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Competitive Market Structure and SegmentationAnalysis with Self-Organizing Feature Maps

(Competitve Paper)

Thomas Reutterer1

Abstract

The simultaneous treatment of two interrelated and well-known tasks from strategic marketingplanning, namely the determination of competitive market structure (CMS) and marketsegmentation, is addressed via application of the ”Self-Organizing (Feature) Map” (SOM)methodology, as originally proposed by Kohonen (1982). In the present paper, some majoraspects of the methodological basis of the SOM method are outlined and an SOM-based jointCMS-(preference-)segmentation analysis is illustrated using individual brand choiceprobabilities derived from diary household panel data.

1. Introduction

The analysis of competitive market structure (CMS) represents an important concept in the(strategic) marketing planning process (see, e.g., Wind and Robertson, 1983; Day, 1984;Myers, 1996). In an excellent exposition of contemporary approaches to the determination ofCMS, DeSarbo, Manrai and Manrai (1993) describe the primary task of CMS analysis asderiving a configuration of products/brands in a (prespecified) product class on the basis ofcompetitive relationships between products/brands. Although not uniquely operationalized, itis widely accepted to conceptualize the degree of inter-brand competition as a measure ofsubstitution as perceived by consumers (as a selection of relevant papers cf. Day, Shocker andSrivastava, 1979; Bourgeois, Haines and Sommers, 1982; Fraser and Bradford, 1983; Lattinand McAllister, 1985).

During the last three decades, numerous models for CMS analysis utilizing various measuresof substitutability have been proposed and some of them nowadays are standard tools inmarketing research (the latter are represented in marketing textbooks such as Bagozzi, 1986,pp. 244; Moore and Pessemier, 1993; Urban and Hauser, 1993, pp. 253; Lilien, Kotler andMoorthy, 1992, pp. 74 or Lilien and Rangaswamy, 1998, pp. 95). The measurement rulesassociated to these models regularly correspond to consumers' evaluative processes involvingperceptions and/or preference formation and thus ultimately rely on at least one explicitexplanatory variable steming from some reasonable theoretical framework of consumerbehavior (Roberts and Lilien, 1993, pp. 43).

1 Vienna University of Economics and Business Administration, Department of Retailing and Marketing; contactaddress: Augasse 2-6, A-1090 Vienna, Austria, Europe; email: reutterer@wu-wien.ac.at

Of course, CMS analysis does not necessarily end with the visual representation of the mutualcompetitive relationship patterns for rival brands, which can be achieved via employment ofmultivariate data reduction techniques. In order to provide marketing managers with moremeaningful decision-oriented information needed for the evaluation of actual brands positionswith respect to those of competing offers, decision-making upon target positions of existingproducts as well as the identification of new product opportunities, CMS analysis is extendedtowards positioning models incorporating actual customers’ purchase information, backgroundcharacteristics and/or reactions to marketing mix variables (see, e.g., DeSarbo and Rao, 1986;Moore and Winer, 1987; Cooper, 1988; a review is provided by Green and Krieger, 1989).

In the present paper, however, the issue of a compact representation of competitiverelationship patterns among brands is brought into focus. CMS analysis requires someprespecification of product-market boundaries in terms of a set of products/brands to beincluded in the study. As Grover and Srinivasan (1987, p. 139) point it out, the definition of astarting set of products/brands is frequently determined by managerial judgment and/orconsiderations about the increasing data manipulation efforts associated with broader productmarket definitions. Closely related to this aspect are technical problems the data analyst isfaced with in the presence of incomplete or “missing” data due to customers' limitedawareness about brands and/or their heterogeneous consideration sets.

Thus, CMS turns out to be a segment specific construct, which imposes the issue of decidingabout the "optimum" level of data aggregation for CMS analysis. With the introduction ofconsumer heterogeneity into the analysis, the interdependency of CMS, or it's decision-oriented extension in positioning analysis respectively, and another well-known managerialtask of strategic marketing planning, namely the concept of market segmentation (acomprehensive overview of current segmentation research is provided by a recent book ofWedel and Kamakura, 1998), becomes obvious. Again, the relationship with marketingplanning literature is straightforward; according to the problem of "defining the business", asoutlined in the seminal work of Abell and Hammond (1979), targeting a firm's offer to meetsome consumers' functions refers to one or more groups of consumers (i.e. market segments),which are more homogeneous in terms of their response patterns to marketing actions ascompared to non-group members.

Several approaches to combine the interrelated tasks of CMS and segmentation analysis havebeen introduced to marketing literature so far. While traditional methods typically work in astepwise manner, some more sophisticated models try to solve the combinedCMS/segmentation problem simultaneously. The remainder of this paper is organized asfollows: First, conventional methodology for analyzing CMS is reviewed in brief and anoutline of models for combined CMS and segmentation analysis is provided. In subsequentsections, the adaptive Self-Organizing (Feature) Map (SOM) methodology according toKohonen (1982) is introduced and adopted to the combined CMS/segmentation task in ademonstration study using household-level panel data.

2. Conventional methodology for analyzing CMS

Once a relevant set of (potentially) rival products/brands is defined, DeSarbo, Manrai andManrai (1993, pp. 194) distinguish between two major approaches to determination of CMS,which both require employment of some data reduction techniques that result in either a

• non-spatial (discrete) or a• spatial (geometric) representation

of the competitive relationships between objects.

Non-spatial configurations are derived by partitioning the relevant set of products/brands intosubsets (or ’submarkets’) in a way that products/brands within a subset are closer substitutes(and thus compete more heavily with each other) than products/brands belonging to differentsubmarkets. Competitive patterns are typically represented as tree-like structures (ultrametrictrees), with products or brands appearing as terminal nodes of a tree. The resulting treeconfiguration of such models organizes highly competitive brands into the same branch of thetree. The distance between brands or products in the tree reflects the degree of competitionbetween the respective objects. One may also say that competition between pairs of productsor brands corresponds to the height of their “lowest common ancestor” node. Such treestructures are typically fit by hierarchical clustering techniques (for overviews see e.g. Punjand Stewart, 1983, Arabie and Hubert, 1994 or Gordon, 1996), which also enables the analystto explore different “levels” of competition or competitive groups of “higher order”. As aselection of models using hierarchical tree structures for the determination of CMS one mightconsult the works of Rao and Sabavala (1981), Srivastava, Leone and Shocker (1981) orSrivastava, Alpert and Shocker (1984). An excellent review of non-spatial models for CMSanalysis is provided by DeSarbo, Manrai and Manrai (1993).

