Multidimensional scaling (MDS) is a commonly used visualization technique to approximate the (dis)similarities between multidimensional estimates in terms of euclidean distances in much less dimensions. For instance, MDS can be used to visualize the spatial similarities between spatially distributed brain activity patterns originating from several fMRI observations in a given anatomical space, such as, e. g., MVPA trial, ICA component or GLM contrast maps, either in the volume space (as volume maps) or in the cortical space (as surface maps). Thereby, in this MDS applications, the initial dimensions correspond to the voxels or vertices of the selected anatomical space. To project fMRI spatial patterns as points on a 2-dimensional space, a linear method like the metric MDS (MMDS) (Torgerson, 1952) can be used. Alternatively, two nonlinear methods called Sammon's projection (Sammon, 1969) and Curvilinear Component Analysis (CCA) (Demartines and Herault, 1997) have been also proposed with some useful features.
Given a set of K spatial maps, a natural measure of similarity between pairs of maps is the absolute value of their spatial correlation coefficients: rij where i,j=1, 2, ... K. Correspondingly, the dissimilarities (distances) can be computed as: dij = {1 - [abs(rij)]p}q , where p=q=1 (linear case) (Everitt, 1993) or p=1,2, 3, ... and q=0.5 (square root) to increase the spread of the distance distribution (and therefore the plotted points) via a non-linear transformation of the distances. For MDS plots, a set of points is obtained from the distance matrix D={dij} following the fundamental criterion that the Euclidean distances of the points in the plot {dij*} are to be as similar as possible to the original distances in the matrix D. In other words, MDS methods look for an optimal configuration of points in a 2-D space by minimizing the mismatch between the distances between the points in the plot and the original distances in the matrix.
In the classical linear metric MDS (MMDS) (Torgeson, 1952), the sum of squared errors of the distances in the original and projected space is to be minimized:
This is achieved by first "double centering" the squared distance matrix (i. e. subtracting the row and column means of the distance matrix from its elements and adding the grand mean multiplied by -1/2) and then performing a singular value decomposition (SVD) of the obtained matrix:
In this formulation, Ik and 1k respectively represent the k-by-k identity matrix and a k-by-k matrix of 1s. After the matrix calculations, the first two columns of matrix X can be used as the two coordinates of a 2-dimensional MMDS projection of all patterns.
Non-linear MDS methods aim at more faithfully preserving the distances in the output space by introducing a non-linear function in the cost function and minimizing it with an iterative approach. This is the case of Sammon projection and CCA. For instance, in CCA, the non-linearity is an inverse exponential function of the difference between the original and projected distances, thereby the distances that are short in the projection will have more weight in the cost function in each iteration. This feature is expected to favor the local topology of points in the projection, making CCA more strictly related to self-organizing map (SOM) theory (Kohonen, 1989). Nonetheless, CCA differs from SOM in that it does not quantize the data and it is not confined to a regular grid in the output space.
Torgerson, W., 1952. Multidimensional scaling: I. Theory and methods. Psychometrica 17, 401-419.
Sammon, J.W., 1969. A nonlinear mapping for data structure analysis. IEEE Trans. Comput., C 18 (5), 401-409.
Kohonen, T., 1989. Self-Organization and Associative Memory, third ed. Springer-Verlag, Berlin.
Mead, A., 1992. Review of the development of multidimensional scaling methods. Statistician 41, 27-39.
Everitt, B., 1993. Cluster Analysis, third ed. Edward Arnold, London.
Demartines, P., Herault, J., 1997. Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of data sets. IEEE Trans. Neural Netw. 8 (1), 148-154.