Vol. Polymer characterization with a fuzzy classification ...downloads.hindawi.com/journals/jamc/1994/359813.pdf · analysis’). Theprimaryaimofacluster analysis is to seek andidentify

Journal of Automatic Chemistry, Vol. 16, No. 5 (September-October 1994), pp. 161-165

Polymer characterizationclassification algorithm

with a fuzzy

D. J. Ramsbottom, M. J. Adams*School of Applied Sciences, University of Wolverhampton, Wulfruna Street, WV69EP, UK

and J. CarrollI.C.I. A/laterials, Wilton, Adiddlesbrough, Cleveland, UK

The classification of polymer samples from their infra-red spectrahas been achieved by the application of a fuzzy c-means clusteralgorithm. The generation of a fuzzy classifier allows thecharacterization of samples which are a combination of more thanone pure polymer.

Introduction

The widespread use of personal computer systems in theanalytical laboratory, and the processing power they offer,has supported the increasing interest in the applicationof chemometrics and multivariate analysis. The develop-ment and use of pattern recognition, PR, techniques forclassifying known samples and identifying new sampleshas received a great deal of attention in the literature. Awide range of PR algorithms have been reported; thesemethods are usually categorized as ’supervised’ PR (inwhich a training set of previously identified samples areanalysed in order to determine some discriminant functionwhich allows subsequent samples to be classified), or’unsupervised PR’ (commonly referred to as ’clusteranalysis’). The primary aim of a cluster analysis is to seekand identify implicit structure in unclassified sets of data.Numerous texts and reviews are available describing theprinciples and applications of such data manipulationtechniques [ 1-4].

While hierarchical cluster analysis is one of the most

popular PR techniques in science, because of the familiardendrogram representation ofassociation between objects,the use of this graphic display can produce considerabledistortion in the data representation and the actual displayobtained is dependent on the method employed. Non-hierarchical techniques do not directly distort the data,but their graphic representation is less obvious and theresults may not be simple to interpret. The aim of thepresent study was to investigate the implementation of anon-hierarchical clustering algorithm, based on fuzzypattern recognition, and its application to the classificationof polymer films by means of their infra-red spectra.

The theory offuzzy sets, in a form suitable for computerizedimplementation and as a formal branch of mathematics,was first proposed by Zadeh [5]. The theory extends crispset theory, in which an item either belongs or does not

belong to a set, to define an object’s membership function,

Correspondence to Dr Adams.

or degree of belonging to a set. Such membershipfunctions define a transition from wholly belonging to not

belonging; they are continuous, monotonically increasingor decreasing functions. Fuzzy set theory is currentlyreceiving considerable attention in a wide range oflaboratory automation and data processing tasks as itprovides a means of expressing uncertainty or imprecise(fuzzy) data in a computer readable format. An overviewoffuzzy set theory and potential applications in analyticalchemistry has been presented by Otto [6].

The concept of a membership function is directlyapplicable to pattern recognition and cluster analysis asit provides a numerical value expressing similaritybetween items rather than the binary yes/no classifier’sassociated with conventional binary discrimination func-tions. Fuzzy pattern recognition algorithms have beenextensively reviewed by Bezdek Ill--the algorithmimplemented in this study is the fuzzy c-means method.The application of the algorithm to synthetic artificialdata and to the classification of a selection of IR spectraof polymers is demonstrated.

Fuzzy classification algorithm

Cluster analysis seeks to achieve classification of a groupofsamples or objects according to the similarity offeaturesmeasured on each object. Clearly defined clusters will becharacterized by member objects being similar or close inthe multidimensional feature space, and well separatedfrom other clusters. A common measure of similarity, andthat used in the present study, is the Euclidean distancebetween objects. The fuzzy c-means clustering techniqueseeks to minimize the within-cluster squared Euclideandistance between the objects and the centre ofeach cluster.The number of clusters, c, is pre-selected by the user.The algorithm to generate the c-partitions and, hence, thec-clusters was proposed initially by Bezdek et al. [3] andthe computerized implementation has been discussed byZadeh [7]. The degree or extent to which an object, i,belongs to a specific cluster, c is referred to as that object’smembership function (#ci). From a preselected numberofclusters and a randomly generated initial fuzzy partitionof the objects such that there are no empty clusters, andthe membership functions for an object with respect toeach cluster sum to unity, the algorithm proceedsiteratively. First, weighted means representing the clustercentres are calculated. From these centre values, newmembership functions for each object with respect to eachcluster are calculated, and new fuzzy partitions of thesample space are derived. This process continues with newcluster means and new membership functions until thetotal change in membership functions is consideredinsignificant and the cluster populations remain un-changed. The mathematical basis of the algorithm and

