Multidimensional Scaling Problem Research Articles

Inverse correspondence analysis

In correspondence analysis (CA), rows and columns of a data matrix are depicted as points in low-dimensional space. The row and column profiles are approximated by minimizing the so-called weighted chi-squared distance between the original profiles and their approximations, see for example, [Theory and applications of correspondence analysis, Academic Press, New York, 1984]. In this paper, we will study the inverse CA problem, that is, the possibilities for retrieving one or more data matrices from a low-dimensional CA solution. We will show that there exists a nonempty closed and bounded polyhedron of such matrices. We also present two algorithms to find the vertices of the polyhedron: an exact algorithm that finds all vertices and a heuristic approach for larger sized problems that will find some of the vertices. A proof that the maximum of the Pearson chi-squared statistic is attained at one of the vertices is given. In addition, it is discussed how extra equality constraints on some elements of the data matrix can be imposed on the inverse CA problem. As a special case, we present a method for imposing integer restrictions on the data matrix as well. The approach to inverse CA followed here is similar to the one employed by De Leeuw and Groenen [J. Classification 14 (1997) 3] in their inverse multidimensional scaling problem.

Methods for binary multidimensional scaling.

Multidimensional scaling (MDS) is the process of transforming a set of points in a high-dimensional space to a lower-dimensional one while preserving the relative distances between pairs of points. Although effective methods have been developed for solving a variety of MDS problems, they mainly depend on the vectors in the lower-dimensional space having real-valued components. For some applications, the training of neural networks in particular, it is preferable or necessary to obtain vectors in a discrete, binary space. Unfortunately, MDS into a low-dimensional discrete space appears to be a significantly harder problem than MDS into a continuous space. This article introduces and analyzes several methods for performing approximately optimized binary MDS.

A New Formulation of the Nonmetric Strain Problem in Multidimensional Scaling

A natural extension of classical metric multidimensional scaling is proposed. The result is a new formulation of nonmetric multidimensional scaling in which the strain criterion is minimized subject to order constraints on the disparity variables. Innovative features of the new formulation include: the parametrization of the p-dimensional distance matrices by the positive semidefinite matrices of rank ≤p; optimization of the (squared) disparity variables, rather than the configuration coordinate variables; and a new nondegeneracy constraint, which restricts the set of (squared) disparities rather than the set of distances. Solutions are obtained using an easily implemented gradient projection method for numerical optimization. The method is applied to two published data sets.

A Nonlinear Programming Approach to Metric Unidimensional Scaling

Classical unidimensional scaling provides a difficult combinatorial task. A procedure formulated as a nonlinear programming (NLP) model is proposed to solve this problem. The new method can be implemented with standard mathematical programming software. Unlike the traditional procedures that minimize either the sum of squared error (L 2 norm) or the sum pf absolute error (L 1 norm), the proposed method can minimize the error based on any L p norm for 1 ≤p < ∞. Extensions of the NLP formulation to address a multidimensional scaling problem under the city-block model are also discussed.

Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional Scaling

We solve the following variational problem: Find the maximum of $E\|X - Y\|$ subject to $E\|X\|^2 \leq 1$, where $X$ and $Y$ are i.i.d. random $n$-vectors, and $\|\cdot\|$ is the usual Euclidean norm on $\mathbb{R}^n$. This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal $X$ is unique and is (1) uniform on the surface of the unit sphere, for dimensions $n \geq 3$, (2) circularly symmetric with a scaled version of the radial density $\rho/(1 - \rho^2)^{1/2}, 0 \leq \rho \leq 1$, for $n = 2$, and (3) uniform on an interval centered at the origin, for $n = 1$ (Plackett's theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real) $n p$ or $p > 2$ in general). Although this is an easy consequence of known results, it appears to be new in a strict sense. In the radial problem, the average distance $D(r_1, r_2)$ between two spheres of radii $r_1$ and $r_2$ is used as a kernel. We derive properties of $D(r_1, r_2)$, including nonnegative definiteness on signed measures of zero integral.

