Showing papers on "Measure (mathematics) published in 2001"

A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems

Abstract: The area under the ROC curve, or the equivalent Gini index, is a widely used measure of performance of supervised classification rules. It has the attractive property that it side-steps the need to specify the costs of the different kinds of misclassification. However, the simple form is only applicable to the case of two classes. We extend the definition to the case of more than two classes by averaging pairwise comparisons. This measure reduces to the standard form in the two class case. We compare its properties with the standard measure of proportion correct and an alternative definition of proportion correct based on pairwise comparison of classes for a simple artificial case and illustrate its application on eight data sets. On the data sets we examined, the measures produced similar, but not identical results, reflecting the different aspects of performance that they were measuring. Like the area under the ROC curve, the measure we propose is useful in those many situations where it is impossible to give costs for the different kinds of misclassification.

Convex measures of risk and trading constraints

Abstract: We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et aL (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust not ion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

Building Better Theory: Time and The Specification of When Things Happen

Abstract: In any investigation of a causal relationship between an X and a Y, the time when X and Y are measured is crucial for determining whether X causes Y, as well as the true strength of that relationship. Using past research and a review of current research, we develop a set of X,Y configurations that describe the main ways that causal relationships are represented in theory and tested in research. We discuss the theoretical, methodological, and analytical issues pertaining to when we measure X and Y and discuss the implications of this analysis for constructing better organizational theories.

Smoothed Analysis of Algorithms: Why the Simplex Algorithm Usually Takes Polynomial Time

Abstract: We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has polynomial smoothed complexity.

Induced measures in the space of mixed quantum states

Abstract: We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N×K composite system, induces a unique measure in the space of N×N mixed states (or in the space of K×K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N×N pure states behaves as lnN-1/2.

Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses

Abstract: In this paper, the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n × n matrices of H? frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitiviiy resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.

Measure and integration theory

Abstract: This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem. The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory. The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation

Abstract: A fundamental concern of spatial analysts is to find patterns in spatial data that lead to the identification of spatial autocorrelation or association. Further, they seek to identify peculiarities in the data set that signify that something out of the ordinary has occurred in one or more regions. In this paper we provide a statistic that tests for local spatial autocorrelation in the presence of the global autocorrelation that is characteristic of heterogeneous spatial data. After identifying the structure of global autocorrelation, we introduce a new measure that may be used to test for local structure. This new statistic Oi is asymptotically normally distributed and allows for straightforward tests of hypothe- ses. We provide several numerical examples that illustrate the performance of this statistic and compare it with another measure that does not account for global structure.

The Distance to SN 1999em from the Expanding Photosphere Method

Abstract: We present optical and IR spectroscopy of the first two months of evolution of the Type II SN 1999em. We combine these data with high-quality optical/IR photometry beginning only three days after shock breakout, in order to study the performance of the ``Expanding Photosphere Method'' (EPM) in the determination of distances. With this purpose we develop a technique to measure accurate photospheric velocities by cross-correlating observed and model spectra. The application of this technique to SN 1999em shows that we can reach an average uncertainty of 11% in velocity from an individual spectrum. Our analysis shows that EPM is quite robust to the effects of dust. In particular, the distances derived from the VI filters change by only 7% when the adopted visual extinction in the host galaxy is varied by 0.45 mag. The superb time sampling of the BVIZJHK light-curves of SN 1999em permits us to study the internal consistency of EPM and test the dilution factors computed from atmosphere models for Type II plateau supernovae. We find that, in the first week since explosion, the EPM distances are up to 50% lower than the average, possibly due the presence of circumstellar material. Over the following 65 days, on the other hand, our tests lend strong credence to the atmosphere models, and confirm previous claims that EPM can produce consistent distances without having to craft specific models to each supernova. This is particularly true for the VI filters which yield distances with an internal consistency of 4%. From the whole set of BVIZJHK photometry, we obtain an average distance of 7.5+/-0.5 Mpc, where the quoted uncertainty (7%) is a conservative estimate of the internal precision of the method obtained from the analysis of the first 70 days of the supernova evolution.