0142-0453/94 $10.00 (C) 1994 Taylor & F ’is Ltd.161

D. J. Ramsbottom et al. Polymer characterization with a fuzzy classification algorithm

proposed stopping conditions have been described byBezdek et at. [33 and Zadeh [73.In this study the program was implemented in C languageon a 386-based microcomputer. The results comprised alist of c-cluster centres, and a value denoting the object’sdegree of membership to each cluster.

Test data

Table 1. The partition coefficients for 2,3, 4 and 5 clusters for the butterfly data,showing two clusters to be an optimum.

Clusters Partition coefficient

2 0"8433 0"6854 O’5925 0"609

Typical results obtained with the fuzzy c-means algorithmcan be illustrated using the ’butterfly’ data set, which isbi-variate and two-dimensional data--see figure l(a). Iftwo clusters are selected, then the membership of eachobservation to its nearest cluster centre (#) is as shown.The selection of c, the number of clusters into which thedata are to be partitioned, is chbsen by the user. As withtraditional clustering techniques, there are no formal rulesto dictate what is optimal or best. Interpreting clusters islargely subjective, meaningful clusters can often beobtained for several values ofc. Several objective functionsto assess cluster validity have been reviewed by Bezdek[t]; a particularly useful one, which is simple toimplement, is the partition coefficient defined by:

k=l i=1

where ik is the degree of membership of the object to thecluster, i, and n is the total number of objects.

(0.87,0.13) / (0.13,0.87)

/J. (0.95,0.05) / (0.05,0.95)

(0.97,0.03)/.J,(1.0,O) /J(0.88,0.12) (0.5,0.5) (’.12,0.ee) , (,1.0) (0.03,0.97)

(0.95,0.05) (0.05,0.95)

(0.87,0.13) /x (0.13,0.87)

2 3 4 5 6 7

x

Figure l(a). The butter.fly data set with membership values #(A, B), where A is the membership to cluster 1 and B is themembership to cluster 2.

The computed value for F(#; c) depends on all observationsand maximizing this function can produce the most validclustering. The butterfly data were examined with valuesfor c 2, 3, 4 and 5, and table shows the partitioncoefficients calculated for each value. As to be expectedc 2 provides an optimal clustering as given by themaximum value for F(#; c), as well as the most intuitivelymeaningful.

Once a satisfactory clustering and partition is achieved,a membership function surface can be created using theoriginal data as a training set, as illustrated in figure (b),for c 2. New, unclassified samples can be subsequentlyclassified by interpolation on the surface.

The subjective nature of cluster analysis can be betterappreciated by reference to a more diffuse data set.

Figure 2 illustrates the scatter plot of a bi-variate dataset [8]. The natural clustering patterns are not so obviousor unambiguous as with the butterfly data. Using fuzzyc-means, the results for c 2, 3, 4 and 5 are illustratedin figure 3, together with the computed partitioncoefficients. High values of F(#; c) are obtained for eachvalue of c tested, a result in agreement with visualinterpretation of the data since each group of partitionshas significance, depending on the context of the inter-pretation. It is of particular interest and value to notethat, unlike hard clustering techniques, fuzzy clusteringdoes not force an object into one or other cluster but canallow it to belong equally to two, or more, groups.

Characterization of polymers

The use and application of the fuzzy c-means algorithmwas investigated further using infra-red spectral data andexamining the ability of the technique to classify spectra.

Figure l(b). The membership profile to 2 clusters of the butterflydata.

2 3 4 5 6

x

Figure 2. The test data employed by Zupan [8].

