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Multidimensional scaling (MDS) is a set of related statistical techniques often used in information visualization
Information visualization

File:HaloVisualizationTechnique.pngInformation visualization the interdisciplinary study of the visualisation Representation of large-scale collections of non-numerical information, such as files and lines of code in software systems, and the use of graphical techniques to help people understand and analyze data....
 for exploring similarities or dissimilarities in data. MDS is a special case of ordination
Ordination (statistics)

In multivariate statistics, ordination is a method complementary to data clustering, and used mainly in exploratory data analysis . Ordination partially ordered set objects that are characterized by values on multiple variables so that similar objects are near each other and dissimilar objects are further from each other....
. An MDS algorithm starts with a matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 of item–item similarities, then assigns a location to each item in N-dimensional space, where N is specified a priori. For sufficiently small N, the resulting locations may be displayed in a graph or 3D visualisation.


e 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? More than 20 is cumbersome.






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    Multidimensional scaling (MDS) is a set of related statistical techniques often used in information visualization
    Information visualization

    File:HaloVisualizationTechnique.pngInformation visualization the interdisciplinary study of the visualisation Representation of large-scale collections of non-numerical information, such as files and lines of code in software systems, and the use of graphical techniques to help people understand and analyze data....
     for exploring similarities or dissimilarities in data. MDS is a special case of ordination
    Ordination (statistics)

    In multivariate statistics, ordination is a method complementary to data clustering, and used mainly in exploratory data analysis . Ordination partially ordered set objects that are characterized by values on multiple variables so that similar objects are near each other and dissimilar objects are further from each other....
    . An MDS algorithm starts with a matrix
    Matrix (mathematics)

    In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
     of item–item similarities, then assigns a location to each item in N-dimensional space, where N is specified a priori. For sufficiently small N, the resulting locations may be displayed in a graph or 3D visualisation.

    Categorization of MDS


    MDS algorithms fall into a taxonomy
    Taxonomy

    Taxonomy is the practice and science of classification. The word comes from the Greek language ', taxis and ', nomos .Taxonomies, or taxonomic schemes, are composed of taxonomic units known as taxa , or kinds of things that are arranged frequently in a hierarchical structure....
    , 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. (pp. 207-212).
    • Metric multidimensional scaling: 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
      Stress majorization

      Stress majorization is an optimization used in multidimensional scaling where, for a set of n, m-dimensional data items, a configuration X of n points in r-dimensional space is sought that minimises the so called stress function ....
      .
    • Non-metric multidimensional scaling: In 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
      Isotonic regression

      In numerical analysis, isotonic regression involves finding a weighted least-squares fit to a Euclidean space with weights vector subject to a set of monotonicity constraints giving a totally ordered set or partial order over the variables....
      .


    Procedure

    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? More than 20 is cumbersome. Less than 8 (4 pairs) will not give valid results. What purpose is the study to be used for?
    2. Obtaining Input Data – Respondents are asked a series of questions. For each product pair they are asked to rate similarity (usually on a 7 point Likert scale
      Likert scale

      A Likert scale is a psychometrics scale commonly used in questionnaires, and is the most widely used scale in survey research. When responding to a Likert questionnaire item, respondents specify their level of agreement to a statement....
       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 which 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 softwares for statistics. Often there is a choice between Metric MDS (which deals with interval or ratio level data), and Nonmetric MDS (which deals with ordinal data). The researchers must decide on the number of dimensions they want the computer to create. The more dimensions, the better the statistical fit, but the more difficult it is to interpret the results.
    4. 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. The dimensions must be labelled by the researcher. This requires subjective judgement and is often very challenging. The results must be interpreted ( see perceptual mapping
      Perceptual mapping

      Perceptual mapping is a graphics technique used by asset marketing that attempts to visually display the perceptions of customers or potential customers....
      ).
    5. 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 .6 is considered the minimum acceptable level. Other possible tests are Kruskal’s Stress, split data tests, data stability tests (i.e., eliminating one brand), and test-retest reliability.


