Multidimensional scaling

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. MDS is used to translate "information about the pairwise 'distances' among a set of objects or individuals" into a configuration of points mapped into an abstract Cartesian space.[1]

An example of classical multidimensional scaling applied to voting patterns in the United States House of Representatives. Each red dot represents one Republican member of the House, and each blue dot one Democrat.

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.[2]

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[3] .

TypesEdit

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

Classical multidimensional scalingEdit

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.[2] For example, given the Euclidean aerial distances   between various cities indexed by i and j, you want to find the coordinates   of the cities such that  . In this example, an exact solution is possible (assuming the Euclidean distances are exact). In practice, this is usually not the case, and MDS therefore seeks to approximate the lower-dimensional representation by minimising a loss function. General forms of loss functions called stress in distance MDS and strain in classical MDS. The strain is given by:  , where   now denote vectors in N-dimensional space,   denotes the scalar product between   and  , and   are the elements of the matrix   defined on step 2 of the following algorithm, which are computed from the distances.

Steps of a Classical MDS algorithm:
Classical MDS uses the fact that the coordinate matrix   can be derived by eigenvalue decomposition from  . And the matrix   can be computed from proximity matrix   by using double centering.[4]
  1. Set up the squared proximity matrix  
  2. Apply double centering:   using the centering matrix  , where   is the number of objects,   is the   identity matrix, and   is an   matrix of all ones.
  3. Determine the   largest eigenvalues   and corresponding eigenvectors   of   (where   is the number of dimensions desired for the output).
  4. Now,   , where   is the matrix of   eigenvectors and   is the diagonal matrix of   eigenvalues of  .
Classical MDS assumes Euclidean distances. So this is not applicable for direct dissimilarity ratings.

Metric multidimensional scaling (mMDS)Edit

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  :   and   for distance. In classical scaling  . Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities.

Non-metric multidimensional scaling (nMDS)Edit

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. The relationship is typically found using isotonic regression: let   denote the vector of proximities,   a monotonic transformation of  , and   the point distances; then coordinates have to be found, that minimize the so-called stress,

 

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. The basic steps in a non-metric MDS algorithm are:
  1. Find a random configuration of points, e. g. by sampling from a normal distribution.
  2. Calculate the distances d between the points.
  3. Find the optimal monotonic transformation of the proximities, in order to obtain optimally scaled data  .
  4. Minimize the stress between the optimally scaled data and the distances by finding a new configuration of points.
  5. Compare the stress to some criterion. If the stress is small enough then exit the algorithm else return to 2.

Louis Guttman's smallest space analysis (SSA) is an example of a non-metric MDS procedure.

Generalized multidimensional scaling (GMD)Edit

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.[5]

DetailsEdit

The data to be analyzed is a collection of   objects (colors, faces, stocks, . . .) on which a distance function is defined,

  distance between  -th and  -th objects.

These distances are the entries of the dissimilarity matrix

 

The goal of MDS is, given  , to find   vectors   such that

  for all  ,

where   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.[6]

In other words, MDS attempts to find a mapping from the   objects into   such that distances are preserved. If the dimension   is chosen to be 2 or 3, we may plot the vectors   to obtain a visualization of the similarities between the   objects. Note that the vectors   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   indicates the set of real numbers, and the notation   refers to the Cartesian product of   copies of  , which is an  -dimensional vector space over the field of the real numbers.)

There are various approaches to determining the vectors  . 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.[citation needed]

ProcedureEdit

There are several steps in conducting MDS research:

