Brereton Chemometrics

272	CHEMOMETRICS

techniques for a particular situation takes a huge 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 separate the methodology that has built up around a small number of spectroscopic methods such as NIR from the general principles applicable throughout chemistry. 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.

There are a whole series of problems in chemistry for which multivariate calibration is appropriate, but each is very different in nature. Many of the most successful applications have been in the spectroscopy or chromatography of mixtures and we will illustrate this chapter with this example, although several diverse applications are presented in the problems at the end.

1.The simplest is calibration of the concentration of a single compound using a spectroscopic or chromatographic method, an example being the determination of the concentration of chlorophyll by electronic absorption spectroscopy (EAS) – sometimes called UV/vis spectroscopy. 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. Each variable measures the same information, but better information is obtained by considering all the wavelengths.

2.A more complex situation is a multi-component mixture where all pure standards are available. 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. Sometimes 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: how well the concentration of a small impurity can 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, polyaromatic hydrocarbons within coal tar pitch volatiles. In 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 providing 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 and experimental design 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 to determine a statistical parameter. There will no longer be pure standards available, and the training set must consist of a sufficiently representative group. An example is to determine the concentration of a class of compounds in food, such as protein in wheat. It is not possible (or desirable) to isolate each single type of protein, and we rely on the original samples

CALIBRATION	273

being sufficiently representative. This situation also occurs, for example, in quantitative structure–property relationships (QSPR) or quantitative structure–activity relationships (QSAR).

There are many pitfalls in the use of calibration models, perhaps the most serious being variability in instrument performance over time. Each measurement technique has different characteristics and on each day and even hour the response can vary. How serious this is for the stability of the calibration model should 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.

This chapter will describe the main algorithms and principles of calibration. We will concentrate on situations in which there is a direct linear relationship between blocks of variables. It is possible to extend the methods to include multilinear (such as squared) terms simply by extended the X matrix, for example, in the case of spectroscopy at high concentrations or nonlinear detection systems.

5.1.2 Case Study

It is easiest to illustrate the methods in this chapter using a small case study, involving recording

•25 EAS spectra at

•27 wavelengths (from 220 to 350 nm at 5 nm intervals) and

•consisting of a mixture of 10 compounds [polyaromatic hydrocarbons (PAHs)].

In reality the spectra might be obtained at higher digital resolution, but for illustrative purposes we reduce the sampling rate. The aim is to predict the concentrations of individual PAHs from the mixture spectra. The spectroscopic data are presented in Table 5.1 and the concentrations of the compounds in Table 5.2.

The methods in this chapter will be illustrated as applied to the spectroscopy of mixtures as this is a common and highly successful application of calibration in chemistry. However, similar principles apply to a wide variety of calibration problems.

5.1.3 Terminology

We will refer to physical measurements of the form in Table 5.1 as the ‘x’ block and those in Table 5.2 as the ‘c’ block. One area of confusion is that users of different techniques in chemometrics tend to employ incompatible notation. In the area of experimental design it is usual to call the measured response ‘y’, e.g. the absorbance in a spectrum, and the concentration or any related parameter ‘x’. In traditional multivariate calibration this notation is swapped around. For the purpose of a coherent text it would be confusing to use two opposite notations; however, some compatibility with the established literature is desirable. Figure 5.1 illustrates the notation used in this text.

274	CHEMOMETRICS

Table 5.1 Case study consisting of 25 spectra recorded at 27 wavelengths (nm) (absorbances in AU).

