Showing papers on "Measure (mathematics) published in 1989"

Sensitivity, Precision, and Linear Aggregation of Signals for Performance Evaluation

Abstract: Several accounting and other signals are generally available for the construction of a managerial performance evaluation measure on which an optimal compensation contract is based. The demand for aggregation in evaluating managerial performance arises because reporting all the basic transactions and other nonfinancial information about performance is costly and impracticable (see Ashton [1982], Casey [1978], and Holmstrom and Milgrom [1987]). We identify necessary and sufficient conditions on the joint density function of the signals under which linear aggregation, a simple and commonly employed way to construct a performance evaluation measure, is optimal. This characterization suggests that the linear form of aggregation is optimal for a large class of situations. Focusing on performance measures that are linear aggregates enables us to determine the relative weights on the individual signals in the optimal linear aggregate, since these weights are invariant for all realizations of the signals. We interpret these weights in terms of statistical characteristics (sensitivity and precision) of the joint distribution of the signals.

Real analysis and probability

Abstract: This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

Spectral representations of infinitely divisible processes

Abstract: The spectral representations for arbitrary discrete parameter infinitely divisible processes as well as for (centered) continuous parameter infinitely divisible processes, which are separable in probability, are obtained. The main tools used for the proofs are (i) a “polar-factorization” of an arbitrary Levy measure on a separable Hilbert space, and (ii) the Wiener-type stochastic integrals of non-random functions relative to arbitrary “infinitely divisible noise”.

The Ergodic Theory of Discrete Groups

Abstract: 1. Preliminaries 2. The limit set 3. A measure of the limit set 4. Conformal densities 5. Hyperbolically harmonic functions 6. The sphere at infinity 7. Elementary ergodic theory 8. The geodesic flow 9. Geometrically finite groups 10. Fuchsian groups.

The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities

Abstract: The overlapping coefficient is defined as a measure of the agreement between two probability distributions. Its relationship to the dissimilarity index and its propertie are described. An extensive treatment of maximum-likelihood estimation of the overlap between two normal distributions is presented as an example of estimating the overlapping coefficient from sample data.

Cantor spectra and scaling of gap widths in deterministic aperiodic systems.

Abstract: The relationship between geometry and physical properties of aperiodic structures is investigated by considering the example of the tight-binding Schr\"odinger equation in one dimension, where the site potentials are given by an arbitrary deterministic aperiodic sequence. In a perturbative analysis of the integrated density of states, the gaps in the energy spectrum can be ``labeled'' by the singularities of the Fourier transform of the sequence of potentials. This approach confirms known properties of quasiperiodic and almost-periodic systems, and suggests an extension of them to more general sequences, such as those with a singular continuous Fourier transform. There is strong evidence that the spectrum is a Cantor set with zero measure for a much larger class of models than quasiperiodic ones. The dependence of the widths of various gaps on the potential strength is also determined: several different kinds of behavior are obtained, such as a power law with a nontrivial exponent, or an essential singularity. These general results are compared with those of various other approaches for four self-similar sequences generated by substitution, namely the Thue-Morse sequence, the period-doubling sequence, a ``circle sequence,'' and the Rudin-Shapiro sequence.

Anticipated utility: a measure representation approach

Abstract: This paper presents axioms which imply that a preference relation over lotteries can be represented by a measure of the area above the distribution function of each lottery. A special case of this family is the anticipated utility functional. One additional axiom implies this theory. This functional is then extended for the case of vectorial prizes.

Voice activity detection

Abstract: Voice activity detector (VAD) for use in an LPC coder in a mobile radio system uses autocorrelation coefficient R 0 , R 1 . . . of the input signal, weighted and combined, to provide a measure M which depends on the power within that part of the spectrum containing no noise, which is thresholded against a variable threshold to provide a speech/no speech logic output. The measure is formula (I), where H i are the autocorrelation coefficients of the impulse response of an Nth order FIR inverse noise filter derived from LPC analysis of previous non-speech signal frames. Threshold adaption and coefficient update are controlled by a second VAD response to rate of spectral change between frames.

Nonlinear parameter estimation by global optimization—efficiency and reliability

Abstract: In this paper we first show that the objective function of a least squares type nonlinear parameter estimation problem can be any non- negative real function, and therefore this class of problems corre- sponds to global optimization. Two non-derivative implementations of a global optimization method are presented; with nine standard test functions applied to measure their efficiency. A new nonlinear test problem is then presented for testing the reliability of global op- timization algorithms. This test function has a countable infinity of local minima and only one global minimizer. The region of attraction of the global minimum is of zero measure. The results of efficiency and reliability tests are given.

