Quadratic Rule Research Articles

Designing Core-Selecting Payment Rules: A Computational Search Approach

Combinatorial auctions are regularly used to allocate resources worth billions of dollars. However, finding optimal payment rules for such auctions is still an open problem. To this end, we develop a new computational search framework for finding payment rules with desirable properties. We show that the rule most commonly used in practice, the quadratic rule, can be improved upon in terms of efficiency, incentives and revenue. Our best-performing rules are so-called large-style rules—that is, they provide better incentives to bidders with larger values. Ultimately, we identify two particularly well-performing rules and suggest that they be considered for practical implementation in place of the currently used rule.

On the numerical treatment and analysis of Hammerstein integral equation

In this paper, we study the quadratic rules for the numerical solution of Hammerstein integral equation based on spline quasi-interpolant. Also the convergence analysis of the methods are given. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.

Component Combination Rules of Wind Load Effects of Building Structures

Under the action of the same wind azimuth, the extreme values of the wind load effect components of building structures are generated in the along-wind, cross-wind, vertical, and torsional directions. In designing the wind-resistant structure, the extreme values of effect components need to be combined to determine the internal force envelope values of members. Complete quadratic combination (CQC) and Turkstra combination rules are often used to determine the combination value of extreme values of wind effect components. The extreme probability distribution expressions of the CQC, and the Turkstra and approximate rules, are derived. The simplified combination Equations and combination coefficients of the CQC and Turkstra approximate rules are proposed in this paper. We use the combination Equations and Monte Carlo simulation method to analyze the accuracy of Turkstra and its approximate rules. The results show that the combination extreme is associated with the correlation coefficients, mean values, ratios of standard deviations, and fluctuating extremes of effect components. The errors between Turkstra and its approximate rules are small when load effect components show a positive correlation. The errors are largest when the standard deviations of components are equal. Our research results provide a theoretical basis for the combination method of wind load effect components of building structures.

Robust and efficient WLS-based dynamic state estimation considering transformer core saturation

For transformer asset management, the dynamic state estimation can track transient phenomena of transformers or extract an unmeasurable quantity such as magnetic flux linkage. Even the extended Kalman filter or unscented Kalman filter, which are widely-used dynamic state estimations, might result in significant estimation errors when dealing with highly nonlinear dynamics such as transformer core saturation. In this sense, this paper proposes a dynamic state estimation based on the weighted least squares, using real-time measurements and the dynamic model induced by numerically integrating differential equations. The proposed method can directly use implicit functions including differential equations while Kalman filters must rearrange the implicit functions into the explicit ones to obtain the prediction process. Further, because the weighted least squares approach is implemented via the Newton iterative method, the estimation accuracy and robustness to various conditions can be improved despite strong nonlinearity. For the numerical integration of dynamics, the computationally efficient trapezoidal rule or the numerically stable quadratic rule is used. This paper presents numerical simulation results, with a comparison with other dynamic state estimators.

How Many Imputations Do You Need? A Two-stage Calculation Using a Quadratic Rule

When using multiple imputation, users often want to know how many imputations they need. An old answer is that 2–10 imputations usually suffice, but this recommendation only addresses the efficienc...

On the numerical treatment and analysis of two-dimensional Fredholm integral equations using quasi-interpolant

In this paper, we study the quadratic rule for the numerical solution of linear and nonlinear two-dimensional Fredholm integral equations based on spline quasi-interpolant. Also the convergence analysis of the method is given. We show that the order of the method is $$O(h_{x}^{m+1})+O(h_{y}^{m^{\prime }+1})$$. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.

Perfect tracking control of discrete-time quadratic TS fuzzy systems via feedback linearisation

ABSTRACTPerfect tracking control is an important and frequently encountered requirement in various industries (e.g. robotic control). We developed a novel systematic framework for designing a fuzzy controller via feedback linearisation to control a class of discrete-time Takagi–Sugeno (TS) fuzzy systems with quadratic rule consequents to achieve such tracking. We established a necessary condition for its local stability and a necessary and sufficient condition for the boundedness of the controller. The feedback linearisation is known to fail to work in certain systems due to the unboundedness of the tracking controller output. To address this issue, we developed a method to check whether any given quadratic TS fuzzy system will cause such a failure. We developed a scheme to ensure that the output of the controller designed for any failure-causing system will be bounded and the resulting controller will attain nearly perfect tracking performance. Applying feedback linearisation to the quadratic fuzzy systems is innovative relative to the literature exclusively dealing with the TS fuzzy systems with linear rule consequents (including our previous results), which are now generalised by the new findings. Two numerical examples are provided to illustrate the effectiveness and utility of our new theoretical results.

