Introduction to multivariate calibration in analytical chemistry.PDF

Introduction to multivariate calibration in analytical

chemistry†

Richard G. Brereton

School of Chemistry, University of Bristol, Cantock’s Close, Bristol, UK BS8 1TS

Received 12th May 2000, Accepted 11th September 2000

First published as an Advance Article on the web 31st October 2000

1 Introduction

1.1 Overview

1.2 Case study 1

1.3 Case study 2

2 Calibration methods

2.1 Univariate calibration

2.2 Multiple linear regression

2.3 Principal components regression

2.4 Partial least squares

2.5 Multiway methods

3 Model validation

3.1 Autoprediction

3.2 Cross-validation

3.3 Independent test sets

3.4 Experimental design

4 Conclusions and discussion

Acknowledgements

A Appendices

A1 Vectors and matrices

A1.1 Notation and definitions

A1.2 Matrix operations

A2 Algorithms

A2.1 Principal components analysis

A2.2 PLS1

A2.3 PLS2

A2.4 Trilinear PLS1

References

1 Introduction

1.1 Overview

Multivariate calibration has historically been a major corner-

stone of chemometrics as applied to analytical chemistry.

However, there are a large number of diverse schools of

thought. To some, most of chemometrics involves multivariate

calibration. Certain Scandinavian and North American groups

have based much of their development over the past two

decades primarily on applications of the partial least squares

(PLS) algorithm. At the same time, the classic text by Massart

and co-workers 1 does not mention PLS, and multivariate

calibration is viewed by some only as one of a large battery of

approaches to the interpretation of analytical data. In Scandina-

via, many use PLS for almost all regression problems (whether

appropriate or otherwise) whereas related methods such as

multiple linear regression (MLR) are more widely used by

mainstream statisticians.

There has developed a mystique surrounding PLS, a

technique with its own terminology, conferences and establish-

ment. Although originally developed within the area of

economics, most of its prominent proponents are chemists.

There are a number of commercial packages on the market-

place that perform PLS calibration and result in a variety of

diagnostic statistics. It is, though, important to understand that

a major historic (and economic) driving force was near infrared

spectroscopy (NIR), primarily in the food industry and in

process analytical chemistry. Each type of spectroscopy and

chromatography has its own features and problems, so much

software was developed to tackle specific situations which may

not necessarily be very applicable to other techniques such as

chromatography or NMR or MS. In many statistical circles NIR

and chemometrics are almost inseparably intertwined. How-

ever, other more modern techniques are emerging even in

process analysis, so it is not at all certain that the heavy

investment on the use of PLS in NIR will be so beneficial in the

future. Despite this, chemometric approaches to calibration

have very wide potential applicability throughout all areas of

quantitative analytical chemistry.

There are very many circumstances in which multivariate

calibration methods are appropriate. The difficulty is that to

develop a very robust set of data analytical techniques for a

particular situation takes a large investment in resources and

time, so the applications of multivariate calibration in some

areas of science are much less well established than in others. It

is important to distinguish the methodology that has built up

around a small number of spectroscopic methods such as NIR,

from the general principles applicable throughout analytical

chemistry. This article will concentrate on the latter. There are

probably several hundred favourite diagnostics available to the

professional user of PLS e.g. in NIR spectroscopy, yet each one

has been developed with a specific technique or problem in

mind, and are not necessarily generally applicable to all

calibration problems. The untrained user may become confused

† Electronic Supplementary Information available. See http://www.rsc.org/

suppdata/an/b0/b003805i/

Richard Brereton performed his undergraduate, postgraduate

and postdoctoral studies in the University of Cambridge, and

moved to Bristol in 1983, where he is now a Reader. He has

published 169 articles, 85 of which are refereed papers, and his

work has been cited over 1100 times. He has presented over 50

public invited lectures. He is currently chemometrics columnist

for the webzine the Alchemist.

He is author of one text, and

editor of three others. His inter-

ests encompass multivariate

curve resolution, calibration,

experimental design and pat-

tern recognition, primarily in

the area of coupled chromatog-

raphy, as applied to a wide

variety of problems including

pharmaceutical impurity mon-

itoring, rapid reaction kinetics,

food and biological chemistry.

