Inverse MDS: Inferring Dissimilarity Structure from Multiple Item Arrangements

Abstract

The pairwise dissimilarities of a set of items can be intuitively visualized by a 2D arrangement of the items, in which the distances reflect the dissimilarities. Such an arrangement can be obtained by multidimensional scaling (MDS). We propose a method for the inverse process: inferring the pairwise dissimilarities from multiple 2D arrangements of items. Perceptual dissimilarities are classically measured using pairwise dissimilarity judgments. However, alternative methods including free sorting and 2D arrangements have previously been proposed. The present proposal is novel (a) in that the dissimilarity matrix is estimated by “inverse MDS” based on multiple arrangements of item subsets, and (b) in that the subsets are designed by an adaptive algorithm that aims to provide optimal evidence for the dissimilarity estimates. The subject arranges the items (represented as icons on a computer screen) by means of mouse drag-and-drop operations. The multi-arrangement method can be construed as a generalization of simpler methods: It reduces to pairwise dissimilarity judgments if each arrangement contains only two items, and to free sorting if the items are categorically arranged into discrete piles. Multi-arrangement combines the advantages of these methods. It is efficient (because the subject communicates many dissimilarity judgments with each mouse drag), psychologically attractive (because dissimilarities are judged in context), and can characterize continuous high-dimensional dissimilarity structures. We present two procedures for estimating the dissimilarity matrix: a simple weighted-aligned-average of the partial dissimilarity matrices and a computationally intensive algorithm, which estimates the dissimilarity matrix by iteratively minimizing the error of MDS-predictions of the subject’s arrangements. The Matlab code for interactive arrangement and dissimilarity estimation is available from the authors upon request.

Representational Similarity of Body Parts in Human Occipitotemporal Cortex

“…The physical shape similarity (shape 1) was computed using the shape context algorithm (Belongie et al, 2002). The perceived shape similarity (shape 2) and physical proximity were rated by an independent group of participants (n ϭ 6) using the multiple object arrangement method (Kriegeskorte and Mur, 2012). Each participant rated a different subset of eight images (one image for each body part condition), taken from the set of images used in the functional neuroimaging study.…”

Unique semantic space in the brain of each beholder predicts perceived similarity

“…All subjects were additionally presented with a standard set of 36 images that were not personally meaningful. Each subject performed similarity judgments on the full set of 72 images (32,33). The subject then was scanned while viewing the images (day 1).…”

“…On subsequent trials, the subject arranged subsets designed adaptively to estimate optimally the potentially high-dimensional dissimilarity structure (see ref. 33 for details). Eye regions are occluded above in the images to protect privacy, but were not occluded in the experiments.…”

Multidimensional Scaling

“…The data were obtained using a spatial arrangement method, originally proposed by Goldstone ( 42 ; see also 43,44 ). This technique involves presenting all stimuli to the participant at once, randomly arranged on a computer screen.…”