Exponential Class Research Articles

COMPETITIVE COMPLEXITY OF MOBILE ROBOT ON-LINE MOTION PLANNING PROBLEMS

This paper classifies common mobile robot on-line motion planning problems according to their competitive complexity. The competitiveness of an on-line algorithm measures its worst case performance relative to the optimal off-line solution to the problem. Competitiveness usually means constant relative performance. This paper generalizes competitiveness to any functional relationship between on-line performance and optimal off-line solution. The constants in the functional relationship must be scalable and may depend only upon on-line information. Given an on-line task, its competitive complexity class is a pair of lower and upper bounds on the competitive performance of all on-line algorithms for the task, such that the two bounds satisfy the same functional relationship. The paper classifies the following on-line motion planning problems into competitive classes: area coverage, navigation to a target, and on-line search for an optimal path. In particular, it is shown that navigation to a target whose position is either apriori known or recognized upon arrival belongs to a quadratic competitive complexity class. The hardest on-line problem involves navigation in unknown variable traversibility environments. Under certain restriction on traversibility, this last problem belongs to an exponential competitive complexity class.

Ratio of the Tail of an Infinitely Divisible Distribution on the Line to that of its Lévy Measure

A necessary and sufficient condition for the tail of an infinitely divisible distribution on the real line to be estimated by the tail of its Lévy measure is found. The lower limit and the upper limit of the ratio of the right tail of an infinitely divisible distribution to the right tail of its Lévy measure are estimated from above and below by reviving Teugels's classical method. The exponential class and the dominated varying class are studied in detail.

Bayes estimators of parameters of Poisson type exponential class software model considering generalized Poisson and gamma priors

The paper deals with the estimation problem of Poisson type Exponential Class model which has wide application in software reliability. This model consists of two parameters namely, the total number of failures and the failure rate. Considering the behaviour of the parameter, total number of failure and limitation of Poisson distribution, a generalized Poisson distribution has been proposed as a prior for this parameter. Further, an inverted Gamma prior has been selected for failure rate in view of its property. The Bayes estimators have been obtained considering these priors under Squared Error Loss Function. To study the performance of obtained estimators, these estimators have been compared with corresponding maximum likelihood estimators on the basis of a Monte Carlo simulation study.

Risk-sensitive dynamic pricing for a single perishable product

We show that the monotone structures of dynamic pricing for a single perishable product under risk-neutrality are preserved under risk-sensitivity with the additive general utility and atemporal exponential utility functions. We also show that the optimal price is decreasing over the degree of risk-sensitivity under the exponential class of both additive and atemporal utility functions.

Low regularity classes and entropy numbers

We note a sharp embedding of the Besov space \(B^{\infty}_{0,q}({\mathbb{T}})\) into exponential classes and prove entropy estimates for the compact embedding of subclasses with logarithmic smoothness, considered by Kashin and Temlyakov.

Tail Extrapolation in MLSE Receivers Using Nonparametric Channel Model Estimation

This paper presents a method for determining the probability of rare events, in particular for probability density function (pdf) and bit error rate (BER) estimation. The derivation of the method is based on the presumption that the pdf is a member of a family of distributions very often named as the generalized exponential (GE) class of distributions. Based on high reliability estimations obtained in short simulation/measurement times, the low probably events are estimated accurately by extrapolation. The suggested method can be applied to some distributions that are different from GE distributions, such as noncentral chi-square distributions, to extrapolate to low probability events, with some extrapolation error. It can also be applied to BER estimation. The method is in particular helpful for estimating channels suffering from both severe signal distortion causing undesired intersymbol interference (ISI) of several symbols, and from severe noise. Such conditions prevail, for example, in metro and long haul high-speed optical fiber communication systems. So the method may be implemented in particular in maximum-likelihood sequence estimation (MLSE) optical receivers using nonparametric channel model estimation. A special use of the extrapolation method is explained for practical systems using trellis branch metrics derived from the estimated pdf to decode the transmitted sequence of symbols.

Polymatroids and mean-risk minimization in discrete optimization

We study discrete optimization problems with a submodular mean-risk minimization objective. For 0–1 problems a linear characterization of the convex lower envelope is given. For mixed 0–1 problems we derive an exponential class of conic quadratic valid inequalities. We report computational experiments on risk-averse capital budgeting problems with uncertain returns.

Detection and classification using the Estimated Ocean Detector

We have developed the Estimated Ocean Detector, a likelihood ratio receiver with an estimator‐correlator structure, and applied it to detection and classification of underwater acoustic signals. The receiver requires that the noise probability density function (pdf) to belong to the exponential class but need not be Gaussian. A composite hypothesis is employed in order to incorporate knowledge (or predictions) of the signal parameter statistics. Previously, receiver performance was demonstrated for Gaussian noise and sinusoidal signals with known frequency and phase and whose amplitude pdfs were predicted using knowledge of the ocean environment and an acoustic propagation program. In order for the receiver to be useful operationally, it must be able to accommodate unknown signal phase and incorporate numerical estimates of noise and signal parameter pdfs. Progress toward satisfying these requirements is reported in this talk. Work supported by the Office of Naval Research Undersea Signal Processing.