Another stream of modeling efforts in the field of CMS analysis concentrates on the compactgeometric (i.e. spatial) representation of competitive patterns between products/brands, whichin the marketing literature is referred to as methods of perceptual and/or preference mapping(see e.g. Hauser and Koppelman, 1979; Urban and Hauser, 1993, p. 201-252; Lilien, Kotlerand Moorthy, 1992, p. 74-97). Spatial models project products/brands as points in a low-dimensional geometric coordinate space so that the degree of observed competitivenessbetween the items in the space is represented by (euclidean) distance measures. Spatialmethods of CMS can further be subdivided into compositional and decompositionalapproaches (see e.g. Lilien, Kotler and Moorthy, 1992, p. 74), which differ with respect to thedata reduction techniques used and respective input data requirements.

Compositional methods are typically concerned with the issue of reducing multidimensionalattributions of product/brand characteristics as perceived by consumers (profile data) intolower-dimensional spaces and employ traditional methods of data reduction, such as principalcomponents (Hauser and Shugan, 1980) or discriminant analysis (Pessemier and Root, 1973).Since standard principal components analysis is not suited for non-metric profile data,Hoffman and Franke (1986) address this problem by using correspondence analysis forcategorical input data. Furthermore, contributions to the psychometric literature provideextensions and/or generalizations of the factor analytic methodology for dichotomous datastructures (see, e.g. Christoffersson, 1977 or Muthén, 1989) and models for three-mode (i.e.simultaneous Q- and R-type factoring) factor analysis (see Kroonenberg, 1983, 1984;Harshman and Lundy, 1984; for marketing specific applications cf. the works of Cooper,1988; Cooper and Inoue, 1996; Cooper, Klapper and Inoue, 1996).

Decompositional approaches are usually based on multidimensional scaling (MDS)techniques and require proximity or dominance data as observed by respondents' globalsimilarity and/or preference statements concerning competing products or brands. Theobjective is to create competitive object spaces of low dimensionality in such a way that the

metric interpoint-distances of the scaled items (products or brands) are fit to the (rank) orderof the observed similarities or dissimilarities (or preference judgments) between any pairs ofbrands or products.

Metric and non-metric MDS procedures differ in the way how this dimensional configurationis derived. In the case of the traditional metric MDS, the reduced space representation isrequired to be a linear function of the original proximity data. Non-metric MDS modelsmerely require a monotonic relationship between proximities and the derived model distances;for an overview of current standard MDS techniques see Green and Rao (1972), Kruskal andWish (1978), Wish and Carroll (1982) or Cox and Cox (1994). Hauser and Koppelmann(1979) as well as Huber and Holbrook (1979) compared the performance of an MDS basedapproach with compositional approaches to competitive product space construction.

Since the first applications of MDS techniques in the context of competitive-market-structureanalysis (e.g. Lehmann, 1971; Green, Wind and Claycamp, 1975), a large number ofimprovements and model extensions concerning the allowed input data formats andestimation techniques have been presented (for an asymmetric MDS model see e.g. Harshman,Green, Wind and Lundy, 1982). Developments important in marketing research are the(external) multidimensional unfolding extensions, like PREFMAP (for a compact descriptionsee, e.g. Green and Rao, 1972, pp. 214) or LINMAP (Shocker and Srinivasan, 1974), whichallow the representation of consumer preferences as vectors or ideal points together withbrands/products in one and the same geometric space (“joint space”), or the INDSCAL modelby Carroll and Chang (1970). INDSCAL allows the diagnosis of individual (consumerspecific) differences in a “common” product space.

Recent contributions comprise internal unfolding models for (binary) brand switching data(e.g. Holbrook, Moore and Winer, 1982; DeSarbo and Hoffman, 1987). In the GENFOLD2model by DeSarbo and Rao (1986), product coordinates are related to marketing mix variablesand product features while consumers' ideal points are related to their (demographic,socioeconomic, etc.) background characteristics. In a similar approach, Moore and Winer(1987) integrate longitudinal dynamics of joint space maps into market response models. Inorder to overcome stability problems of deterministic MDS and to allow for statisticsignificance testing of the resulting maps, stochastic procedures like PROSCAL by MacKayand Zinnes (1986) have been proposed.

3. Approaches of combined CMS and segmentation analysis

As already mentioned in the introduction to this paper, the presence of heterogeneousconsideration sets and/or varying preference (or profile) values across individual consumersclosely link CMS analysis together with the concept of market segmentation. In fact, asGrover and Srinivasan (1987) point it out, especially the utilization of consumers' brandchoice probabilities as basis for segmentation turns CMS and market segmentation out to be"reverse sides of the same analysis".

Formally speaking, the basic difference of these two "sides" of combined CMS/segmentationanalysis lies in the kind of how the observed data matrix is processed. According to theterminology from data theory as discussed by Young (1987), the shape of a data matrix isdefined by the number of ways (i.e. the dimensions of the matrix), the number of modes (setsof different entities that represent the ways of the matrix; e.g. consumers, products/brands,

product attributes or time periods represent different data modes) and symmetry conditions(symmetric/asymmetric data). Furthermore, each way of a data matrix has it’s own number oflevels (corresponding to the number of entities in the respective object set). Thus, the waysdefine the overall shape of a data matrix, modes determine the interpretation of the objects,and the levels specify the size of the matrix (see Jacoby, 1991).

