Data Analysis Machine Learning and Applications Episode 3 Part 8 doc

692 Palumbo et al. In the early 80’s, Tanaka proposed the first fuzzy linear regression model, moving on from fuzzy sets theory and possibility theory (Tanaka et al., 1980). The functional relation between dependent and independent variables is represented as a fuzzy linear function whose parameters are given by fuzzy numbers. Tanaka proposed the first Fuzzy Possibilistic Regression (FPR) using the following fuzzy linear model with crisp input and fuzzy parameters: ˜y n = ˜ E 0 + ˜ E 1 x n1 + + ˜ E p x np ,+ + ˜ E P x nP (4) where the parameters are symmetric triangular fuzzy numbers denoted by ˜ E p = (c p ;w p ) L with c p and w p as center and the spread, respectively. Differently from statistical regression, the deviations between data and linear models are assumed to depend on the vagueness of the parameters and not on measurement errors. The basic idea of Tanaka’s approach was to minimize the uncertainty of the estimates, by minimizing the total spread of the fuzzy coefficients. Spread minimiza- tion must be pursued under the constraint of the inclusion of the whole given data set, which satisfies a degree of belief D (0 < D < 1) defined by the decision maker. The estimation problem is solved via a mathematical programming approach, where the objective function aims at minimizing the spread parameters, and the constraints guarantee that observed data fall inside the fuzzy interval: minimize N  n=1 P  p=0 w p |x np | (5) subject to the following constraints:  c 0 +  P p=1 c p x np  +(1−D)  w 0 +  P p=1 w p |x np |  ≥ y n  c 0 +  P p=1 c p x np  −(1−D)  w 0 +  P p=1 w p |x np |  ≤ y n w p ≥ 0, c p ∈ R, x n0 = 1, n =(1, ,N), p =(1, ,P) where x n0 = 1 (n = 1, ,N), w p ≥ 0andc p ∈ R (p = 1, ,P). 2.2 The F-PLSPM algorithm The F-PLSPM follows the component based approach SEM-PLS, alternatively de- fined PLS Path Modeling (PLS-PM) (Tenenhaus et al., 2005). The reason is that fuzzy regression and PLS path modeling share several characteristics. They are both soft modeling and data oriented approaches. Specifically, fuzzy regression joins PLS-PM in its final step, allowing for a fuzzy structural model (see, Figure 1) but a still crisp measurement model. This connection implies a two stage estimation procedure: • stage 1: latent variables are estimated according to the PLS-PM estimation pro- cedure (Wold, 1982); Fuzzy PLS Path Modeling 693 Fig. 1. Fuzzy path model representation • stage 2: FPR on the estimated latent variables is performed so that the following fuzzy structural model is obtained: [ h = ˜ E h0 +  h  ˜ E hh  [ h  (6) where ˜ E hh  refers to the generic fuzzy path coefficient, [ h and [ h  are adjacent latent variables and h,h  ∈ [1, ,H] vary according to the model complexity. It is worth noticing that the structural model from this procedure is different with respect to the traditional structural model. Here the path coefficients are fuzzy num- bers and there is no error term, as a natural consequence of a FPR. In the analysis of a statistical model one should always, in one way or another, take into account the goodness of fit, above all in comparing different models. The proposal is then to use the FPR. The estimation of fuzzy parameters, instead of single-valued (crisp) param- eters, permits us to gather both the structural and the residual information. The char- acteristic to embed the residual in the model via fuzzy parameters (Tanaka and Guo, 1999) permits to evaluate the differences between assessors (panel performance) as well as the reproducibility of each assessor (assessor performance) (Romano and Palumbo, 2006b). 3 Application The data set comes from sensory profiling of 14 cheese samples by a panel of 12 assessors on the basis of twelve attributes in two replicates. The final data matrix consists of 336 rows (12 assessors × 14 samples × 2 repli- cates) and 12 columns (attributes: intensity odour, acidic odour, sun odour, rancid odour, intensity flavour, acidic flavour, sweet flavour, salty flavour, bitter flavour, sun flavour, metallic flavour, rancid flavour). Two blocks of variables describe the latent variables odour and flavour. First the hierarchical PLS model proposed by Tenen- haus and Vinzi (2005) will be used to estimate a global model after averaging over the assessors and the replicates (see, Figure 2). Thus, collapsing the data structure into a two-way table (samples × attributes). Then fuzzy PLS path modeling will 694 Palumbo et al. provide two sets of synthesized assessments: the overall latent scores for each prod- uct and the partial latent scores for the different blocks of attributes. The synthesis of scores into a global assessment permits to investigate differences between products. However, in such a way, we lose all the information on the individual differences between assessors. At this aim, as many path models as assessors will be considered and compared in terms of fuzzy path coefficients so as to detect eventual hetero- geneity in the panel. Figure 2 shows the global path model. As can be seen, the latent variable global depends on the two latent variables odour and flavour. The F-PLSPM Fig. 2. Global model algorithm is used to estimate the fuzzy path coefficients ( ˜ E 1 and ˜ E 2 ). Crisp path co- efficients in Table 1 show that the global quality of the products mostly depends on the flavour rather than on the odour. Furthermore, fuzzy path coefficients describe a worse panel performance for the flavour emphasized by a more imprecise estimate (wider fuzzy interval). Therefore, the F-PLSPM algorithm enriches the results of the classical PLSPM crisp approach by providing information on the imprecision of path coefficients. At the same time, the coherence of results is granted as the crisp estimates are comprised within the fuzzy intervals. Table 1. Global Model Path Coefficients Latent Variable crisp path coefficients fuzzy path coefficients Odour 0.4215 [0.3952;0.4517] Flavour 0.6283 [0.6043;0.7817] The most interesting result coming from the proposed approach is in Figure 3, which compares the interval valued estimates on the different assessors. Figure 3 reports the fuzzy path coefficients for the 12 local models referred to each assessor. By looking within each plot (flavour and odour) separately, the asses- sor performance and the coherence between assessors can be evaluated: a) the wider Fuzzy PLS Path Modeling 695 Fig. 3. Local fuzzy path coefficients the interval, the less consistent is the assessor; b) the closer the intervals between them, the more coherent are the assessors. In the example, for the odour, assessor 7 is the least consistent assessor while assessor 12, being positioned far away from the rest of the assessors, is the least coherent as compared to the panel. Finally, by comparing the two plots, differences in the way each assessor perceives flavour and odour may be detected: for instance, assessor 7 is the most imprecise for the odour while it is extremely consistent for the flavour; assessor 12 is similarly consistent for both flavour and odour but, in both cases, it is in clear disagreement with the panel (a much higher influence of the odour as opposed to a much lower influence of the flavour). 4 Conclusion The joint use of PLS component-based approach to structural equation modeling and fuzzy possibilistic regression has yielded promising results in the framework of sensory data analysis. Namely, while taking into account the multi-block feature of sensory data, the proposed Fuzzy-PLSPM leads to a fuzzy estimation of the path coefficients. Such an estimation provides information on the precision of the classi- cal estimates and allows a thorough comparison of the sensory evaluations between assessors and within assessors for different products. Future directions of research aim to extend the fuzzy approach also to the measurement model by introducing an appropriate fuzzy possibilistic regression in the external estimation phase of the PLSPM algorithm. This further development has a twofold interest: allowing for fuzzy input data; yielding fuzzy estimates of the loadings, of the outer weights and, as a consequence, of the latent variable scores, thus embedding the measurement error that naturally affects sensory assessments. 