Efficient procedures for the weighted squared tardiness permutation flowshop scheduling problem

This paper addresses a permutation flowshop scheduling problem, with the objective of minimizing total weighted squared tardiness. The focus is on providing efficient procedures that can quickly solve medium or even large instances. Within this context, we first present multiple dispatching heuristics. These include general rules suited to various due date-related environments, heuristics developed for the problem with a linear objective function, and procedures that are suitably adapted to take the squared objective into account. Then, we describe several improvement procedures, which use one or more of three techniques. These procedures are used to improve the solution obtained by the best dispatching rule. Computational results show that the quadratic rules greatly outperform the linear counterparts, and that one of the quadratic rules is the overall best performing dispatching heuristic. The computational tests also show that all procedures significantly improve upon the initial solution. The non-dominated procedures, when considering both solution quality and runtime, are identified. The best dispatching rule, and two of the non-dominated improvement procedures, are quite efficient, and can be applied to even very large-sized problems. The remaining non-dominated improvement method can provide somewhat higher quality solutions, but it may need excessive time for extremely large instances.

Linear components of quadratic classifiers

We obtain a decomposition of any quadratic classifier in terms of products of hyperplanes. These hyperplanes can be viewed as relevant linear components of the quadratic rule (with respect to the underlying classification problem). As an application, we introduce the associated multidirectional classifier; a piecewise linear classification rule induced by the approximating products. Such a classifier is useful to determine linear combinations of the predictor variables with ability to discriminate. We also show that this classifier can be used as a tool to reduce the dimension of the data and helps identify the most important variables to classify new elements. Finally, we illustrate with a real data set the use of these linear components to construct oblique classification trees.

How Many Imputations Do You Need? A Two-stage Calculation Using a Quadratic Rule.

When using multiple imputation, users often want to know how many imputations they need. An old answer is that 2-10 imputations usually suffice, but this recommendation only addresses the efficiency of point estimates. You may need more imputations if, in addition to efficient point estimates, you also want standard error (SE) estimates that would not change (much) if you imputed the data again. For replicable SE estimates, the required number of imputations increases quadratically with the fraction of missing information (not linearly, as previous studies have suggested). I recommend a two-stage procedure in which you conduct a pilot analysis using a small-to-moderate number of imputations, then use the results to calculate the number of imputations that are needed for a final analysis whose SE estimates will have the desired level of replicability. I implement the two-stage procedure using a new SAS macro called %mi_combine and a new Stata command called how_many_imputations.

Probabilistic Forecasting with Stationary VAR Models

I use a set of vector autoregressive models to forecast some of the main macroeconomic variables in a wide range of countries. The goal is to provide some insight about different forecast accuracy measures in a probabilistic forecasting framework. The countries are selected based on their available data in the International Financial Statistics (IFS) database. The target variables are GDP, CPI and exchange rate growth in yearly, quarterly and monthly frequencies. I specify the model sets by using a list of potentially relevant variables from IFS database. The results are available up to four forecast horizons and for two different classes of forecast accuracy measures: The probabilistic class which contains linear, logarithmic, quadratic, Hyvarinen and continuous ranked probability score rules, and the non-probabilistic class which contains mean absolute and mean square error measures. The results show that the forecasts are sensitive to the choice of a forecast accuracy class, in sense of Hellinger distance. However, within each class, the choice is not critical. Among different probabilistic measures, the logarithmic and quadratic rules are more reliable in practice. The results also show that the forecast power of stationary multivariate models is higher than the univariate ones, for both probabilistic and non-probabilistic accuracy measures.

The Model and Quadratic Stability Problem of Buck Converter in DCM

Quadratic stability is an important performance for control systems. At first, the model of Buck Converter in DCM is built based on the theories of hybrid systems and switched linear systems primarily. Then quadratic stability of SLS and hybrid feedback switching rule are introduced. The problem of Buck Converter’s quadratic stability is researched afterwards. In the end, the simulation analysis and verification are provided. Both experimental verification and theoretical analysis results indicate that the output of Buck Converter in DCM has an excellent performance via quadratic stability control and switching rules.

Discriminant Analysis of Interval Data: An Assessment of Parametric and Distance-Based Approaches

Building on probabilistic models for interval-valued variables, parametric classification rules, based on Normal or Skew-Normal distributions, are derived for interval data. The performance of such rules is then compared with distancebased methods previously investigated. The results show that Gaussian parametric approaches outperform Skew-Normal parametric and distance-based ones in most conditions analyzed. In particular, with heterocedastic data a quadratic Gaussian rule always performs best. Moreover, restricted cases of the variance-covariance matrix lead to parsimonious rules which for small training samples in heterocedastic problems can outperform unrestricted quadratic rules, even in some cases where the model assumed by these rules is not true. These restrictions take into account the particular nature of interval data, where observations are defined by both MidPoints and Ranges, which may or may not be correlated. Under homocedastic conditions linear Gaussian rules are often the best rules, but distance-based methods may perform better in very specific conditions.

Investigation of aberration characteristics of eyes at a peripheral visual field by individual eye model.

