Introduction to multivariate methodology, an ... - Science Direct

Chemometrics and intelligent laboratory systems ELSEVIER

Chemometrics

and Intelligent Laboratory

Systems 29 (1995) 1777188

Tutorial

Introduction to multivariate methodology,

an alternative way?

Olav H.J. Christie Research

Rogaland

of Stm~angcr,

and Unwersity

Received 20 January

POB 2503 N-4034 Star,anger,

1995; acccptcd

1 February

Noway

1995

Abstract This is a tutorial for the tutor and summarises an approach to teaching the principles of principal components decomposition that apears to lower the student’s conceptual entrance barrier. It starts with summarising the fate of dissemination of the methodology that explains the scepticism with the methodology and defines the role distribution between expert, tutor, and student. The link of items presented are: generation of data structures, an alternative concept of correlation related to the elements of the score and loading vectors, formation of the model matrix by multiplication of the correlation vectors, interpretation of the loading vector, the latent variable, and decomposition of the centred observation matrix. One example of decomposition into three meaningful components is given, and, finally, the widespread confusion of interpreting sandwiched correlation structures is exemplified. Keywords:

Multi-component

analysis;

Multivariate

analysis;

Data structure;

Yeltsin example;

Correlations;

Principal

components

Teaching

Contents .

1. Introduction 2. How it started

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4. The basic feature.

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178

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5. The first cornerstone.

.

and the principal component

7. In search of real data structures: 8. What sort of information 9. Decomposition

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.............................

3. Setting the stage

6. Correlations

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. . . . .

.

. .

and heat conductivity

of sedimentary

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.

. . . .

179 179

.

. .

. . .

rocks

0169.7439/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSLII 0169-7439(95)00024-O

. .

179 181

. .. ..

is present in the loading and score matrices?

.. .

. .

178

.

.. . .

the matrix decomposition

into multiple principal components

10. Mineral composition

. .

.

. . .

.. .

182

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183

. . . .

184

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185

analysis;

0.H.J. Christie/

178 Il.

Prediction and the latent variable

12. Isolating correlation Acknowledgements. References..

Chemometncs

and Intelligent Laboratory

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1. Introduction The sketched tutorial starts with introducing processes as generators of correlation structures, like water-gunning or growing up of children. It aims at making the student familiar with different correlation structures being scrambled in real life, as exemplified with the Yeltsin and the Negro/Eskimo example. The tutorial uses the concept of score and loading vectors as object and variable correlation vectors. The basic principle of principal components decomposition directly refers to the correlation structures described by the object and loading vectors. After familiarising the student with this correlation concept through running a number of tutorial examples the attention is focused on projection techniques and the latent variable. Finally the difference between the generalised approach by principal components analysis and the focusing approach by projection techniques is thoroughly exemplified, because this appears to be the area of highest entrance barrier in vectorial multivariate data analysis.

2. How it started Every novelty needs its time. It needs time for consolidation and time for dissemination and acceptance. For multivariate data analysis the road to full acceptance in the technical and scientific community is still unfinished, even after 25 years of continuous efforts by path breakers in chemometrics. The pioneers laid the cornerstones of multivariate data analysis at an early stage, and the major steps into a fully-fledged methodology were taken around 1980. Still, multivariate methodology has not achieved the status in science and technology expected by the early pioneers despite the fact that computer force became

Systems 29 (19951 177-188 186 187 188 188

generally available through desktop PC’s ten years ago. The replacement of many conventional methods of statistics by vectorial methods was a revolution in data analysis. Looking back, it is evident that the principles of chemometrics were presented to a totally unprepared scientific community. Rather than being received with open arms, the application of multivariate methodology to chemistry often met with a baffled, ignorant and rejecting audience. Possibly, people felt they were talked to, face to face at too close a distance and this made them feel chemometricians as offending intruders into their own private domain. For some, multivariate methodology turned from a minor nuisance to a straightforward threat to their own professional integrity. Referees to wellknown international journals made statements to some of the basic chemometric papers of the mid1980’s that rank from absolute nonsense to deep personal affronts. There are reasons for this. The major issue is possibly that multivariate data analysis has a barrier of concepts and conventions that is not easily overcome by the layman. Even though the simple and straightforward illustrations of basic principles of chemometrics in early chemometric papers are lucid to the mast_er, they did not trigger the right sequence of associations of the audience (see, for example, [l-3] and the more updated papers by Wold and co-workers 14,511. The majority of people in science and industry have little formal training and even far less everyday experience in matrix algebra that is the backbone of multivariate data analysis. Consequently the scientific and industry community did not have the right receiving apparatus for many of the early presentations of multivariate data analysis. However, it is not necessary to have extensive knowledge to matrix algebra in order to become a competent user of multivariate data analysis. What is needed is knowledge to

