Open AccessAN INNER PRODUCT MODEL FOR THE MULTIDIMENSIONAL SCALING OF SYMMETRIC LAYOUTS
Author(s) -
Bechtel Gordon G.
Publication year - 1970
Publication title -
ets research bulletin series
Language(s) - English
Resource type - Journals
eISSN - 2333-8504
pISSN - 0424-6144
DOI - 10.1002/j.2333-8504.1970.tb00417.x
Subject(s) - multidimensional scaling , pairwise comparison , mathematics , scaling , set (abstract data type) , product (mathematics) , least squares function approximation , class (philosophy) , representation (politics) , matrix (chemical analysis) , space (punctuation) , explained sum of squares , combinatorics , algorithm , mathematical optimization , computer science , statistics , artificial intelligence , geometry , political science , law , composite material , programming language , operating system , materials science , estimator , politics
A multidimensional scaling analysis is presented for replicated symmetric layouts of pairwise compositions. The replicates may represent individuals, situations, or blocks which contain all pairwise responses generated from a distinct set of objects or treatments. The class of replicates and the class of objects are scaled in a joint space by means of an inner product model which weights each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for the model. The solution involves the decomposition of a three‐way least squares problem into three subproblems; two of these are trivial, and the third is solvable by classical least squares matrix factorization.