  1. Formulating the problem – What variables do you want to compare? How many variables do you want to compare? What purpose is the study to be used for?
  2. Obtaining input data – For example, :- Respondents are asked a series of questions. For each product pair, they are asked to rate similarity (usually on a 7-point Likert scale from very similar to very dissimilar). The first question could be for Coke/Pepsi for example, the next for Coke/Hires rootbeer, the next for Pepsi/Dr Pepper, the next for Dr Pepper/Hires rootbeer, etc. The number of questions is a function of the number of brands and can be calculated as   where Q is the number of questions and N is the number of brands. This approach is referred to as the “Perception data : direct approach”. There are two other approaches. There is the “Perception data : derived approach” in which products are decomposed into attributes that are rated on a semantic differential scale. The other is the “Preference data approach” in which respondents are asked their preference rather than similarity.
  3. Running the MDS statistical program – Software for running the procedure is available in many statistical software packages. Often there is a choice between Metric MDS (which deals with interval or ratio level data), and Nonmetric MDS[7] (which deals with ordinal data).
  4. Decide number of dimensions – The researcher must decide on the number of dimensions they want the computer to create. Interpretability of the MDS solution is often important, and lower dimensional solutions will typically be easier to interpret and visualize. However, dimension selection is also an issue of balancing underfitting and overfitting. Lower dimensional solutions may underfit by leaving out important dimensions of the dissimilarity data. Higher dimensional solutions may overfit to noise in the dissimilarity measurements. Model selection tools like AIC/BIC, Bayes factors, or cross-validation can thus be useful to select the dimensionality that balances underfitting and overfitting.
  5. Mapping the results and defining the dimensions – The statistical program (or a related module) will map the results. The map will plot each product (usually in two-dimensional space). The proximity of products to each other indicate either how similar they are or how preferred they are, depending on which approach was used. How the dimensions of the embedding actually correspond to dimensions of system behavior, however, are not necessarily obvious. Here, a subjective judgment about the correspondence can be made (see perceptual mapping).
  6. Test the results for reliability and validity – Compute R-squared to determine what proportion of variance of the scaled data can be accounted for by the MDS procedure. An R-square of 0.6 is considered the minimum acceptable level.[citation needed] An R-square of 0.8 is considered good for metric scaling and .9 is considered good for non-metric scaling. Other possible tests are Kruskal’s Stress, split data tests, data stability tests (i.e., eliminating one brand), and test-retest reliability.
  7. Report the results comprehensively – Along with the mapping, at least distance measure (e.g., Sorenson index, Jaccard index) and reliability (e.g., stress value) should be given. It is also very advisable to give the algorithm (e.g., Kruskal, Mather), which is often defined by the program used (sometimes replacing the algorithm report), if you have given a start configuration or had a random choice, the number of runs, the assessment of dimensionality, the Monte Carlo method results, the number of iterations, the assessment of stability, and the proportional variance of each axis (r-square).

ImplementationsEdit

See alsoEdit

ReferencesEdit

  1. ^ Mead, A (1992). "Review of the Development of Multidimensional Scaling Methods". Journal of the Royal Statistical Society. Series D (The Statistician). 41 (1): 27–39. JSTOR 234863. Abstract. Multidimensional scaling methods are now a common statistical tool in psychophysics and sensory analysis. The development of these methods is charted, from the original research of Torgerson (metric scaling), Shepard and Kruskal (non-metric scaling) through individual differences scaling and the maximum likelihood methods proposed by Ramsay.
  2. ^ a b Borg, I.; Groenen, P. (2005). Modern Multidimensional Scaling: theory and applications (2nd ed.). New York: Springer-Verlag. pp. 207–212. ISBN 978-0-387-94845-4.
  3. ^ Genest, Christian; Nešlehová, Johanna G.; Ramsay, James O. (2014). "A Conversation with James O. Ramsay". International Statistical Review / Revue Internationale de Statistique. 82 (2): 161–183. JSTOR 43299752. Retrieved 30 June 2021.
  4. ^ Wickelmaier, Florian. "An introduction to MDS." Sound Quality Research Unit, Aalborg University, Denmark (2003): 46
  5. ^ Bronstein AM, Bronstein MM, Kimmel R (January 2006). "Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching". Proc. Natl. Acad. Sci. U.S.A. 103 (5): 1168–72. Bibcode:2006PNAS..103.1168B. doi:10.1073/pnas.0508601103. PMC 1360551. PMID 16432211.
  6. ^ Kruskal, J. B., and Wish, M. (1978), Multidimensional Scaling, Sage University Paper series on Quantitative Application in the Social Sciences, 07-011. Beverly Hills and London: Sage Publications.
  7. ^ Kruskal, J. B. (1964). "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis". Psychometrika. 29 (1): 1–27. doi:10.1007/BF02289565. S2CID 48165675.

BibliographyEdit

  • Cox, T.F.; Cox, M.A.A. (2001). Multidimensional Scaling. Chapman and Hall.
  • Coxon, Anthony P.M. (1982). The User's Guide to Multidimensional Scaling. With special reference to the MDS(X) library of Computer Programs. London: Heinemann Educational Books.
  • Green, P. (January 1975). "Marketing applications of MDS: Assessment and outlook". Journal of Marketing. 39 (1): 24–31. doi:10.2307/1250799. JSTOR 1250799.
  • McCune, B. & Grace, J.B. (2002). Analysis of Ecological Communities. Oregon, Gleneden Beach: MjM Software Design. ISBN 978-0-9721290-0-8.
  • Young, Forrest W. (1987). Multidimensional scaling: History, theory, and applications. Lawrence Erlbaum Associates. ISBN 978-0898596632.
  • Torgerson, Warren S. (1958). Theory & Methods of Scaling. New York: Wiley. ISBN 978-0-89874-722-5.