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given object lies in the direct product of these spaces, in which a further metric has to be defined. In many methods the real n-dimensional vector space together with the Euclidean metric is the first representation space which is then projected into the Euclidean plane to visualize some image of the data structure. These methods use in addition to the interpretation principle distance the interpretation principle coordinatization which indicates here the use of the real n-dimensional vector space, bases, coordinates and standard matrix algebra. These classical tools can be applied successfully in the case of measurements on the absolute scale level (in the sense of measurement theory), especially for the analysis of contingency tables which are usually obtained from measurement data by nominal scaling. It is well-known that some types of data (e.g. ordinal data) can´t be represented satisfactorily by metric methods, especially not by metric vector space methods. Therefore more general methods have been developed, using the interpretation principle hierarchy. A simple tool in this realm is the concept of a chain (where a chain is defined as an ordered set in which any two elements are comparable with respect to the given order relation). If the domain structure of each attribute of the given data structure is interpreted as a chain, then these data can be represented in a direct product of chains. A very general and fruitful interpretation principle is that of a concept. This has a long philosophical tradition in which the connection between objects and attributes is investigated. The classical concept of concept and the conceptual hierarchy has been formalized in Formal Concept Analysis, a new method in data analysis. Now we collect in List 1 some graphical data analysis methods and for each of these methods several key-words to give an overview over the main interpretation principles used in the selected methods. List 1: Method Interpretation principles FA Factor Analysis distance, coordinatization, absolute scale PCA Principal Component Analysis distance, coordinatization, absolute scale CA Correspondence Analysis distance, coordinatization, nominal scaling, contingency table, absolute scale MCA Multiple Correspondence Analysis distance, coordinatization, nominal scaling PRINCALS Principal Component Analysis distance, coordinatization by Alternating Least Squares CLUS Cluster Analysis distance, hierarchy MDS Multidimensional Scaling distance POSA Partial Order Scalogram Analysis hierarchy, product of chains FCA Formal Concept Analysis concept, hierarchy Before going into details we give a graphical representation of the information in this list using Formal Concept Analysis. 3
Diagram 1: Interpretation principles of graphical data analysis methods Diagram 1 contains exactly the information given in List 1. How to read such a line diagram? Each graphical data analysis method and each key-word of List 1 is represented as a point in Diagram 1. Each method has exactly those key-words in List 1 whose points are reachable by an upwards leading path from the point representing the given method (e.g. FCA has the keywords concept and hierarchy). For further details about Formal Concept Analysis the reader is referred to WILLE (1984, 1987) and WOLFF (1988, 1994a,b). Now we use Diagram 1 to understand List 1: Diagram 1 shows that the key-words distance and hierarchy play a prominent role, since all methods mentioned are covered by these two key-words. Cluster analysis is the only method having both key-words. MDS is the only method having only the key-word distance. What´s the reason why we didn´t give MDS the key-word coordinatization though the MDS calculations use matrix algebra? Answer: the formulation of the MDS-problem doesn´t need the structure of the vector space, but only the set R n together with the Euclidean metric. Among the methods using coordinatization PRINCALS is the most general one (in the sense that is has no further key-word under coordinatization), while MCA transforms the data using nominal scaling and FA and PCA work usually with data of absolute scale type. Simple Correspondence Analysis (CA) represents contingency tables whose values are of absolute scale type, but these values are usually obtained from data using nominal scaling. Besides the hierachic cluster analysis methods there are two hierarchical methods which use neither distance nor coordinatization: POSA represents the data in direct products of chains and FCA uses the concept of concept and the conceptual hierarchy which is represented by line diagrams in the plane. POSA and FCA don´t have the key-word coordinatization in the narrow sense explained above though both methods use ordinal coordinates in direct products of chains as an important tool. 4
Page 1 and 2: COMPARISON OF GRAPHICAL DATA ANALYS
Page 3: esulting structure of the data tabl
Page 7 and 8: 3.2 Explanation of Table 1: Now we
Page 9 and 10: • Multiple Correspondence Analysi
Page 11 and 12: direct product of two chains looses
Page 13 and 14: References Benzécri, J.-P.: L´ana

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