\documentclass[a4paper]{article}
%\VignetteIndexEntry{Short manual for the chemCal package}
\newcommand{\chemCal}{{\tt chemCal}}
\newcommand{\calplot}{{\tt calplot}}
\newcommand{\calpredict}{{\tt calpredict}}
\newcommand{\R}{{\tt R}}
\usepackage{hyperref}

\title{Basic calibration functions for analytical chemistry}
\author{Johannes Ranke}

\usepackage{/usr/share/R/share/texmf/Sweave}
\begin{document}
\maketitle

The \chemCal{} package was first designed in the course of a lecture and lab
course on "analytics of organic trace contaminants" at the University of Bremen
from October to December 2004. In the fall 2005, an email exchange with 
Ron Wehrens led to the belief that it could be heavily improved if the
inverse prediction method given in \cite{massart97} would be implemented,
since it also covers the case of weighted regression.

At the moment, the package only consists of two functions, working
on univariate linear models of class \texttt{lm} or \texttt{rlm}.

When calibrating an analytical method, the first task is to generate
a suitable model. If we want to use the \chemCal{} functions, we
will have to restrict ourselves to univariate, possibly weighted, linear
regression so far.

Once such a model has been created, the calibration can be graphically
shown by using the \texttt{calplot} function:

\begin{Schunk}
\begin{Sinput}
> library(chemCal)
> data(massart97ex3)
> attach(massart97ex3)
> yx <- split(y, factor(x))
> ybar <- sapply(yx, mean)
> s <- round(sapply(yx, sd), digits = 2)
> w <- round(1/(s^2), digits = 3)
> weights <- w[factor(x)]
> m <- lm(y ~ x, w = weights)
> calplot(m)
\end{Sinput}
\end{Schunk}
\includegraphics{chemCal-001}

This is a reproduction of Example 8 in \cite{massart97}. We can
see the influence of the weighted regression on the confidence
and prediction bands of the calibration.

If we now want to predict a new x value from measured y values,
we use the \texttt{inverse.predict} function:

\begin{Schunk}
\begin{Sinput}
> inverse.predict(m, 15, ws = 1.67)
\end{Sinput}
\begin{Soutput}
$Prediction
[1] 5.865367

$`Standard Error`
[1] 10.66054

$Confidence
[1] 29.59840

$`Confidence Limits`
[1] -23.73303  35.46376
\end{Soutput}
\end{Schunk}

The weight \texttt{ws} assigned to the measured y value has to be 
given by the user in the case of weighted regression. By default, 
the mean of the weights used in the linear regression is used.

\section*{Theory}
Equation 8.28 in \cite{massart97} gives a general equation for predicting the
standard error $s_{\hat{x_s}}$ for an x value predicted from measurements of y
according to the linear calibration function $ y = b_0 + b_1 \cdot x$:

\begin{equation}
s_{\hat{x_s}} = \frac{s_e}{b_1} \sqrt{\frac{1}{w_s m} + \frac{1}{\sum{w_i}} +
    \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
        {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - {\left( \sum{ w_i x_i } \right)}^2 \right) }}
\end{equation}

with

\begin{equation}
s_e = \sqrt{ \frac{\sum w_i (y_i - \hat{y_i})^2}{n - 2}}
\end{equation}

where $w_i$ is the weight for calibration standard $i$, $y_i$ is the mean $y$
value (!) observed for standard $i$, $\hat{y_i}$ is the estimated value for
standard $i$, $n$ is the number calibration standards, $w_s$ is the weight
attributed to the sample $s$, $m$ is the number of replicate measurements of
sample $s$, $\bar{y_s}$ is the mean response for the sample, 
$\bar{y_w} = \frac{\sum{w_i y_i}}{\sum{w_i}}$ is the weighted mean of responses
$y_i$, and $x_i$ is the given $x$ value for standard $i$.

The weight $w_s$ for the sample should be estimated or calculated in accordance
to the weights used in the linear regression. 

I adjusted the above equation in order to be able to take a different precisions
in standards and samples into account. In analogy to Equation 8.26 from \cite{massart97}
we get

\begin{equation}
s_{\hat{x_s}} = \frac{1}{b_1} \sqrt{\frac{{s_s}^2}{w_s m} + 
    {s_e}^2 \left( \frac{1}{\sum{w_i}} +
        \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
            {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - {\left( \sum{ w_i x_i } \right)}^2 \right) } \right) }
\end{equation}

\begin{thebibliography}{1}
\bibitem{massart97}
Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J.,
Smeyers-Verbeke, J. 
\newblock Handbook of Chemometrics and Qualimetrics: Part A,
\newblock Elsevier, Amsterdam, 1997
\end{thebibliography}

\end{document}