696 Palumbo et al. References ALEFELD, G. and HERZENBERGER, J. (1983): Introduction to Interval computation. Aca- demic Press, New York. BOLLEN, K. A. (1989): Structural equations with latent variables. Wiley, New York. COPPI, R., GIL, M.A. and KIERS, H.L. (2006): The fuzzy approach to statistical analysis. Computational statistics & data analysis, 51 (1), 1–14. J ¨ ORESKOG K. (1970): A general method for analysis of covariance structure. Biometrika, 57, 239–251. ROMANO, R. (2006): Fuzzy Regression and PLS Path Modeling: a combined two-stage ap- proach for multi-block analysis. Doctoral Thesis, Univ. of Naples, Italy. ROMANO, R. and PALUMBO, F. (2006a): Fuzzy regression and least squares regression: the relationship between two different fitting criteria. Abstracts of the SIS2006 Conference, 2, 693–696. ROMANO, R. and PALUMBO, F. (2006b): Classification of SEM based on fuzzy regression. In: Esposito-Vinzi et al. (Eds.): Knowledge Extraction and Modeling. Tilapia, Anacapri, 67-68. TANAKA, H., UEIJIMA, S. and ASAI, K. (1980): Fuzzy linear regression model. IEEE Transactions Systems Man Cybernet, 10, 2933–2938. TANAKA, H. and GUO, P. (1999) Possibilistic Data Analysis for Operations Research. Physica-Verlag, Wurzburg. TENENHAUS, M. and ESPOSITO VINZI, V. (2005): PLS regression, PLS path modeling and generalized Procrustean analysis: a combined approach for multiblock analysis. Journal of Chemometrics, 19 (3), 145–153. TENENAHUS, M., ESPOSITO VINZI, V., CHATELIN, Y M. and LAURO, C. (2005): PLS path modeling Comp. Stat. and Data Anal. 48, 159–205. WOLD, H. (1982) Soft modeling: the basic design and some extensions. In: K.G. Joreskog and H. Wold (Eds.): Systems under Indirect Observation, Vol. Part II. North-Holland, Amsterdam, 1-54. ZADEH, L. (1965): Fuzzy Sets. Information and Control, 8, 338–353. ZADEH, L. (1973): Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems Man and Cybernet, 1, 28–44. Scenario Evaluation Using Two-mode Clustering Approaches in Higher Education Matthias J. Kaiser, Daniel Baier Institute of Business Administration and Economics, Brandenburg University of Technology Cottbus, Postbox 101344, 03013 Cottbus, Germany {mjkaiser, daniel.baier}@tu-cottbus.de Abstract. Scenario techniques have become popular tools for dealing with possible futures. Driving forces of the development (the so-called key factors) and their possible projections into the future are determined. After a reduction of the possible combinations of projections to a set of consistent and probable candidates for possible futures, traditionally one-mode cluster analysis is used for grouping them. In this paper, two-mode clustering approaches are proposed for this purpose and tested in an application for the future of eLearning in higher education. In this application area, scenario techniques are a very young and promising methodology. 1 Introduction: Scenario analysis Since its first applications for business prognostication (e.g., Kahn, Wiener (1967), Meadows et al. (1972), Schwartz (1991)), scenario techniques have become popular tools for governmental and corporate planners in order to deal with possible futures (“scenarios”) and to support decisions in the face of uncertainty. Nowadays, in many research areas scenario analysis is an attractive tool with a huge variety of applica- tions (e.g., Götze (1993), Mißler-Behr (2002), Welfens et al. (2004), van der Heij- den (2005), Pasternack (2006), Ringland (2006)). However, for higher education, the application of scenario analysis is new (e.g., Sprey (2003)). Different methodologi- cal approaches have been proposed, most of them using (roughly) four stages (e.g., Coates (2000), Phelps et al. (2001)): •Inafirst stage, the scope of the scenario analysis has to be defined including the focal issues (e.g. influence areas) and the driving forces for them (social, economic, political, environmental, technological factors). After a reduction of these driving forces with respect to relevance, importance, and inter-connection, a list of so-called key factors results (e.g., A, B, C). • Then, in the second stage, alternative projections (possible levels) for these key factors (e.g., A1, A2, A3, B1, B2) have to be determined. By combining these projections, a database of candidates for possible futures (e.g., (A1,B1,C1, ), (A1,B2,C1, )) is available. Additionally, the consistency for pairs of projections 666 Matthias J. Kaiser, Daniel Baier (e.g., (A1,B1), (A1,B2)) and the probability/realism of single projections within the time span under research has to be rated. • Then, in a third stage, the candidates in the database have to be evaluated on basis of their projections’ pairwise consistency and probability. Using rankings and/or cut-off values or similar approaches, the database is reduced to a set of consistent and probable candidates. Finally, the reduced set of candidates (the so-called first mode), described by their projections w.r.t. the key factors (the so-called second mode), is grouped via cluster analysis into a small number of