Use of DCO ratios for spin determination in γ-γ coincidence measurements

Abstract: The use of large detector arrays allows to measure γ-γ angular correlations and to determine spins and multipolarities, even of weak and unresolved transitions The method utilizing directional correlations of γ-rays deexciting oriented states (DCO ratio method) has been described and applied to experimental data on 178 W taken with the γ-γ coincidence spectrometer OSIRIS The spin of the 35 ns isomer in 178 W has been established to be 14

Approximation algorithms for the shortest common superstring problem

Abstract: The object of the shortest common superstring problem (SCS) is to find the shortest possible string that contains every string in a given set as substrings. As the problem is NP-complete, approximation algorithms are of interest. The value of an aproximate solution to SCS is normally taken to be its length, and we seek algorithms that make the length as small as possible. A different measure is given by the sum of the overlaps between consecutive strings in a candidate solution. When considering this measure, the object is to find solutions that make it as large as possible. These two measures offer different ways of viewing the problem. While the two viewpoints are equivalent with respect to optimal solutions, they differ with respect to approximate solutions. We describe several approximation algorithms that produce solutions that are always within a factor of two of optimum with respect to the overlap measure. We also describe an efficient implementation of one of these, using McCreight's compact suffix tree construction algorithm. The worstcase running time is O ( m log n ) for small alphabets, where m is the sum of the lengths of all the strings in the set and n is the number of strings. For large alphabets, the algorithm can be implemented in O ( m log m ) time by using Sleator and Tarjan's lexicographic splay tree data structure.

Remarks on Chacon’s biting lemma

Abstract: Chacon's Biting Lemma states roughly that any bounded sequence in LI possesses a subsequence converging weakly in LI outside a decreasing family Ek of measurable sets with vanishingly small measure. A simple new proof of this result is presented that makes explicit which sets Ek need to be removed. The proof extends immediately to the case when the functions take values in a reflexive Banach space. The linit function is identified via the Young measure and approximations. The desci iption of concentration provided by the lemma is discussed via a simple example.

Multiloop amplitudes in the theory of quantum strings and complex geometry

Abstract: The evaluation of multiloop amplitudes in the theory of closed oriented bosonic strings is reduced to the problem of finding the measure on the moduli space of Riemann surfaces. It is shown that the measure is equal to the product of the square of the modulus of a holomorphic function and the determinant of the imaginary part of the period matrix, raised to the power 13. A consequence of this theorem is that the measure can be expressed in terms of theta-functions. A variant of the holomorphy theorem, in the form of Quillen's theorem, is used to evaluate the dependence of the determinants of the Laplace operator on a Riemann surface on the boundary conditions. When the Riemann surface is represented by a branched covering of a plane, the measure is expressed in terms of the coordinates of the branch points, and to each branch point there corresponds a vertex operator. The measure is the correlation function of these operators, and this can be used to represent the sum over all the higher loops as the partition function of a certain two-dimensional conformal field theory.

The space of interactions in neural networks: Gardner's computation with the cavity method

Abstract: Gardner's (1987,1988) computation of the number of N-bit patterns which can be stored in an optical neural network used as an associative memory is derived without replicas, using the cavity method. This allows for a unified presentation whatever the basic measure in the space of coupling constants, but above all it gives the clear physical content of the assumption of replica symmetry. TAP equations are also derived.

Local properties of Lévy processes on a totally disconnected group

Abstract: This paper is the first study of the sample path behavior of processes with stationary independent increments taking values in a nondiscrete, locally compact, metrizable, totally disconnected Abelian group. After some preparatory results of independent interest we give a general integral criterion for a deterministic function to be a local modulus of right-continuity for the paths of the process and then study the sets of “fast” and “slow” points where the local growth of the process is anomalously large or small. We establish the lim sup behavior for the sequence of first exit times from a collection of concentric balls for an arbitrary process and show that no deterministic function can act as an exact lower envelope. Under appropriate conditions similar results hold for the related sojourn time sequence. We consider various candidates for measuring the variation of the paths of the process, show that they exist and coincide in our situation, and then determine the common value for a general process. Using earlier results we calculate the Hausdorff and packing dimensions of the image of an interval, exhibit the correct Hausdorff measure for this set, and establish a dichotomy that classifies measure functions into those that lead to a zero packing measure for the image and those that lead to an infinite packing measure. Lastly, we prove some uniform dimension results, which bound the dimension of the image of a set in terms of the dimension of the set itself. These results hold almost surely for all sets simultaneously.

Uncertainty, belief, and probability

Abstract: We introduce a new probabilistic approach to dealing with uncertainty, based on the observation that probability theory does not require that every event be assigned a probability. For a nonmeasurable event (one to which we do not assign a probability), we can talk about only the inner measure and outer measure of the event. Thus, the measure of belief in an event can be represented by an interval (defined by the inner and outer measure), rather than by a single number. Further, this approach allows us to assign a belief (inner measure) to an event E without committing to a belief about its negation E (since the inner measure of an event plus the inner measure of its negation is not necessarily one). Interestingly enough, inner measures induced by probability measures turn out to correspond in a precise sense to Dempster-Shafer belief functions. Hence, in addition to providing promising new conceptual tools for dealing with uncertainty, our approach shows that a key part of the important Dempster-Shafer theory of evidence is firmly rooted in classical probability theory.

