MDS algorithms fall into a
taxonomy, depending on the meaning of the input matrix:
Classical multidimensional scaling It is also known as
Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a
loss function called
strain, :# Set up the squared proximity matrix D^{(2)}=[d_{ij}^2] :# Apply double centering: B=-\frac{1}{2}CD^{(2)}C using the
centering matrix C=I-\frac{1}{n}J_n, where n is the number of objects, I is the n \times n identity matrix, and J_{n} is an n\times n matrix of all ones. :# Determine the m largest
eigenvalues \lambda_1,\lambda_2,...,\lambda_m and corresponding
eigenvectors e_1,e_2,...,e_m of B (where m is the number of dimensions desired for the output). :# Now, X=E_m\Lambda_m^{1/2} , where E_m is the matrix of m eigenvectors and \Lambda_m is the
diagonal matrix of m eigenvalues of B. :Classical MDS assumes metric distances. So this is not applicable for direct dissimilarity ratings.
Metric multidimensional scaling (mMDS) It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called
stress, which is often minimized using a procedure called
stress majorization. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares:\text{Stress}_D(x_1,x_2,...,x_n)=\sqrt{\sum_{i\ne j=1,...,n}\bigl(d_{ij}-\|x_i-x_j\|\bigr)^2}. Metric scaling uses a power transformation with a user-controlled exponent p: d_{ij}^p and -d_{ij}^{2p} for distance. In classical scaling p=1. Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities.
Non-metric multidimensional scaling (NMDS) In contrast to metric MDS, non-metric MDS finds both a
non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. Let d_{ij} be the dissimilarity between points i, j. Let \hat d_{ij} = \| x_i - x_j\| be the Euclidean distance between embedded points x_i, x_j. Now, for each choice of the embedded points x_i and is a monotonically increasing function f, define the "stress" function: :S(x_1, ..., x_n; f)=\sqrt{\frac{\sum_{i The factor of \sum_{i in the denominator is necessary to prevent a "collapse". Suppose we define instead S=\sqrt{\sum_{i, then it can be trivially minimized by setting f = 0, then collapse every point to the same point. A few variants of this cost function exist. MDS programs automatically minimize stress in order to obtain the MDS solution. The core of a non-metric MDS algorithm is a twofold optimization process. First the optimal monotonic transformation of the proximities has to be found. Secondly, the points of a configuration have to be optimally arranged, so that their distances match the scaled proximities as closely as possible. NMDS needs to optimize two objectives simultaneously. This is usually done iteratively: :# Initialize x_i randomly, e. g. by sampling from a normal distribution. :# Do until a stopping criterion (for example, S ) :## Solve for f = \arg\min_f S(x_1, ..., x_n ; f) by
isotonic regression. :## Solve for x_1, ..., x_n = \arg\min_{x_1, ..., x_n} S(x_1, ..., x_n ; f) by gradient descent or other methods. :#Return x_i and f
Louis Guttman's smallest space analysis (SSA) is an example of a non-metric MDS procedure.
Generalized multidimensional scaling (GMD) An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another.
Super multidimensional scaling (SMDS) An extension of MDS, known as Super MDS, incorporates both distance and angle information for improved source localization. Unlike traditional MDS, which uses only distance measurements, Super MDS processes both distance and angle-of-arrival (AOA) data algebraically (without iteration) to achieve better accuracy. The method proceeds in the following steps: •
Construct the Reduced Edge Gram Kernel: For a network of N sources in an \eta-dimensional space, define the edge vectors as v_{i} = x_{m} - x_{n}. The dissimilarity is given by k_{i,j} = \langle v_i, v_j \rangle. Assemble these into the full kernel K = VV^T, and then form the reduced kernel using the N-1 independent vectors: \bar{K} = [V]_{(N-1)\times\eta}\ [V]_{(N-1)\times\eta}^T, •
Eigen-Decomposition: Compute the eigen-decomposition of \bar{K}, •
Estimate Edge Vectors: Recover the edge vectors as \hat{V} = \Bigl( U_{M \times \eta}\, \Lambda^{\odot \frac{1}{2}}_{\eta \times \eta} \Bigr)^T , •
Procrustes Alignment: Retrieve \hat{V} from V via Procrustes Transformation, •
Compute Coordinates: Solve the following linear equations to compute the coordinate estimates \begin{pmatrix} 1 \vline \mathbf{0}_{1 \times N-1} \\ \hline \mathbf{[C]}_{N-1 \times N} \end{pmatrix} \cdot \begin{pmatrix}\mathbf{x}_{1} \\ \hline[\mathbf{X}]_{N-1 \times \eta} \end{pmatrix}=\begin{pmatrix} \mathbf{x}_{1} \\ \hline[\mathbf{V}]_{N-1 \times \eta} \end{pmatrix}, This concise approach reduces the need for multiple anchors and enhances localization precision by leveraging angle constraints. ==Details==