Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n objects in a set into a configuration of n points mapped into an abstract Cartesian space. More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction. Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot. Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis.

Types

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

Classical multidimensional scaling

It is also known as Principal Coordinates Analysis (PCoA), 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, which is given by where x_{i} denote vectors in N-dimensional space, x_i^T x_j denotes the scalar product between x_{i} and x_{j}, and b_{ij} are the elements of the matrix B defined on step 2 of the following algorithm, which are computed from the distances.

Metric multidimensional scaling (mMDS)

It is a superset of classical MDS that generalizes the optimization 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. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares:"" Metric scaling uses a power transformation with a user-controlled exponent p: d_{ij}^p and for distance. In classical scaling p=1. Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities.

Non-metric multidimensional scaling (NMDS)

In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. Let d_{ij} be the dissimilarity between points i, j. Let be the Euclidean distance between embedded points x_i, x_j. Now, for each choice of the embedded points x_i and is a monotonically increasing function f, define the "stress" function: The factor of in the denominator is necessary to prevent a "collapse". Suppose we define instead, then it can be trivially minimized by setting f = 0, then collapse every point to the same point. A few variants of this cost function exist. MDS programs automatically minimize stress in order to obtain the MDS solution. The core of a non-metric MDS algorithm is a twofold optimization process. First the optimal monotonic transformation of the proximities has to be found. Secondly, the points of a configuration have to be optimally arranged, so that their distances match the scaled proximities as closely as possible. NMDS needs to optimize two objectives simultaneously. This is usually done iteratively: Louis Guttman's smallest space analysis (SSA) is an example of a non-metric MDS procedure.

Generalized multidimensional scaling (GMD)

An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another.

Details

The data to be analyzed is a collection of M objects (colors, faces, stocks, . . .) on which a distance function is defined, These distances are the entries of the dissimilarity matrix The goal of MDS is, given D, to find M vectors such that where |\cdot| is a vector norm. In classical MDS, this norm is the Euclidean distance, but, in a broader sense, it may be a metric or arbitrary distance function. For example, when dealing with mixed-type data that contain numerical as well as categorical descriptors, Gower's distance is a common alternative. In other words, MDS attempts to find a mapping from the M objects into such that distances are preserved. If the dimension N is chosen to be 2 or 3, we may plot the vectors x_i to obtain a visualization of the similarities between the M objects. Note that the vectors x_i are not unique: With the Euclidean distance, they may be arbitrarily translated, rotated, and reflected, since these transformations do not change the pairwise distances. (Note: The symbol \mathbb{R} indicates the set of real numbers, and the notation refers to the Cartesian product of N copies of \mathbb{R}, which is an N-dimensional vector space over the field of the real numbers.) There are various approaches to determining the vectors x_i. Usually, MDS is formulated as an optimization problem, where is found as a minimizer of some cost function, for example, A solution may then be found by numerical optimization techniques. For some particularly chosen cost functions, minimizers can be stated analytically in terms of matrix eigendecompositions.

Procedure

There are several steps in conducting MDS research:

Implementations