SMACOF

SMACOF is an iterative optimization algorithm used in multidimensional scaling (MDS) to produce a configuration of points in a low-dimensional Euclidean space that best represents a given set of pairwise dissimilarities among objects. The method seeks to minimize a stress function that measures the discrepancy between the observed proximities and the distances in the configuration, often Kruskal’s stress-1 or a related weighted stress.

The algorithm relies on majorization, a principle from optimization. At each iteration SMACOF constructs a simple surrogate function that upper bounds the current stress and is easier to minimize. By minimizing this surrogate, SMACOF produces a new configuration that guarantees a nonincreasing stress. Repeating the process yields convergence to a local minimum of the original stress function. The coordinates are updated in a closed-form step, which contributes to the method’s stability and efficiency.

SMACOF is flexible and robust. It supports metric and nonmetric MDS, allows weights for observations, and can handle missing data. It is compatible with different distance metrics, although Euclidean distances are standard in many applications. Its monotone improvement property and straightforward implementation have contributed to its popularity in applied settings.

Applications of SMACOF span various fields, including social sciences, psychology, biology, and information visualization, wherever a faithful low-dimensional visualization of dissimilarities is useful. Software implementations are widely available, notably in the R package smacof, as well as in MATLAB and Python ecosystems.

Historically, SMACOF is associated with the majorization-based optimization approach for MDS developed in the 1990s and discussed in the literature and comprehensive volumes on modern multidimensional scaling. The method’s key appeal is monotone stress reduction and practical performance on moderate to large datasets.