DOI: 10.1039/b003805i

Analyst , 2000, 125 , 2125–2154

2125

This journal is © The Royal Society of Chemistry 2000

by these statistics; indeed he or she may have access to only one

specific piece of software and assume that the methods

incorporated into that package are fairly general or well known,

and may even inappropriately apply diagnostics that are not

relevant to a particular application.

There are a whole series of problems in analytical chemistry

for which multivariate calibration is appropriate, but each is

very different in nature.

1. The simplest is calibration of the concentration of a single

compound using a spectroscopic or chromatographic method,

an example being determining the concentration of chlorophyll

by EAS (electronic absorption spectroscopy). 2 Instead of using

one wavelength (as is conventional for the determination of

molar absorptivity or extinction coefficients), multivariate

calibration involves using all or several of the wavelengths.

2. A more complex situation is a multi-component mixture

where all pure standards are available, such as a mixture of four

pharmaceuticals. 3 It is possible to control the concentration of

the reference compounds, so that a number of carefully

designed mixtures can be produced in the laboratory. Some-

times the aim is to see whether a spectrum of a mixture can be

employed to determine individual concentrations, and, if so,

how reliably. The aim may be to replace a slow and expensive

chromatographic method by a rapid spectroscopic approach.

Another rather different aim might be impurity monitoring, 4

how well the concentration of a small impurity may be

determined, for example, buried within a large chromatographic

peak.

3. A different approach is required if only the concentration

of a portion of the components is known in a mixture, for

example, the polyaromatic hydrocarbons within coal tar pitch

volatiles. 5 In the natural samples there may be tens or hundreds

of unknowns, but only a few can be quantified and calibrated.

The unknown interferents cannot necessarily be determined and

it is not possible to design a set of samples in the laboratory

containing all the potential components in real samples.

Multivariate calibration is effective provided that the range of

samples used to develop the model is sufficiently representative

of all future samples in the field. If it is not, the predictions from

multivariate calibration could be dangerously inaccurate. In

order to protect against samples not belonging to the original

dataset, a number of approaches for determination of outliers

have been developed.

4. A final case is where the aim of calibration is not so much

to determine the concentration of a particular compound but a

group of compounds, for example protein in wheat. 6 The criteria

here become fairly statistical and the methods will only work if

a sufficiently large and adequate set of samples are available.

However, in food chemistry if the supplier of a product comes

from a known source that is unlikely to change, it is often

adequate to set up a calibration model on this training set.

There are many pitfalls in the use of calibration models,

perhaps the most serious being variability in instrument

performance over time. Each instrument has different character-

istics and on each day and even hour the response can vary. How

serious this is for the stability of the calibration model needs to

be assessed before investing a large effort. Sometimes it is

necessary to reform the calibration model on a regular basis, by

running a standard set of samples, possibly on a daily or weekly

basis. In other cases multivariate calibration gives only a rough

prediction, but if the quality of a product or the concentration of

a pollutant appears to exceed a certain limit, then other more

detailed approaches can be used to investigate the sample. For

example, on-line calibration in NIR can be used for screening a

manufactured sample, and any dubious batches investigated in

more detail using chromatography.

There are many excellent articles and books on multivariate

calibration which provide greater details about the algo-

rithms. 7–14 This article will compare the basic methods,

illustrated by case studies, and will also discuss more recent

developments such as multiway calibration and experimental

design of the training set. There are numerous software

packages available, including Piroutte, 15 Unscrambler, 16

SIMCA 17 and Matlab Toolkit 18 depending on the user’s

experience. However, many of these packages contain a large

number of statistics that may not necessarily be relevant to a

particular problem, and sometimes force the user into a

particular mode of thought. For the more computer based

chemometricians, using Matlab for developing applications

allows a greater degree of flexibility. It is important to recognise

that the basic algorithms for multivariate calibration are, in fact,

extremely simple, and can easily be implemented in most

environments, such as Excel, Visual Basic or C.