O.H.J. Christie/

Chemometrics

and Intelligent Laboratory Systems 29 (1995) 177-188

its fundamental philosophy and a limited set of basic rules that over time lead to a feeling for data structures. The conventional tutorial lay-out of chemometrics is to start with simple principles and equations of matrix algebra, explaining all that is needed for understanding the practices of chemometrics. It is the intention of this essay to re-orchestrate the score in such a fashion that the processes generating and scrambling data structures are mentioned in the first movement, and that the unscrambling methods are explained in the following movements. Before doing this, we need to set the stage of tutor and students. The stage of reconstitution of the concept of correlation.

3. Setting the stage Human being perceives its environment in terms of multi-dimensionality, mostly by seeing and hearing. But the impression is so complex that we often need a simplified picture to be able to make quick decisions - a basic must of the survival. This basic must leads us to filter the visual and audible perceptions, often by translation of the multivariate into the univariate. We seldom realise that the filtering itself is a biological masterpiece of ingenuity, putting us in the position to hear the difference between Pavarotti and Domingo singing the same aria, or to identify the white colour of the paper of a newspaper, even though it is darker in artificial light than the black ink at sunshine. Filtering is so common that the concept of blackand-white thinking is a basic issue in politics as well as in human relationships. We are trained to think in terms of simple either-or correlations, like good-orevil or black-or-white. The major issue of the tutor of data analysis is to reform the student’s concept of correlation and to change the black-and-white thinking into colours and shade. There is a vast literature on learning and mnemotechnical techniques to improve the learning process, and the leading argument is that we do not remember in terms of numbers or words, we remember in terms of pictures. After one week the student may have forgotten as much as 80% of the content

179

of an oral presentation. Translation of words into pictures is one way of remembering forever, and selection of good pictures is a major tutorial task, even in chemometrics. However, the tutor’s attitude to roles during the tutorial is decisive to success. Basically, the tutor knows more than the student and has two alternative roles, that of the expert and that of the tutor. That sets the stage. The success of the tutor in transferring expertise is probably dependent on motivation to forget the role as expert. It is the student who plays the major part on the stage, not the expert. The student is the target of importance, and each tutor being tempted to expose his or her brilliance produces a barrier of ridicule to the audience. No student is interested in the pride of the tutor. The focus of interest of the student is to acquire the knowledge of the expert. Thus, in the learning process the tutor is not the source of knowledge, the tutor is the bridge over which his or her excellence as expert is conveyed to the students. This means a conceptual shift of traditional roles, a constructive and rational shift without touching the professional excellence of the expert or the integrity of the tutor.

4. The basic feature A birds eye glance over the multivariate building construction unveils some basic features: The foundation stone is the fact that Nature mixes data structures into a composite of correlation sets, and that multivariate data analysis aims at decomposing this composite into its components.

5. The first cornerstone The first cornerstone of multivariate data analysis tells about how data structures are generated: if a process influences a system, then it generates a data structure, and fills in later that if it does not generate a data structure, then it does not influence the system.

0.H.J. Christie/

Chemometrics

and Intelligent Laboratory Systems 29 (1995) 177-188

Fig. 1. People are listening to Mr. Yeltsin holding a speech a few days before he took over the leadership in Russia.

The ‘Yeltsin example’ offers a simple explanation to how different processes or properties generate different data structures. Imagine that you are in a helicopter and have taken a picture of Boris Yeltsin talking to a crowd in a square in Moscow just before he took over leadership from Mr. Gorbatschov (Fig. 1). Suddenly, the police come with their water canons, and in a jiffy the people have shifted their position, as you photographed them from the helicopter (Fig. 2). What you observe is that a process, the watergunning, produces a data structure, and people are running in one direction. A correlation pattern is generated and this pattern is an important element of the data structure of the crowd. If you make a table of location co-ordinates of persons relative to two imaginary co-ordinate axes, the data structure generated by the water-gunning is consistent with a principal component of that table. After a short while, people start running into a nearby street, and the running along the escape route generates another data structure, independent of the water-gunning process. And your last snap-shot from the helicopter looks like Fig. 3. Principal components describe the important data structures generated in the crowd (Fig. 3) and have been superimposed in Fig. 4 which shows that two independent correlations do exist. Along the first

Fig. 2. The result of the process of water-gunning. Zubrovka and Mr. Smirnoff added.

