A Note on Two-Phase Method for a Class of Metric Models in Individual Scaling

In this note, we discuss the typical problem in individual scaling viz., finding a common configuration and weights attached to dimensions for each individual from the given interpoint distances or scalar products. Trucker and Messick (1963), Horan (1969) and others have developed procedures for solving the problem. Carroll and Chang (1970) defined a minimization criterion (STRAIN) in terms of product moments computed from raw data. They use an alternative least square (ALS) method for estimating the configuration dn weights. Within the STRAIN frame work, Schonemann (1972) presented an algebraick solution in the case of exact data. Takane, Young and De Leeuw (1977) proposed a procedure called, ALSCAL in which the criterion function (SSTRESS) is in terms of distances obtained from raw data. The configuration and weights are obtained by solving certain normal equations in the least square method alternately. In this note, we consider the problem within the STRA in framework and propose a two-phase method. In the first phase, the problem of determining the optimal weights (Wi) for a given configuration (X) is posed as a standard quadratic programming problem for which efficient finitely convergent algorithms are available. In the second phase, for a given set of weights (Wi), a system of equations is developed for obtaining the configuration X. the relation to the quadratic programming problem to obtain Wi and the approach to obtain X appear to be new. An explicit solution to the problem is obtained for one dimensional case and an approach is described for the two dimensional problem. Numerical examples are given for one and two dimensions cases. The solution obtained by the proposed method is also compared with the solution obtained by Schonemann (1972) for the two-dimensional problem.