162


2CLUSTERS PARTITION-0.824 ,_3 CLUSTERS PARTITION 0.792

4CLUSTERS PARTITION-0.802 5 CLUSTERS PARTITION -0.749

Figure 3. Membership profiles and partitions for 2, 3, 4 and 5 clusters of the test data from figure 2.

Polymer samples tbr characterization were acquired froma number of sources, including a commercial polymeridentification kit [9] and an industrial polymer analysislaboratory (ICI Materials Research Centre, Middles-brough, Cleveland, UK). Several homopolymers werechosen with differing copolymers, copolymer levels,additives and additive levels. These were prepared foranalysis as thin films, using standard methods of pressingunder hot plates or casting from solution.

Table 2. The variance and cumulative variance from a principlecomponent analysis ofthe polymer spectra.

Principal components Variance cumulative variance

53"46 53"462 25"2O 78"663 12"6O 91"264 3"60 94-865 2"06 96"92

Data acquisition and pre-processing

The infra-red spectra were acquired on a Philips PU9624FTIR spectrometer from 4000 to 666 cm -1 (2"5-15microns) at a resolution of 4 cm- x, to provide absorbancespectra containing 867 points each. All spectra werenormalized on the most intense absorption band to reducefilm thickness effects. As infra-red spectral data are highlycorrelated, data reduction was carried out in two steps todecrease the number of variables required to representthe spectra. First, the spectra were reduced to 216 valuesby tbur-point averaging, and then a principal componentanalysis (PCA) was performed.

It can be seen from table 2 that the first three principalcomponents account tbr more than 91 of the totalvariance within the spectral data. A scatter plot of thepolymer data loaded onto these components is shown in

figure 4. It is evident that J.hese three components providefor effective and clear separation between groups ofsamples. The first component, PC1, forms a partitionbetween acrylic polymers and others. PC2 forms two

partitions, nylon and PVC polymers are separated fromstyrene and acrylic polymers and PC3 provides for apartition between styrene and the others.

Examination of the principal component loading spectra(figure 5) highlights the weighting given to each datapoint in the spectra. Where each partition is formed, themajority of bands in the corresponding spectra receivedeither a strong positive or negative weighting. Where PC2produces two partitions it can be seen that the bands inthe nylon spectra receive positive weighting and the bandsin the PVC spectra receive negative weighting. Wherethe bands in a spectrum do not conflict, as in the carbonylband at 1700cm -a for acrylic in figure 5(a), the

163


Scatter Plot of first 5 PC’s

Groups

NyAcrylicStyrene

Figure 4. A scatter plot of the first three principal components oflhe four polymer groups.

weightings are high. However, where common bandsexist, for example, about 3000 cm -1 in figures 5(b) and5(c), the weighings are small. As there are subtledifferences between spectra in each group, only thecommon base polymer bands receive weightings.

The occurrence of copolymers and blends of polymers isquite common in plastics, and the application of fuzzyc-means methods to cluster the data set has an intuitiveappeal as unknown samples may indeed be a mixture oftwo groups, and may therefore have a relatively highmembership to the cluster of each pure polymer group.

Subsequent characterization of an unknown samplerequires the same data reduction steps to be carried outas for the test samples, i.e. reduction to 216 points andapplication of the calculated principal component vectorsto give three principal components. These are used tocalculate the membership of the unknown sample to each

PCA1 and Acrylic pealacrylic

1.0-

0.8

0.6

0.4

0.2

0.0

-0.4 30’04000 0 2000 1000

-0.2

Figure 5(a). The weightings given by PC1 plotted with an

infra-red spectrum of an acrylic sample. The major bandf in thespectrum receive negative weightings, causing the samples to bepartitioned into two groups.

PCA2 and Nylon

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.44000 30’00 2000

cm-1

pca2nylon

10’00

Figure 5(b). The weightings given by PC2 plotted with an

infra-red spectrum of nylon, showing the nylon bands receivingpositive weightings.

PCA2 and PVC pca2pvc

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.44000 30’00 20’00 1000

cm-1

Figure 5(c). The weightings given by PC2 plotted with an

infra-red spectrum of PVC, showing the PVC bands receivingnegative weightings.

