diff options
Diffstat (limited to 'trunk/inst')
| -rw-r--r-- | trunk/inst/doc/Makefile | 33 | ||||
| -rw-r--r-- | trunk/inst/doc/Rplots.ps | 1970 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal-001.eps | 1762 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal-001.pdf | 1727 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal-002.eps | 288 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal-002.pdf | 275 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.Rnw | 162 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.aux | 18 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.bbl | 0 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.blg | 46 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.log | 362 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.out | 0 | ||||
| -rw-r--r-- | trunk/inst/doc/chemCal.pdf | bin | 0 -> 123694 bytes | |||
| -rw-r--r-- | trunk/inst/doc/chemCal.tex | 169 |
14 files changed, 6812 insertions, 0 deletions
diff --git a/trunk/inst/doc/Makefile b/trunk/inst/doc/Makefile new file mode 100644 index 0000000..8eca69e --- /dev/null +++ b/trunk/inst/doc/Makefile @@ -0,0 +1,33 @@ +# Makefile for Sweave documents containing both Latex and R code +# Author: Johannes Ranke <jranke@uni-bremen.de> +# Last Change: 2006 Jun 23 +# based on the Makefile of Nicholas Lewin-Koh +# in turn based on work of Rouben Rostmaian +# SVN: $Id: Makefile.rnoweb 62 2006-05-24 08:30:59Z ranke $ + +RNWFILES = $(wildcard *.Rnw) +TARGETS = $(patsubst %.Rnw,%.tex,$(RNWFILE)) $(patsubst %.Rnw,%.pdf,$(RNWFILES)) + +RERUN = "(There were undefined references|Rerun to get (cross-references|the bars) right|Table widths have changed. Rerun LaTeX.|Linenumber reference failed)" +RERUNBIB = "No file.*\.bbl|Citation.*undefined" + +%.tex: %.Rnw + echo 'Sweave("$<")' | R --no-save --no-restore + +%.pdf: %.tex + pdflatex $< + egrep -c $(RERUNBIB) $*.log && (bibtex $*;pdflatex $<); true + egrep $(RERUN) $*.log && (pdflatex $<) ; true + egrep $(RERUN) $*.log && (pdflatex $<) ; true + +all: all-recursive $(TARGETS) + +clean: clean-recursive + rm -f *.aux *.log *.bbl *.blg *.brf *.cb *.ind *.idx *.ilg \ + *.inx *.ps *.dvi *.toc *.out *.lot *~ *.lof *.ttt *.fff + +all-recursive: + for dir in $(wildcard *); do if [ -d $$dir ] && [ -f $$dir/Makefile ]; then cd $$dir; $(MAKE) all; cd ..; fi; done + +clean-recursive: + for dir in $(wildcard *); do if [ -d $$dir ] && [ -f $$dir/Makefile ]; then cd $$dir; $(MAKE) clean; cd ..; fi; done diff --git a/trunk/inst/doc/Rplots.ps b/trunk/inst/doc/Rplots.ps new file mode 100644 index 0000000..c5e68ec --- /dev/null +++ b/trunk/inst/doc/Rplots.ps @@ -0,0 +1,1970 @@ +%!PS-Adobe-3.0 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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: + +<<echo=TRUE,fig=TRUE>>= +library(chemCal) +data(massart97ex3) +m0 <- lm(y ~ x, data = massart97ex3) +calplot(m0) +@ + +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: + +<<echo=TRUE,fig=TRUE>>= +plot(m0,which=3) +@ + +Therefore, in Example 8 in \cite{massart97} weighted regression +is proposed which can be reproduced by + +<<>>= +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) +@ + +If we now want to predict a new x value from measured y values, +we use the \texttt{inverse.predict} function: + +<<>>= +inverse.predict(m, 15, ws=1.67) +inverse.predict(m, 90, ws = 0.145) +@ + +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}. + +\section*{Fitting and using a variance function} + +In the R package \texttt{nlme} variance functions are used for weighted +regressions. But since the \texttt{predict.nlme} method does not calculate +prediction intervals, this is not useful for the \texttt{calplot} function. + +Two approaches could be used for fitting variance functions, one based on +residuals from an unweighted fit, and one based on just the variances +of the different samples along the x axis. If we used the residuals for +fitting, a bias of the model in a certain area would result in a higher +variance, so it seems preferable to choose the second approach. Of course, +a prerequisite is to have sufficient repetitions for each sample in any +case. + +Let's take the above example and estimate a variance function + +<<>>= +massart97ex3 +massart97ex3$x <- factor(massart97ex3$x) +summary <- summaryBy(y~x, data = massart97ex3,FUN=c(mean,sd,var)) +summary$x <- as.numeric(as.vector((summary$x))) +plot(summary$x, summary$y.var, + xlim=c(0,50), + ylim=c(0,max(summary$y.var))) +varModel <- lm(y.var ~ I(x^2) + x, data=summary) +varCurve <- predict(varModel, newdata=data.frame(x=0:5000/100)) +lines(0:5000/100,varCurve) + + + + + +\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} diff --git a/trunk/inst/doc/chemCal.aux b/trunk/inst/doc/chemCal.aux new file mode 100644 index 0000000..20bfc98 --- /dev/null +++ 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b/trunk/inst/doc/chemCal.out diff --git a/trunk/inst/doc/chemCal.pdf b/trunk/inst/doc/chemCal.pdf Binary files differnew file mode 100644 index 0000000..c7a01d7 --- /dev/null +++ b/trunk/inst/doc/chemCal.pdf diff --git a/trunk/inst/doc/chemCal.tex b/trunk/inst/doc/chemCal.tex new file mode 100644 index 0000000..9617cda --- /dev/null +++ b/trunk/inst/doc/chemCal.tex @@ -0,0 +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 + +$`Standard Error` +[1] 0.892611 + +$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} |