The Choice of Initial Configurations in Multidimensional Scaling: Local Minima, Fit, and Interpretability

Abstract

Multidimensional scaling (MDS) algorithms can easily end up in local minima, depending on the starting configuration. This is particularly true for 2-dimensional ordinal MDS. A simulation study shows that there can be many local minima that all have an excellent model fit (i.e., small Stress) even if they do not recover a known latent configuration very well, and even if they differ substantially among each other. MDS programs give the user only one supposedly Stress-optimal solution. We here present a procedure for analyzing all MDS solutions resulting from using a variety of different starting configurations. The solutions are compared in terms of fit and configurational similarity. This allows the MDS user to identify different types of solutions with acceptable Stress, if they exist, and then pick the one that is best interpretable.

Brain parcellation selection: An overlooked decision point with meaningful effects on individual differences in resting-state functional connectivity

“…Note that to solve for an MDS solution, an initial starting configuration is required that is then optimized. We elected to use a Torgerson configuration (classical MDS) as the initial starting configuration for the unconstrained MDS (instead of a random start), as this initial start generally leads to among the lowest stress value and further produces non-stochastic MDS solutions ( Borg and Mair, 2017 ; Mair et al, 2016 ). We then fit a mixed-effects model to examine whether stress differences (dependent variable) varied significantly across parcellations (independent variable), with observations nested within subject (ie.…”

Is reanalysis selective when regressions are consciously controlled?

“…The matrix of pairwise scanpath similarities produced by Scasim was transformed into a two-dimensional Euclidean map using nonmetric multidimensional scaling as implemented in the vegan package (Oksanen et al, 2020). Given that multidimensional scaling algorithms are nondetermistic and may converge to different local stress optima, 6 we repeated the procedure several times and chose the configuration with the lowest stress and the best interpretability (Borg & Mair, 2017). Mixture-based clustering was applied to the result using the mclust package (Scrucca, Fop, Murphy, & Raftery, 2016), and the number of clusters was reduced using the entropy criterion of Baudry, Raftery, Celeux, Lo, and Gottardo (2010).…”

Ecological and conceptual consequences of Arctic pollution

“…Parameterisation of the model aims to find the best fit between the simulated TRW, normalised with respect to their 1901–1942 values, and a target TRW chronology, the average of the above‐mentioned L. sibirica , P. obovata and L. gmelinii chronologies, by adjusting the values for the 12 site parameters in the VS‐Lite model that tune the model to local conditions (Tychkov et al ., 2019). The solution to tuning the model by direct mathematical optimisation of the multidimensional parameter space is problematic due to a high probability of reaching the local optimum that generates an artificial decision (Etschberger and Hilbert, 2003; Borg and Mair, 2017). The value for any one parameter values should not conflict with the biological constraints on growth and/or site conditions observed in the field data.…”