350	0.053	0.077	0.048	0.046	0.067	0.056	0.030	0.056	0.057	0.066	0.016	0.025	0.028	0.028	0.044	0.047	0.031	0.032	0.070	0.036	0.037	0.030	0.063	0.070	0.058
345	0.075	0.105	0.068	0.060	0.094	0.078	0.041	0.078	0.074	0.093	0.030	0.036	0.043	0.040	0.067	0.069	0.045	0.047	0.096	0.051	0.059	0.042	0.093	0.101	0.086
340	0.110	0.140	0.089	0.107	0.146	0.115	0.089	0.113	0.098	0.134	0.071	0.053	0.078	0.064	0.114	0.081	0.061	0.072	0.115	0.080	0.105	0.064	0.118	0.145	0.116
335	0.165	0.178	0.102	0.191	0.239	0.178	0.193	0.164	0.129	0.193	0.154	0.065	0.144	0.114	0.211	0.087	0.077	0.106	0.119	0.130	0.182	0.095	0.138	0.219	0.147
330	0.127	0.150	0.095	0.134	0.170	0.135	0.129	0.127	0.104	0.157	0.100	0.056	0.094	0.079	0.143	0.081	0.071	0.081	0.114	0.098	0.133	0.070	0.124	0.163	0.128
325	0.130	0.155	0.115	0.130	0.158	0.129	0.122	0.122	0.109	0.156	0.100	0.065	0.082	0.075	0.125	0.087	0.084	0.085	0.120	0.103	0.127	0.074	0.126	0.155	0.145
320	0.162	0.180	0.136	0.179	0.201	0.167	0.183	0.144	0.137	0.186	0.154	0.077	0.110	0.107	0.168	0.095	0.104	0.107	0.123	0.141	0.164	0.106	0.138	0.199	0.175
315	0.137	0.159	0.123	0.144	0.162	0.141	0.142	0.123	0.126	0.160	0.118	0.070	0.076	0.087	0.128	0.089	0.096	0.092	0.116	0.119	0.122	0.094	0.125	0.160	0.150
310	0.135	0.157	0.134	0.138	0.151	0.138	0.136	0.120	0.136	0.153	0.114	0.076	0.062	0.089	0.113	0.093	0.105	0.095	0.118	0.123	0.103	0.104	0.127	0.150	0.151
305	0.161	0.185	0.168	0.172	0.170	0.175	0.165	0.155	0.167	0.173	0.139	0.094	0.076	0.122	0.133	0.116	0.139	0.119	0.144	0.148	0.120	0.146	0.158	0.189	0.179
300	0.208	0.254	0.224	0.212	0.213	0.241	0.178	0.220	0.213	0.226	0.154	0.123	0.103	0.158	0.171	0.174	0.188	0.151	0.219	0.182	0.160	0.199	0.239	0.260	0.237
295	0.290	0.405	0.292	0.295	0.311	0.390	0.215	0.292	0.298	0.358	0.211	0.194	0.161	0.204	0.277	0.287	0.260	0.219	0.375	0.290	0.242	0.275	0.380	0.386	0.359
290	0.706	1.008	0.717	0.573	0.827	0.868	0.363	0.794	0.713	0.910	0.345	0.334	0.424	0.360	0.658	0.781	0.524	0.500	0.924	0.573	0.602	0.443	0.934	0.983	0.846
285	0.633	0.934	0.639	0.516	0.749	0.810	0.343	0.731	0.649	0.847	0.322	0.309	0.361	0.331	0.596	0.710	0.477	0.458	0.866	0.524	0.525	0.418	0.881	0.897	0.785
280	0.682	0.993	0.692	0.562	0.798	0.869	0.369	0.783	0.680	0.911	0.355	0.338	0.396	0.362	0.638	0.751	0.515	0.474	0.921	0.566	0.571	0.466	0.934	0.948	0.837
275	0.755	1.050	0.728	0.668	0.866	0.938	0.493	0.829	0.716	0.938	0.469	0.366	0.461	0.438	0.727	0.772	0.576	0.519	0.907	0.641	0.667	0.552	0.956	1.079	0.892
270	0.741	0.965	0.790	0.704	0.775	0.925	0.604	0.716	0.724	0.836	0.616	0.353	0.421	0.474	0.641	0.693	0.647	0.501	0.790	0.718	0.640	0.623	0.801	1.036	0.916
265	0.678	0.870	0.728	0.656	0.681	0.859	0.560	0.658	0.642	0.713	0.561	0.344	0.387	0.449	0.581	0.627	0.625	0.451	0.722	0.671	0.601	0.611	0.731	0.948	0.840
260	0.673	0.869	0.697	0.670	0.692	0.860	0.510	0.646	0.624	0.675	0.554	0.383	0.415	0.439	0.611	0.607	0.603	0.442	0.750	0.669	0.619	0.656	0.737	0.894	0.813
255	0.837	1.162	0.892	0.883	0.955	1.087	0.549	0.842	0.815	0.732	0.724	0.599	0.647	0.487	0.757	0.740	0.669	0.569	1.035	0.835	0.775	0.931	0.907	1.031	0.914
250	0.768	1.121	0.805	0.838	0.907	1.013	0.473	0.771	0.734	0.642	0.661	0.613	0.637	0.436	0.748	0.671	0.597	0.525	1.009	0.810	0.729	0.908	0.855	0.908	0.837
245	0.671	0.911	0.622	0.738	0.820	0.882	0.463	0.671	0.614	0.577	0.593	0.504	0.562	0.395	0.708	0.539	0.488	0.449	0.807	0.694	0.660	0.751	0.701	0.791	0.726
240	0.587	0.738	0.504	0.655	0.750	0.724	0.508	0.618	0.515	0.563	0.538	0.389	0.489	0.375	0.687	0.429	0.404	0.408	0.599	0.601	0.601	0.584	0.586	0.716	0.663
235	0.537	0.638	0.514	0.558	0.640	0.612	0.448	0.605	0.450	0.550	0.433	0.331	0.392	0.319	0.564	0.386	0.377	0.375	0.528	0.496	0.535	0.447	0.559	0.622	0.681
230	0.658	0.737	0.689	0.662	0.732	0.732	0.547	0.799	0.543	0.803	0.450	0.402	0.441	0.409	0.586	0.501	0.487	0.432	0.644	0.593	0.660	0.427	0.711	0.723	0.894
225	0.714	0.826	0.847	0.679	0.801	0.850	0.577	0.866	0.577	0.866	0.454	0.450	0.512	0.478	0.583	0.588	0.557	0.489	0.713	0.655	0.714	0.492	0.852	0.835	0.989
220	0.771	0.951	0.912	0.688	0.873	0.953	0.613	0.927	0.585	0.835	0.477	0.496	0.594	0.512	0.662	0.768	0.635	0.575	0.768	0.811	0.827	0.673	0.949	0.939	1.055
No.	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

