Abstract
Some intriguing relationships among four methods of metric and nonmetric multidimensional scaling (MDS) are explicated. It is shown that all four methods of MDS considered here amount to solving, explicitly or implicitly, the stationary point of a matrix which can be generally represented as A’HA, where A is a difference matrix (to be defined), and where H depends on a particular criterion being optimized. H may be a matrix of fixed constants or of functions of unknown parameters (stimulus coordinates) of the representation model. A conceptual distinction is made as to the scale level of measurement in reference to MDS methods and solutions.
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Takane, Y. On The Relations Among Four Methods of Multidimensional Scaling. Behaviormetrika 4, 29–43 (1977). https://doi.org/10.2333/bhmk.4.29
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DOI: https://doi.org/10.2333/bhmk.4.29
Key Words and Phrases
- multidimensional scaling (MDS)
- metric and nonmetric MDS
- quantification method IV (Q4)
- Torgerson-Gowcr type principal coordinates method (the T-G method)
- the Young-Housholder transformation
- scalar product optimization
- distance optimization
- squared distance optimization
- optimal scaling
- alternating least squares (ALS)
- monotone convergence
- scale levels of measurement
- admissible transformation