Analysis for Applied Mathematics

Abstract: Normed Linear Spaces * Hilbert Spaces * Calculus in Banach Spaces * Approximate Methods of Analysis * Distributions * The Fourier Transform * Additional Topics * Measure and Integration * References * Index

Multifractal Products of Cylindrical Rules

Abstract: A new class of random multiplicative and statistically self-similar measures is defned on IR It is the limit of measure-valued martingales constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process in a strip of the plane Several fundamental problems are solved, including the non-degeneracy and the distribution of the limit measure, mu; the finiteness of the (positive and negative) moments of the total mass of mu restricted to bounded intervals Compared to the familiar canonical multifractals generated by multiplicative cascades, the new measures and their multifractal analysis exhibit strikingly novel features which are discussed in detail

A Coupling Approach to Randomly Forced Nonlinear PDE's. II

Abstract: We consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure. The result obtained applies, in particular, to the 2D Navier-Stokes system and multidimensional complex Ginzburg-Landau equation with random kick-force.

Mushrooms and other billiards with divided phase space.

Abstract: We present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a “chaotic sea” (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary.

Nonlinear measures: a new approach to exponential stability analysis for Hopfield-type neural networks

Abstract: In this paper, a new concept called nonlinear measure is introduced to quantify stability of nonlinear systems in the way similar to the matrix measure for stability of linear systems. Based on the new concept, a novel approach for stability analysis of neural networks is developed. With this approach, a series of new sufficient conditions for global and local exponential stability of Hopfield type neural networks is presented, which generalizes those existing results. By means of the introduced nonlinear measure, the exponential convergence rate of the neural networks to stable equilibrium point is estimated, and, for local stability, the attraction region of the stable equilibrium point is characterized. The developed approach can be generalized to stability analysis of other general nonlinear systems.

A measure of data collapse for scaling

Abstract: Data collapse is a way of establishing scaling and extracting associated exponents in problems showing self-similar or self-affine characteristics as, for example, in equilibrium or non-equilibrium phase transitions, in critical phases, in dynamics of complex systems and many others. We propose a measure to quantify the nature of data collapse. Via a minimization of this measure, the exponents and their error-bars can be obtained. The procedure is illustrated by considering finite-size-scaling near phase transitions and quite strikingly recovering the exact exponents.

Statistical measures of DNA sequence dissimilarity under Markov chain models of base composition.

Abstract: In molecular biology, the issue of quantifying the similarity between two biological sequences is very important. Past research has shown that word-based search tools are computationally efficient and can find some new functional similarities or dissimilarities invisible to other algorithms like FASTA. Recently, under the independent model of base composition, Wu, Burke, and Davison (1997, Biometrics 53, 1431 1439) characterized a family of word-based dissimilarity measures that defined distance between two sequences by simultaneously comparing the frequencies of all subsequences of n adjacent letters (i.e., n-words) in the two sequences. Specifically, they introduced the use of Mahalanobis distance and standardized Euclidean distance into the study of DNA sequence dissimilarity. They showed that both distances had better sensitivity and selectivity than the commonly used Euclidean distance. The purpose of this article is to extend Mahalanobis and standardized Euclidean distances to Markov chain models of base composition. In addition, a new dissimilarity measure based on Kullback-Leibler discrepancy between frequencies of all n-words in the two sequences is introduced. Applications to real data demonstrate that Kullback-Leibler discrepancy gives a better performance than Euclidean distance. Moreover, under a Markov chain model of order kQ for base composition, where kQ is the estimated order based on the query sequence, standardized Euclidean distance performs very well. Under such a model, it performs as well as Mahalanobis distance and better than Kullback-Leibler discrepancy and Euclidean distance. Since standardized Euclidean distance is drastically faster to compute than Mahalanobis distance, in a usual workstation/PC computing environment, the use of standardized Euclidean distance under the Markov chain model of order kQ of base composition is generally recommended. However, if the user is very concerned with computational efficiency, then the use of Kullback-Leibler discrepancy, which can be computed as fast as Euclidean distance, is recommended. This can significantly enhance the current technology in comparing large datasets of DNA sequences.

Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise

Abstract: We consider the stochastic Ginzburg–Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg–Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for thelow-lying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure.

A Sweep-Line Method for State Space Exploration

Abstract: We present a state space exploration method for on-the-fly verification. The method is aimed at systems for which it is possible to define a measure of progress based on the states of the system. The measure of progress makes it possible to delete certain states on-the-fly during state space generation, since these states can never be reached again. This in turn reduces the memory used for state space storage during the task of verification. Examples of progress measures are sequence numbers in communication protocols and time in certain models with time. We illustrate the application of the method on a number of Coloured Petri Net models, and give a first evaluation of its practicality by means of an implementation based on the DESIGN/CPN state space tool. Our experiments show significant reductions in both space and time used during state space exploration. The method is not specific to Coloured Petri Nets but applicable to a wide range of modelling languages.

Space bounds for resolution

Abstract: We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition (1994, B. H. Kleine and T. Lettman, "Aussangenlogik: Deduktion und Algorithmen, Teubner, Stuttgart) the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows us to relate the space needed in a resolution proof of a formula to other well-studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula and, as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n-c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general pigeonhole principle with n holes and more than n pigeons need space n+1 independent of the number of pigeons. Since a matching space upper bound of n+1 for these formulas exists, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations. 2001 Elsevier Science.

Time-Changed Lévy Processes and Option Pricing∗

Abstract: We apply stochastic time change to Levy processes to generate a wide variety of tractable option pricing models. In particular, we prove a fundamental theorem that transforms the characteristic function of the time-changed Levy process into the Laplace transform of the stochastic time under appropriate measure change. We extend the traditional measure theory into the complex domain and define the measure change by a class of complex valued exponential martingales. We provide extensive examples to illustrate its applications and its link to existing models in the literature. JEL Classification: G10, G12, G13.

Measure representation and multifractal analysis of complete genomes

Abstract: This paper introduces the notion of measure representation of DNA sequences. Spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes. The main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria. From the measure representations and the values of the Dq spectra and related Cq curves, it is concluded that these complete genomes are not random sequences. In fact, spectral analyses performed indicate that these measure representations, considered as time series, exhibit strong long-range correlation. Here the long-range correlation is for the K-strings with dictionary ordering, and it is different from the base pair correlations introduced by other people. For substrings with length K58, the Dq spectra of all organisms studied are multifractal-like and sufficiently smooth for the Cq curves to be meaningful. With the decreasing value of K, the multifractality lessens. The Cq curves of all bacteria resemble a classical phase transition at a critical point. But the ‘‘analogous’’ phase transitions of chromosomes of nonbacteria organisms are different. Apart from chromosome 1 of C. elegans, they exhibit the shape of double-peaked specific heat function. A classification of genomes of bacteria by assigning to each sequence a point in two-dimensional space (D21 ,D1) and in three-dimensional space ( D21 ,D1 ,D22) was given. Bacteria that are close phylogenetically are almost close in the spaces ( D21 ,D1) and (D21 ,D1 ,D22).

Way-finding and landmarks: the multiple-bearings hypothesis.

Abstract: Nucifraga columbiana) are capable of very precise searching using the metric relationships between a goal and multiple landmarks to relocate the goal location. They can judge the direction more accurately than the distance to a landmark when the landmark is distant from the goal. On the basis of these findings, we propose that nutcrackers use a set of bearings, each a measure of the direction from the goal to a different landmark, when searching for that goal. The results of a simulation demonstrate that increasing the number of landmarks used results in increasingly precise searching. This multiple-bearings hypothesis makes a series of detailed predictions about how the distribution of searches will vary as a function of the geometry of the locations of the relevant landmarks and the goal. It also suggests an explanation for inconsistencies in the literature on the effects of clock-shifts on searching and on homing. Summary