CALIBRATION	275

Table 5.2 Concentrations of the 10 PAHsa in the data in Table 5.1.

Spectrum No.					PAH concentration mg l−1
		Py	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	0.188
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	0.188
8	0.456		0.080	0.224	0.160	0.112	2.160	0.120	1.000	0.040	0.188
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	0.188
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	0.188
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	used in this chapter:			Py = pyrene;	Ace = acenaphthene;		Anth = anthracene;		Acy = acenaphthylene;		Chry =

chrysene; Benz = benzanthracene; Fluora = fluoranthene; Fluore = fluorene; Nap = naphthalene; Phen = phenanthracene.

Response e.g.				Measurement
Spectroscopic
				e.g. Spectroscopic
	Y			X	Calibration
		Experimental design

X	C
Independent	Predicted
variable, e.g.	parameter, e.g.
Concentration	Concentration

Figure 5.1

Different notations for calibration and experimental design as used in this book

276	CHEMOMETRICS

5.2 Univariate Calibration

Univariate calibration involves relating two single variables to each other, and is often called linear regression. It is easy to perform using most data analysis packages.

5.2.1 Classical Calibration

One of the simplest problems is to determine the concentration of a single compound using the response at a single detector, for example a single spectroscopic wavelength or a chromatographic peak area.

Mathematically a series of experiments can be performed to relate the concentration to spectroscopic measurements as follows:

x ≈ c.s

where, in the simplest case, x is a vector consisting, for example, of absorbances at one wavelength for a number of samples, and c is of the corresponding concentrations. Both vectors have length I , equal to the number of samples. The scalar s relates these parameters and is determined by regression. Classically, most regression packages try to find s.

