General Weight Function Research Articles

A weighted localization of halfspace depth and its properties

Statistical depth functions are well-known nonparametric tools for analysing multivariate data. Halfspace depth is most frequently used, and while it has many desirable properties, it is dependent on global characteristics of the underlying distribution. In some circumstances, however, it may be desirable to take into account more local and intrinsic characteristics of the data. To this end, we introduce weighted halfspace depths in which the indicator function of closed halfspace is replaced by a more general weight function. Our approach, which calls in part on functions associated with conic sections, encompasses as special cases the notions of sample halfspace depth and kernel density estimation. We give several illustrations and prove the strong uniform consistency of weighted halfspace depth incorporating mild conditions on the weight function.

Generalized Bounded Tree Cover of a Graph

A tree cover of a graph is a collection of subgraphs such that each vertex is a part of at least one subgraph and each subgraph is a tree. The bounded tree cover problem (BTC) finds a tree cover with minimum number of trees of bounded weight. This paper considers several generalized versions of BTC. The first problem deals with graphs having multiple metric weight functions. Strong and weak tree cover problems are two variations of the first problem. In strong tree cover, every tree must be bounded with respect to all weight functions, whereas in weak tree cover, each tree must be bounded with respect to at least one weight function. A 4-approximation algorithm for strong tree cover and an $O(\log n)$-approximation algorithm for weak tree cover problem are proposed. The objective of the second problem is to find a tree cover where bounds of the trees in the tree cover are not necessarily same. It is proved that this problem cannot be approximated within a constant factor unless P=NP. A constant factor approximation algorithm is proposed when the ratio of maximum and minimum bounds is bounded by a constant. The third problem considers BTC for a graph with a general weight function which is not necessarily metric. A 3-approximation algorithm is proposed for this problem.

Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.

A generalized class of form-invariant bivariate weighted distributions

ABSTRACTWeighted random variables are used to model sampling methods with unequal sampling probabilities proportional to non negative weight functions, that is, when we want to study original distributions under weighted samples. Recently, Alavi and Chinipardaz [Alavi, S.M.R., Chinipardaz, R. (2009). Form-invariance under weighted sampling. Statistics. 43(1):81–90] have introduced two new classes of form-invariant bivariate weighted distributions. In this paper these classes have been generalized, under a more general weight function . The class includes some of custom continuous bivariate models. Some properties of the class are explained and maximum likelihood estimates of parameters are obtained.

Oscillation of Second Order Nonlinear Differential Equations with a Damping Term

A class of second-order nonlinear differential equations with a damping term is investigated in this paper. By using the Riccati transformation technique and general weight functions, we obtain some new sufficient conditions for the oscillation of the equation. Our results improve and extend some known results. Two examples are given to illustrate the main results.

IMPROVED BACKPROPAGATION ALGORITHMS BY EXPLOITING DATA REDUNDANCY IN LIMITED-ANGLE DIFFRACTION TOMOGRAPHY

Filtered backpropagation (FBPP) is a well-known technique used in Diffraction Tomography (DT). For accurate reconstruction of a complex-valued image using FBPP, full 360 ◦ angular coverage is necessary. However, it has been shown that by exploiting inherent redundancies in the projection data, accurate reconstruction is possible with 270 ◦ coverage. This is called the minimal-scan angle range. This is done by applying weighting functions (or filters) on projection data of the object to eliminate the redundancies. There could be many general weight functions. These are all equivalent at 270 ◦ coverage but would perform differently at lower angular coverages and in presence of noise. This paper presents a generalized mathematical framework to generate weight functions for exploiting data redundancy. Further, a comparative analysis of different filters when angular coverage is lower than minimal-scan angle of 270 ◦ is presented. Simulation studies have been done to find optimum weight filters for sub-minimal angular coverage. The optimum weights generate images comparable to a full 360 ◦ coverage FBPP reconstruction. Performance of the filters in the presence of noise is also analyzed. These fast and deterministic algorithms are capable of correctly reconstructing complex valued images even at angular coverage of 200 ◦ while still under the FBPP regime.