On the three-dimensional spaces which admit a continuous group of motions

Abstract: namely the law by which we measure infinitesimal arclengths in the space Sn, from which the law of measure for finite arclengths follows. We consider n independent real variables x1, x2, . . . , xn and assume that the coefficients aik of the quadratic differential form (1) as well as their first and second derivatives are real, finite and continuous functions of the x for the entire range of values which we consider. We also assume that the discriminant of expression (1) is always nonzero and that the coefficients aik fulfill the well known inequalities which make this differential form positivedefinite. It is well known how the law for measuring angles and the entire geometry of the space Sn is determined by equation (1). If two spaces Sn, S′ n can be put into a one-to-one correspondence in such a way that the line elements are the same, the two spaces will be called isometric and the two geometries will be identical. When the line elements of the two spaces only differ by a constant factor or can be reduced to this relationship by a transformation of coordinates, the two spaces will be called similar, and we will consider them as belonging to the same type. Their geometries are essentially identical; the only thing which changes from one to the other is the unit of linear measure. An isometry of a space Sn into itself will be called a motion of this space. We will consider the spaces which admit continuous motions into themselves, namely, such that in the corresponding equations of the transformation appear some arbitrary parameters. The set of all these motions for a given Sn clearly forms a group. Simple geometrical considerations show that the number of parameters of this group is necessarily finite, which is in fact easily demonstrated analytically as we will see. If r is the number of these parameters in the complete group of motions, in every case this group will consist of a Original title: Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, Tomo XI, pp. 267–352 (1898). Printed with the kind permission of the Accademia Nazionale delle Scienze, detta dei XL, in Rome, the current copyright owner. Translated by Robert Jantzen, Department of Mathematical Sciences, Villanova University, Villanova, Pa 19085, USA. This paper was also reprinted in: Opere [The Collected Works of Luigi Bianchi], Rome, Edizione Cremonese, 1952, vol. 9, pp. 17-109.

Asymptotic relative entropy of entanglement.

Abstract: We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.

Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms

Abstract: The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q_1, q_2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S_{\delta}(q_1,q_2)=0. This results in expressions involving bilinear theta-constants M. The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4,Z) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds.

Measure representation and multifractal analysis of complete genomes

Abstract: This paper introduces the notion of measure representation of DNA sequences. Spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes. The main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria. From the measure representations and the values of the Dq spectra and related Cq curves, it is concluded that these complete genomes are not random sequences. In fact, spectral analyses performed indicate that these measure representations, considered as time series, exhibit strong long-range correlation. Here the long-range correlation is for the K-strings with dictionary ordering, and it is different from the base pair correlations introduced by other people. For substrings with length K=8, the Dq spectra of all organisms studied are multifractal-like and sufficiently smooth for the Cq curves to be meaningful. With the decreasing value of K, the multifractality lessens. The Cq curves of all bacteria resemble a classical phase transition at a critical point. But the ‘‘analogous’’ phase transitions of chromosomes of nonbacteria organisms are different. Apart from chromosome 1 of C. elegans, they exhibit the shape of double-peaked specific heat function. A classification of genomes of bacteria by assigning to each sequence a point in two-dimensional space (D_{-1} ,D1) and in three-dimensional space (D_{-1} ,D1 ,D_{-2}) was given. Bacteria that are close phylogenetically are almost close in the spaces (D_{-1} ,D1) and (D_{-1} ,D1 ,D_{-2}).

A polynomial time computable metric between point sets

Abstract: Measuring the similarity or distance between sets of points in a metric space is an important problem in machine learning and has also applications in other disciplines e.g. in computational geometry, philosophy of science, methods for updating or changing theories, \(\ldots\). Recently Eiter and Mannila have proposed a new measure which is computable in polynomial time. However, it is not a distance function in the mathematical sense because it does not satisfy the trian gle inequality. We introduce a new measure which is a metric while being computable in polynomial time. We also present a variant which computes a normalised metric and a variant which can associate different weights with the points in the set.