From this point of view, both the concepts of market segmentation and CMS may beconsidered as formally identical data reduction problems. The only (formal) differencebetween the two concepts concerns the mode of data reduction. Traditional approaches toCMS analysis as discussed in the previous section often consider the variation ofproduct/brand evaluation across consumers in a two step manner, with the data reduction taskof the product/brand mode (CMS analysis) being performed for each of the (a priori)constructed market segments separately.

Principal components analysis (PCA), for example, is confronted with three-way, two-modedata (attribute ratings × brands × consumers); however, in standard PCA individual attributeratings × brands matrices are either aggregated for (a priori) known homogeneous segments oracross the total number of respondents and thus the consumer mode is neglected.Decompositional models for CMS analysis, like INDSCAL or the INDCLUS model (seeCarroll and Arabie, 1983) reduce three-way, two mode proximity data (m × m proximitymeasures of m brands for individual consumers) to a low-dimensional brands configurationfor the aggregate of consumers with dimensional weights accounting for individualdifferences that might be used as a basis for segmentation.

The task of a combined CMS/segmentation approach, however, is to reduce the consumer andproduct/brand mode in one single model, which requires the model to perform the task ofclassification and representation of the two data modes simultaneously. According to thecompilation of conventional analysis provided in Figure 1 such models might by termed as“hybrid” approaches.

Figure 1: Models for CMS/segmentation analysis cross-classified by data modes and representation types of results

Discrete("Non-Spatial")

Geometric("Spatial")

RepresentationType:

Mode ofData Reduction:

Products/Brands

Subjects(Consumers)

Hierarchical (Tree Models)/Non-Hierarchical

Classification Techniques

Compositional/Decompositional

Positioning Analysis

Preference ScalingModels

A PosterioriMarket Segmentation

"Hybride" Approaches(e.g. LCMDS models)

Approaches towards this "hybrid" direction are presented by authors like Hruschka (1986),Grover and Srinivasan (1987), Kamakura and Russell (1989), Wedel and Steenkamp (1991),Ramaswamy and DeSarbo (1990) or Zenor and Srivastava (1993). More recently, DeSarbo,Manrai and Manrai (1994) reviewed a special class of simultaneous modeling approaches tosegmentation and CMS analysis which they referred to as ’’Latent Class MultidimensionalScaling’’ (LCMDS). LCMDS models typically represent one data mode (e.g. products/brands)in a continuous or dimensional MDS-like fashion (representation task), while the otherreduced mode (e.g. consumers) is compressed to discrete partitions (classification task), whichare projected as vectors in the reduced space of the first data mode. Therefore, LCMDSmodels may be referred to as ”hybrid approaches” as indicated in Figure 1.

Current LCMDS methodology is possibly best represented by the MULTICLUS model ofDeSarbo, Howard and Jedidi (1991), which simultaneously performs MDS and clusteranalysis. The MULTICLUS model is designed to process profile or dominance data andexplicitly models segment heterogeneity via maximum likelihood estimates for segment-specific ideal vectors in a joint space of brand coordinates. However, both the dimensionalityand the number of segments have to be predetermined prior to the analysis. Another limitationof the MULTICLUS model arises from the "missing data" problem caused by heterogeneoussets of consumers' brands consideration, which becomes especially dominant if choice dataare collected from consumer or household panel surveys. In a more recent paper, DeSarbo andJedidi (1995) try to circumvent this problem by taking consumers' consideration sets explicitlyas a basis for segmentation.

However, "missing data", restrictive input data requirements and parametric modelassumptions still impose problems associated with contemporary approaches to the datareduction tasks of simultaneous CMS and segmentation analysis which remain not yetadequately solved by the use of "classical" statistical methodology. The following sectionintroduces an alternative approach to multimode data compression that — when adopted tothe combined CMS/segmentation problem — addresses similar tasks like LCMDS models,but imposes less restrictive assumptions on input data conditions and thus might serve as acandidate to overcome some of the above mentioned issues.

4. The Self-Organizing (Feature) Maps (SOM) methodology

As a special class of artificial neural network models, the Self-Organizing (Feature) Map(SOM) algorithm as originally proposed by Kohonen (1982) is designed for multidimensionaldata reduction with topology-preserving properties. Previous applications of SOMmethodology resulted in some successful implementations for solving a variety ofcategorization and pattern recognition tasks in speech, image or signal processing (for anoverview cf. Kohonen, 1995, pp. 191). Introductions to the methodological foundation ofSOM models are provided by Kohnen (1982, 1984, 1995), Pao (1989, pp. 182-196), Freemanand Skapura (1991, pp. 263-289), Ritter, Martinetz and Schulten (1991), Gallant (1993),Haykin (1994) or Martinetz and Schulten (1994).

Like other partitioning methods, the SOM methodology is based on an unsupervised learningscheme, i.e. there is no outside information that denotes a correct classification of the inputdata vectors. Similar to PCA or MDS, an SOM network constructs a low- (usually two-)dimensional mapping in order to detect the inherent structure of high-dimensional input datain a visually easily inspectable manner. In spite of this analogy to conventional data reduction

techniques for CMS analysis, SOM models posses some distinctive features which make themcomparable to models classified as "hybrid" approaches in the previous section: SOMs arefacing the dual problem of constructing a classification of input data space and simultaneoslyto this mapping the class centers onto a prespecified grid of units so that the topologicalstructure of input data is respresented as accurate as possible (i.e. the class centers are orderedin the low-dimensional grid according to their similarity in the high-dimensional data space).In contrast to the geometric joint space configurations estimated by the maximum likelihoodbased MULTICLUS procedure, the SOM methodology arrives at a non-linear topologicalprojection of input data space onto a two-dimensional discrete map of ordered class centersthrough an adaptive procedure — i.e. a sequential presentation of input data vectors frequentlyreferred to as "network training" (see Kohonen, 1982, p. 139; Haykin, 1994, pp. 414).