PCA3 and Polystyrene

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.44000 3000 20100 10’00

cm-I

Figure 5(d). The weightings given by PC3 plotted with an

infra-red spectrum of polystyrene, showing the polystyrene bandsreceiving positive weightings.

164


Table 3. The membership values for a series of plastic samples,A-F. Where A acrylic; B polystyrene; C polystyrene;D nylon; E PVC; and F acrylic/styrene blend.

Standard polymers

Sample Actualcode Acrylic Styrene PVC Nylon main polymer

A 0"40 0"19 0"25 0"16 AcrylicB 0"005 0"97 0"015 0"01 StyreneC 0"01 0"97 0"015 0"005 StyreneD 0"032 0" 155 0"089 0"724 NylonE 0"18 0"30 0"33 0"18 PVCF 0"33 0"25 0"24 0" 18 Acrylic and

styrene

based applications. For example the development ofexpert systems for the interpretation of analytical datahas indicated that fuzzy logic represents an excellentmeans of developing and propagating uncertainty withinsuch systems [10]. A serious problem associated withintegrating uncertainty in expert systems lies withgenerating numerical values of uncertainty from analyticaldata. The fuzzy c-means algorithm provides these valuesand the algorithm can be linked directly to an expertsystem shell. As applications of artificial intelligencecontinue to grow in analytical science, the effect ofuncertainty and its measurement and integration intodata analysis schemes will become more important.

Acknowledgements

cluster. The test plastics examined were ground to powderand their spectra obtained as compressed KBr discs. Thisproduced spectra with lower signal-to-noise ratios thanspectra from thin films and sloping base-lines due to

scattering and diffraction effects. However, with base-linecorrection, it is evident from the results shown in table 3that correct identification and classification ofeach samplewas possible.

Conclusion

A pattern recognition program based on the fuzzy c-meansalgorithm has been implemented, and evaluated forartificial data sets and data obtained from an infra-redanalysis of a range of polymer samples. The resultsobtained illustrate that the method is capable ofprovidingmeaningful clustering and classification, without theproblems associated with producing traditional binaryclassification decisions.

The calculations, and generation, of membership functionvalues as obtained from the algorithm discussed above,are of interest not only for their intrinsic value in directclassification, but also for their potential use in broader

The authors wish to acknowledge I.C.I. Wilton, UK andUnicam Analytical Systems, Cambridge, UK for samplesand their support in this project.

References

1. BF.ZDF.I, J. C., Pattern Recognition with Fuzzy Objective Function

Algorithm (Plenum Press, New York, 1987).2. SEyy, P., Kemija u Industro’i, 39 (1990), 177.3. BF.ZDEK, J. C., EHRLmH, R. and FtJLL, W., Computers and Geosciences,

10 (1984), 191.4. MIYAMOTO, S., Fuzzy gels in Information Retrieval and Cluster Analysis

(Kluwer Academic Publishers, The Netherlands, 1990).5. ZADF+I, L. A., Information and Control, 8 (1965), 338.6. OTTO, M., Chemometrics and Intelligent Laboratory Systems, 4 (1988), 101.7. ZAD.H, L. A., in Classification and Clustering, edited by Van Ryzin, J.

(Academic Press, New York, 1977).8. ZUPAY, J., Clustering of Large Data Sets (Wiley and Sons, Chichester,

1982).9. The Resin Kit Company, 1112 River St., PO Box 509, Woonsocket,

RI 02895.10. RAMSBOTTOM, D. J., ADAMS, M. J., SUMIGA, J. and CARROLL, J.,

Chemometrics and Intelligent Laboratory Systems, 19 (1993), 53.11. PERKINS, J. H., HASF.YOEmL, E. J. and GRIFFITHS, P. R., Analytical

Chemistry, 63 1991 ), 1747.

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Vol. Polymer characterization with a fuzzy classification ...downloads.hindawi.com/journals/jamc/1994/359813.pdf · analysis’). Theprimaryaimofacluster analysis is to seek andidentify