A simple method for solving this equation is to use the pseudo-inverse (see Chapter 2, Section 2.2.2.3, for an introduction):

c .x ≈ (c .c).s

so

(c .c)−1.c .x ≈ (c .c)−1.(c .c).s

or

I

xi ci

s ≈ (c .c)−1.c .x = i=1

I

ci2

i=1

Many conventional texts express regression equations in the form of summations rather than matrices, but both approaches are equivalent; with modern spreadsheets and matrix oriented programming environments it is easier to build on the matrix based equations and the summations can become rather unwieldy if the problem is more complex. In Figure 5.2, the absorbance of the 25 spectra at 335 nm is plotted against the concentration of pyrene. The graph is approximately linear, and provides a best fit slope calculated by

I

xi ci = 1.916

i=1

CALIBRATION	277

Absorbance (AU)

0.30

0.25

0.20

0.15

0.10

0.05

0.00
0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80

Concentration mg l−1

Figure 5.2

Absorbance at 335 nm for the PAH case study plotted against concentration of pyrene

and

I

ci2 = 6.354

i=1

so that xˆ = 0.301c. The predictions are presented in Table 5.3. The spectra of the 10 pure standards are superimposed in Figure 5.3, with pyrene indicated in bold. It can be seen that pyrene has unique absorbances at higher wavelengths, so 335 nm will largely be characteristic of this compound. For most of the other compounds in these spectra, it would not be possible to obtain such good results from univariate calibration.

The quality of prediction can be determined by the residuals (or errors), i.e. the difference between the observed and predicted, i.e. x − xˆ , the smaller, the better. Generally,

the root mean error is calculated:
E	=		I	(xi	xi )2	/d
		i		1	− ˆ

=

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 − 1 = 24 (see Chapter 2, Section 2.2.1), so that

E = 0.0279/24 = 0.0341

278 CHEMOMETRICS

Table 5.3 Concentration of pyrene, absorbance at 335 nm and predictions of absorbance, using single parameter classical calibration.

Concentration	Absorbance at	Predicted
(mg l−1)	335 nm	absorbance
0.456	0.165	0.137
0.456	0.178	0.137
0.152	0.102	0.046
0.760	0.191	0.229
0.760	0.239	0.229
0.608	0.178	0.183
0.760	0.193	0.229
0.456	0.164	0.137
0.304	0.129	0.092
0.608	0.193	0.183
0.608	0.154	0.183
0.152	0.065	0.046
0.608	0.144	0.183
0.456	0.114	0.137
0.760	0.211	0.229
0.152	0.087	0.046
0.152	0.077	0.046
0.304	0.106	0.092
0.152	0.119	0.046
0.456	0.130	0.137
0.608	0.182	0.183
0.304	0.095	0.092
0.304	0.138	0.092
0.760	0.219	0.229
0.304	0.147	0.092

220

240

260

280

300

320

340

Figure 5.3

Spectra of pure standards, digitised at 5 nm intervals. Pyrene is indicated in bold

CALIBRATION	279

This error can be represented as a percentage of the mean, E% = 100(E/x) = 24.1 % in this case. Sometimes the percentage error is calculated relative to the standard deviation rather than mean: this is more appropriate if the data are mean centred (because the mean is 0), or if the data are all clustered at high values, in which case an apparently small error relative to the mean still may imply a fairly large deviation; there are no hard and fast rules and in this chapter we will calculate errors relative to the mean unless stated otherwise. It is always useful, however, to check the original graph (Figure 5.2) just to be sure, and this percentage appears reasonable. Provided that a consistent measure is used throughout, all percentage errors will be comparable.