Subsequently, the adaptive SOM procedure is outlined in more detail. A typical SOM networkarchitecture consists of the following components (see Figure 2 for a graphical illustration ofan SOM with nine units):

• An input layer of dimensionality m which represents input data vectors xk = [ξk1, ξk2, ... ,ξkm]T out of a set of k = (1, ... ,K) training data vectors (e.g., in CMS analysis xk mightstand for preference ratings or choice probabilities of consumer k for the mproducts/brands under investigation).

• A usually two-dimensional competitive layer (also called "SOM-" or "Kohonen-Layer")organized as a grid of units uij, where i represents the row index and j the column index ofthe respective unit position in the layer.

• An m-dimensional weights vector for each of SOM unit uij: wij = [µij1, µij2, ... , µijm]T.

Figure 2: Graphical illustration of an 3 × 3 SOM architecture

u31

. . .. . .ξ11 ξ12

µ 31,1

(3× 3 SOM units)

(m-dimensional weights vectorsfor each SOM-Unit)

(m-dimensionalinput vectors)

Notice that the (dotted) connections between each of the SOM units and its respective(horizontal, vertical and lateral) neighbors as shown in Figure 2 are illustrated for intuitivereasons only, as they do not represent explicit weight vectors between units but are implicitlymodeled by the updating function which is responsible for the topological ordering of theclass centers in the SOM layer as described below.

The SOM algorithm can be briefly described by the following iterative procedure (seeKohonen, 1984):

(1) Initialization of Model Parameters: Weights vectors wij (usually with random numberswithin the interval [–0.1;+0.1]), definition of the "neighborhood function" hc(i)c(j)(t), thefunctional form of the "learning rate" α(t) as well as assignment of the starting valuesfor hc(i)c(j)(1) and α(1); set counter t : = 1.

(2) Computation of Distances: Randomly choose an input vector xk from the set of trainingvectors and compute the (usually euclidean) distances || xk – wij(t) || between the inputvector and each of the present values for the units' weights vectors.

(3) Similarity-Matching: Determination of the "winning" or "best matching" (the one withthe smallest values for the distances from stage 2) unit (ci,cj) according to the rule:

{ } )1(.)(min)( twxtwx ijkij

cck ji−=−

(4) Weights-Updating: Updating of the "winning" unit’s (ci,cj) and it’s adjacent weightsvectors according to the update rule:

[ ]w t w t t h t x t w tij ij c i c j k ij( ) ( ) ( ) ( ) ( ) ( ) , ( )( ) ( )+ = + −1 2α

where hc(i)c(j)(t) defines the allowed interaction between the "winning" unit and it’sneighborhood and α(t) being the value of the learning rate at iteration t.

(5) Set t = t + 1, decrease the neighborhood parameters hc(i)c(j)(t) and the learning rate α(t).

(6) Repeat steps 2 – 5 until the change of the weights values ∆wij(t–1, t) is less than aprespecified threshold value or the maximum number T of iterations is reached.

The adoptions imposed to the weights vectors through the combination of steps (3) and (4) arevital for the ultimate objective of SOM learning algorithm. Step (4) tells us that the inputvector xk presented to the SOM layer at iteration t is "mapped" onto unit (ci,cj) — i.e. the"winning" unit determined in step (3) — by allowing a "maximum" adjustment of thecorresponding weights vector for the amount of wc(i)c(j)(t) + α(t)[xk – wc(i)c(j)(t)]. Withproceeding sequences of input vector presentations (i.e. one input vector at each iteration), theweights vectors corresponding to respective winning units rapidly become prototypes or"representatives" of a specific type of input data patterns. Since according to step (5) of theSOM algorithm the learning rate α(t) is forced to decrease with time, a classification of theinput data space would be achieved via stochastic approximation of the minimum (innergroup) variance partition (cf. Bock, 1997).

However, as the formulation of the weights updating rule in step (4) of the above algorithmsuggests, there is another feature of SOM learning which differs from other clusteringtechniques: Not only the "winning" unit (ci,cj) but also those units, which are located in thegrid topographically close to the "winner" up to a certain distance as defined by the presentvalue of hc(i)c(j)(t) will be activated to learn gradually from the same input vector. This effect ofneighborhood updating is responsible for the above mentioned topological ordering of clusterprototypes (i.e. weights vectors representing distinctive input data patterns) onto the discretemap of units and therefore provides the SOM methodology with an appealing property for thetask of combined CMS/segmentation analysis.

As indicated above, both the learning rate α(t) and the neighborhood function hc(i)c(j)(t) aremonotonically shrinking functions of time. In order to assure convergence of the SOMconfiguration to a stable mapping of input data space, it can be shown that certain necessaryand sufficient conditions as known from stochastic approximation must hold for the learningrate factor (see Kohonen, 1984, p. 138; Lo, Yu and Bavarian, 1993):

α α α( ) ; ( ( )) ; ( ) ( )t t tt t=

∞

=

∞∑ ∑= ∞ < ∞ ≤ ≤0

2

00 1 3

Useful and frequently suggested functional forms for the learning rate comprise time-inversefunctions like α(t) = α(1)(a/(t+b)), where a and b are some suitably chosen constants, orexponential functions of the type α(t) = α(1)ct-c, with c being a constant within the interval 0< c ≤ 1 (see e.g. Kohonen, 1984, pp. 138; Hertz, Krogh and Palmer, 1991, p. 237).

