Showing papers on "Euclidean distance published in 1974"
Book•
01 Jan 1974
TL;DR: Inverse and Implicit Function Theorems and Related Topics Fubini's Theorem and the Change of Variables Formula Fourier Analysis Answers to Selected Exercises Index as mentioned in this paper.
Abstract: Introduction: Sets and Functions The Real Line and Euclidean Space Topology of Euclidean Space Compact and Connected Sets Continuous Mappings Uniform Convergence Differentiable Mappings The Inverse and Implicit Function Theorems and Related Topics Fubini's Theorem and the Change of Variables Formula Fourier Analysis Answers to Selected Exercises Index
427 citations
TL;DR: In this paper, the authors discuss several methods used in the clustering of objects which can be represented as points in Euclidean space, and only those procedures are considered that lead to partition of a sample into a preestablished number of groups, although some ideas are also valid for hierarchical schemes.
Abstract: In this work we discuss several methods used in the clustering of obj ects which can be represented as points in Euclidean space. Moreover, only those procedures are considered that lead to partition of a sample into a preestablished number of groups, although some ideas are also valid for hierarchical schemes (Lance and Williams [1966]). There are two major approaches to this problem. In the first one (which could be called "metric") a measure of distance is defined between each point x of the sample and each subset G thereof (an (i, k)-measure in the terminology of Lance and Williams [1968]). An initial partition is generated (for instance at random, or based on some external considerations) and then the points are successively reassigned to the "nearest" group, until no more reallocations are possible (or desired). In the second approach, each partition is assigned a numerical value, usually measuring the reduction of uncertainty due to grouping, and a search is made for partitions that optimize this functional (Friedman and Rubin [1967], Rubin [1967]). The simplest and most commonly used metric is the Euclidean distance between x and the mean of G (Ball and Hall [1967], Forgy [1965], MacQueen [1967]). In spite of its practical advantages, it has the drawback that the results it produces are not invariant under nonsingular linear transformations of the data (e.g., changes in the measurement units). One could avoid this drawback if he knew the within groups covariance matrix W corresponding to the "correct" partition he was looking for. Then the most natural metric would be Mahalanobis distance based on W. A possible solution to this dilemma is at each step to use the Mahalanobis metric induced by the matrix W corresponding to the current partition. A second conceivable improvement is obtained when one tries to overcome the fact that in the two former cases the metric is the same for all groups. If he suspects that the "true" clusters may have very different covariance structures, it would be reasonable to let each cluster have its own metric, based on its current covariance matrix. This was advocated by Chernoff [1970] and Rohlf [1970].
58 citations
IBM1
TL;DR: In this article, an algorithm for finding the point of smallest Euclidean norm in the convex hull of a given finite point set in a convex space is presented, with particular attention paid to the description of the procedure in geometric terms.
Abstract: An algorithm is developed for the problem of finding the point of smallest Euclidean norm in the convex hull of a given finite point set in a Euclidean space, with particular attention paid to the description of the procedure in geometric terms.
45 citations
TL;DR: In this article, algebraic criteria for complete Euclidean space controllability and for complete and null controllable control systems with delay were obtained for a special class of differential-difference equations.
Abstract: Algebraic criteria for complete Euclidean space controllability and for Euclidean space null controllability are obtained for control systems with delay. This is achieved by the use of a new form for the fundamental solution of differential-difference equations. In addition, it is shown that the new results reduce to known results for special classes of differential-difference equations.
41 citations
TL;DR: For the eigenvalues λi of an n × n matrix A the inequality ∑ i |λ i | 2 (A 4 − 1 2 ‖D 2 ) 1 2 is proved, where D ≔ AA ∗ − A ∗ A and ‖ ● ‖; denotes the euclidean norm.
27 citations
TL;DR: In this paper, the authors obtained an upper bound for the number of spherical segments of angular radius α that lie without overlapping on the surface of an n-dimensional sphere, and a lower bound on the density of filling ndimensional euclidean space with equal spheres.
Abstract: In this article we obtain an upper bound for the number of spherical segments of angular radius α that lie without overlapping on the surface of an n-dimensional sphere, and an upper bound for the density of filling n-dimensional euclidean space with equal spheres. In these bounds, the constant in the exponent of n is less than the corresponding constant in previously known bounds. Bibliography: 8 items.
21 citations
TL;DR: In this article, the notions of flatness and dimension for metric spaces were introduced and sufficient conditions for embedding a given metric space in Euclidean space were given, and it was shown that a metric space can be embedded in Euclidian n-space if and only if the metric space is flat and of dimension less than or equal to n.
