Discrepancy Function Research Articles

Inversive congruential generator over a Galois ring of dimension p l

We consider the construction of of an inversive congruential generator over a Galois ring of odd dimension pl, whichwas proposed by Sole and Zinoviev for p = 2. Using the estimates of trigonometric sums on the sequences of pseudorandom numbers, we obtain the estimates of a discrepant function, a generated sequence of pseudorandom numbers, and the associated sequence of two-dimensional “overlapping” points.

Quadrature Errors, Discrepancies, and Their Relations to Halftoning on the Torus and the Sphere

This paper deals with continuous-domain quantization, which aims to create the illusion of a gray-value image by appropriately distributing black dots. For lack of notation, we refer to the process as halftoning, which is usually associated with the quantization on a discrete grid. Recently a framework for this task was proposed by minimizing an attraction-repulsion functional consisting of the difference of two continuous, convex functions. The first one of these functions describes attracting forces caused by the image gray values, the second one enforces repulsion between the dots. In this paper, we generalize this approach by considering quadrature error functionals on reproducing kernel Hilbert spaces (RKHSs) with respect to the quadrature nodes, where we ask for optimal distributions of these nodes. For special reproducing kernels these quadrature error functionals coincide with discrepancy functionals, which leads to a geometric interpretation. It turns out that the original attraction-repulsion fu...

A decentralized control scheme for an effective coordination of phasic and tonic control in a snake-like robot

Autonomous decentralized control has attracted considerable attention because it enables us to understand the adaptive and versatile locomotion of animals and facilitates the construction of truly intelligent artificial agents. Thus far, we have developed a snake-like robot (HAUBOT I) that is driven by a decentralized control scheme based on a discrepancy function, which incorporates phasic control. In this paper, we investigate a decentralized control scheme in which phasic and tonic control are well coordinated, as an extension of our previous study. To verify the validity of the proposed control scheme, we apply it to a snake-like robot (HAUBOT II) that can adjust both the phase relationship between its body segments and the stiffness at each joint. The results indicate that the proposed control scheme enables the robot to exhibit remarkable real-time adaptability over various frictional and inclined terrains. These findings can potentially enable us to gain a deeper insight into the autonomous decentralized control mechanism underlying the adaptive and resilient locomotion of animals.

Spatially adapted regularization parameter selection based on the local discrepancy function for Poissonian image deblurring

The total variation (TV) model with a fidelity term of the generalized Kullback–Leibler (KL) divergence is a classical method for Poissonian image deblurring. In this paper, we propose a new TV-KL model with a spatially dependent regularization parameter. This model is able to preserve small details of images while homogeneous regions still remain sufficiently smooth. The automated selection of the regularization parameter is based on the local discrepancy function. The corresponding minimization problem with a spatially adapted regularization parameter can be solved efficiently by the split Bregman method. Numerical experiments demonstrate that the proposed algorithm has the potential to enhance regions of images containing detail and remove Poisson noise simultaneously, which leads to an improvement in the signal-to-noise ratio and the mean absolute error for deblurring results.

On the Mitra–Wan forestry model: A unified analysis

Abstract We present a substantive and far-reaching generalization of the principal results in the economics of forestry, as formalized by Mitra and Wan (1986). Rather than a polarized dichotomy of linear and strictly concave, differentiable benefit (felicity) functions, we develop the theory in the context of functions that are supported at the golden-rule consumption and are not necessarily concave. Through a non-interiority condition on the set of zeroes of a resulting “discrepancy function,” we show the equivalence of finitely-maximal, maximal, minimal value-loss and optimal programs, and thereby answer questions left open by Brock and Mitra. Our synthesizing criterion is new to the capital theory literature, and in the concave setting, proves to be necessary and sufficient for the asymptotic convergence of good programs.

Quantifying Simulator Discrepancy in Discrete-Time Dynamical Simulators

When making predictions with complex simulators it can be important to quantify the various sources of uncertainty. Errors in the structural specification of the simulator, for example due to missing processes or incorrect mathematical specification, can be a major source of uncertainty, but are often ignored. We introduce a methodology for inferring the discrepancy between the simulator and the system in discrete-time dynamical simulators. We assume a structural form for the discrepancy function, and show how to infer the maximum-likelihood parameter estimates using a particle filter embedded within a Monte Carlo expectation maximization (MCEM) algorithm. We illustrate the method on a conceptual rainfall-runoff simulator (logSPM) used to model the Abercrombie catchment in Australia. We assess the simulator and discrepancy model on the basis of their predictive performance using proper scoring rules. This article has supplementary material online.