Location of Mrs.

Fig. 3. The result of the process running along an escape route street is added to the structure of the water-gunning result.

O.H.J. Christie/

Chemometrics

181

and Intelligent Laboratory Systems 29 (1995) 177-188

principal component (water-gunning) the locations of Mrs. Zubrovka and Mr. Smirnoff are similar: their projection onto the arrow marked PC1 fall on roughly the same point and their first principal component scores are, therefore, equal. Along the second principal component (PC2, escape route), however, their locations and, thus, their second principal component scores are different. In this fashion, principal component decomposition unscrambles the effect of two processes: watergunning and flight along an escape route, and allows an accurate description of each object in relation to those processes. The tutor would tell that all these properties are important in the multivariate sense, and that they are

sandwiched together to form structure of the party.

6. Correlations

the composite

and the principal

data

component

The second cornerstone describes the data structure as composed of two sets of correlations: one among the variables and one among the objects. It might be illustrated by the diagram of Fig. 4, displaying that the direction of the principal component is defined by the correlation between x and y (the variables) and by how the objects score along this component. The correlation between age of children and fea-

Y

Fig. 4. First and second principal components describe the result of the two processes Location of Mr. Antonov, Mrs. Zubrovka, Mr. Smirnoff and Mr. Kouznetsov added.

water-gunning

and running along the escape route.

O.H.J. Christie / Chemometrics

182

Table I The model matrix (bold itaiic) is produced by multiplying vector by the loading vector Height Weight Experience 6 10 3 Wendy 2 20 John 4 40 Michael 6 60 120 Mary 12 Scores

6 12 18 36

12 24 36 72

Metabolism - 10

and Intelligent Laboratory

the score

Loadings

- 20 - 40 - 60 - 120

tures like instance height, weight, experience, and metabolic rate may serve as a first example of how a data structure in multi-dimensional terms is generated. The tutorial continues by the following line of arguments: as an exercise we shall mimic Nature and act as the process of children growing up. In the column vector we insert the age as the score of each of the five children. In the row vector we arrange the features (that we call variables) in terms of numbers representing them (loadings).The model matrix is formed simply by multiplication of the column (score> vector by the row (loading) vector (Table 1) and the loading values are selected so that the elements in the model matrix are reasonably similar to what we would have measured in real life. The model matrix consists of the numbers in bold italic of Table 1 and that was generated by multiplication of the score vector by the loading vector. The loading values of height, weight, and experience are all positive. This means that they increase by increasing age. The metabolism loading value is negative, which means that by increasing age the metabolic rate decreases. We can read that Michael (age 6) weighs 18 kg (score times loading, 6 X 3 = 181, and that John’s metabolic rate is - 40 (4 X (-- lo)), which means forty less than some arbitrary start value that, by the way, is not given in Table 1. For the moment we are not worried that Wendy’s height is only 20 cm at an age of 2 years, because we have only produced a model of the real world. The main thing is that by multiplication of a column vector by a row vector we have produced a model matrix. Now we lift the white rabbit by its ears from the

Systems 29 (1995) I77-188

top hat: the combined column and row vector is the principal component. We stress that the elements in the column score vector characterise correlation between the children in terms of age, and, consequently, the score vector is a correlation vector of the objects, In the same fashion, the loading vector characterises the correlation between different properties of the children. For instance, there is a negative correlation between the childrens’ height and their metabolic rate, as seen from the corresponding loadings which have opposite signs. There is positive correlation between height and weight, as seen from the height loading and the weight loading having the same sign. The main issue of the second cornerstone is that the score uector represents the correlation between the objects, and the loading Llector represents the correlation between the variables, and a model matrix can be produced by multiplying the score uector by the loading vector.