candidate groups, the so-called “scenarios”. In an unrelated second step these candidate groups have to be analyzed to find out which projections best characterize them. Recently, new fuzzy clustering approaches have been proposed for dealing with this identification problem (see e.g. Mißler-Behr (1993), (2002)). • Finally, in a fourth stage, strategic options how to deal with the selected possible futures (“scenarios”) have to be developed. In this paper we develop new two-mode clustering approaches for simultaneously grouping candidates and projections in the third stage. The new approach bases on Baier et al. (1997)’s two-mode additive clustering procedure for simultaneous market segmentation and structuring with overlapping and non-overlapping cases. 2 Two-Mode clustering (for scenario evaluation) 2.1 The model As in Baier et al. (1997), the following notation is used (see Krolak-Schwerdt, Wiedenbeck (2006) for a recent comparison of similar additive clustering approaches): i=1, ,I is an index for first mode objects (e.g., preselected consistent and probable candidates (A1,B1,C1, ) or (A1,B2,C1, ) from stage two). j=1, ,J is an index for second mode objects (e.g., projections A1, A2, A3, ). k=1, ,K is an index for first mode clusters (cluster of candidates) and l=1, ,L an index for second mode clusters (clusters of projections). S =(s ij ) I×J is a matrix of (observed) associations between first and second mode objects (s ij ∈ IR ∀i, j). With association values of 1 – if the projection is part of the candidate – or 0 – if the projection is not part of the candidate –, S is a binary data matrix (see, e.g., Li (2005) for an analysis of binary data using two-mode clustering). Model parameters are the following: P=(p ik ) I×K is a binary matrix describing first mode cluster membership with p ik =1 if first mode object i belongs to first mode cluster k and =0 otherwise. Q=(q jl ) J×L is a binary matrix describing second mode cluster membership with q jl =1 if second mode object j belongs to second mode cluster l and =0 otherwise. W=(w kl ) K×L is a matrix of weights (w kl ∈ IR ∀k,l). In order to provide results where candidates are members of one and only one scenario whereas projections are allowed to be member of none, one, or more than one scenario, additional assumptions are necessary: The first mode membership ma- trix P is restricted to be non-overlapping (i.e.  K k=1 p ik = 1 ∀i) whereas for the Scenario Evaluation Using Two-mode Clustering Approaches 667 second mode membership matrix Q no such restrictions hold. Q is allowed to be overlapping. 2.2 Parameter estimation The parameters are determined in order to minimize the objective function Z = I  i=1 J  j=1 (s ij − ˆs ij ) 2 with ˆs ij = K  k=1 L  l=1 p ik w kl q jl ∀i, j, (1) or, equivalently, to maximize the variance accounted for VA F = 1−Z/ I  i=1 J  j=1 (s ij − ¯s) 2 with ¯s = I  i=1 J  j=1 s ij /(IJ) (2) on the basis of the underlying model S = PWQ’ + error. In our approach, an alternating least squares procedure is applied. The different sets of model parameters (P, W, and Q) are initialized and alternatingly improved w.r.t. Z. Alternatively, a Bayesian model formulation could be used (see DeSarbo et al. (2005) in a market structuring setting). However, for our approach, we first discuss the iterative steps for obtaining improved estimates for selected model parameters when estimates for the remaining sets of model parameters are given. Finally, the complete procedure is presented. a) Estimation of P for given W and Q:Set p ik = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1if J  j=1 (s ij – L  l=1 w kl q jl ) 2 = min 1≤N≤K { J  j=1 (s ij – L  l=1 w Nl q jl ) 2 } 0 otherwise ∀i,k. (3) b) Estimation of Q and W for given P: Using (for l=1, ,L selected) Z = I  i=1 J  j=1 (s ij − K  k=1 L  l  =1∧l  =l p ik w kl  q jl     =: s ijl − K  k=1 p ik w kl q jl ) 2 (4) (s ijl is constant w.r.t. q 1l , ,q Jl ,w 1l , ,w Kl ), estimates of Q and W can be obtained by starting from initial values and alternatingly improving the parameter estimates for second mode cluster l = 1, ,L via q jl = ⎧ ⎪ ⎨ ⎪ ⎩ 1if I  i=1 (s ijl − K  k=1 p ik w kl ) 2 < I  i=1 (s ijl ) 2 0 otherwise ∀j (5) 668 Matthias J. Kaiser, Daniel Baier and minimizing I  i=1 J  j=1 (s ijl − K  k=1 p ik w kl q jl ) 2 via OLS w.r.t. {w 1l , ,w Kl } (6) (OLS=ordinary least squares regression). Thus, our estimation procedure can be described as follows: 1. Determine initial estimates of P, W,andQ. Compute Z. 2. Repeat Improve the estimates of P using a). Improve the estimates of Q and W using b) . Until Z cannot be improved any more. For applying the above model and algorithms for scenario evaluation, additionally, the first and second mode clusters can be linked by setting K=L and restricting W to an identity matrix. This can be achieved by initialization and by omitting the cor- responding algorithmic steps where W is updated. In the following section, this ap- proach (with K=L and W restricted to an identity matrix) is applied in stage three of a scenario analysis in higher education. 