Multidimensional scaling (MDS) is a set of related statistical techniques often used in data visualization for exploring similarities or dissimilarities in data. MDS is a special case of ordination. An MDS algorithm starts with a matrix of item-item similarities, and then assigns a location of each item in a low-dimensional space, suitable for graphing or 3D visualization.
MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:
Classical multidimensional scaling
Also known as Torgerson Scaling or Torgerson-Gower scaling – takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain.
Metric multidimensional scaling
A superset of classical MDS that generalizes the optimisation procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress which is often minimized using a procedure called Stress Majorization.
Generalized multidimensional scaling (GMDS)
A superset of metric MDS that allows for the target distances to be non-Euclidean.
Non-metric multidimensional scalingIn contrast to metric MDS, non-metric MDS both finds a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distance between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression.
MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:
Classical multidimensional scaling
Also known as Torgerson Scaling or Torgerson-Gower scaling – takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain.
Metric multidimensional scaling
A superset of classical MDS that generalizes the optimisation procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress which is often minimized using a procedure called Stress Majorization.
Generalized multidimensional scaling (GMDS)
A superset of metric MDS that allows for the target distances to be non-Euclidean.
Non-metric multidimensional scalingIn contrast to metric MDS, non-metric MDS both finds a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distance between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression.
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