Kloosterman sums in residue classes

We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any xed primitive congruence class up to bounds towards the Ramanujan conjecture.

The exactness of a general Skoda complex

We show that a Skoda complex with a general plurisubharmonic weight function is exact if its 'degree' is sufficiently large. This answers a question of Lazarsfeld and implies that not every integrally closed ideal is equal to a multiplier ideal even if we allow general plurisubharmonic weights for the multiplier ideal, extending the result of Lazarsfeld and Lee \cite{LL}.

Stress intensity factors calculation for surface crack in cylinders under longitudinal gradient pressure using general point load weight function

ABSTRACTGeneral point load weight function (GPLWF) for a longitudinal semi‐elliptical crack in a thick‐walled cylinder with outer radius to inner radius ratio of 1.25 is derived. GPLWF is used to calculate stress intensity factors (SIFs) for any point along the crack front with the crack faces subjected to uniform and longitudinal gradient internal pressure. The comparison between the results obtained from the GPLWF and those available in the literature is presented, indicating an acceptable accuracy. The proposed GPLWF can be easily used in order to calculate SIFs for surface crack in cylinders subjected to any longitudinal gradient mechanical and thermal load to determine the pattern of crack growth and estimate the fatigue life.

General point load weight function for semi-elliptical crack in finite thickness plates

General point load weight function for semi-elliptical crack in finite thickness plates is derived. It is used to calculate stress intensity factors for any point along the crack front with the crack faces under two dimensional stress distributions. The comparison between the results obtained from the derived weight function and those available in literature is presented, indicating a good accuracy. In order to determine the pattern of crack growth, and to estimate fatigue life, this weight function can be easily used in plates as applied in industry under complicated stress fields such as residual stress field due to welding.

Fast construction of Fejér and Clenshaw–Curtis rules for general weight functions

The main purpose of this paper is to compute the weights of Clenshaw–Curtis and Fejér type quadrature rules via DCT and DST, for general weight functions w. The approach is different from that used by Waldvogel (2006) in [10], where the author considered the computation of these sets only for the Legendre weight w≡1 using DFT arguments.

Point Load Weight Functions for Semi-Elliptical Cracks in Finite Thickness Plate

Abstract This paper presents the application of the weight function method for the calculation of stress intensity factors for surface semi-elliptical cracks in finite thickness plates subjected to arbitrary two-dimensional stress fields. A new general mathematical form of point load weight function has been formulated by taking advantage of the knowledge of a few specific weight functions for two-dimensional planar cracks available in the literature and certain properties of the weight function in general. The existence of the generalized form of the weight function simplifies the determination of a specific weight function for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalized weight function expression. These unknown parameters can be determined from reference stress intensity factor solutions. This method is used to derive the weight functions for semi-elliptical surface cracks in finite thickness plates. The derived weight functions are then validated against stress intensity factor solutions for several linear and non-linear two-dimensional stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planar surface cracks subjected to fluctuating nonlinear stress fields resulting from surface treatment (shot peening), stress concentration, or welding (residual stress)

Bias Correction in Logarithmic Regression and Comparison with Weighted Regression for Nonlinear Models

AbstractNon-linear models with heteroscedasticity are commonly used in ecological and forestry modeling, and logarithmic regression and weighted regression are usually employed to estimate the parameters. Using the single-tree biomass data of three large samples, the bias correction in logarithmic regression for non-linear models was studied and comparison between logarithmic regression and weighted regression was discussed in this paper. Firstly, the imminent cause producing bias in logarithmic regression was analyzed, and a new correction factor was presented with which three commonly used bias correction factors were examined together, and the results showed that the correction factors presented here and derived by Baskerville (1972) should be recommended, which could insure the corrected model to be asymptotically consistent with that fitted by weighted regression. Secondly, the fitting results of weighted regression for non-linear models, using the weight function based on residual errors of the model estimated by ordinary least squares (OLS) and the general weight function (w=1/ƒ(x)2) presented by Zeng (1998) respectively, were compared with each other that showed two weight functions worked well and the general function was more applicable. It was suggested that the best way to fit non-linear models with heteroscedasticity would be using weighted regression, and if the total relative error of the estimates from the model fitted by the general weight function was more than a special allowance such as ±3%, a better weight function based on residual errors of the model fitted by OLS should be used in weighted regression.