Implications of Parcel-Allocation Variability for Comparing Fit of Item-Solutions and Parcel-Solutions

This article relates a still-popular motivation for using parceling to an unrecognized cost. The still-popular motivation is improvement in fit with respect to the item-solution. The cost is uncertainty in fit due to the selection of one out of many possible item-to-parcel allocations. A theoretical framework establishes the reason for this relationship: The same mechanisms that cause larger item- versus parcel-solution differences in the minimized discrepancy function also cause larger allocation to allocation variability in the parcel-solution's minimized discrepancy function. Study 1 illustrates that these shared causal mechanisms lead to a strong positive association between average item–parcel differences in minimized discrepancy function values and parcel-allocation variability in those values. Study 2 extends these results from discrepancy function values to fit indexes, showing that the association remains positive, but varies in magnitude depending on what quantities other than the discrepancy function are involved in computing the fit index. The important implication for practice is that when item–parcel fit differences are large enough to alter conclusions about model adequacy, parcel-allocation variability tends to be large enough for parcel-solution model adequacy to depend on the particular allocation chosen.

Function spaces and optimal currents in impedance tomography

Abstract The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j, which contain the most information about the difference between the unknown physical conductivity σ* and a given approximation σ 0. The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.

On the applicability of the decentralized control mechanism extracted from the true slime mold: a robotic case study with a serpentine robot

A systematic method for an autonomous decentralized control system is still lacking, despite its appealing concept. In order to alleviate this, we focused on the amoeboid locomotion of the true slime mold, and extracted a design scheme for the decentralized control mechanism that leads to adaptive behavior for the entire system, based on the so-called discrepancy function. In this paper, we intensively investigate the universality of this design scheme by applying it to a different type of locomotion based on a ‘synthetic approach’. As a first step, we implement this design scheme to the control of a real physical two-dimensional serpentine robot that exhibits slithering locomotion. The experimental results show that the robot exhibits adaptive behavior and responds to the environmental changes; it is also robust against malfunctions of the body segments due to the local sensory feedback control that is based on the discrepancy function. We expect the results to shed new light on the methodology of autonomous decentralized control systems.

A New Calibrated Bayesian Internal Goodness-of-Fit Method: Sampled Posterior p-Values as Simple and General p-Values That Allow Double Use of the Data

BackgroundRecent approaches mixing frequentist principles with Bayesian inference propose internal goodness-of-fit (GOF) p-values that might be valuable for critical analysis of Bayesian statistical models. However, GOF p-values developed to date only have known probability distributions under restrictive conditions. As a result, no known GOF p-value has a known probability distribution for any discrepancy function.Methodology/Principal FindingsWe show mathematically that a new GOF p-value, called the sampled posterior p-value (SPP), asymptotically has a uniform probability distribution whatever the discrepancy function. In a moderate finite sample context, simulations also showed that the SPP appears stable to relatively uninformative misspecifications of the prior distribution.Conclusions/SignificanceThese reasons, together with its numerical simplicity, make the SPP a better canonical GOF p-value than existing GOF p-values.

The effects of social networks on choice set dynamics: Results of numerical simulations using an agent-based approach

Activity-based analysis has slowly shifted gear from the analysis of daily activity patterns to the analysis and modeling of dynamic activity-travel patterns. In this paper, we address one type of dynamics: the formation and adaptation of location choice sets under influence of dyad relationships within social networks. It extends the dynamic model developed in earlier work, which simulates habitual behavior versus exploitation and exploration as a function of discrepancies between dynamic, context-dependent aspiration levels and expected outcomes. Principles of social comparison and knowledge transfer are used in modeling the impact of social networks through information exchange, adaptations of spatial choice sets and formation of common aspiration levels. We demonstrate model properties using numerical simulation with a case study of shopping activities.

Ridge structural equation modelling with correlation matrices for ordinal and continuous data

This paper develops a ridge procedure for structural equation modelling (SEM) with ordinal and continuous data by modelling the polychoric/polyserial/product-moment correlation matrix R. Rather than directly fitting R, the procedure fits a structural model to R(a) =R+aI by minimizing the normal distribution-based discrepancy function, where a > 0. Statistical properties of the parameter estimates are obtained. Four statistics for overall model evaluation are proposed. Empirical results indicate that the ridge procedure for SEM with ordinal data has better convergence rate, smaller bias, smaller mean square error, and better overall model evaluation than the widely used maximum likelihood procedure.

Bayesian Temporal Source Attribution of Foodborne Zoonoses:Campylobacterin Finland and Norway

Statistical source attribution approaches of food-related zoonoses can generally be based on reported diagnosed human cases and surveillance results from different food sources or reservoirs of bacteria. The attribution model, or probabilistic classifier, can thus be based on the (sub)typing information enabling comparison between human infections and samples derived from source surveillance. Having time series of both data allows analyzing temporal patterns over time providing a repeated natural experiment. A Bayesian approach combining both sources of information over a long time series is presented in the case of Campylobacter in Finland and Norway. The full model is transparently presented and derived from the Bayes theorem. Previous statistical source attribution approaches are here advanced (1) by explicit modeling of the cases not associated with any of the sources under surveillance over time, (2) by modeling uncertain prevalence in a food source by bacteria type over time, and (3) by implementing formal model fit assessment using posterior predictive discrepancy functions. Large proportion of all campylobacteriosis can be attributed to broiler, but considerable uncertainty remains over time. The source attribution is inherently incomplete if only the sources under surveillance are included in the model. All statistical source attribution approaches should include a model fit assessment for judgment of model performance with respect to relevant quantities of interest. It is especially relevant when the model aims at a synthesis of several incomplete information sources under significant uncertainty of explanatory variables.