As for the neighborhood function, a relatively wide neighborhood in the beginning anddecreasing values with time are recommended. At least the following two fundamentallydifferent types of neighborhood functions can be distinguished (see also Kohonen, 1995, pp.79 or Mao and Jain, 1995, pp. 302):

1. Discrete neighborhood function with lateral interactions ("neighborhood set updating"),

2. Continuous neighborhood function ("neighborhood kernel updating").

The choice of the first mentioned version of neighborhood functions leads to the definition ofa set of array points around the respective "winning" unit’s node index (ci,cj) at each iteration(t). Denoting this index set as Nc(i)c(j)(t) and introducing a scalar-valued interaction parameterβ(t), the (discrete) updating rule can be formulated as follows:

h tt i j N t

c i c jc i c j

( ) ( )( ) ( )( )

( ),

,

( , ) ( ),( )=

∉

β0

4if

otherwise,

where the interaction term β(t) defines the degree of participation in the weights-updatingaccording to step (4) of the SOM algorithm allowed for the set of adjacent units around the"winner" location index (ci,cj). Figure 3a) illustrates the degree of allowed neighborhoodupdating for varying location distances of units adjacent to the "winner" for an interactionvalue of β(t) = 0.3 and a neighborhood set Nc(i)c(j)(t) of three units around the winner location(ci,cj). It is clear from figure 3a) that all the units within the neighborhood set participate to anequal amount (i.e. 0.3 times less than the "winning" unit) in the updating of their respectiveweights vectors. In order to achieve convergence, both β(t) and the radius of Nc(i)c(j)(t) are

usually decreasing monotonically in time. Values of the interaction parameter are frequentlyrestricted to the interval 0 ≤ β(t) ≤ α(t) ≤ 1.

As another widely used concept for accomplishing neighborhood updating, a "continuous"neighborhood function leads to a smoother neighborhood kernel, which can be written interms of the following Gaussian function:

( ) ( )[ ])5(, exp)( 2

22

)(2)()(

−= −+−

t

cjci

jcicjith

σ

where i (j) denote the row (column) indices of an SOM unit uij and ci (cj) the correspondingindices of the actual "winning" unit. The denominator of expression (5) therefore measures thesquared euclidean location distance between each of the unit uij and the "winner" (ci,cj). Thekernel parameter σ(t) here defines the width of the neighborhood kernel at iteration (t). Again,σ(t) is defined as some monotonically shrinking function of time. As we can see from figure3b), the smoother concept of neighborhood kernel weights-updating does not discriminate thatradically between members and non-members of a discrete set of neighboring units in terms ofallowed neighborhood updating, but allows a continuously reduced degree of weightsupdating with increasing location distances around the "winner" (the hc(i)c(j)(t) values in figure3b) are computed for a kernel width of σ(t) = 1.5).

Figure 3: Weights-updating in the presence of alternative neighborhood functions

Since the learning process involved for SOM formation is of a stochastic nature, the accuracyof the final mapping is critically dependent on the number of iterations of the SOM algorithmas well as on how (starting values and functional forms) the main parameters of the algorithm,i.e. the learning rate α(t) and the neighborhood function hc(i)c(j)(t), are selected. The lack of atheoretical basis leads to the parallel existence of several heuristics for the selection of theseparameters.

Kohonen (1984, p.139) himself states that the proper choice of the two time dependentparameters may be best determined by a process of trial and error. Nevertheless, inventoriedexperiences tell us that the following decomposition of the training process into two stagesmay provide a useful "rule of thumb" for selection of the learning parameters (see Kohonen,1984, p. 139; Haykin, 1994, pp. 412):

a) Neighborhood set updating

0

0.2

0.4

0.6

0.8

1

Nei

ghbo

rhoo

d-U

pdat

ing

Winner(w)

(w+1) (w+2) (w+3) (w+4)(w-1)(w-2)(w-3)(w-4)

a) Neighborhood kernel updating

0

0.2

0.4

0.6

0.8

1

Nei

ghbo

rhoo

d-U

pdat

ing

Winner(w)

(w+1) (w+2) (w+3) (w+4)(w-1)(w-2)(w-3)(w-4)

• Ordering Phase:During the initial phase of training (say 1000 iterations or so), the topological ordering ofthe weight vectors wij takes place. In order to achieve this, the learning-rate factor α(t)should start at a value close to unity and thereafter shrink gradually to a value of about 0.1.The initial radius of the neighborhood around respective “winners” also should be fairlywide (even more than half of the diameter of the SOM layer) and, when topologicalordering of the prototype vectors take place, be permitted to monotonically shrink to asmall value of a couple of neighbors (depending of the total size of the SOM layer).Depending on the kind of neighborhood function hc(i)c(j)(t) used, this can be achieved viathe (joint) modification of the neighborhood set Nc(i)c(j)(t) and the interaction parameterβ(t) (in case of a set-update-function like expression (4)) or by reduction of the kernelparameter σ(t) (in case of a kernel-type update function according to expression (5)).

• Convergence Phase:Illustratively speaking, during this ordering phase each of the SOM units rapidly getresponsible for a specific pattern of input data vectors and subsequently representprototypes for symptomatic combinations of vector components. After this ordering phase,the relatively high (10 and even more times of iterations than the ordering phase) numberof iterations are primarily needed for fine tuning of the map. For good statistical accuracyof the final (converging) map, the learning rate should remain at fairly small values in theorder of α(t) = 0.01 or less. During convergence phase, hc(i)c(j)(t) should allow only for thenearest neighbors of the respective “winning” unit to participate in the (anyway verylimited) updating of weights vectors (by adjusting Nc(i)c(j)(t) to a value of eventually oneunit location distance or setting kernel parameter σ(t) to very small values). In order toarrive at a mapping of input data space onto the previously topologically ordered weightvectors with (nearly) centroid properties, updating for the last few iterations can also berestricted to a neighborhood size of zero units (i.e. only the weight vector of the “winner”remains to be gradually adjusted).

Experimental findings for SOM training with varying combinations of learningparameterizations suggest that special caution is required in the choice of a neighborhooddefinition (initial neighborhood size and dynamic change of size). In order to achieve atopological ordering of the map, especially for SOMs of very small dimensionality (e.g. using36 units in an 6 × 6 SOM layer or less as we will typically use in CMS analysis) the choice ofa smooth neighborhood kernel update rule seems to be more appropriate than the discreteneighborhood set (see e.g. Reutterer, 1997).

5. Application of SOMs for combined CMS/segmentation analysis: Ademonstration study using household-level panel data

From the above illustration of the SOM learning algorithm it is straightforward that the SOMmethodology performs the following two tasks which are considered as importantcharacteristics for simultaneous CMS and segmentation analysis:

1. Partitioning of input data space on the basis of some features of the training vectors(classification task), and

2. topological ordering of the map so that input data vectors with symptomatic patterns offeature combinations tend to produce a response in units that are close to each other in thegrid of SOM units (representation task).

