, cubic, fourth and fifth order polynomials and record the root mean squared errors. Based solely on the information from this third dataset, speculate on the form of the underlying system. Compare these results with those from dataset A. Justify your answer.

4.

Assuming that both datasets came from the same source, what is your best guess of the underlying system, f(x)? Justify your answer.

5.

The following is a true story. My neighbor, Mrs. X [not her real name, of course], approached me for some help in a lawsuit which she had brought against her former employers. She had suffered serious health problems and attributed them to a faulty furnace at her former place of employment. The furnace had cracked and was leaking carbon monoxide (CO) into the building. Continual exposure to carbon monoxide can cause a wide variety of health problems. An environmental consulting firm hired by the defendants measured the extent of the leak in the following way. They turned off the furnace and allowed the building to vent for many hours.

 

Table: Tables of first (left) and second (right) carbon monoxide measurements.

$\parbox{3in}{\begin{tabular}{\vert l\vert l\vert} \hline Time (min) ... ...line 60 & 19.1 \\ \hline 70 & 19.8 \\ \hline 80 & 20.5 \\ \hline \end{tabular}}$ $\parbox{3in}{\begin{tabular}{\vert l\vert l\vert} \hline Time (min) ... ...line 50 & 21.9 \\ \hline 60 & 22.2 \\ \hline 70 & 22.3 \\ \hline \end{tabular}}$

Then, they turned on the furnace and measured the carbon monoxide levels at a duct to a common area at regular time intervals (see Table 1). After this test, they turned off the furnace, waited a few hours, and ran a second test (see Table 1 again). I ask you the same questions that my neighbor asked me at the time.

• The second set of data appears to be very different from the first which would imply that there is some nonsystematic behavior in the furnace. Do the two data sets reflect the same physical phenomena or not?

• The CO levels are increasing in all measurements. The environmental firm claims that they stopped measuring CO levels because they would not have increased substantially more. (How would they know if they stopped measuring???) Was the environmental firm correct when the stopped taking measurements? If not, how high would the CO levels eventually go?

In addition, I ask you the following question: Assuming the employer repaired the furnace after the tests so that no further measurements were possible, what additional information would you want or need to have to confidently answer the questions above?