7. In search of real data structures: composition

the matrix de-

The ‘age of children’ example exemplifies how a model matrix is generated by multiplication of two correlation matrices: the column score vector and the row loading vector. The student learns that in multivariate data analysis we start by doing this operation the other way around. We start with an observation matrix, 0, where the properties of the objects are given as rows in the matrix. The loading and score values in T and B, respectively, are stepwise changed in iterated computations in such a fashion that the model matrix M becomes as similar as possible to the observation matrix 0. The diagram of Fig. 5 illustrates that the difference between the model matrix and the observation matrix goes into to the residual matrix R. The fitting of the principal component to the centred observation matrix means that the residual matrix R is minimised. The less the values in the residual matrix, the better is the observation matrix described by the principal component.

O.H.J. Christie/ Chemometricsand IntelligentLaboratorySystems29 (1995)177-188 Variables

=

‘II Principal component

+

R

1

M

Fig. 5. The centred observation matrix 0 is decomposed into the principal component TB and the residual matrix R. For practical reasons, the observation matrix is initially centred. This means that mean values are calculated for each variable, and the original observation matrix is replaced by the matrix of raw data less mean value for each variable.

8. What sort of information ing and score matrices?

is present in the load-

The leading questions in multivariate data analysis are: 1. How can the set of loading values be interpreted? 2. How do the objects relate to that information? The graphical presentation given in Fig. 6 is a very simplified display of a loading pattern which illustrates the type of information in a loading vector.

Intuitively, we see that there is a positive correlation between height, weight and experience, and a negative correlation of these three variables to metabolic rate. Thus, Fig. 6 displays the correlation between the variables. And this is where the experience of the analyst comes in. We make it clear to the student that multivariate data analysis tells about data structures and correlation patterns, but it does not tell the name of the processes that generated them. So, the student will have to ask: what is it among children that increases their height, weight and experience, but reduces their metabolic rate at the same time. In this case the answer is obvious: the process is ageing, or rather: growing up. The student has spotted the identity of the principal component from evaluation of the correlation presented by the loading diagram, and will learn that such interpretation of loading patterns is a central exercise in multivariate data analysis. The next thing is that one can read the correlation between the children from the score vector. This means that the correlation between the children in terms of growing up is given by their score value which has a systematic relation to their age, as illustrated in Fig. 7. It demonstrates that Michael (6) is closer by age to John (4) than to Mary (12). Extending this to general validity, the student realises that score values can be used as measures of properties that are not measured or listed among our

IO

a 6 4

3

2--

l-----l

F P O a !j -2 --

Height

Weight

183

Exprnce

Fig. 6. Loading diagram of the ‘age of children’ example

Metabol

184

O.H.J. Christie / Chemomerrics

and Intelligent Laboratov

observations (age is not a variable in the child matrix). A new and realistic example is now appropriate: a data matrix of good and bad objects where no quality variable is included. Let us presume it is decomposed into one principal component, and it turns out that all the bad objects have low scores and all the good ones have high scores. Then the score value is a measure of quality, and this opens for setting up a forceful quality predictor which could be very useful if the actual measurement of quality is complicated and expensive. In this simple fashion, the student has got a taste for the latent cariable. At this moment the student has familiarised with the fact that the loading row vector got its numerical values from the process that acted on the system and expresses the correlation between the variables, and, furthermore, that the score column vector shows the correlation between the objects and contains information about to what extent the process has influenced the individual objects.

9. Decomposition nents

into multiple

principal

compo-

In the pious hope to reduce the height of the entrance barrier to multivariate data analysis we have skipped some of the precision in terminology and

Wendy

John

Systems 29 (199.5) 177-188

concepts. Now the time has come to explain elements of multivariate data analysis. The technique of decomposing an observation matrix into two or more principal components is simple. We start with decomposition of the centred observation matrix of the Yeltsin crowd in the usual way (Fig. 5) to form the first principal component as well as the first residual matrix. By this operation we have extracted the most prominent data structure, which in the Yeltsin example corresponds to the result of the water-gunning. But the information of flight along escape routes is still kept in the residual matrix R. Consequently, we subject the residual matrix to a principal components decomposition, and these two steps are illustrated in the diagram of Fig. 8. In the Yeltsin example, the first principal component contains information about the result of the first process (water-gunning) and the second principal component about the result of the second process (running along escape routes). B, tells the direction of the water-gunning and T, tells how far each person has run along that direction. B, tells the direction of the escape street, and T, tells how far each person has run along that direction. After this simple presentation the student is ready for the ‘European food’ example, well known to every chemometrics tutor. The first principal component separates the Mediterranean countries from the

Michael

Fig. 7. Score diagram of the ‘age of children’ example (non-centred)

Mary

O.H.J. Christie / Chemometrics

and Intelligent Laboratory

Systems 29 (1995) 177-188

185

misplaced adds to the tutorial value of the data set: some important food items, like red pepper, and the whole range of beverages are not included in the data set. Missing important variables may lead to an incomplete picture.

