1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
|
\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}
\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}.
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:
<<echo=TRUE,fig=TRUE>>=
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)
@
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:
<<>>=
inverse.predict(m,15,ws=1.67)
@
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 x
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})^2}{n - 2}}
\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}
|