3 Example: Scenario evaluation in higher education 3.1 Stage One: Defining the scope of the analysis Currently, at many universities, the concrete future of higher education and how to deal with this uncertainty is unclear. Whereas some developments like the demo- graphics (older and fewer Germans), the ongoing of the Bologna-process (more stan- dardization and Europe-wide exchange in higher education), the importance of better and life-long education, or the higher competition between universities for funds and talented students seem to be predictable, other developments are highly uncertain (see, e.g., Michel (2006), Opaschowski (2006), Schulmeister (2006)). Especially for universities that plan to invest in technical teaching and learn- ing environments and/or plan to attract more students for distance learning - this is unbearable. Therefore, our main research question deals with the future of higher education. As a focal time point we use the year 2020. Also, this analysis is used as an application example for our new two-mode clustering approach. In the first stage of our scenario analysis, basing on a Delphi-study on the future of eLearning, acceptance and preferences surveys, and other research projects at our institute (e.g. Göcks (2006)) as well as from other research institutes (e.g. Cuhls et al. (2002), Opaschowski (2006)) (university) internal as well as (university) external influencing factors on higher education were identified and possible projections for the near future were described. Moreover, using expert workshops with teachers, students, people from univer- sity administration and government, these lists and descriptions were extended and Scenario Evaluation Using Two-mode Clustering Approaches 669 modified, resulting in six areas of influence and thirty influencing factors (see figure 1) with a total of 73 detailed described projections w.r.t. these influencing factors. Fig. 1. Influencing factors overview 3.2 Stage Two: Creating a database of candidates In the second stage of scenario analysis, these thirty influencing factors were reduced to 12 key factors for the ongoing analysis. We did this by filtering redundant aspects and indirect dependencies. Additionally, we used scoring methods and evaluation as- pects from a group of scientific experts and analyzed relevant scientific sources (see, e.g., Kröhnert et al. (2004), Michel (2006)). Furthermore, the alternative projections for each key factor were reduced and specified in detail (resulting in one page text for each projection). As a result, a database of 2 11 3 1 =6,144 candidates (all possi- ble combinations of the 2-3 projections for each of the 12 key factors) for possible futures was available. Additionally, the pairwise consistency of these projections was evaluated using values ranging from 1=“totally inconsistent” to 9=“totally consistent”. Consequently, as discussed in the theoretical introduction, a consistency value was calculated for each candidate (e.g. (A1,B2,C3, )) as the mean pairwise consistency of its pairs of projections (e.g. (A1,B2), (A1,C3), (B2,C3), ). 3.3 Stage Three: Evaluating, selecting, and clustering candidates In a third stage the database was first reduced and then clustered. For reduction, the so-called "‘complete combination scanning"’ was used, what means that for each pair of projections that candidate with the highest mean pairwise consistency was kept for further analysis. The reduction resulted into 286 candidates. [...]... Component Analysis, 1 93 Normal Mixtures, 127 Notation, 697 Number Expression Extraction, 5 53 R, 33 5, 38 9, 569 Rank Data, 681 Recommender Systems, 525, 533 , 541, 619 Record Linkage, 33 5 Reference Modelling, 37 3 Regression, 36 3 Relationships, 36 3, 629 Return Prediction, 499 Robust Estimation, 1 03 Robust Regression, 277 Robustness, 127 Object Labeling, 2 93 Object-Identification, 171 Ontology Learning, 577, 585 ... 