Chapter 4.2, Problem 9CP

The file scrippsm.txt, available from the textbook website, is a list of 180 numbers which represent the concentration of atmospheric carbon dioxide, in parts per million by volume (ppv), recorded monthly at Mauna Loa from Jan, 1996 to Dec. 2010, taken from the same Scripps study as Computer Problem 8.

a. Carry out a least squares fit of the ${\text{CO}}_{\text{2}}$ data using the model $f(t)=c_1+c_{2}t+c_3\cos 2\pi t+c_4\sin 2\pi t$ where t is measured in months. Report the best fit coefficients $c_i$ and the RMSE of the fit. Plot the continuous curve from Jan. 1989 to the end of this year. including the 180 data points in the plot.

b. Use your model to predict the ${\text{CO}}_{\text{2}}$ concentration in May 2004, Sept. 2004. May 2005, and Sept. 2005. These months tend to contain the yearly maxima and minima of the ${\text{CO}}_{\text{2}}$ cycle. The actual recorded values are 380.63, 374.06, 382.45, and 376.73 ppv. respectively. Report the model error at these four points.

c. Add the extra term $c_5\cos 4\pi t$ and redo parts (a) and (b). Compare the new RMSE and four model errors.

d. Repeat part (c) using the extra term $c^5{t}^2$ . Which term leads to more improvement in the model, part (c) or (d)?

e. Add both terms from (c) and (d) and redo parts (a) and (b). Prepare a table summarizing your results from all parts of the problem, and try to provide an explanation for the results. See the website http://scrippsco2.ucsd.edu for much more data and analysis of the Scripps carbon dioxide study.