A NOTE ON THE CLOSURE OF CONVOLUTION POWER MIXTURES (RANDOM SUMS) OF EXPONENTIAL DISTRIBUTIONS

AbstractWe make a correction to an important result by Cline [D. B. H. Cline, ‘Convolutions of distributions with exponential tails’, J. Austral. Math. Soc. (Series A)43 (1987), 347–365; D. B. H. Cline, ‘Convolutions of distributions with exponential tails: corrigendum’, J. Austral. Math. Soc. (Series A)48 (1990), 152–153] on the closure of the exponential class $\mathcal {L}(\alpha )$ under convolution power mixtures (random summation).

An Outer-Measure Approach for Resource-Bounded Measure

Lutz developed a general theory of resource-bounded measurability and measure on suitable complexity classes C⊆C (see Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 236–248, 1998), where Cantor Space C is the class of all decision problems, and classes C include various exponential time and space complexity classes, the class of all decidable languages, and the Cantor space C itself. In this paper, a different general theory of resource-bounded measurability and measure on those complexity classes is developed. Our approach is parallel to the Caratheodory outer measure approach in classical Lebesgue measure theory. We shall show that many nice properties in the classical Lebesgue measure theory hold in the resource-bounded case also. The Caratheodory approach gives short and easy proofs of theorems in the resource-bounded case as well as in the classical case. The class of measurable sets in our paper is strictly larger than that of Lutz, and the two resource-bounded measures assign the same measure for a set if the set is measurable in the sense of Lutz.

Identifying and integrating likelihood functions and signal parameter probability density functions (PDFs) for sonar signal processing

A likelihood ratio receiver developed using an estimator-correlator structure requires that the noise probability density function (pdf) belong to the exponential class, but need not be Gaussian. The receiver employs a composite hypothesis that takes into account knowledge (or predictions) of the signal parameter statistics. In order to implement the receiver, we must be able to integrate the a priori pdf for the random parameter times the likelihood function over the domain of the random parameter, i.e., ∫p(r−θ)p(θ)dθ. The signal fluctuation models used by Swerling (1960) for radar detection theory belong to the chi-squared family of distributions, which are part of the exponential class. Likewise, the maximum entropy method with moments lnθ and θ could be used to fit p(r−θ) with a chi-squared pdf. If both noise and signal parameters are chi-squared distributed, a closed-form solution may exist. This paper considers approaches to fitting pdfs to noise and signal parameter samples and to solving the integral described above. [Work sponsored by ONR Undersea Signal Processing.]

Brownian super-exponents

We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a supermartingale even in cases where the original process is a martingale. We determine a necessary and su-cient condition for the transform to be a martingale process. The condition links expected values of the transformed stochastic exponential to the distribution function of certain time-integrals.

Algebraic properties of a class of p-adic exponentials

In this Note we give an algebraic construction of a class of p-adic exponentials of Artin–Hasse type which are convergent in the disk D − ( 0 , 1 ) . Moreover we have a control for the field of coefficients of power series that defines such functions. Such objects were used by Christol and Robba to calculate the irregularity of a rank 1 p-adic differential operator, under the restriction of spherical completeness for the field of coefficients, and recently by Pulita, in order to classify the same equations. To cite this article: D. Chinellato, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Locally Most Powerful Sequentially Planned Tests in Continuous Time

In this article we study the problem of finding a locally most powerful sequentially planned test in a one-sided sequentially planned testing problem for stochastic processes of the exponential class with continuous time parameter, thereby applying the theory of optimal sampling in continuous time (cf. Roters, 1995b1997). The theory presented here extends the discrete-time results of Berk (1975) and Schmitz (1993).

Rank one solvable p-adic differential equations and finite Abelian characters via Lubin–Tate groups

We introduce a new class of exponentials of Artin–Hasse type, called π-exponentials. These exponentials depend on the choice of a generator π of the Tate module of a Lubin–Tate group \(\mathfrak{G}\) over \(\mathbb{Z}_p\). They arise naturally as solutions of solvable differential modules over the Robba ring. If \(\mathfrak{G}\) is isomorphic to \(\widehat{\mathbb{G}}_m\) over \(\mathbb{Z}_p\), we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin–Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of \(\epsilon^\dagger_{K_{\infty}}\), whose residue field is a given Artin–Schreier extension of \(k(\!(t)\!)\), where k is the residue field of K. We then compute explicitly the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one φ-module over \(\epsilon^\dagger_{K_\infty}\) attached to a rank one representation of \(\mathrm{Gal}(k(\!(t)\!)^{\mathrm{sep}}/k(\!(t)\!))\), defined by an Artin–Schreier character.