Thus, the concept of SOM analysis becomes comparable with LCMDS models in a sense that,e.g., the consumer mode can be regarded as entities of input data to be “clustered” and thebrand mode assisting as "feature variables" in the interpretation of the resulting topologicallyordered representation. However, an SOM model merely classifies consumers or brands butproject the distribution of the "feature variables" (which e.g. can be brand profiles asperceived by consumers, purchase probabilities, brand preferences or choice sets ofcustomers) onto the grid of SOM units in such a way that each unit represents symptomaticpatterns of the feature observations.

The subsequently presented case study follows to recent pioneering applications of SOMmethodology for CMS analysis as provided by Mazanec (1995a) in the context of exploringcompetitive relationship patterns between cities as tourist destinations and Mazanec (1995b)in a positioning study on company images of luxury hotels.

5.1. Description of input data

Input data are derived from household-level purchase histories in the product class ofmargarine brands for 781 non-brandloyal panel households. Panel data are provided by theAustrian affiliate of the pan-European marketing research agency GfK (Gesellschaft fürKonsum-, Markt- und Absatzforschung). There are nine major margarine brands (six nationalbrands and three nation-wide distributed private labels) which accounted for more than 90 percent of total market share (see table 1).

Table 1: Characteristics of brands used in the case study

Brand NameFeatures

(brand type, fat content)Market Share

(% of purchase occasions)

Rama (national/normal) 31.2Osana (private/normal) 11.6Thea (national/normal) 11.3Feine Thea (national/normal) 10.4Vita (national/reduced) 8.2Becel (national/diet) 8.1Du darfst (national/diet) 6.7Bellasan (private/normal) 5.4Bellasan Diät (private/diet) 3.0Others 4.2

Analogous to similar approaches to CMS (e.g., Bucklin and Srinivasan, 1991; Krishnamurthiand Raj, 1991; Balakrishnan, Cooper, Jacob and Lewis, 1996), households’ (unconditional)brand choice probabilities are used as input data for SOM-training. Thus, each individualbrand-switching household k may be described by a nine-component data vector pk which isindicative for various degrees of preference with respect to the brands considered in this study.Element pmk measures the relative purchase frequency of brand index m ∈ {1, … , 9} in the

two years observation period, where ∑m pmk = 1. Note that differences of consideration sets

across households (the average set contains about three brands) lead to a relatively largenumber of zero-entries.

Nevertheless it is widely used practice in CMS analysis (see e.g. the works of Rao andSabavala, 1981 or Grover and Srinivasan, 1987), it should be noticed that aggregation ofchoice probabilities over time assumes a homogeneous brand choice process of "zero-order",i.e. stability of the market.

5.2. Properties of alternative SOM results

Based on the above recommendations and own experimental findings, the followingcombination of parameters turned out to be appropriate for SOM training in the presentapplication context: Starting with a relatively high α(1) = 0.9, both the learning rate and theinitially wide neighborhood size were reduced exponentially (to achieve topological orderingof the weights vectors and convergence of the map) during the total training process of T =10,000 iterations. Thus, the learning rate tends to converge asymptotically against a final valueof α(T) → 0 and the smooth neighborhood kernel update function were designed to end at akernel parameter of σ(T) = 1, i.e. adjacent units to the ”winner” are allowed to (however veryslightly) participate in the weights updating until the end of the training process.

Once SOM training is completed, the final classification of input data space can bedetermined by a sequential recall run through the set of input data vectors. Now the weightvectors are used to identify the "best matching" SOM unit (i.e. the minimum distance unitaccording to step (3) in the above described training algorithm) for each single preferencevector. Thus the final SOM configuration corresponds to a partitioning of input data, which isdefined via vectors' assignments to their respective (final) "winning" units.

As already mentioned in previous sections, the adaptive SOM algorithm performs a stochasticapproximation of topologically correct ordered feature maps and (due to the lack of any clearobjective function) may get stuck in an only locally optimum mapping of input data space.Moreover, comparable to other techniques of exploratory data analysis, the appropriatenumber and format of SOM units has to be determined heuristically. In analogy to similarproblems as well-known in principal components, cluster analysis or MDS, the plot of someinformation or "goodness of fit" measures against descending sizes of SOM layers provides auseful basis for an examination of alternative topologically ordered cluster solutions (see e.g.in factor analysis the "scree plot" according to Cattell, 1966).

Figure 4 contains sequential plots for the heterogeneity (MSSE) and simplicity (MSSIPD)measure used in the study as well as for the sum of both measures (MSSE + MSSIPD). Asknown from cluster analysis (see e.g. Anderberg, 1973, p. 165), heterogeneity of the resultingclassification is defined as the ”Mean Sum of Squared Errors” (MSSE), which summarizes thesquared euclidean distances between each data point and its corresponding mean or centroidvector (i.e. the inner group variance) divided by the total number of data points ("meanquantization error"; cf. Kohonen, 1995, pp. 35). As more and more households with dissimilarbrand preference patterns are resembled together, it is quite clear from figure 4 thatheterogeneity of cluster solutions increases with decreasing numbers of prototypes (i.e. unitsin SOM analysis).

Figure 4: Heterogeneity and similarity scores for alternative SOM solutions

The simplicity measure is designed to summarize deviations of the final weight vectors fromthe prespecified rectangular grid of SOM units ("badness of fit"). Simplicity is defined as the"Mean Sum of Squared euclidean Inter-Prototype Distances" (MSSIPD) for adjacent SOMunits of order one, i.e. distances are computed between each of the prototype vectors and thoseof its immediately adjacent units in the SOM layer. In order to make the measure comparableacross different grid formats, the sum of distances is divided by the total number of possibleneighborhood relations. The conceptual idea underlying to this simplicity measurecorresponds to the aimed topology-representing property of an SOM. Thus, small similarityvalues indicate a "good" topological representation of input data in a sense that "similar"prototype vectors should be adjacent in the grid of SOM units.