An Alternate Measure of Consistency

Abstract: The AΗΡ provides a decision maker with a way of examining the consistency of entries in a pairwise comparison matrix and the hierarchy as a whole through the consistency ratio measure. It has always seemed to us that this commonly-used measure could be improved upon. The purpose of this paper is to present an alternate measure of consistency and demonstrate how it might be applied. The contributions and limitations of the new measure are discussed.

An existence result for scalar conservation laws using measure valued solutions.

Abstract: (1989). An existence result for scalar conservation laws using measure valued solutions. Communications in Partial Differential Equations: Vol. 14, No. 10, pp. 1329-1350.

A new description of the Bowen—Margulis measure

Abstract: The Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.

Laws in Logic and Combinatorics

Abstract: This is a survey of logical results concerning random structures. A class of relational structures on which a (finitely additive) probability measure has been defined has a 0–1 law for a particular logic if every sentence of that logic has probability either 0 or 1. The measure may be an asymptotic probability on finite structures or generated on a class of infinite structures by assigning fixed probabilities to independently occurring properties. Conditions under which all sentences of a logic have a probability, and under which 0–1 laws occur, are examined. Also, the complexity of computing probabilities of sentences is considered.

Finding tailored partitions

Abstract: We consider the following problem: given a planar set of points S, a measure m acting on S, and a pair of values m1 and m2, does there exist a bipartition S = S1 U S2 satisfying m(Si) ≤ mi for i = 1,2? We present algorithms of complexity O(n log n) for several natural measures, including the diameter (set measure), the area, perimeter or diagonal of the smallest enclosing axes-parallel rectangle (rectangular measure), and the side length of the smallest enclosing axes-parallel square (square measure). The problem of partitioning S into k subsets, where k ≥ 3, is known to be NP-complete for many of these measures.

Spectrum and extended states in a harmonic chain with controlled disorder: Effects of the Thue-Morse symmetry

Abstract: Along the lines of previous work, we give the general framework together with a detailed and rigorous study of the spectrum and Born-von Karman eigenstates of a 1D harmonic chain with controlled disorder determined by the Thue-Morse sequence. The spectrum is a Cantor-like set; we prove numerically that its measure is zero and calculate its Bouligand-Minkowski dimension (box dimension). We prove that the value of the IDS on each of the gaps is (2k+1)/(3·2 p ), withk andp integers. Finally, we also prove that points in a dense subset of the spectrum give rise to extended states, an exceptional property due to the symmetry of the Thue-Morse substitution which can have important applications to multilayered structures, and we illustrate this situation.

The Stein Paradox in the Sense of the Pitman Measure of Closeness

Abstract: The dominance and related optimality properties of the usual Stein-rule or shrinkage estimators are typically developed for quadratic error loss functions. It is shown that under the classical Pitman closeness criterion the Stein-rule estimators possess a similar dominance property when the "closeness" measure is based on suitable quadratic norms.

The dynamics of Rayleigh quotient iteration

Abstract: This paper analyzes the Rayleigh Quotient Iteration algorithm for symmetric matrices. Dynamical systems techniques are employed to characterize the sets of points for which the algorithm will converge to an eigenvector. It is shown that these sets have full measure and that their geometric nature is related to the spacing of the eigenvalues.

Existence theorems for measures on continous posets, with applications to random set theory.

Abstract: We state conditions on a partially ordered set L and a mapping λ, defined on a class of filters on L, under which λ extends to a measure on the minimal σ-field over this class. By applying this result to the case when L is a continuous lattice, all locally finite measures on L are identified as well as all Levy-Khinchin measures. We then characterize these kinds of measures on continuous semilattices and continuous posets. The correspondence between probability measures on the line and distribution function is a particular case of this result. We give a simple proof of the Daniell-Kolmogorov existence theorem for probability measures on products of continuous lattices

Forcings with ideals and simple forcing notions

Abstract: Using generic ultrapower techniques we prove the following statements: (1) for every sequence 〈μn |n <ω〉 of 0–1σ-additive measures over the set of reals, there exists a set which is nonmeasurable in eachμn, (2) there is no nowhere primeσ-complete ℵ0-dense ideal, (3) ifI is a nowhere prime ideal over a setX then add (I) ≦d(I), (4) suppose thatμ is aσ-additive total nowhere prime probability measure over a setX, then add (μ)

An exact formula for the measure dimensions associated with a class of piecewise linear maps

Abstract: An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.