The estimated ocean detector: Predicted performance for continuous time signals in a random/uncertain ocean

This paper addresses implementation of the maximum likelihood (ML) detector for passive SONAR detection of continuous time stochastic signals that have propagated through a random or uncertain ocean. We have shown previously that Monte Carlo simulation and the maximum entropy method can make use of knowledge of environmental variability to construct signal and noise parameter probability density functions (pdf’s) belonging to the exponential class. For these cases, the ML detector has an estimator-correlator and noise-canceller implementation. The estimator-correlator detector computes the conditional mean estimate of the signal conditioned on the received data and correlates it with a function of the received data, hence the name estimated ocean detector (EOD). Here we derive the detector structure for continuous time stochastic signals and Gaussian noise and present receiver operating characteristic (ROC) curves for the detector as a function of the signal-to-noise ratio. [Work supported by ONR Undersea Signal Processing Code 321US.]

Performance of the estimated ocean detector (EOD) against non-Gaussian signal and noise

Our overall goal is to develop a method for incorporating knowledge of environmental variability into a passive sonar detector. An earlier talk presented the derivation of the estimated ocean detector (EOD), which is the maximum-likelihood detector for the exponential class of received signal parameter probability density functions (pdfs). A particular implementation of the EOD for Gaussian signal and noise (members of the exponential class) was shown and performance bounded. This talk evaluates the performance of the Gaussian EOD under nonideal conditions, including inaccuracies in the estimates of signal and noise variance, and examines robustness under received signal distributions that contain nonzero skew and kurtosis. To assess detection performance for non-Gaussian signals, receiver operating characteristics are computed for received signal pdfs obtained by applying the maximum entropy method to fit an exponential class pdf to experimental data. Gaussian EOD performance for non-Gaussian signal and noise conditions is compared to the classical envelope and correlation detectors, which form the limiting cases of the EOD. [Work supported by The Undersea Signal Processing Program, Office of Naval Research.]

The estimated ocean detector (EOD) for Gaussian signal and noise

Our overall goal is to develop a method for incorporating knowledge of the environment, and in particular, environmental variability, into the signal processor. This talk begins with the assumption that all available information about the environment has been used to compute the probability density functions (pdfs) of relevant received signal parameters (e.g., amplitude or time spread). For the exponential class of pdfs that arises from applying the maximum entropy method to random signal parameters, the maximum likelihood detectors (MLD) can be implemented as an estimator-correlator [Schwartz, ‘‘The estimator-correlation for discrete-time problems,’’ IEEE Trans. on Information Theory. 23, 93–100 (1977)]. The MLD detector correlates received data with the conditional mean estimate of the signal; hence we have named it the estimated ocean detector (EOD). Here, we present the general structure of the EOD and explore EOD performance for Gaussian signals and noise, which are members of the exponential class. The resulting implementation produces a detection statistic that is the weighted sum of an energy detector and a correlation receiver, with the weights calculated from the variances of the noise and signal. EOD performance bounds are calculated using completely known and completely unknown oceans. [Work supported by The Undersea Signal Processing Program, Office of Naval Research.]

Using the maximum entropy method to estimate probability density functions for oceanographic and acoustic parameters

The maximum entropy (MaxEnt) method offers a powerful means of estimating the probability density functions (pdf’s) of measured data. Traditionally, pdf’s have been estimated by fitting data to particular distributions (e.g., Gaussian, log-normal, Rayleigh, chi-squared, etc.). This approach can lead to misleading results, especially if the data are highly skewed or kurtotic. First published in 1957 by Jaynes [E. T. Jaynes, Phys. Rev., 106, 620–630 (1957)], MaxEnt has gained wide use in recent years. The method uses Lagrangian multipliers to maximize the entropy, a measure of uncertainty, subject to specified constraints. The result is a pdf that Jaynes describes as being the ‘‘least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information.’’ In other words, all possibilities under the given constraints are considered equally. This method is particularly useful in determining prior pdf’s based on limited measurements. MaxEnt pdf’s, which belong in the exponential class of distributions, can also be used to derive optimum processors in the maximum likelihood sense. This talk will discuss the theory and implementation of the MaxEnt method and emphasize its utility for modeling acoustic propagation in uncertain environments. [Work supported by The Undersea Signal Processing Program, Office of Naval Research.]

Overview of signal processing in uncertain and random environments

This paper is an overview of optimum and/or robust signal-processing approaches for detecting signals in random and uncertain propagation and interference environments, a topic that is intertwined with statistical modeling of the medium and signals. The classical approach to processing of signals that have propagated through randomly fluctuating media is based on the concepts of randomly time-varying impulse responses, spreading functions, and scattering functions. This approach uses second-order statistics and gives insight into performance of active sonar. Implementation of maximum likelihood (ML) detectors for signals that have propagated through random environments requires knowledge of the probability density functions (pdf’s) of the random signal and noise parameters. Random signal parameters that are typically affected by medium randomness are amplitude, phase, and arrival time; pulse and Doppler spread; arrival angle bias and spread. Signal pdf’s can be derived from ocean acoustic models using the maximum entropy (ME) principle, which exploits what is known but is maximally noncommittal of what is uncertain. The ME method can be used to generate densities that belong to the exponential class, and for this class, ML detectors can be implemented as an estimator-correlator and estimator-noise canceler structure. [Work supported by The Undersea Signal Processing Program Office of Naval Research.]