Abstract: In this paper, we give necessary and sufficient conditions for embedding a given metric space in Euclidean space. We shall introduce the notions of flatness and dimension for metric spaces and prove that a metric space can be embedded in Euclidean n-space if and only if the metric space is flat and of dimension less than or equal to n.
21 citations
17 citations
13 citations
7 citations
TL;DR: In this article, it was shown that a linear transformation on the complex matrices that preserves the Euclidean norm of the rank r matrices must be a unitary transformation.
Abstract: Let $1\leqq r\leqq \,\min \{ s,t \}$. It is shown that a linear transformation on the $s \times t$ complex matrices that preserves the Euclidean norm of the rank r matrices must be a unitary transformation. This was conjectured by M. Marcus for $r = 1$ and $s = t$. An analogous result is shown to hold for real matrices and orthogonal transformations if $r > 1$, and an example is given of a nonorthogonal linear transformation on the real $s \times t$ matrices that preserves the Euclidean norm of the rank 1 matrices.
01 Jun 1974
TL;DR: The analysis is used to predict classifier performance on both artificially generated terrain with Gauss-Markov statistics and on real-world terrain, and gives performance as a function of signal/noise ratio and as afunction of terrain roughness.
Abstract: : A theoretical analysis of the performance of a nearest-prototype classifier (nearest in the sense of Euclidean distance) is presented as an approximate analysis of the performance of the Terrain Contour Matching Navigation System (TERCOM) The analysis is used to predict classifier performance on both artificially generated terrain with Gauss-Markov statistics and on real-world terrain Over 150 simulation runs were performed over such terrain having added Gauss-Markov noise (independent of the terrain process) with 6 different noise levels Results of the simulation are presented, giving performance as a function of signal/noise ratio and as a function of terrain roughness Measured miss-distances are also presented The simulation also includes the performance of a nearest-prototype classifier based on mean absolute distance (the first Minkowski metric), which is the actual metric employed in TERCOM (Modified author abstract)
TL;DR: A model has recently been proposed for the (sites x species x times) case in marine ecology; it can be regarded as based either on Euclidean distance or on the analysis of variance.
Abstract: The analysis of three-dimensional data sets has received considerable attention in ecology, but relatively little in agriculture, in which such data sets are equally common. A model has recently been proposed for the (sites x species x times) case in marine ecology; it can be regarded as based either on Euclidean distance or on the analysis of variance. The model is recapitulated in outline, its properties are somewhat extended, and its application to agronomic experiments is discussed. A brief account is given of the use of the model in the analysis of two real-life agronomic experiments. Finally, the relationship of such methods to those of classical statistics is briefly discussed.
TL;DR: The user is provided with the choice of four nonarbitrary initial configurations (two are metric; two are nonmetric) from which to commence the iterative process of APLIMDA, which is in most respects identical to the SSA-I algorithm.
Abstract: where dij is the rank linage (Guttman, 1968) of the Euclidean distance, dij . Commencing with the initial configuration of the user's choice as the first estimate of the matrix of coordinates, X, the iterative sequences indicated above are utilized. There is no guarantee that the algorithm always avoids local minima traps; however, as one step toward alleviating the problem, we provide the user with the choice of four nonarbitrary initial configurations (two are metric; two are nonmetric) from which to commence the iterative process. By utilizing the Guttman-Lingoes initial configuration, APLIMDA is in most respects identical to the SSA-I algorithm. The only significant difference is that we have set the number of inner iterations in the 1970; Lingoes & Roskam, 1973) suggest are optimal in terms of speed of convergence and robustness with respect to local minima, as well as further options that a researcher may have some felt need to consider. Commencing with the input of a correlation matrix, R, the initial configuration can be based on the principal axes decomposition of Torgerson's (1958, p.256) B* matrix of scalar products formed over the numerical values of R. Optionally, the user may select the principal axes of Torgerson's B* matrix formed on the rank order of the elements of R; the Guttman-Lingoes initial configuration (Lingoes & Roskam, 1973); or alternatively, the principal components of R. From the initial configuration, Guttman-Lingoes's double-phase iterations (Guttman, 1968) based on rank-images guide the configuration of points to a position where the loss function is, in most circumstances, close to the overall or global minimum. After the termination of the double-phase procedure, single-phase iterations also based on rank images tend to bring about convergence at the global minimum. The pivotal equation (cast into matrix notation) of the APLIMDA iterative process is based on the Guttman-Lingoes SSA-I algorithm (Guttman, 1968): and JAMES E. DANNEMILLER Survey Research Office An APL/360 program for interactive monotone distance analysis*