Showing papers on "Metric (mathematics) published in 1975"

Fuzzy metrics and statistical metric spaces

Abstract: The adjective "fuzzy" seems to be a very popular and very frequent one in the contemporary studies concerning the logical and set-theoretical foundations of mathematics. The main reason of this quick development is, in our opinion, easy to be understood. The surrounding us world is full of uncertainty, the information we obtain from the environment, the notions we use and the data resulting from our observation or measurement are, in general, vague and incorrect. So every formal description of the real world or some of its aspects is, in every case, only an approxima­ tion and an idealization of the actual state. The notions like fuzzy sets, fuzzy orderings, fuzzy languages etc. enable to handle and to study the degree of uncertainty mentioned above in a purely mathematic and formal way. A very brief survey of the most interest­ ing results and applications concerning the notion of fuzzy set and the related ones can be found in [l]. The aim of this paper is to apply the concept of fuzziness to the clasical notions of metric and metric spaces and to compare the obtained notions with those resulting from some other, namely probabilistic statistical, generalizations of metric spaces. Our aim is to write this paper on a quite self-explanatory level the references being necessary only for the reader wanting to study these matters in more details.

Minimal Mutation Trees of Sequences

Abstract: Given a finite tree, some of whose vertices are identified with given finite sequences, we show how to construct sequences for all the remaining vertices simultaneously, so as to minimize the total edge-length of the tree. Edge-length is calculated by a metric whose biological significance is the mutational distance between two sequences.

Mechanical program analysis

Abstract: One means of analyzing program performance is by deriving closed-form expressions for their execution behavior. This paper discusses the mechanization of such analysis, and describes a system, Metric, which is able to analyze simple Lisp programs and produce, for example, closed-form expressions for their running time expressed in terms of size of input. This paper presents the reasons for mechanizing program analysis, describes the operation of Metric, explains its implementation, and discusses its limitations.

Optimally conditioned optimization algorithms without line searches

Abstract: New variable metric algorithms are presented with three distinguishing features: (1) They make no line searches and allow quite arbitrary step directions while maintaining quadratic termination and positive updates for the matrixH, whose inverse is the hessian matrix of second derivatives for a quadratic approximation to the objective function. (2) The updates fromH toH+ are optimally conditioned in the sense that they minimize the ratio of the largest to smallest eigenvalue ofH−1H+. (3) Instead of working with the matrixH directly, these algorithms represent it asJJT, and only store and update the Jacobian matrix J. A theoretical basis is laid for this family of algorithms and an example is given along with encouraging numerical results obtained with several standard test functions.

The use of cluster analysis in physical data base design

Abstract: The physical structure and relative placement of information elements within a data base is critical for the efficient design of a computerized information system which is shared by a community of users. Traditionally the selection among alternative structural designs has been handled largely via heuristics. Recent research has shown that a number of significant design problems can be stated mathematically as nonlinear, integer, zero-one programming problems. In concept, therefore, mathematical programming algorithms can be used to determine "optimal" data base designs. In practice, one finds that realistic problems of even modest size are computationally infeasible. This paper presents a means for overcoming this difficulty in the design of data base records. A metric with which to measure the similarity of usage among data items is developed and used by a clustering algorithm to reduce the space of alternative designs to a point where solution is economically feasible.

A Generalization of Ornstein's $\bar d$ Distance with Applications to Information Theory

Abstract: Ornstein's $\bar{d}$ distance between finite alphabet discrete-time random processes is generalized in a natural way to discrete-time random processes having separable metric spaces for alphabets. As an application, several new results are obtained on the information theoretic problem of source coding with a fidelity criterion (information transmission at rates below capacity) when the source statistics are inaccurately or incompletely known. Two examples of evaluation and bounding of the process distance are presented: (i) the $\bar{d}$ distance between two binary Bernoulli shifts, and (ii) the process distance between two stationary Gaussian time series with an alphabet metric $|x - y|$.

The Structure of Superspace

Abstract: Publisher Summary In classical geometrodynamics, one is led to a configuration space, S(M), which is a stratified Frechet manifold. In quantum geometrodynamics, one must allow the underlying topology to fluctuate. In fact, the fluctuations of the topology and metric will be so violent that the resulting spaces will seem foam like. This chapter discusses the superspace S of isometry classes of compact metric spaces and discusses that one can define a natural metric D on S. It also describes some of the properties of the metric space (S,D). A usual assumption of classical cosmology is that space is a Riemannian 3-sphere whose metric is close to the standard metric. If one assumes that in quantum cosmology the probability measure is concentrated near S3 in the metric D, then one has the prediction that, with very high probability, space will be a foamy 3-sphere.

