More on Multi-dimensional Scaling

Multi-dimensional scaling (MDS) was originally developed for use in the social science domain (for example, Kruskal & Hill (1964)). It is often based upon data where human subjects rate n objects for similarity. In the metric form of MDS, the closeness of each pair of objects is represented in numerical form. From these ratings a dissimilarity (or proximity) matrix is generated using the mean values of the similarity ratings. In the non-metric implementation the dissimilarity matrix is based on the rank-order of the inter- object dissimilarities. The aim is to uncover any structure or pattern that may be present in the dissimilarity data. The basis of analysis is the n dimensional spatial model where each point corresponds to one of the n objects. The indicator of dissimilarity for any two objects is the distance between the corresponding two points. A number of between point distance measures are used but the most common is the Euclidean distance. For n

dimensions, the distance between object i and object j is given by

dij =

n X k=1

(xik−xjk)2 (3.22)

where xi1, ..., xin and xj1, ..., xjn are elements of vectors xi and xj respectively (Everitt & Dunn 1993).

In the same way as principal component analysis seeks to reduce the dimensionality of data using the covariance matrix, the multidimensional scaling technique seeks a spacial model where the distances between points are represented as accurately as possible with as few dimensions as possible. The algorithm moves the ‘objects’ around in space for a given dimensionality and checks how well the distances between objects can be reproduced for a particular configuration of axes. More precisely, it uses a function minimization algorithm that uses the eigenvalues of the dissimilarity matrix to give the best configuration in terms of goodness of fit (Statsoft 2003). The process is sometimes called metric scaling or principal co-ordinate analysis (Krzanowski 1988). As with principal component analysis, multidimensional scaling can indicate structure in the data that is not observable from the raw dissimilarity data. The multidimensional scaling technique was adapted for use in analysis of musical sounds by Grey (1977) who found the technique very fruitful in his studies of musical timbre.

In general, with MDS the beginning point is a dissimilarity matrix where each ofn objects is rated on a similarity basis with each other object. Except in special cases (for example,

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the distances between cities on a map) the ratings cannot be exactly represented in n di- mensional space. An alternative approach is where each object is defined by n attributes or variables. In this case each object can be exactly represented by a point in n dimen- sional space. The dissimilarity between objects can be measured by the distance (usually Euclidean) between corresponding points. To reduce the dimensionality and to explore the structure of the data and expose meaningful relationships between the variables the technique of factor analysis or principal component analysis can be applied. Each of the axes, after the transformation, is called a principal component or a factor and corresponds to the most important features contained in the data.

As we described earlier, a similar approach was used in the 1990’s for analysis of musical timbre (Hourdin & Charbonneau (1997), Kaminskyj (1999)) but using physical data based on spectral analysis of each tone. In a plot of this data, each point in space represented a spectral snapshot of the sound at a particular time. The technique allowed the changes in musical sound quality to be plotted over time and the timbre of a musical tone to be represented as a sequence of points in n dimensional space - a multidimensional scaling ‘trajectory path’. By using the physical properties of a tone this approach provides an objective approach to measuring timbre in contrast to the subjectivity of data based on human judgements. To reduce the dimensionality of these n-dimensional plots and uncover the structure of the timbre principal component analysis or factor analysis was applied. The underlying premise of this approach is that the n-dimensional trajectory plot of the evolution of a tone over time represents the timbre of the tone. It follows that if the n- dimensional trajectory paths of two tones directly correspond, then the timbre of the tones is identical and visa versa. See figure 3.9 for a plot of the MDS trajectory paths using the first three PC’s of two guitar tones at the same pitch from the same instrument and for a plot of the trajectories of two guitar tones from different instruments. We can easily see that the pair of paths from the same instrument correspond more closely than the pair from different instruments. If we can find a way of measuring the closeness of the paths then this will provide a means for classifying tones based on information from the whole tone.

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Figure 3.9: MDS trajectory paths for two guitar tones at the same pitch from the same guitar and from different guitars - tracing their evolution over time.