4 63 Customer Equity Management, 479 Customer Segmentation, 479 Data Analysis, 31 9 Data Augmentation, 111 Data Depth, 455 Data Integration, 33 5 Data Mining, 421 Data Quality, 33 5 Data Transformation, 681 Decision Trees, 38 9 Dendrograms, 95 Design Rationale, 155 714 Keywords Dewey Decimal Classification (DDC), 697 Dialectology, 647 Dimensionality Reduction, 619 Discriminant Analysis, 245 Dispersion, 1 63. .. Relational Learning, 269 Structure Features, 237 Subgrouping, 629 Supervised Classification, 55, 421 Support Vector Machines, 3, 11, 55, 77, 245, 515 Supreme Administrative Court, 569 Survival Analysis, 5 93 Swarm Intelligence, 139 Tagged Data, 6 73 Taxonomies, 37 3 Temporal Data Mining, 2 53 Text Analysis, 637 Text Categorization, 655 Text Classification, 637 Text Cleaning, 39 7 Text Mining, 569 Textmining, 4 13 Top-down... Mining Algorithms, 32 7 Calibration of Classifier Scores, 29 Canonical Form, 229 Capability Indices, 405 Cartography, 647 Categorical Data, 1 63 Centrality, 38 1 Certification Model, 507 Characteristic Vector, 38 1 Choice-theory, 541 Chunking, 601 Classification, 45, 77, 1 83 , 237 Classifier Fusion, 19 Classifiers, 95 Cluster Analysis, 85 , 681 Cluster-Trees, 1 63 Clustering, 119, 139 , 489 , 647, 6 73 Clustering with... Report 46, CS Department, University Marburg, Germany, 2005 ZHAO, Y., KARYPIS, G Criterion Functions for Document Clustering Experiments and analysis Machine Learning, in press, 20 03 Keywords Adaptive Conjoint Analysis, 447 Additive Clustering, 38 1 Additive Spline, 1 93 ADSL, 34 3 Alphabet, 285 Ambiguity, 611 Analysis, 697 Analytic Hierarchy Process, 447 Ancient Watermarks, 237 Artificial Life, 139 Artificial... Algorithms, 32 7 Moduli Spaces, 95 Multiple Correspondence Analysis, 1 83 Multiple Factor Analysis, 219 Multivariate Additive Partial Least Squares, 201 Multivariate Control Charts, 1 93 Multivariate EWMA Control Charts, 455 Multivariate Outliers, 1 03 Music, 261 Musical Time Series, 285 Pipelining Environment, 31 9 Place Labeling, 2 93 PLS Path Modelling, 1 83 PLS-PM, 689 Positive-Definite Matrices, 3 Posterior... Data, 689 Indefinite Kernels, 37 Indo-European, 629 Information Criteria, 61 Information Extraction, 5 53, 577 Information Integration, 171 Information Retrieval, 261 Information Systems, 37 3 Integer Partitions, 541 Interpretability of Components, 209 Interval Data, 705 Interval Sequences, 2 53 Invariance, 37 Invention, 4 13 K-way Classification, 29 KDT, 4 13 Kernel Methods, 37 Kernels, 3 KNIME, 31 9 Knowledge... 629 Large data sets, 11 Law, 569 Learning Vector Quantization, 55 Local Models, 69 Logit Models, 1 83 Loyalty, 4 63 Margin–based Classification, 29 Marketing, 489 Maximum Likelihood Estimator, 30 1 MCDA, 561 MCMC, 285 Meta -learning, 421 MIMIC-Model, 4 63 Missing Data, 111 Mixed Logistic Regression, 471 Keywords 715 Mixture Modeling, 119 Mixture Regression, 61 Model Selection, 61, 30 1 Modular Data Mining... Software, 38 9 Outliers, 127 P-adic Numbers, 95 Parameter Estimation, 36 3 Partial Least Squares regression, 507 Pattern Classification, 245 Patterns in Data Mining Process, 32 7 Phylogeny, 629 Qn, 277 Quantitative Linguistics, 637 , 655 Question Answering, 5 53 Satisfaction-Retention Link, 471 Scenarios, 665 Self-organizing Maps, 34 3 Self-organizing Neural Networks, 45 Semi-Supervised Learning, 139 Semi-Supervised-Clustering,... Detection, 35 5 Frequent Graph Mining, 229 Frequent Patterns, 2 53 Fuzzy, 647 Fuzzy Model, 689 Gaussian Mixture Model, 1 03 Generalized Method of Moments Estimators, 30 1 Geographical Information Science, 31 1 Hard and Soft Partitions, 147 Hierarchical Bayes Estimation, 431 Higher Education, 665 Hodges-Lehmann Estimator, 277 Homograph, 611 ICA, 245 Idea, 4 13 Image Analysis, 245 Image Retrieval, 237 Imprecise Data, . Information and Control, 8, 33 8 35 3. ZADEH, L. (19 73) : Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems Man and Cybernet, 1, 28 44. Scenario. Extraction and Modeling. Tilapia, Anacapri, 67- 68. TANAKA, H., UEIJIMA, S. and ASAI, K. (1 980 ): Fuzzy linear regression model. IEEE Transactions Systems Man Cybernet, 10, 2 933 –2 9 38 . TANAKA, H. and GUO,. (for linking first- and second-mode clusters). The resulting VAF-values from analyses with totals of K=L=1 to 8 clusters (VAF=0.056, 0.2 43, 0 .32 5, 0 .36 2, 0 .36 3, 0 .39 4, 0.4 48, 0.452) indicate via