

\subsection{\bf Transformation of M to Eigendirections}

\begin{slide}
\begin{itemize}
\item Put $\Underline{$\beta$} = \mbox{\bf P} \Underline{$\alpha$}$.

\item Then $\chi^2 = \chi^2_0 +\ \mbox{linear terms}\ + \lambda_1 \left(
\beta_1 - \beta_{01} \right)^2 +\ \lambda_2 \left( \beta_2 - \beta_{01}
\right)^2$ $\ + \ldots$.  

\item Now the shifts $\Delta\beta_j$ can be calculated independently
and do not affect each other.

\item Assume that we order the eigenvalues
\[
\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n \ge 0
\]

\item Then clearly the first new shifts $\Delta\beta_j$ corresponding
to $\lambda_1 \ldots \lambda_L$, say are reliable and the remainder
$j=L+1\ldots n$ unreliable.

\item Suppose we ignore the predictions for these last and set
\begin{eqnarray*}
\Delta\beta_j & = & \mbox{Predicted value}\ j \le L \\
& = & 0 \hspace{1em} j>L
\end{eqnarray*}

\item We need a criterion for choosing $L$ but subject to this, this
method is intuitively reasonable.
\end{itemize}
\end{slide}

