M.W. TROSSET. Applications of multidimensional scaling to molecular conformation. Technical report, Rice Unversity, Houston, Texas, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....are surprisingly many applications for this problem, sometimes called the molecule problem. These applications include NMR data, determination of protein structure, surveying, satellite ranging, and molecular conformation; e.g. the survey [23] and the discussion in [50] and the related papers [49, 89, 114, 117, 44]. We now consider the approximate EDMCP and follow the approach in [1] where the reader will nd all the proofs and details omitted here. Let A be a pre distance matrix and let H be an n n symmetric matrix with nonnegative elements (weights) Consider the objective function f(D) kH (A ....

M.W. TROSSET. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29:148-152, 1998.

....the form of an observed dissimilarity matrix Delta 0 = ffi 0 ij ) The best known use of this information is made by nonmetric MDS, an approach to MDS that was motivated by psychometric considerations and which is often quite useful for nonlinear dimension reduction. As described by Trosset [42, 38], nonmetric MDS can be modified in a natural way that replaces its characteristic order constraints with the bound constraints that more typically arise when attempting to determine molecular structure. Let D n (p) denote the closed cone of Euclidean distance matrices, i.e. D = d ij ) 2 D n (p) ....

....has been developing an approach that employs the classical Strain error criterion, parametrized by squared interatomic distances, and a variable reduction approach to optimization. The latter approach is, of course, the foundation of the present proposal. Our presentation of each is adapted from [38]. 2.3.1 The Data Box Algorithm The Stress criterion for metric MDS is ae(D; Delta) kD Gamma Deltak 2 2 . Suppose that we adopt the Stress criterion and reparametrize Problem (1) by writing D = D(X) obtaining minimize oe(X; Delta) kD(X) Gamma Deltak 2 2 subject to Delta 2 [L; U ] ....

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M. W. Trosset. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29, 1997. To appear.

....algorithms for EDMCP appears in [41] The point is made that there is no definitive general algorithm for EDMCP, i.e. one cannot provide an efficient decision rule for the question of whether a completion exists or not. However, there are many algorithms that find approximate completions. In [41] [39], 40] the author presents results on finding EDM completions based on spectral decompositions. In particular, the computationally hard problem of fixing the rank (the embedding dimension) is discussed. Some work on finding the closest EDM to a given symmetric matrix appears in [12] 47] 2] ....

M.W. TROSSET. Applications of multidimensional scaling to molecular conformation. Technical report, Rice Unversity, Houston, Texas, 1997.

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Trosset, M. W. (1998a). Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29:148-152.

....to the so called additive constant problem by Saito [19] de Leeuw and Heiser [5] and Trosset, Baggerly and Pearl [29] subsequently suggested using the variable projection functional F p ffi . Trosset [27] proposed a formulation of nonmetric multdimensional scaling using F p ffi and Trosset [26, 28, 25] suggested variable reduction for distance geometry problems with bound constraints. 4 Optimization by Variable Alternation Henceforth, we restrict attention to B n = Omega n (K) for K a closed convex subset of (3) i.e. we study minimize f ( Delta; B) k ( Delta) Gamma Bk 2 F subject to ....

M. W. Trosset. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29, 1997. To appear.

....also investigated an example that was designed to evoke problems in which one attempts to infer molecular conformation from information about a molecule s interatomic distances. Our approach is similar to the the Data Box Algorithm of Glunt, Hayden, and Raydan [4] however, as discussed by Trosset [15, 12], we use a different objective function and a different optimization strategy. The following example is intentionally artificial, as molecular conformation problems have considerable structure that is beyond the scope of this report. Efforts to apply the methods of this report to such problems are ....

M. W. Trosset. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29, 1997. To appear.

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M.W. TROSSET. Applications of multidimensional scaling to molecular conformation. Technical report, Rice Unversity, Houston, Texas, 1997.

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M.W. Trosset. Applications of Multidimensional Scaling to Molecular Conformation. Computing Science and Statistics,

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M.W. TROSSET. Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29:148--152, 1998.

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Trosset, M. W. (1998a), \Applications of Multidimensional Scaling to Molecular Conformation, " Computing Science and Statistics, 29, pp. 148-152.

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