Turning back to figure 4, similarity of SOM solutions slightly improves with shrinkingdimensionality of SOM layers, but remains at a relatively constant level. However, it isimportant to notice that the fairly "good" topological solutions are due to the fact that adjacentunits of respective "winners" are included in the weights adjustment until the end of SOMlearning. Although not reported here, it can be shown that training of SOMs with finalneighborhood updating size of zero order (i.e. only the "winner’s" weights are adjusted for thepresented input vector) tends to result in a distortion of the map’s topological ordering. As aconsequence of neglecting the topological neighborhood structure during training, the weightsvectors are not forced to be squeezed to the "corset" of the SOM grid anymore — thussimplicity increases, however at the benefit of improved centroid properties of the weightsvectors (reduction of heterogeneity).

Roughly speaking, the occurrence of this trade-off relationship requires the user to decidebetween a "good" topological representation and an accurate (i.e. minimum within groupvariance) classification of input data vectors. The process can be controlled via specificationof the neighborhood updating function. In the demonstration study presented here, favor isgiven to the topological property of the brand preference map at cost of some additionalquantization error for classification of households.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 2 3 4 5 6 7

Format of SOM-Layers (Rows*Columns Units)

Heterogeneity (MSSE)

Simplicity (MSSIPD)

MSSE + MSSIPD

5*5 4*44*5 3*4 3*3 2*4 2*3

5.3. Visualization and interpretation of an SOM configuration

According to the plotted final heterogeneity and similarity values for alternative formats ofSOM layers in the above figure 4, a certain "elbow" of MSSEs augmentation can be observedwhen reducing an 3 × 3 format to an 2 × 4 SOM layer; i.e. heterogeneity "levels off" if thenumber of household classes is reduced from nine to eight while simplicity of the mapsremains at an almost stable low level. For this reason we may wish to accept the 3 × 3 SOMsolution for further investigation.

As we know from the above described SOM architecture, irrespectively of the two-dimensional organization of the SOM layer the weights vectors (which now representtopologically ordered prototypes of the household classes) of SOM units are of the samedimensionality as the (here) nine components brand choice probability vectors presented asinput data. For easy visual inspection of the final map, the prototypes therefore have to beprojected onto a lower dimensional space first.

In order to derive a portrayal of the prototypes in a two-dimensional space, the non-linear dataprojection technique provided by Sammon (1969) is applied. Similar to MDS techniques, theSammon algorithm reduces the nine-dimensional prototypes into two dimensions byapproximating the original structure of euclidean inter-prototype distances between datapoints via steepest gradient decent minimization of an error term indicating reproductionquality. The Sammon visualization of the prototype vectors for a 3 × 3 SOM solution isportrayed in figure 5.

Figure 5: Sammon’s projection of an 3 × 3 segment specificcompetitive map for margarine brands

3*3 SOM-LayerS-Stress: .0073

23

4

1

5 6

7

8 9

-0.4

-0.3

-0.2

-0.1

0

0.1

-0.35 -0.25 -0.15 -0.05 0.05 0.15

Abbrev.: R=Rama, O=Osana, T=Thea, F=Feine Thea, V=Vita,B=Becel, D=Du darfst , BN=Beallasan Normal, BD=Bellasan Diet

(#HH: 6%)

O (.2), R (.2)

B (.1), BN (.1)

(#HH: 18 %)

O ( .3 ) ,

R (.2),

BN (.1) (#HH: 17 % )

B (.2), R (.2), F(.2)

V (.1), D (.1)

(#HH: 2.6%)

R ( .4 ) ,

O (.2)

(#HH: 1%)

R ( .3 ) ,

T (.1), F (.1), O (.1)

(#HH: 2 8 % )

R ( .5 ) ,

F (.1)

(#HH: 5.6%)

R ( .4 ) ,

T (.2), F (.2)

(#HH: 19% )

T ( .3 ) , F ( .3 ) ,

R (.2)

(#HH: 3.5%)

R (.2),F (.2),

T (.2), B (.1)

A small value of .0073 for the error term (also referred to as "Sammon’s Stress" measure)indicates an excellent two-dimensional reconstruction of the original euclidean inter-prototypedistances. Moreover, the "well-behaved" shape of the two-dimensional map of the prototypesis indicative of a "good" topological solution.

Each of the nine prototypes in figure 5 is labeled with it's SOM unit number, the percentage ofhouseholds mapped onto the unit ("#HH: percentage" indicated in brackets) and some keycharacteristics of the respective vector components (abbreviations of those brand names withweights values ≥ 0.1; for brands with weights ≥ 0.3, the label is underscored). From theprevious discussion we know that with proceeding weights adjustment (i.e. training of thenetwork) each of the weights vectors is adapted to symptomatic patterns of featurecombinations and becomes a representative (prototype) for a specific type of input datavectors. Since training is completed now, the conserved prototype information can be recalledfor assistance in interpreting the competitive relationship patterns between major brands in themargarine product class. rigor

The "hybrid" (see section 3) character of SOM based CMS analysis suggests a segmentspecific interpretation of brand competition. Each of the prototypes positioned in figure 5represents one segment (or submarket) of households with it’s typical pattern of brandpreferences. The exact weights for each segment are listed in the columns denoted “Prot.” oftable 2 (again only brands with weights ≥ 0.1 are displayed).