Robust Economic Implications of Nonlinear Pricing Kernels

Based on a family of discrepancy functions, we derive nonparametric stochastic discount factor (SDFs) bounds that naturally generalize variance (Hansen and Jagannathan, 1991), entropy (Backus, Chernov and Martin, 2011), and higher-moment (Snow, 1991) bounds. These bounds are especially useful to identify how parameters affect pricing kernel dispersion in asset pricing models. In particular, they allow us to distinguish between models where dispersion comes mainly from skewness from models where kurtosis is the primary source of dispersion. We analyze the admissibility of disaster, disappointment aversion and long-run risk models with respect to these bounds.

Proper Orthogonal Decomposition and Radial Basis Functions in material characterization based on instrumented indentation

For the mechanical characterization of structural materials non-destructive tests combined with computer simulations and inverse analyses are more and more frequently and advantageously employed in various engineering fields. The contribution to such development presented in this paper can be outlined as follows. With reference to isotropic elastic–plastic material models, indentation test simulations are done preliminarily, once-for-all, by a conventional finite element forward operator. Results of these simulations are employed in a procedure which is centered on Proper Orthogonal Decomposition and Radial Basis Functions approximation and is used for fast interpolations which replace further finite element analyses in the parameter identification process. Comparative computational exercises are presented in order to point out the consequent significant reduction of computing times in test simulations and, hence, in the minimization of the discrepancy function by the Trust Region Algorithm, namely by a traditional first-order mathematical programming method. Such a parameter identification procedure may be carried out routinely and economically on small computers for in situ structural diagnoses. Both the force–penetration relationship (provided by an instrumented indenter) and the average imprint profile (achievable by laser profilometer) are considered as sources of measurable response quantities or experimental data.

On lower bounds for the [formula omitted]-discrepancy

The L 2 -discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the L 2 -discrepancy of arbitrary N -point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L 2 -discrepancy in dimension 2 are well known, and are of the sharp order log N . Nevertheless, the gap in the constants between the best-known lower and upper bounds is unsatisfactorily large for a two-dimensional problem. Our lower bound improves upon this situation considerably. The main method is an adaption of Roth’s method, using the Fourier coefficients of the discrepancy function with respect to the Haar basis. We obtain the same improvement in the quotient of lower and upper bounds in the general d -dimensional case. Our lower bounds are also valid for the weighted discrepancy.

The Concavity Assumption on Felicities and Asymptotic Dynamics in the RSS Model

An analysis of the RSS model in mathematical economics involves the study of an infinite-horizon variational problem in discrete time. Under the assumption that the felicity function is upper semicontinuous and “supported” at the value of the maximally-sustainable level of a production good, we report a generalization of results on the equivalence, existence and asymptotic convergence of optimal trajectories in this model. We consider two parametric specifications, and under the second, identify a “symmetry” condition on the zeroes of a “discrepancy function” underlying the objective function that proves to be necessary and sufficient for the asymptotic convergence of good programs. With a concave objective function, as is standard in the antecedent literature, we show that the symmetry condition reduces to an equivalent “non-interiority” condition.

Minimum distance estimation in normed linear spaces with Donsker-classes

We consider minimum distance estimators where the discrepancy function is defined in terms of a supremum-norm based on a Donsker-class of functions. If the parameter set is contained in a normed linear space we prove a Portmanteau-type theorem. Here, the limit in general is not a probability measure, but an outer measure given by the hitting family of the set of all minimizing points of a certain stochastic process. In case there is exactly one minimizer one obtains traditional weak convergence.

Multimodality Imaging of Peripheral Neuropathies of the Upper Limb and Brachial Plexus

The peripheral nerves of the upper limb are affected by a number of entrapment and compression neuropathies. These discrete syndromes involve the brachial plexus as well as the musculocutaneous, axillary, suprascapular, ulnar, radial, and median nerves. Clinical examination and electrophysiologic studies are the traditional mainstay of diagnostic work-up; however, ultrasonography and magnetic resonance imaging provide spatial information regarding the affected nerve and its surroundings, often assisting in narrowing the differential diagnosis and guiding treatment. Imaging is particularly valuable in complex cases with discrepant nerve function test results. Familiarity with the clinical features of various peripheral neuropathies of the upper extremity, the relevant anatomy, and the most common sites and causes of nerve entrapment assists in diagnosis and treatment.

Bayesian Estimation of Prediction Error and Variable Selection in Linear Regression

Summary An important statistical application is the problem of determining an appropriate set of input variables for modelling a response variable. In such an application, candidate models are characterized by which input variables are included in the mean structure. A reasonable approach to gauging the propriety of a candidate model is to define a discrepancy function through the prediction error associated with this model. An optimal set of input variables is then determined by searching for the candidate model that minimizes the prediction error. In this paper, we focus on a Bayesian approach to estimating a discrepancy function based on prediction error in linear regression. It is shown how this approach provides an informative method for quantifying model selection uncertainty.