Multidimensional Scaling Multidimensional Scaling , Similarity Judgments , Proximity data , Geometric Models , Knowledge Representation , data visualization , dimension reduction , multivariate analysis
Multidimensional Scaling (MDS) describes a family of techniques for the analysis of proximity data on a set of stimuli to reveal the hidden structure underlying the data. The proximity data can come from similarity judgments, identification confusion matrices, grouping data, same-different errors or any other measure of pairwise similarity. The main assumption in MDS is that stimuli can be described by values along a set of dimensions that places these stimuli as points in a multidimensional space and that the similarity between stimuli is inversely related to the distances of the corresponding points in the multidimensional space. The Minkowski distance metric provides a general way to specify distance in a multidimensional space: r n k r jk ik ij x x d 1 1       − = ∑ = , where n is the number of dimensions, and x ik is the value of dimension k for stimulus i. With r =2, the metric equals the Euclidian distance metric while r=1 leads to the city-block metric. A Euclidian metric is appropriate when the stimuli are composed of integral or perceptually fused dimensions such as the dimensions of brightness and saturation for colours. The city-block metric is appropriate when the stimuli are composed of separable dimensions such as size and brightness (Attneave, 1950). In practice, the Euclidian distance metric is often used because of mathematical convenience in MDS procedures. MDS can be applied with different purposes. One is exploratory data analysis; by placing objects as points in a low dimensional space, the observed complexity in the original data matrix can often be reduced while preserving the essential information in the data. By a representation of the pattern of proximities in two or three dimensions, researchers can visually study the structure in the data.