We propose a method of constructing an individual eye model with a large visual field, and then investigate aberration characteristics of eyes in peripheral fields with constructed models. Twelve eyes of different aberrations are selected from 89 myopic eyes. It is shown that astigmatism increases as visual field in a quadratic manner. The variation tendency of defocus can be expressed by the cubic curve for 50% of eyes. For most of the eyes, the variation of spherical aberration shows a quadratic rule within ±24° visual field. Coma exhibits obvious individual differences. The impact of high-order aberrations on vision is mainly at a smaller visual field, and it becomes negligible beyond 24° visual field.

Centralised cooperative spectrum sensing under correlated shadowing

The authors consider centralised cooperative spectrum sensing under correlated shadowing in this study. Formulating the spectrum sensing problem as a Gauss–Gauss hypothesis test, they use a linear quadratic rule and show that it is the optimal detector under Bayesian criterion. They derive the upper and lower Bhattacharyya bounds and investigate the error performance of spectrum sensing by studying the behaviour of these upper and lower bounds. They also study the asymptotic error performance in two different scenarios of finite and infinite area networks. They show that by increasing the number of nodes the sensing error probability approaches zero in both cases but with different decay rates. The lower the correlation between nodes or the larger the network area, the faster the decay.

Classification of Higher-order Data with Separable Covariance and Structured Multiplicative or Additive Mean Models

Although devised in 1936 by Fisher, discriminant analysis is still rapidly evolving, as the complexity of contemporary data sets grows exponentially. Our classification rules explore these complexities by modeling various correlations in higher-order data. Moreover, our classification rules are suitable to data sets where the number of response variables is comparable or larger than the number of observations. We assume that the higher-order observations have a separable variance-covariance matrix and two different Kronecker product structures on the mean vector. In this article, we develop quadratic classification rules among g different populations where each individual has κth order (κ ≥2) measurements. We also provide the computational algorithms to compute the maximum likelihood estimates for the model parameters and eventually the sample classification rules.

Studies of the hybrid effect in mechanical properties of tencel blended ring-, rotor- and air-jet spun yarns

This paper presents a study of hybrid effect on mechanical properties of tencel blended ring-, rotor- and Murata jet spinning yarns. The tensile strength, breaking elongation, flexural rigidity and abrasion resistance of tencel-polyester and tencel-cotton blended yarns are examined and compared with the value predicted using linear and quadratic rule of mixture (ROM). It is observed that there is no significant hybrid effect on abrasion resistance of tencel-polyester mix yarns and tenacity and breaking extension of tencel-cotton mix yarns; hence, a linear ROM is found better in predicting the yarn properties. Further, for all yarn types, there is a significant hybrid effect on tenacity, breaking extension and flexural rigidity of tencel-polyester mixes and flexural rigidity and abrasion resistance of tencel-cotton mixes. A quadratic ROM model suits well for these yarns. Overall, it is found that there is a negative hybrid effect on tenacity and breaking extension whereas there is a positive hybrid effect on flexural rigidity for all the yarn samples irrespective of fibre and yarn type. The abrasion resistance has a mixed trend. Further, tencel-polyester yarns yield more satisfactory results than the tencel-cotton yarns in terms of strength and breaking extension but behave poorly during abrasion test for 200 cycles. Increasing tencel content both in tencel-polyester and tencel-cotton fibre-mix produces a rigid yarn with reduced abrasion resistance.

A Quadratic Classification Rule with Equicorrelated Training Vectors for Non Random Samples

This article proposes a quadratic extension of the traditional classification rules for multiple classes based on equicorrelated training vectors by using multivariate equicorrelation. This new classification rule can incorporate within sample dependencies, thus appropriate for non random samples.

Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations

Rule of mixtures are an essential feature of the modeling of plastic deformation in complex materials in which more than one strain-hardening mechanism is involved. In this work, use is made of dislocation dynamics simulations to characterize the individual and the superposed contributions of two major mechanisms of crystal plasticity, i.e. Orowan strengthening and forest hardening. Based on a formal description of each hardening mechanism, evidence is presented to show that a quadratic rule of mixtures has the ability to predict quantitatively the flow stress of complex materials such as reactor pressure vessel steel.

An Extended Projection Data Depth and Its Applications to Discrimination

This article investigates the possible use of our newly defined extended projection depth (abbreviated to EPD) in nonparametric discriminant analysis. We propose a robust nonparametric classifier, which relies on the intuitively simple notion of EPD. The EPD-based classifier assigns an observation to the population with respect to which it has the maximum EPD. Asymptotic properties of misclassification rates and robust properties of EPD-based classifier are discussed. A few simulated data sets are used to compare the performance of EPD-based classifier with Fisher's linear discriminant rule, quadratic discriminant rule, and PD-based classifier. It is also found that when the underlying distributions are elliptically symmetric, EPD-based classifier is asymptotically equivalent to the optimal Bayes classifier.