This approach to calibration, although widely used throughout most branches of science, is nevertheless not always appropriate in all applications. We may want to answer the question ‘can the absorbance in a spectrum be employed to determine the concentration of a compound?’. It is not the best approach to use an equation that predicts the absorbance from the concentration when our experimental aim is the reverse. In other areas of science the functional aim might be, for example, to predict an enzymic activity from its concentration. In the latter case univariate calibration as outlined in this section results in the correct functional model. Nevertheless, most chemists employ classical calibration and provided that the experimental errors are roughly normal and there are no significant outliers, all the different univariate methods should result in approximately similar conclusions.

For a new or unknown sample, however, the concentration can be estimated (approximately) by using the inverse of the slope or

cˆ = 3.32x

5.2.2 Inverse Calibration

Although classical calibration is widely used, it is not always the most appropriate approach in chemistry, for two main reasons. First, the ultimate aim is usually to predict the concentration (or independent variable) from the spectrum or chromatogram (response) rather than vice versa. The second relates to error distributions. The errors in the response are often due to instrumental performance. Over the years, instruments have become more reproducible. The independent variable (often concentration) is usually determined by weighings, dilutions and so on, and is often by far the largest source of errors. The quality of volumetric flasks, syringes and so on has not improved dramatically over the years, whereas the sensitivity and reproducibility of instruments has increased manyfold. Classical calibration fits a model so that all errors are in the response [Figure 5.4(a)], whereas a more appropriate assumption is that errors are primarily in the measurement of concentration [Figure 5.4(b)].

Calibration can be performed by the inverse method whereby

c ≈ x.b

or

I

xi ci

b ≈ (x .x)−1.x .c = i=1

I

xi2

i=1

280	CHEMOMETRICS

(a)

(b)

Figure 5.4

Difference between errors in (a) classical and (b) inverse calibration

giving for this example, cˆ = 3.16x, a root mean square error of 0.110 or 24.2 % relative to the mean. Note that b is only approximately the inverse of s (see above), because each model makes different assumptions about error distributions. The results are presented in Table 5.4. However, for good data, both models should provide fairly similar predictions, and if not there could be some other factor that influences the data, such as an intercept, nonlinearities, outliers or unexpected noise distributions. For heteroscedastic noise distributions there are a variety of enhancements to linear calibration. However, these are rarely taken into consideration when extending the principles to the multivariate calibration.

The best fit straight lines for both methods of calibration are given in Figure 5.5. At first it looks as if these are a poor fit to the data, but an important feature is that the intercept is assumed to be zero. The method of regression forces the line through the point (0,0). Because of other compounds absorbing in the spectrum, this is a poor approximation, so reducing the quality of regression. We look at how to improve this model below.

5.2.3 Intercept and Centring

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

c ≈ b0 + b1x

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 b0 (the intercept) and the second to b1 (the slope). X is now a matrix of two columns, the first of which is a column of ones and the second the absorbances.

CALIBRATION	281

Table 5.4 Concentration of pyrene, absorbance at 335 nm and predictions of concentration, using single parameter inverse calibration.

Concentration	Absorbance at	Predicted concentration
(mg l−1)	335 nm	(mg l−1)
0.456	0.165	0.522
0.456	0.178	0.563
0.152	0.102	0.323
0.760	0.191	0.604
0.760	0.239	0.756
0.608	0.178	0.563
0.760	0.193	0.611
0.456	0.164	0.519
0.304	0.129	0.408
0.608	0.193	0.611
0.608	0.154	0.487
0.152	0.065	0.206
0.608	0.144	0.456
0.456	0.114	0.361
0.760	0.211	0.668
0.152	0.087	0.275
0.152	0.077	0.244
0.304	0.106	0.335
0.152	0.119	0.377
0.456	0.130	0.411
0.608	0.182	0.576
0.304	0.095	0.301
0.304	0.138	0.437
0.760	0.219	0.693
0.304	0.147	0.465

	0.30
	0.25
(AU)	0.20				Inverse
Absorbance	0.15
	0.10
	0.05					Classical
	0.00								0.80
	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70
				Concentration mg l−1

Figure 5.5

Best fit straight lines for classical and inverse calibration: data for pyrene at 335 nm, no intercept