Preference Structures. II: Distances Between Asymmetric Relations

Abstract: In this paper the results of Preference structures I are extended to not necessarily transitive relations. The extension of the city block metric of [3] leads to a discussion of the relation of majority rule to the median of a collection of relations. The extension of the Euclidean metric of [3] leads to a discussion of the mean of a collection of relations and methods for finding means. Finally, the extension of the Euclidean metric leads to a discussion of statistical inference about random variables whose ranges are preference relations.

Generalized “transition probability”

Abstract: An operationally meaningful symmetric function defined on pairs of states of an arbitrary physical system is constructed and is shown to coincide with the usual “transition probability” in the special case of systems admitting a quantum-mechanical description. It can be used to define a metric in the set of physical states. Conceivable applications to the analysis of certain aspects of Quantum Mechanics and to its possible modifications are mentioned.

On all kinds of homogeneous spaces

Abstract: Several open questions on homogeneous spaces are answered. A few of the results are: (1) An n-homogeneous metric continuum, which is not the circle, is strongly n-homogeneous. (2) A 2-homogeneous metric continuum is locally connected. (3) If X is a homogeneous compact metric space or a homogeneous locally compact, locally connected separable metric space, then X is a coset space. (4) If G is a complete separable metric topological group with is n-connected, then G is locally n-connected.

Predicting Preferences on Experimental Bundles of Attributes: A Comparison of Models

Abstract: This study tests the ability of various models to predict individual preferences on stimuli defined by physical characteristics. Within all models, metric routines were found to be superior to nonm...

Metric updater for maximum likelihood decoder

Abstract: In a convolutional code decoder, a current received input corresponds to a last data state of a received sequence. Path metrics, respective correlations of the received sequence to each of a limited number of possible sequences, are determined to choose the "correct" possible sequence. Decoding then proceeds by known means. The current input addresses "look-up" memories, each associated with a possible input data state and providing a "branch metric address" output. "Update" memories, each associated with one of the data states, are each addressed by path metrics of prior sequences which can enter its associated state and by one "branch metric address." In functional effect, each "update" memory adds a separate branch metric to update each prior path metric leading into each state. Path metrics for the limited number of possible sequences are thus provided. An indication of which updated path metric is larger for each state and its value are output from each "update" memory. The largest of the updated path metrics provided by the memories identifies the "correct" possible sequence.

A heuristic problem solving design system for equipment or furniture layouts

Abstract: The Designer Problem Solver (DPS) demonstrates that the computer can perform simple design tasks. In particular, it designs furniture and equipment layouts. This task was chosen because it is simple, well defined, and characteristic of many design tasks in architecture, engineering, urban planning, and natural resource management. These space planning tasks usually involve manipulating two-dimensional representations of objects to create feasible or optimal solutions for problems involving topological and metric spatial constraints. The paper describes extensive tests performed on the program.DPS is a heuristic problem solver with a planning phase prefixed to it. It uses the planning process to give it a sense of direction, diagnostic procedures to locate difficulties, and remedial actions to recover from difficulties. It uses a convex polygon representation to accurately describe the objects and the layout. This representation allows topological and metric constraints to be tested and the design to be easily updated.DPS has been applied to 50 problems. While it is slow and limited in scope, the ideas behind it are general. It demonstrates the need for selectivity in controlling search and the methods used to achieve it: task-specific information, planning, diagnostic procedures, remedial actions, and selective alternative generators.

Taxonomic congruence in Eskimoid populations.

Abstract: The study compares distance relationships in Eskimoid populations based on metric and attribute data with linguistic relationships based on structural and lexicostatistical data. Taxonomic congruence and the non-specificity hypothesis are investigated by matrix correlations and by a clustering procedure. The matrix correlation approaches employed are the Pearson product-moment correlation coefficient and the Spearman rank-order correlation coefficient. An unweighted pair-group clustering procedure provides a visual comparison of biological and linguistic relationships. Data consist of 74 craniometric measurements and 28 cranial observations taken on 12 Eskimoid populations. Mahalanobis' D2 and Balakrishnan and Sanghvi's B2 were used to compute the metric and attribute distances, respectively. The results indicate that a strict adherence to the non-specificity hypothesis is untenable. Also, there is better concordance between the sexes for metric distances than for attribute distances, and the metric data are more concordant with linguistic relationships than are the attribute data.