10. Mineral composition sedimentary rocks

Fig. 8. Stepwise principal component decomposition: first of the observation matrix 0 into the column score vector T, and the loading row vector B, plus the residual matrix R,, and next, decomposition of the residual matrix R, to form the second score column vector T,, the second loading row vector B,, and the second residual matrix R,.

North European ones by olive oil and garlic, as nicely displayed by score and loading plots, and the second principal component separates the crisp-bread chewing and real-coffee drinking Scandinavians from the English and French. The first-to-second score plot delights the student by finding that Holland lies between Germany and England, and Belgium between Germany and France. The fact that Austria seems

and heat conductivity

of

Multivariate data analysis was an important tool for the development of a commercial data base for heat conductivity of sedimentary rocks. The loading diagrams produced during that development offers another example of how principal components decomposition unscrambles different processes influencing the same set of variables, which in the present case are mineral data, porosity and total organic carbon (TOC). Again we emphasise that interpretation of loading diagrams is a key issue to any multidimensional data analysis. If the loading pattern is meaningless, then the source data are probably confusing as well. The first principal component loading diagram of Fig. 9 demonstrates that there is negative correlation of clay minerals to both quartz and sum carbonates, and positive correlation of clay minerals to pyrite and

and carbonates

-25 1

Fig. 9. First principal shales/mudstones,

component

as represented

balance shales loadings

emphasise

by clay minerals.

the importance

of the balance

where pyrite and TOC is found of elastic rocks (quartz),

carbonates

(sumcarb),

and

186

O.H.J. Christie/

Chemometrics

and Intelligent Laboratory S,vstems 29 (1995) 177-188

Increase in quartz

e expense of calcite

-30 Fig. 10. Second principal

component

loadings

are characterised

by negative correlation

total organic carbon. In plain geology text this means that elastic rocks (as represented by quartz) and carbonates balance shales/mudstones where pyrite and total organic carbon is found, and that the balance of these rock types is important to the heat conductivity of sedimentary rocks. The second principal component loading diagram of Fig. 10 shows that increase in quartz at the expense of calcite reduces the porosity, which, like calcite and sum carbonates, is negatively correlated to quartz. Immediately, the sedimentologist would refer to the process quartz cementation of chalk. The loading diagram of Fig. 11 shows that there is negative correlation between silicates and dolomite.

of quartz to sum of clays, sum of carbonates

It is also negative correlation between dolomite and porosity. Again, in plain geology text, this loading diagram tells that when dolomite increases at the relative expense of silicates - which is called dolomitisation - it reduces the porosity. This is also important to heat conductivity of sedimentary rocks.

11. Prediction

and the latent variable

Once the student has become familiar with the meaning of scores and loadings, the road to classification and prediction is easy. The tutor has made the student confident in scores and loadings by adding

when dolomite increases

20 15 1 70 i

2

0

& .6

Silicates are reduced

5 0

t

-5

3 -10 -15 -20

se

-25 Fig. 11. Third principal component

loadings demonstrate

and

that there is negative correlation

of dolomite to silicates and porosity

O.H.J. Christie/

Chemometrics

and Intelligent Laboratory Systems 29 (1998) 177-188

other examples of correlation unscrambling by aid of principal components decomposition. Now is the time for putting a name on the variable that was not entered in the raw data, but that is present as a ghost in the correlation structure: the latent rariable. Age is not a variable in the loading vector of the example of properties of children. Still, the major correlation structure is related to age, and the scores are closely correlated to age - in fact we started the whole example by entering age in the score vector. The student is well prepared to take the techniques of PCR and PLS. The tutor tells that the latent variable age can be estimated by multiplying the score value by a constant, and the student may harvest from a multitude of examples in the literature.