1.2 Case study 1

The first and main case study for this application is of the

electronic absorption spectra (EAS) of ten polyaromatic

hydrocarbons (PAHs). Table 1 is of the concentrations of these

PAHs in 25 spectra (dataset A) recorded at 1 nm intervals

between 220 and 350 nm, forming a matrix which is often

presented as having 25 rows (individual spectra) and 131

columns (individual wavelengths). The spectra are available as

Electronic Supplementary Information (ESI Table s1†). The

aim is to determine the concentration of an individual PAH in

the mixture spectra.

A second dataset consisting of another 25 spectra, whose

concentrations are given in Table 2, will also be employed

where necessary (dataset B). The full data are available as

Electronic Supplementary Information (ESI Table s2†). Most

calibration will be performed on dataset A.

1.3 Case study 2

The second case study is of two-way diode array detector

(DAD) HPLC data of a small embedded peak, that of

3-hydroxypyridine, buried within a major peak (2-hydroxypyr-

idine). The concentration of the embedded peak varies between

1 and 5% of the 2-hydroxypyridine, and a series of 14

chromatograms (including replicates) are recorded whose

concentrations are given in Table 3.

The chromatogram was sampled every 1 s, and a 40 s portion

of each chromatogram was selected to contain the peak cluster,

and aligned to the major peak maximum. Fifty-one wavelengths

between 230 and 350 nm (sampled at 2.4 nm intervals) were

recorded. Hence a dataset of dimensions 14 3 40 3 51 was

obtained, the aim being to use multimode calibration to

determine the concentration of the minor component. Further

experimental details are reported elsewhere. 4

The dataset is available in ESI Table s3†. It is arranged so that

each column corresponds to a wavelength and there are 14

successive blocks, each of 40 rows (corresponding to successive

points in time). Horizontal lines are used to divide each block

for clarity. The chromatograms have been aligned.

2 Calibration methods

We will illustrate the methods of Sections 2.1–2.4 with dataset

A of case study 1, and the methods of Section 2.5 with case

study 2.

2.1 Univariate calibration

2.1.1 Classical calibration. There is a huge literature on

univariate calibration. 19–23 One of the simplest problems is to

determine the concentration of a single compound using the

2126

Analyst , 2000, 125 , 2125–2154

response of a single detector, for example a single spectroscopic

wavelength or a chromatographic peak area.

Mathematically a series of experiments can be performed to

give

vectors have length I , equal to the number of samples. The

scalar s relates these parameters and is determined by the

experiments.

A simple method for solving this equation is as follows:

c A .x

c . s

c A.c . s

where, in the simplest case, x is a vector consisting of

absorbances at one wavelength for a number of samples (or the

response), and c is of the corresponding concentrations. Both

( c A.c) 2 1 . c A. x

( c A.c) 2 1 . ( c A.c). s

Table 1 Concentrations of polyarenes in dataset A for case study 1 a

Polyarene conc./mg L 2 1

Spectrum

Ace

Anth

Acy

Chry

Benz

Fluora

Fluore

Nap

Phen

1 0.456 0.120 0.168 0.120 0.336 1.620 0.120 0.600 0.120 0.564

2 0.456 0.040 0.280 0.200 0.448 2.700 0.120 0.400 0.160 0.752

3 0.152 0.200 0.280 0.160 0.560 1.620 0.080 0.800 0.160 0.118

4 0.760 0.200 0.224 0.200 0.336 1.080 0.160 0.800 0.040 0.752

5 0.760 0.160 0.280 0.120 0.224 2.160 0.160 0.200 0.160 0.564

6 0.608 0.200 0.168 0.080 0.448 2.160 0.040 0.800 0.120 0.940

7 0.760 0.120 0.112 0.160 0.448 0.540 0.160 0.600 0.200 0.118

8 0.456 0.080 0.224 0.160 0.112 2.160 0.120 1.000 0.040 0.118

9 0.304 0.160 0.224 0.040 0.448 1.620 0.200 0.200 0.040 0.376

10 0.608 0.160 0.056 0.160 0.336 2.700 0.040 0.200 0.080 0.118

11 0.608 0.040 0.224 0.120 0.560 0.540 0.040 0.400 0.040 0.564

12 0.152 0.160 0.168 0.200 0.112 0.540 0.080 0.200 0.120 0.752

13 0.608 0.120 0.280 0.040 0.112 1.080 0.040 0.600 0.160 0.376

14 0.456 0.200 0.056 0.040 0.224 0.540 0.120 0.800 0.080 0.376

15 0.760 0.040 0.056 0.080 0.112 1.620 0.160 0.400 0.080 0.940

16 0.152 0.040 0.112 0.040 0.336 2.160 0.080 0.400 0.200 0.376

17 0.152 0.080 0.056 0.120 0.448 1.080 0.080 1.000 0.080 0.564

18 0.304 0.040 0.168 0.160 0.224 1.080 0.200 0.400 0.120 0.118

19 0.152 0.120 0.224 0.080 0.224 2.700 0.080 0.600 0.040 0.940

20 0.456 0.160 0.112 0.080 0.560 1.080 0.120 0.200 0.200 0.940

21 0.608 0.080 0.112 0.200 0.224 1.620 0.040 1.000 0.200 0.752

22 0.304 0.080 0.280 0.080 0.336 0.540 0.200 1.000 0.160 0.940

23 0.304 0.200 0.112 0.120 0.112 2.700 0.200 0.800 0.200 0.564

24 0.760 0.080 0.168 0.040 0.560 2.700 0.160 1.000 0.120 0.376

25 0.304 0.120 0.056 0.200 0.560 2.160 0.200 0.600 0.080 0.752

a Abbreviations for PAHs: Py = pyrene; Ace = acenaphthene; Anth = anthracene; Acy = acenaphthylene; Chry = chrysene; Benz = benzanthracene;

Fluora = fluoranthene; Fluore = fluorene; Nap = naphthalene; Phen = phenanthrene.

