×

Least squares metric, unidimensional scaling of multivariate linear models. (English) Zbl 0716.92028

Summary: The squared error loss function for the unidimensional metric scaling problem has a special geometry. It is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed. This approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables. The algorithms shown in the paper are linear in the number of parameters. They always descend and, at convergence, every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed. Convergence is very rapid and Monte Carlo tests show that the basic procedure almost always converges to the overall global minimum.

MSC:

91C15 One- and multidimensional scaling in the social and behavioral sciences
62P15 Applications of statistics to psychology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Asher, Herbert B., & Herbert F. Weisberg. (1978). Voting change in congress: Some dynamic perspectives on an evolutionary process.American Journal of Political Science, 22, 391–425. · doi:10.2307/2110622
[2] Carrol, J. Douglas, Pruzansky, Sandra, & Kruskal, Joseph B. (1980). CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters.Psychometrika, 45, 3–24. · Zbl 0478.62050 · doi:10.1007/BF02293596
[3] Clausen, Aage. (1973).How congressmen decide: A policy focus. New York: St. Martin’s Press.
[4] Defays, D. (1978). A short note on a method of seriation.British Journal of Mathematical and Statistical Psychology, 31, 49–53.
[5] de Leeuw, Jan. (1984). Differentiability of Kruskal’s stress at a local minimum.Psychometrika, 49, 111–114. · doi:10.1007/BF02294209
[6] DeSarbo, Wayne, & Carroll, J. Douglas. (1984). Three-way metric unfolding via alternating weighted least squares.Psychometrika, 50, 275–300. · Zbl 0597.62113 · doi:10.1007/BF02294106
[7] Eckart, Carl, & Young, Gale. (1936). The approximation of one matrix by another of lower rank.Psychometrika, 1, 211–218. · JFM 62.1075.02 · doi:10.1007/BF02288367
[8] Fenno, Richard F. (1978).Home style: House members in their districts Boston: Little, Brown and Co.
[9] Fiorina, Morris. (1974).Representatives, roll calls, and constituencies. Lexington, MA: Heath.
[10] Heiser, Willem J. (1981).Unfolding analysis of proximity data. Leiden: University of Leiden, Department of Psychology.
[11] Hinich, Melvin J., & Roll, Richard. (1981). Measuring nonstationarity in the parameters of the market model.Research in Finance, 3, 1–51.
[12] Hubert, Lawrence, & Arabie, Phipps. (1986). ”Unidimensional Scaling and Combinatorial Optimization.” In de Leeuw et al. (Eds.),Multidimensional data analysis. Leiden: DSWO Press. · Zbl 0925.62009
[13] Hubert, Lawrence, & Arabie, Phipps. (1988). Relying on necessary conditions for optimization: Unidimensional scaling and some extensions. InClassification and related methods of data analysis. Amsterdam: North-Holland.
[14] Kritzer, Herbert M. (1978). Ideology and American political elites.Public Opinion Quarterly, 42, 484–502. · doi:10.1086/268475
[15] Kruskal, Joseph B. (1964a) Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 29, 1–28. · Zbl 0123.36803 · doi:10.1007/BF02289565
[16] Kruskal, Joseph B. (1964b). Nonmetric multidimensional scaling: A numerical method.Psychometrika, 29, 115–130. · Zbl 0123.36804 · doi:10.1007/BF02289694
[17] Peltzman, Sam. (1984). Constituent interest and congressional voting.Journal of Law and Economics, 27, 181–210. · doi:10.1086/467062
[18] Poole, Keith T. (1981). Dimensions of interest group evaluation of the U.S. Senate, 1969–1978.American Journal of Political Science, 25, 49–67. · doi:10.2307/2110912
[19] Poole, Keith T. (1984). Least squares metric, unidimensional unfolding.Psychometrika, 49, 311–323. · Zbl 0557.62062 · doi:10.1007/BF02306022
[20] Poole, Keith T., & Daniels, R. Steven. (1985). Ideology, party, and voting in the U.S. Congress, 1959–80.American Political Science Review, 79, 373–399. · doi:10.2307/1956655
[21] Poole, Keith T., & Rosenthal, Howard. (1986). The dynamics of interest group evaluations of Congress (GSIA Working Paper 1-86-87). Pittsburgh, PA: Carnegie Mellon University.
[22] Ramsay, James O. (1977). Maximum likelihood estimation in multidimensional scaling.Psychometrika, 42, 241–266. · Zbl 0362.92019 · doi:10.1007/BF02294052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.