    Applications

    Applications include scientific visualisation and data mining
    Data mining

    Data mining is the process of extracting hidden patterns from data. As more data is gathered, with the amount of data doubling every three years, data mining is becoming an increasingly important tool to transform this data into information....
     in fields such as cognitive science
    Cognitive science

    Cognitive science may be concisely defined as the study of the nature of intelligence. It draws on multiple empirical disciplines, including psychology, philosophy, neuroscience, linguistics, anthropology, computer science, sociology and biology....
    , information science
    Information science

    Information science is an interdisciplinarity science primarily concerned with the collection, Categorization, manipulation, storage, information retrieval and dissemination of information....
    , psychophysics
    Psychophysics

    Psychophysics is a subdiscipline of psychology dealing with the relationship between physical stimulus and their subjectivity correlates, or percepts....
    , psychometrics
    Psychometrics

    Psychometrics is the field of study concerned with the theory and technique of educational and psychological measurement, which includes the measurement of knowledge, abilities, attitudes, and Wiktionary:personality traits....
    , marketing
    Marketing

    Marketing is defined by the American Marketing Association as the activity, set of institutions, and processes for creating, communicating, delivering, and exchanging offerings that have value for customers, clients, partners, and society at large....
     and ecology
    Ecology

    Ecology is the science study of the distribution and Abundance of life and the interactions between organisms and their nature environment ....
    . New applications arise in the scope of autonomous wireless nodes which populate a space or an area. MDS may apply as a real time enhanced approach to monitoring and managing such populations.

    Marketing

    In marketing
    Marketing

    Marketing is defined by the American Marketing Association as the activity, set of institutions, and processes for creating, communicating, delivering, and exchanging offerings that have value for customers, clients, partners, and society at large....
    , MDS is a statistical technique for taking the preferences and perceptions of respondents and representing them on a visual grid, called perceptual maps
    Perceptual mapping

    Perceptual mapping is a graphics technique used by asset marketing that attempts to visually display the perceptions of customers or potential customers....
    .

    Comparison and advantages

    Potential customers are asked to compare pairs of product
    Product (business)

    The noun product is defined as a "thing produced by labor or effort" or the "result of an act or a process", and stems from the verb produce from the Latin produce, lead or bring forth....
    s and make judgements about their similarity. Whereas other techniques (such as factor analysis
    Factor analysis

    Factor analysis is a statistics method used to describe variance among observed variables in terms of fewer unobserved variables called factors....
    , discriminant analysis, and conjoint analysis
    Conjoint analysis (in marketing)

    Conjoint analysis is a statistical technique used in market research to determine how people value different features that make up an individual product or service....
    ) obtain underlying dimensions from responses to product attributes identified by the researcher, MDS obtains the underlying dimensions from respondents’ judgements about the similarity of products. This is an important advantage. It does not depend on researchers’ judgments. It does not require a list of attributes to be shown to the respondents. The underlying dimensions come from respondents’ judgments about pairs of products. Because of these advantages, MDS is the most common technique used in perceptual mapping.

    See also

    • positioning (marketing)
      Positioning (marketing)

      In marketing, positioning has come to mean the process by which marketers try to create an image or identity in the minds of their target market for its product , brand, or organization....
    • locating
      Locating

      Locating is a process used to determine the location of one position relative to other defined positions. Locating is applied for location metering of moving objects and is not to be confused with surveying, which is utilized in reference to stationary terrestrial objects....
    • perceptual mapping
      Perceptual mapping

      Perceptual mapping is a graphics technique used by asset marketing that attempts to visually display the perceptions of customers or potential customers....
    • product management
      Product management

      Product management is an organizational lifecycle function within a company dealing with the planning or marketing of a product or products at all stages of the product lifecycle....
    • marketing
      Marketing

      Marketing is defined by the American Marketing Association as the activity, set of institutions, and processes for creating, communicating, delivering, and exchanging offerings that have value for customers, clients, partners, and society at large....
    • Generalized multidimensional scaling (GMDS)
      Generalized multidimensional scaling (GMDS)

      Generalized multidimensional scaling is an extension of metric multidimensional scaling, in which the target space is non-Euclidean. In case when 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....
    • Data clustering
      Data clustering

      Clustering is the assignment of objects into groups so that objects from the same cluster are more similar to each other than objects from different clusters....
    • Factor analysis
      Factor analysis

      Factor analysis is a statistics method used to describe variance among observed variables in terms of fewer unobserved variables called factors....
    • Discriminant analysis


    Bibliography

    • Bronstein, A. M, Bronstein, M.M, and Kimmel, R. (2006), Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching, Proc. National Academy of Sciences (PNAS), Vol. 103/5, pp. 1168-1172.
    • 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. (1975) Marketing applications of MDS: Assessment and outlook, Journal of Marketing, vol 39, January 1975, pp 24-31.
    • Kruskal, J. B.
      Joseph Kruskal

      Joseph Bernard Kruskal, Jr. is an United States mathematician, statistician, and psychometrician. He was a student at the University of Chicago and at Princeton University, where he completed his Doctor of Philosophy in 1954, nominally under Albert W....
      , 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.
    • Torgerson, W. S. (1958). Theory & Methods of Scaling. New York: Wiley.


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