Multidimensional scaling

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. It refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. An MDS algorithm aims to place each object in N-dimensional space such that the between-object distances are preserved as well as possible. Each object is then assigned coordinates in each of the N dimensions. The number of dimensions of an MDS plot N can exceed 2 and is specified a priori. Choosing N=2 optimizes the object locations for a two-dimensional scatterplot.

MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:

It is also known as Principal Coordinates Analysis, Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain: For example, given the aerial distances between many cities in a matrix ${\textstyle D=[d_{ij}]}$, where ${\textstyle d_{ij}}$ is the distance between the coordinates of ${\textstyle i^{th}}$ and ${\textstyle j^{th}}$ city, given by ${\textstyle d_{ij}={\sqrt {(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}}}}$. Now, you want to find the coordinates of the cities. This problem is addressed in classical MDS.

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