Implementation and Applications of Bivariate Gaussian Mixture Decomposition

Abstract: An interactive method for decomposing mixtures consisting of an arbitrary number of bivariate Gaussian components is described, which can handle problems currently attacked by cluster analysis methods. In contradistinction to most clustering methods, this procedure does not require selection of a metric or distance function with sample element arguments. Instead, estimates of population bivariate contours are examined graphically to yield estimates of subpopulation parameters. This approach is based on properties of the underlying population rather than on heuristic measures of distance between elements of a sample. Besides discussing the theory underlying this new class of procedures, several examples involving real and simulated data are presented.

Ordmet: A general algorithm for constructing all numerical solutions to ordered metric structures

Abstract: The algorithm is applicable to structures such as are obtained from additive conjoint measurement designs, unfolding theory, general Fechnerian scaling, some special types of multidimensional scaling, and ordinal multiple regression. A description is obtained of the space containing all possible numerical representations which can satisfy the structure, the size and shape of which is informative. The Abelson-Tukey maximinr2 solution is provided.

Studies in the Kerr--Newman metric

Abstract: The Kerr−Newman metric is analyzed according to the null tetrad formalism. The components of the Weyl and the Ricci tensors are calculated and these tensors are then projected on a suitable null−tetrad basis. The spin coefficients of Newman and Penrose are also calculated. These results are applied to obtain the equations of gravitational and neutrino perturbations in the Kerr−Newman metric.

A Nonuniform Bound on Convergence to Normality

Abstract: Various asymptotically correct bounds on the uniform metric for distance between distribution functions in the central limit theorem for sums of independent and identically distributed random variables have previously been given. It is shown in the present paper that corresponding nonuniform bounds can be given for the difference between distribution functions. These results have much wider applicability, such as for obtaining probabilities of moderate deviation or for dealing with L p metrics, 1 ≤ p≤ ∞.

Kurtosis and Departure from Normality

Abstract: The coefficients of skewness and kurtosis are traditional measures of departure from normality which have been widely used, particularly in an empirical context. Theoretical disadvantages of the quantities have been often mentioned in the literature; these are here emphasized by an example of a family of non-symmetric distributions, all of whose odd order moments vanish, which have the same moments, the first four coinciding with those of the unit normal law. Nevertheless, if one restricts the class of distributions under consideration to a class L 2 of mixtures of normals, then the kurtosis appears as a distance in a metric space setting Keilson and Steutel (1974)]. It is shown here that, in this metric space setting, the kurtosis can also be used in a bound on both the uniform metric for the distance between a distribution function and the unit normal distribution function and a non-uniform bound. A similar, and simpler, bound is also given in the case of more general mixtures.

Hierarchical Clustering and the Concept of Space Distortion.

Abstract: Using an occupancy model developed from combinatorics, the prototypic single-link and complete-link hierarchical clustering methods are considered to be at the two extremes of a space distortion clustering continuum. Two approaches for attacking the space distortion problem are suggested: (i) using an intermediate r-diameter criterion that includes the single-link and complete-link methods as special cases, and (ii) preprocessing the original proximity measures to force a metric structure on the input data that will lead to a better correspondence between the results produced by the two extreme clustering strategies. In addition to several numerical examples that typify the effect of using an r-diameter criterion or, alternatively, an initial preprocessing of a given set of proximity measures, Monte Carlo results are presented illustrating empirically the space distortion properties of the single-link and complete-link methods.

A bound to the condition number of canonical rank-two corrections and applications to the variable metric method

Abstract: Properties of rank-two symmetric corrections to a positive definite matrix are considered in the context of canonical forms. The conditions for maintenance of positive definitess are given and a new bound to the condition number of the corrected matrix is obtained. Some applications to the variable metric method are presented and the performance of the new bound is compare numerically with that of a bound proposed by Brodlie, Gourlay and Greenstadt.

Connexions and prolongations

Abstract: Computation of the velocity of a given motion depends on measurement of nearby position changes only. Computation of acceleration, on the other hand, depends on measurement of nearby changes in velocity. But since velocity vectors are attached to positions so that even nearby ones are not a priori comparable, acceleration is not computable until a rule for comparison of vectors along a curve is given. Such a rule-parallel translation or linear connexion - exists automatically in Euclidean spaces. For motions in more general manifolds, for example (semi-) Riemannian ones, parallel translation is a less obvious consequence of the metric properties.