12. Isolating

correlation

Chemometrics offers a thrilling opportunity to isolate correlations by target rotation. There are two major items of isolation: data laundering and variable focusing. In some cases the data are contaminated by the influence of an undesired process, for instance amount of precipitation upon acidification of lakes. If causality and characterisation of air pollution influence upon lake acidification is the goal of the study, the raw data are contaminated by the fact that there is more precipitation in some areas than in others. That effect can be removed by target rotation to the variable amount of annual precipitation, and the actual information about pollution influence is kept in the residual matrix. This example is also easily understood by the student, but it there are conceptual problems to come. The fight for dissemination of chemometrics is the fight against inborn prejudice and black-and-white reasoning. The trace element distribution in crude oil offers an example of the conceptual problem. There are many different characterising parameters of crude oils. The so-called biomarkers are a lump group chemical constituents, most of them related either to steranes (like cholestane - formed from cholesterol) or to hopanes being related to components of bacterial cell-walls. Another hot item of oil geochemistry is trace elements in oils. Some trace elements, like the light transition elements, seem to be related to the

187

living conditions of the organisms that later were buried in sediments and constitute the source of oil. The living conditions are indicators of geological conditions of major interest to oil explorationists. The chemometrician meets the following challenge: is there a systematic correlation between a given trace element and the distribution pattern of biomarkers? The answer lies in the loading pattern of the biomarker molecules by target rotation to the given trace element. The correlation between a given biomarker and trace elements is obtained by target rotation to that biomarker molecule. The tutor target rotates to molybdenum and finds that variation in molybdenum is positively correlated to the so-called bis-norhopane and extended hopanes (having 28 and more than 30 carbon atoms, respectively) and negatively correlated to one C27 hopane isomer and the compound gammacerane. Furthermore, the target rotation demonstrates that molybdenum is strongly correlated to antimony. The tutor tries another target rotation: this time to gammacerane. The target rotation shows that there is no correlation between molybdenum and antimony whatsoever, and the situation of confusion is complete. The student once more has met the wall of black-and-white thinking, chemometrics seems to be a source of confusion and the irritated students loses self-confidence. What appeared so simple appears to result in rubbish. This is the battlefield of the tutorial, it takes much effort and many examples to restore the student’s self-confidence and trust. The tutor returns to the Negro-Eskimo example of superimposed correlations: Fifty Negroes and fifty Eskimos have been invited to a garden party. When asked, one student would say that the major data structure among the guests is skin colour and language. The tutor might argue that an equally important issue is sex, and another student might argue that age, separating children from adults is the important issue. The tutor makes it clear that there is a basic difference between principal components decomposition one the one hand and latent variable projections and target rotations on the other: principal components decomposition is a method of generalisation of correlation structures. Latent variable projection and target rotation are methods of focusing. With principal components eyes you se the diver-

188

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Chemometrics

and Intelligent Laboratory Systems 29 (1995) 177-188

sity of the guests of the garden party, with target rotation you focus on either skin colour, sex or age. You do not change skin colour by changing focus from men to women. This is one of the conceptual difficulties of the student. The whole party plays the role of oil chemistry, skin colour might play the part of trace elements, and age plays that of biomarker concentrations. Focusing on trace elements does not change the biomarker identities, and vice versa. All methods are meaningful, but the different chemometric tools are used for different things. Practice helps and once the tutor has the right battery of examples the student becomes a master.

Acknowledgements My acknowledgement goes first of all to my tutor in the 1970’s and 1980’s: Svante Weld. He strongly influenced my professional character and profile by

the brilliant, simple and careful way he introduced myself, being ignorant of matrix algebra and advanced statistics, to chemometrics. I am also greatly indebted to the many gifted students I met 15 years ago and who later contributed so significantly to the perfection of chemometrics and its dissemination. Last, but not least, I am full of gratitude to my dear wife who tolerated my physical presence and mental absence at the home-PC with endless love.

References [l] [2] [3] [4]

S. Weld, Pattern Recognition, 8 (1976) 127. S. Wold, Technometrics, 20 (1978) 397. S. Wold and M. SjiistrGm, ACS Symp. Ser., 52 (1977) 243. S. Wold, C. Albano, W.J. Dunn III, 0. Edlund, K. Esbcnsen, P. Geladi, S. Hellberg, E. Johansson, W. Lindberg and M. SjiistrGm, in B.R. Kowalski (Ed), Chemomctrics, Mathematics and Statistics in Chemistry, Reidel, Dordrccht, 1984, p. 17. [5] S. Wold, K. Esbensen and P. Geladi, Chemom. Intell. Lab. Syst., 2 (1987) 37.