Table 2 Concentration of the polyarenes in the dataset B for case study 1

Polyarene conc./mg L 2 1

Spectrum

Ace

Anth

Acy

Chry

Benz

Fluora

Fluore

Nap

Phen

0.456

0.120

0.168

0.120

0.336

1.620

0.120

0.600

0.120

0.564

0.456

0.040

0.224

0.160

0.560

2.160

0.120

1.000

0.040

0.188

0.152

0.160

0.224

0.200

0.448

1.620

0.200

0.040

0.376

0.608

0.160

0.280

0.160

0.336

2.700

0.040

0.200

0.080

0.188

0.608

0.200

0.224

0.120

0.560

0.540

0.040

0.400

0.040

0.564

0.760

0.160

0.168

0.200

0.112

0.540

0.080

0.200

0.120

0.376

0.608

0.120

0.280

0.040

0.112

1.080

0.040

0.600

0.080

0.940

0.456

0.200

0.056

0.040

0.224

0.540

0.120

0.400

0.200

0.940

0.760

0.040

0.056

0.080

0.112

1.620

0.080

1.000

0.200

0.752

0.152

0.040

0.112

0.040

0.336

1.080

0.200

1.000

0.160

0.940

0.152

0.080

0.056

0.120

0.224

2.700

0.200

0.800

0.200

0.564

0.304

0.040

0.168

0.080

0.560

2.700

0.160

1.000

0.120

0.752

0.152

0.120

0.112

0.200

0.560

2.160

0.200

0.600

0.160

0.376

0.456

0.080

0.280

0.200

0.448

2.700

0.120

0.800

0.080

0.376

0.304

0.200

0.280

0.160

0.560

1.620

0.160

0.400

0.080

0.188

0.760

0.200

0.224

0.200

0.336

2.160

0.080

0.400

0.040

0.376

0.760

0.160

0.280

0.120

0.448

1.080

0.080

0.200

0.080

0.564

0.608

0.200

0.168

0.160

0.224

1.080

0.040

0.400

0.120

0.188

0.760

0.120

0.224

0.080

0.224

0.540

0.080

0.600

0.040

0.752

0.456

0.160

0.112

0.080

0.112

1.080

0.120

0.200

0.160

0.752

0.608

0.080

0.112

0.040

0.224

1.620

0.040

0.800

0.160

0.940

0.304

0.080

0.056

0.080

0.336

0.540

0.160

0.800

0.200

0.752

0.304

0.040

0.112

0.120

0.112

2.160

0.160

1.000

0.160

0.564

0.152

0.080

0.168

0.040

0.448

2.160

0.200

0.800

0.120

0.940

0.304

0.120

0.056

0.160

0.448

2.700

0.160

0.600

0.200

0.188

Analyst , 2000, 125 , 2125–2154

2127

= Â (

x x

ˆ )/

ª ¢

( cc c x

) . =

where d is called the degrees of freedom. In the case of

univariate calibration this equals the number of observations ( N )

minus the number of parameters in the model ( P ) or in this case,

25 2 1 = 24, so that

where the A is the transpose as described in Appendix A1.

Many conventional texts use summations rather than matri-

ces for determination of regression equations, but both

approaches are equivalent. In Fig. 1, the absorbance of the

spectra of case study 1A at 336 nm is plotted against the

concentration of pyrene (Table 1). The graph is approximately

linear, and provides a best fit slope calculated by

0.0289/24 0.0347

This error can be represented as a percentage of the mean E % =

100 ( E / x ) = 24.1% in this case. It is always useful to check the

original graph (Fig. 1) just to be sure, which appears a

reasonable answer. Note that classical calibration is slightly

illogical in analytical chemistry. The aim of calibration is to

determine concentrations from spectral intensities, and not vice

versa yet the calibration equation in this section involves fitting

a model to determine a peak height from a known concentra-

tion.

For a new or unknown sample, the concentration can be

estimated (approximately) by using the inverse of the slope or

c = 3.44 x

The spectrum of pure pyrene is given in Fig. 2, superimposed

over the spectra of the other compounds in the mixture. It can be

seen that the wavelength chosen largely represents pyrene, so a

reasonable model can be obtained by univariate methods. For

most of the other compounds in the mixtures this is not possible,

so a much poorer fit to the data would be obtained.

= Â

1 849

Â =

and

c i

6 354

so that x = 0.291 c . Note the hat (ˆ) symbol which indicates a

prediction. The results are presented in Table 4.

The quality of prediction can be determined by the residuals

(or errors) i.e. the difference between the observed and

predicted, i.e. x 2 x ; the less this is the better. Generally the root

mean error is calculated,

Table 3 Concentrations of 3-hydroxypyridine in the chromatograms of

case study 2

2.1.2 Inverse calibration. Although classical calibration is

widely used, it is not always the most appropriate approach in

analytical chemistry, for two main reasons. First, the ultimate

aim is usually to predict the concentration (or factor) from the

spectrum or chromatogram (response) rather than vice versa .

There is a great deal of technical discussion of the philosophy

behind different calibration methods, but in other areas of

chemistry the reverse may be true, for example, can a response

Sample

Conc./mM

0.0158

0.0315

0.0473

Table 4 Results of regression of the concentration of pyrene (mg L 2 1 )

against the intensity of absorbance at 336 nm

0.0473

Predicted

absorbance

0.0631

Concentration

Absorbance

Residual

0.0631

0.456

0.161

0.133

0.028

0.0789

0.456

0.176

0.133

0.043

0.0789

0.152

0.102

0.044

0.058

0.760

0.184

0.221

2 0.037

0.760

0.231

0.221

0.010

0.608

0.171

0.176

2 0.006

0.760

0.183

0.221

2 0.039

0.456

0.160

0.133

0.027

0.304

0.126

0.088

0.038

0.608

0.186

0.177

0.009

0.608

0.146

0.177

2 0.031

0.152

0.064

0.044

0.020

0.608

0.139

0.177

2 0.038

0.456

0.110

0.133

2 0.023

0.760

0.202

0.221

2 0.019

0.152

0.087

0.044

0.043

0.152

0.076

0.044

0.032

0.304

0.104

0.088

0.016

0.152

0.120

0.044

0.076

0.456

0.125

0.133

2 0.008

0.608

0.173

0.177

2 0.004

0.304

0.092

0.088

0.004

0.304

0.135

0.088

0.046

0.760

0.212

0.221

2 0.009

0.304

0.142

0.088

0.054

Fig. 1 Absorption at 336 nm against concentration of pyrene.