Table 2: 3 × 3 competitive map for margarine brands with weights and/or segmentspecific brand choice probability means ≥ 0.1

1 (18.2) Prot. Cent. 2 (6.00) Prot. Cent. 3 (16.7) Prot. Cent.O .32 .53 O .19 .10 B .20 .32R .18 .09 R .18 .06 R .18 .14

BN .14 .20 — B .12 .13 — V .13 .17BN .10 .20 D .13 .20BD .07 .35 F .20 .06

| × | × |

4 (2.60) Prot. Cent. 5 (1.00) Prot. Cent. 6 (3.50) Prot. Cent.R .37 .42 R .28 .27 R .19 .07O .18 .20 T .14 .11 F .18 .28

BN .09 .19 — F .12 .19 — T .18 .15O .12 .16 B .12 .08V .08 .17 V .09 .24

D .09 .14| × × |

7 (27.8) Prot. Cent. 8 (5.60) Prot. Cent. 9 (18.7) Prot. Cent.R .52 .67 R .37 .43 T .29 .44F .11 .08 T .18 .21 F .25 .32

— F .18 .27 — R .20 .09

Abbreviations: R = Rama, O = Osana, T = Thea, F = Feine Thea, V = Vita, B = Becel,D = Du darfst, BN = Bellasan Normal, BD = Bellasan Diet;Prot. = Prototypes (weights) vector values, Cent. = Centroid (class mean) vector values.

Notice that the weights values are not identical to the average segment-specific brand choiceprobabilities. In table 2 the mean or centroid values of choice probabilities are displayedbeside the corresponding weights (see the “Cent.” denoted columns). As it is clear from theabove discussion, prototypes with weights close to the class centroids would indicate "good"(i.e. at least locally minimum within group heterogeneity) classification results. However, asfavor was devoted to the quality of the topological solution this is not the case here.Incorporation of the centroid vectors into the two-dimensional Sammon mapping would resultin projections of the group means for segment-specific brand preference vectors which lie far"outside" its respective peripheral prototype position (with the exception of the "central"prototype no. 5).

For the final task of interpreting the competitive map representation, it is the combination ofhigher weights and/or centroid values, which denotes the intensity of competition betweenrival brands in a specific sub-market or segment. The members of segment no. 1, which arepositioned in the upper right corner of the map and represent about 18 % of the panelhouseholds, for example, may be characterized by high preferences for the private labelsOsana and Bellasan as well as for the national market leader Rama. It is more likely that brandswitching occurs among these three brands than with any other margarine brand. Moreover,the high choice probability levels for private labels suggest that store loyalty seems to play amuch more important role in the purchase decision process of segment members than theactual brand choice.

In contrast to this, the opposite (lower left-hand) corner of the competitive map is occupied bysegment members (about 19 % of the panel households) with strong Thea, Feine Thea andagain the leading Rama preferences. Since both Thea and Feine Thea are traditionally used forcooking and baking purposes, purchase behavior of segment no. 9 members is likely to bedominated by respective usage situations. Another segment of panel households withrelatively strong product attribute preferences is represented by the prototype of SOM unit no.3. Members of this market segment tend to select margarine brands among fat reduced or dietnational brands like Becel, Vita or Du darfst. Finally, the largest segment accounting for about28 % of the panel households (labeled as no. 7 in the competitive map) may be denoted byhigh loyalty patterns at the favor of the general-purpose brand Rama.

As a consequence, this submarket is characterized by considerably smaller choice setscompared to other segments of households. The competitive map illustrated above also seemsto discriminate between submarkets with fairly reduced choice sets (along the "corner" formedby the sequence of units no. 1-4-7-8-9) and larger choice sets around the diet-margarine pronesegment no. 3.

From inspection of the segment-specific competitive map (figure 5) and associatedprototype/centroid vectors (table 2) we can summarize the properties of SOM based combinedCMS/segmentation results as follows:

• Segments of households with distinctive patterns of brand choice characteristics (i.e.different types of CMS) are positioned in opposite directions or corners of the discretecompetitive map. Peripheral positions are by far more frequently occupied by households(about four fifth of the households are mapped onto the peripheral units 1, 3, 6 and 9) thanSOM units around the “center” of the map.

• Prototypes positioned between these extreme types of brand choice behavior representsmaller clusters of households with brand preference structures that turn out to be"mixtures" of those from adjacent prototypes (e.g. segment 2 seems to be a mixture type ofthe store-loyal segment 1 and the diet margarine buyers from market segment 3).

• The brand mode of input data is typically distributed over the entire map and serves forinterpretation of the segment positions. High prototype values denote strong brandpreferences of respective segment members. The level of competition between rivalbrands is also a segment-specific concept. Furthermore, low weights values of anindividual brand throughout the map indicate a “fuzzy” preference position that might callfor repositioning considerations.

In order to provide a more complete description of the various sub-markets and to make themaccessible for target marketing, segment members can be associated to meaningfulbackground characteristics like demographic, socio-economic and/or psychographic variablesvia standard methodology as known from segmentation analysis (e.g. discriminant or logisticregression techniques).

6. Discussion

Using household-level brand choice probabilities, SOMs have been shown to be capable tosimultaneously reduce both the consumer and the brand mode of input data into a discretemap with topologically ordered patterns of brand choice characteristics denoting competitiverelationships between brands on a market segment-level.

Due to the adaptive nature of SOM training, only local information (i.e. only one input datavector) is required to be present during each iteration of the SOM algorithm. SOM learningtherefore does not rely on batch processing of input data and can be performed for data sets ofunlimited size and/or "online" (in real-time mode) for continuously incoming data, such asscanner data from check-out systems in retail outlets (see also Mazanec, 1998).

In the demonstration study presented in this paper brand choice data aggregated over timehave been used. However, with the introduction of "time" as one further data mode (e.g.weekly household-specific choice data) it is possible to extend the approach towards adisaggregated level of analysis.

Using stacked brand choice information as input data, the SOM network learns to representpatterns of brand competition under average time conditions. Time period specific recall runsfor the household mode will allow for the analysis of segment membership changes with time,thus enabling the analyst to uncover time-dependent variations in brand choice behavior,which furthermore may be linked to marketing variables.

However, there are also some unsolved deficiencies, which by far restrict the application ofSOMs to exploratory data analysis only. Apart from the problems already mentioned insection 4, especially the lack of appropriate statistical test scores necessary for significancetesting of alternative topologically ordered feature maps require further research activities.

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