\documentclass[11pt]{article}
\topmargin-.5in
\textwidth6.6in
\textheight9in
\oddsidemargin0in

\usepackage{graphicx}

\def\ds{\displaystyle}
\def\d{\partial}

\begin{document}
\pagestyle{empty}

\centerline{\large \bf MA 573 --- PROJECT 2}

\bigskip
\noindent
{\large \bf Due: Friday, September 15}

\bigskip
\noindent
{\large \bf Problem 1.}

\medskip
In this problem, we will model heat generated during the hardening of cement.  Data from \cite{Hald}
is compiled in Table~\ref{table_Hald}.  Here $y$ denotes heat with units of calories/gram cement and $x_1-x_4$ respectively
denote the percentage of tricalcium aluminate, tricalcium silicate, tetracalcium aluminoferite and dicalcium phosphate.  

\medskip
\noindent
(a) Consider first the linear model
$$
  Y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \varepsilon.
$$
Estimate the parameters, plot the residual, and determine confidence intervals of two standard
deviations as well as $95\%$ confidence intervals. 

\bigskip
\noindent
(b) Perform the same analysis using linear models that incorporate only $x_1$ as
well as $x_1$ and $x_2$.  How do your results compare with those obtained in (a)? 


\begin{table}[!hb]
\begin{center}
\begin{tabular}{crrrrr}
\hline
Obs. No. & $x_1$ & $x_2$ & $x_3$ & $x_4$ & y \\ \hline
  1 & 7 & 26 & 6 & 60 & 78.5 \\
  2 & 1 & 29 & 15 & 52 & 74.3 \\
  3 & 11 & 56 & 8 & 20 & 104.3 \\
  4 & 11 & 31 & 8 & 47 & 87.6 \\
  5 & 7 & 52 & 6 & 33 & 95.9 \\
  6 & 11 & 55 & 9 & 22 & 109.2 \\
  7 & 3 & 71 & 17 & 6 & 102.7 \\
  8 & 1 & 31 & 22 & 44 & 72.5 \\
  9 & 2 & 54 & 18 & 22 & 93.1 \\
  10 & 21 & 47 & 4 & 26 & 115.9 \\
  11 & 1 & 40 & 23 & 34 & 83.8 \\
  12 & 11 & 66 & 9 & 12 & 113.3 \\
  13 & 10 & 68 & 8 & 12 & 109.4 \\ \hline
\end{tabular}
\end{center}

\vspace{-.15truein}
\caption{Cement data from \cite{Hald}.}
\label{table_Hald}
\end{table}


\bigskip
\bigskip
\noindent
{\large \bf Problem 2.}

\medskip
This is a variation of the Project on pages 58 and 59 of your book but with data
having unknown parameters and statistical properties.  This data is contained in
the file {\tt Data} where the first and second columns respectively contain simulated
temporal and displacement measurements.  After downloading the file, 
you can load it into MATLAB using the command {\tt >> load Data}. 

Consider the model
$$
\begin{array}{l}
  {\ds \frac{d^2 y}{d t^2} + C \frac{d y}{d t} + K y = 0 } \\
  \noalign{\medskip}
  {\ds y(0) = 2 \ , \ \frac{d y}{d t}(0) = 10 .}
\end{array}
$$

\begin{description}
\item{(i)} Estimate the parameters $C$ and $K$ using the ordinary least squares method.  You should
include a plot of your model fit to the data.  Use a continuous line type for the model and a
discrete line type for the data. 

\item{(ii)} Plot the residuals using a discrete line type (e.g., {\tt x}).  Do your errors appear to be iid?

\item{(iii)} Determine an estimate
$\hat{s}^2$ for the variance $\sigma_0^2$ and compute the covariance matrix
$\widehat{\mbox{\rm cov}(Q)}$ using both the analytic expressions and a finite
difference approximation.  What are the standard errors
$SE_k(\hat{q}) = \sqrt{\widehat{\mbox{\rm cov}(Q)_{kk}}}$ for $k=1,2$.  Do
the parameters appear to be correlated?

\item{(iv)} Determine the 95\% confidence intervals for each parameter.  \end{description}


\begin{thebibliography}{}

\bibitem{Hald} G. Hald, {\em Statistical Theory with Engineering Applications},
John Wiley and Sons, New York, 1952.

\end{thebibliography}


\end{document}


