From 7889cda8d4133e68b07e7b204cd75295e0b348ee Mon Sep 17 00:00:00 2001 From: ranke Date: Fri, 23 Jun 2006 16:06:54 +0000 Subject: Some cosmetic changes, and the vignette created during package building. git-svn-id: http://kriemhild.uft.uni-bremen.de/svn/chemCal@17 5fad18fb-23f0-0310-ab10-e59a3bee62b4 --- inst/doc/chemCal.tex | 338 +++++++++++++++++++++++++-------------------------- 1 file changed, 169 insertions(+), 169 deletions(-) (limited to 'inst/doc/chemCal.tex') diff --git a/inst/doc/chemCal.tex b/inst/doc/chemCal.tex index 0469848..32c5143 100644 --- a/inst/doc/chemCal.tex +++ b/inst/doc/chemCal.tex @@ -1,169 +1,169 @@ -\documentclass[a4paper]{article} -%\VignetteIndexEntry{Short manual for the chemCal package} -\usepackage{hyperref} - -\title{Basic calibration functions for analytical chemistry} -\author{Johannes Ranke} - -\usepackage{/usr/share/R/share/texmf/Sweave} -\begin{document} -\maketitle - -The \texttt{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 would be desirable to implement the -inverse prediction method given in \cite{massart97} since it also covers the -case of weighted regression. Studies of the IUPAC orange book and of DIN 32645 -as well as publications by Currie and the Analytical Method Committee of the -Royal Society of Chemistry and a nice paper by Castillo and Castells provided -further understanding of the matter. - -At the moment, the package consists of four functions, working on univariate -linear models of class \texttt{lm} or \texttt{rlm}, plus to datasets for -validation. - -A \href{http://bugs.r-project.org/cgi-bin/R/wishlst-fulfilled?id=8877;user=guest}{bug -report (PR\#8877)} and the following e-mail exchange on the r-devel mailing list about -prediction intervals from weighted regression entailed some further studies -on this subject. However, I did not encounter any proof or explanation of the -formula cited below yet, so I can't really confirm that Massart's method is correct. - -When calibrating an analytical method, the first task is to generate a suitable -model. If we want to use the \texttt{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) -> m0 <- lm(y ~ x, data = massart97ex3) -> calplot(m0) -\end{Sinput} -\end{Schunk} -\includegraphics{chemCal-001} - -As we can see, the scatter increases with increasing x. This is also -illustrated by one of the diagnostic plots for linear models -provided by R: - -\begin{Schunk} -\begin{Sinput} -> plot(m0, which = 3) -\end{Sinput} -\end{Schunk} -\includegraphics{chemCal-002} - -Therefore, in Example 8 in \cite{massart97} weighted regression -is proposed which can be reproduced by - -\begin{Schunk} -\begin{Sinput} -> attach(massart97ex3) -> yx <- split(y, 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) -\end{Sinput} -\end{Schunk} - -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] 0.7288138 - -$Confidence -[1] 2.023511 - -$`Confidence Limits` -[1] 3.841856 7.888878 -\end{Soutput} -\begin{Sinput} -> inverse.predict(m, 90, ws = 0.145) -\end{Sinput} -\begin{Soutput} -$Prediction -[1] 44.06025 - -$`Standard Error` -[1] 2.310001 - -$Confidence -[1] 6.41359 - -$`Confidence Limits` -[1] 37.64666 50.47384 -\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, or alternatively, -the approximate variance \texttt{var.s} at this location. - -\section*{Theory for \texttt{inverse.predict}} -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} - -where I interpret $\frac{{s_s}^2}{w_s}$ as an estimator of the variance at location -$\hat{x_s}$, which can be replaced by a user-specified value using the argument -\texttt{var.s} of the function \texttt{inverse.predict}. - -\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} +\documentclass[a4paper]{article} +%\VignetteIndexEntry{Short manual for the chemCal package} +\usepackage{hyperref} + +\title{Basic calibration functions for analytical chemistry} +\author{Johannes Ranke} + +\usepackage{d:/Programme/R/R-2.3.1/share/texmf/Sweave} +\begin{document} +\maketitle + +The \texttt{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 would be desirable to implement the +inverse prediction method given in \cite{massart97} since it also covers the +case of weighted regression. Studies of the IUPAC orange book and of DIN 32645 +as well as publications by Currie and the Analytical Method Committee of the +Royal Society of Chemistry and a nice paper by Castillo and Castells provided +further understanding of the matter. + +At the moment, the package consists of four functions, working on univariate +linear models of class \texttt{lm} or \texttt{rlm}, plus to datasets for +validation. + +A \href{http://bugs.r-project.org/cgi-bin/R/wishlst-fulfilled?id=8877;user=guest}{bug +report (PR\#8877)} and the following e-mail exchange on the r-devel mailing list about +prediction intervals from weighted regression entailed some further studies +on this subject. However, I did not encounter any proof or explanation of the +formula cited below yet, so I can't really confirm that Massart's method is correct. + +When calibrating an analytical method, the first task is to generate a suitable +model. If we want to use the \texttt{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) +> m0 <- lm(y ~ x, data = massart97ex3) +> calplot(m0) +\end{Sinput} +\end{Schunk} +\includegraphics{chemCal-001} + +As we can see, the scatter increases with increasing x. This is also +illustrated by one of the diagnostic plots for linear models +provided by R: + +\begin{Schunk} +\begin{Sinput} +> plot(m0, which = 3) +\end{Sinput} +\end{Schunk} +\includegraphics{chemCal-002} + +Therefore, in Example 8 in \cite{massart97} weighted regression +is proposed which can be reproduced by + +\begin{Schunk} +\begin{Sinput} +> attach(massart97ex3) +> yx <- split(y, 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) +\end{Sinput} +\end{Schunk} + +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] 0.892611 + +$Confidence +[1] 2.478285 + +$`Confidence Limits` +[1] 3.387082 8.343652 +\end{Soutput} +\begin{Sinput} +> inverse.predict(m, 90, ws = 0.145) +\end{Sinput} +\begin{Soutput} +$Prediction +[1] 44.06025 + +$`Standard Error` +[1] 2.829162 + +$Confidence +[1] 7.855012 + +$`Confidence Limits` +[1] 36.20523 51.91526 +\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, or alternatively, +the approximate variance \texttt{var.s} at this location. + +\section*{Theory for \texttt{inverse.predict}} +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} + +where I interpret $\frac{{s_s}^2}{w_s}$ as an estimator of the variance at location +$\hat{x_s}$, which can be replaced by a user-specified value using the argument +\texttt{var.s} of the function \texttt{inverse.predict}. + +\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} -- cgit v1.2.1