2128

Analyst , 2000, 125 , 2125–2154

( e.g. a synthetic yield) be predicted from the values of the

independent factors ( e.g. temperature and pH)? The second

relates to error distributions. The errors in the response are often

due to instrumental performance. Over the years, instruments

have become more reliable. The independent variable (often

concentration) is usually determined by weighings, dilutions

and so on, and is often the largest source of error. The quality of

volumetric flasks, syringes and so on has not improved

dramatically over the years. Classical calibration fits a model so

that all errors are in the response [Fig. 3(a)], whereas with

improved instrumental performance, a more appropriate as-

sumption is that errors are primarily in the measurement of

concentration [Fig. 3(b)].

Calibration can be performed by the inverse method where

x . b

Most chemometricians prefer inverse methods, but most

traditional analytical chemistry texts introduce the classical

approach to calibration. It is important to recognise that there

are substantial differences in terminology in the literature, the

most common problem being the distinction between ‘ x ’ and ‘ y ’

variables. In many areas of analytical chemistry, concentration

is denoted by ‘ x ’, the response (such as a spectroscopic peak

height) by ‘ y ’. However, most workers in the area of

multivariate calibration have first been introduced to regression

methods via spectroscopy or chromatography whereby the

experimental data matrix is denoted as ‘ X ’, and the concentra-

tions or predicted variables by ‘ y ’. In this paper we indicate the

experimentally observed responses by ‘ x ’ such as spectroscopic

absorbances of chromatographic peak areas, but do not use ‘ y ’

in order to avoid confusion.

2.1.3 Including the intercept. In many situations it is

appropriate to include extra terms in the calibration model. Most

commonly an intercept (or baseline) term is included to give an

inverse model of the form

= ¢

xx x

. )

.c =

b 0 + b 1 x

which can be expressed in matrix/vector notation by

c ≈ X . b

for inverse calibration where c is a column vector of

concentrations and b is a column vector consisting of two

numbers, the first equal to b 0 (the intercept) and the second to b 1

(the slope). X is now a matrix of two columns, the first of which

is a column of 1’s, the second the spectroscopic readings, as

presented in Table 5.

Exactly the same principles can be employed for calculating

the coefficients as in Section 2.1.2, but in this case b is a vector

rather than scalar, and X is a matrix rather than a vector so

that

giving for this example, c = 3.262 x . Note that b is only

approximately the inverse of s (see above), because each model

makes different assumptions about error distributions. How-

ever, for good data, both models should provide fairly similar

predictions, if not there could be some other factor that

influences the data, such as an intercept, non-linearities, outliers

or unexpected noise distributions. For heteroscedastic noise

distributions 24 there are a variety of enhancements to linear

calibration. However, these are rarely taken into consideration

when extending the principles to the multivariate calibration.

b = ( X A. X ) 2 1 . X A . c

c = 2 0.178 + 4.391 x

Note that the coefficients are different from those of Section

2.1.2. One reason is that there are still a number of interferents,

from the other PAHs, in the spectrum at 336 nm, and these are

modelled partly by the intercept term. The models of Sections

2.1.1 and 2.1.2 force the best fit straight line to pass through the

Table 5 X matrix for example of Section 2.1.3

0.456

0.152

0.760

0.608

0.760

Fig. 2 Spectrum of pyrene superimposed over the spectra of the other pure

PAHs.

0.456

0.304

0.608

0.152

0.608

0.456

0.760

0.152

0.304

0.152

0.456

0.608

0.304

0.760

0.304

Fig. 3 Errors in (a) Classical and (b) Inverse calibration.

Analyst , 2000, 125 , 2125–2154

2129

(

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: