Scaling and seriation of objects have a long history in psychology, anthropology, and statistics. Data in the form of asymmetric proximity matrices occur when we have observations from paired comparisons, round-robin tournaments, and networks. Skew-symmetric matrices are obtained by taking the differences between complementary pairs of objects from asymmetric matrices. Reciprocal matrices are computed from asymmetric matrices by taking the ratios of complementary elements. The general dynamic programming paradigm (GDPP) has been shown to be a useful method for obtaining an optimal seriation using various criteria while reducing computing costs. However, matrices larger than 25 x 25 are still difficult to handle in an optimal way. Unidimensional scaling methods have the ability to attain precise coordinate estimates of the objects. The results of this study show that coordinate estimation methods allow for global optimization of larger matrices than seriation, and can be more accurate. Coordinate estimation may be preferred to seriation especially when robust methods such as L1 are used. Coordinates derived from skew-symmetric matrices generally outperform those obtained from reciprocal matrices. The performance of each method is illustrated with several simulations, and two classic data sets are analyzed using the three best scaling methods. These methods are implemented with an easy-to-use Matlab toolbox created especially for these kinds of data. The Asymmetric Proximity Toolbox takes advantage of modern computing advances (e.g., interior point methods for linear programming) to make the use of L1 optimization practical for data analysis and simulation studies.
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A Comparison of Methods for Seriation and Unidimensional Scaling of Asymmetric Proximity Data
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