MULTI-DIMENSIONAL SCALING

Multi-dimensional Scaling



Multi-dimensional Scaling

Introduction

Multidimensional scaling - is not just a particular procedure, but rather the way the most efficient placement of objects roughly preserves the observed distance between them. In other words, MNSH allocates objects in the space of a given dimension and examines how well the resulting configuration preserves the distance between objects. In more technical terms, MNSH uses an algorithm to minimize a function that evaluates the quality of the display options.

Discussion

Multidimensional scaling (MNSH) can be considered as an alternative to factor analysis. The purpose of the latter, speaking, is the search for an interpretation of "latent (i.e. not directly observable) variables", which allow the user to explain the similarities between objects, given points in the original feature space (Arabie, 2007, p. 34). For clarity and brevity, then, as a rule, we only talk about the similarities of objects, meaning that, in practice, it may be a difference, distance, or degree of connection between them. In the factor analysis, of similarity between objects (e.g., variables) are expressed by the matrix (table) correlation coefficients. The method additionally MNSH to the correlation matrix as input you can use any type of similarity matrix objects. Thus, the input of all the algorithms used MNSH matrix, whose element at the intersection of the i-th row and j-th column contains information on the pair wise similarity of the analyzed objects (object [i] and object [j]). At the output, of the algorithm MNSH obtained numerical values of the coordinates, which are assigned to each object in a coordinate system (in the "subscales" associated with the latent variables, hence the name MNSH), and the dimension of the new feature space significantly less than that of the original (for a proper and there is a struggle).

MNSH logic is illustrated by the following basic example. Suppose we have a matrix of pairwise distances (i.e., the similarity of some features) between major U.S. cities. Analyzing the matrix tends to place a point at cities in the two-dimensional space (the plane), to preserve the real distance between them (Torgerson, 2002, p. 17). The resulting distribution of points on the plane can then be used as an approximate geographical map of the United States.

In general, the method allows MNSH to arrange "objects" (the city in our example) in a compact dimension (in this case it is equal to two) in order to reproduce the observed distance adequately between them. As a result, you can "measure" these distances in terms of latent variables found. Thus, in this example can be explained by the distance in terms of a pair of geographic coordinates North / South and East / West.

As in factor analysis, the orientation of the axes can be chosen arbitrarily. Returning to our example, you can rotate the map in an arbitrary way the United States, but the distance between cities does not change (Shepard, 2002, p. 67). Thus, the final orientation of the axes in the plane or space is largely the result of a substantive decision in ...