An optimization problem on subsets of the symmetric positive-semidefinite matrices

Motivated by the metricsstress problem in multidimensional scaling, the authors consider the more general problem of minimizing a strictly convex function on a particular subset ofR n × n . The subset in question is the intersection of a linear subspace with the symmetric positive-semidefinite matrices of rank ≤p. Because of the rank restriction, this subset is not convex. Several equivalent formulations of this problem are derived, and the advantages and disadvantages of each formulation are discussed.

尺度法とその関連領域:多次元尺度法の研究に関する最近の動向 (2)

This review covers metric and nonmetric multidimensional scaling(MDS)studies. Topics reviewed include metric vs. nonmetric methods, similarities, distances and the Minkowski power metrics, distance vs. content methods, the additive constant problem in the2-way and the3-way MDS, computational problems(gradient vs. rank image methods, the alternating least squares method, the N-way Eckart-Young decomposition, maximum likelihood, convex function analysis, continuous transformation, lincar programming), computer programs, individual difference scaling of similarity and preference data(the3-way scaling and factor analysis, the unfolding vs. vector models), Monte Carlo studies, incmplete designs, interactive scaling, integrated approaches, books on MDS, and a quantitative look at main contributors to the MDS literature. Because of the limited space, applications of MDS are not reviewed, and an extensive bibliography on MDS had to be deleted from(2).

A New Algorithm of MDA-OR

Dr. Hayashi developed one of methods for multidimensional scaling problem, named MDA. This method is a very useful one, but its computation is very difficult. In the present paper, the author shows a modification of its computation with some numerical examples in one dimensional case.

The intrinsic dimensionality of signal collections

In view of the trend toward the representation of signals as physical observables characterized by vectors in an abstract signal space, rather than as time or frequency functions, it is desirable that dimensionality be defined in a manner independent of the choice of basis on which the vectors are projected. The intrinsic dimensionality of a collection of signals is defined to be equal to the number of free parameters required in a hypothetical signal generator capable of producing a close approximation to each signal in the collection. Thus defined, the dimensionality becomes a relationship between the vectors representing the signals. This relationship need not be a linear one and does not depend on the basis onto which the vectors are projected in the signal-measuring process. A digital computer program for estimating this dimensionality from the signal coefficients on an arbitrary basis has been developed. The program makes use of some results obtained from a multi-dimensional scaling problem in experimental psychology and utilizes an inverse relationship between the variance in interpoint distances within a hypersphere and the dimensionality of the hypersphere. Using this method, the results are believed to be independent of the choice of orthogonal basis, and no prior knowledge of the analytical form of the signals is assumed. The validity of the program is tested and verified by using it to estimate the dimensionality of signals of known structure.

THE MULTIDIMENSIONAL ANALYSIS OF ‘ELASTIC’ DISTANCES

A new orientation to the problem of multidimensional scaling is presented using the notion of ‘elastic’ distances. Making use of a rational theory of judgement variability (in line with Thurstone's remarks about discriminal processes); a monotonic distance function (in line with Shepard's suggestions); an algorithm for determining the monotonic transformations at each iteration (as provided by Kruskal); and a criterion which yields information about the tolerance for variability of judgement that any solution demands, a hybrid analytical approach is proposed as a practical multidimensional scaling procedure. The approach is illustrated using experimental data.

The additive constant problem in multidimensional scaling

The problem of choosing the correct additive constant to convert relative interstimulus distances to absolute interstimulus distances in multidimensional scaling is investigated. An artificial numerical example is constructed, and various trial values of the constant are inserted to demonstrate the effect on the multidimensional map of making a variety of incorrect choices. Finally, a general solution to the problem, suggested by Dr. Ledyard R Tucker, is presented; each of the computational steps in this solution is set down for easy reference.