Probability inequalities for weighted distributions

In this paper, properties of weighted distributions for general weight functions are investigated. We establish the subadditivity [superadditivity] property of weighted distributions for log-concave [log-convex] weight functions in the sense of the usual stochastic order. The main result generalizes Lemma 2.3 in Brown (2006). Several interesting moment inequalities are presented.

Weighted CUSUM Control Charts for Monitoring Poisson Processes with Varying Sample Sizes

Monitoring inhomogeneous Poisson processes with varying sample sizes has attracted considerable attention recently in both industrial quality control and public-health surveillance. One of the key challenges in these applications is that a change in incidence rate may be masked by changes of sample sizes. The conventional Poisson cumulative sum (CUSUM) chart designed for detecting a prespecified change in the incidence rate could perform poorly when the actual change is different from the prespecified one. This phenomenon is especially prominent when the sample size is nonconstant and varies over time. To efficiently detect changes in the incidence rate with varying sample sizes, this paper proposes a class of weighted CUSUM (WCUSUM) schemes with general weight functions applied to the likelihood-ratio statistic. The comparison results with the traditional Poisson CUSUM method and other alternatives favor the proposed method. An example in health-care surveillance is used to illustrate the application of the proposed method.

Green-Naghdi theory, part A: Green-Naghdi (GN) equations for shallow water waves

In this work, Green-Naghdi (GN) equations with general weight functions were derived in a simple way. A wave-absorbing beach was also considered in the general GN equations. A numerical solution for a level higher than 4 was not feasible in the past with the original GN equations. The GN equations for shallow water waves were simplified here, which make the application of high level (higher than 4) equations feasible. The linear dispersion relationships of the first seven levels were presented. The accuracy of dispersion relationships increased as the level increased. Level 7 GN equations are capable of simulating waves out to wave ve number times depth kd < 26. Numerical simulation of nonlinear water waves was performed by use of Level 5 and 7 GN equations, which will be presented in the next paper.

On Estimation of the Intensity Function of a Point Process

In this paper we review techniques for estimating the intensity function of a spatial point process. We present a unified framework of mass preserving general weight function estimators that encompasses both kernel and tessellation based estimators. We give explicit expressions for the first two moments of these estimators in terms of their product densities, and pay special attention to Poisson processes.

Principal component analysis for interval‐valued observations

AbstractOne feature of contemporary datasets is that instead of the single point value in the p‐dimensional space ℜp seen in classical data, the data may take interval values thus producing hypercubes in ℜp. This paper studies the vertices principal components methodology for interval‐valued data; and provides enhancements to allow for so‐called ‘trivial’ intervals, and generalized weight functions. It also introduces the concept of vertex contributions to the underlying principal components, a concept not possible for classical data, but one which provides a visualization method that further aids in the interpretation of the methodology. The method is illustrated in a dataset using measurements of facial characteristics obtained from a study of face recognition patterns for surveillance purposes. A comparison with analyses in which classical surrogates replace the intervals, shows how the symbolic analysis gives more informative conclusions. A second example illustrates how the method can be applied even when the number of parameters exceeds the number of observations, as well as how uncertainty data can be accommodated. © 2011 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 4: 229–246 2011

Analysis of dynamic stress intensity factors of three-point bend specimen containing crack

A new formula is obtained to calculate dynamic stress intensity factors of the three-point bending specimen containing a single edge crack in this study. Firstly, the weight function for three-point bending specimen containing a single edge crack is derived from a general weight function form and two reference stress intensity factors, the coefficients of the weight function are given. Secondly, the history and distribution of dynamic stresses in uncracked three-point bending specimen are derived based on the vibration theory. Finally, the dynamic stress intensity factors equations for three-pointing specimen with a single edge crack subjected to impact loadings are obtained by the weight function method. The obtained formula is verified by the comparison with the numerical results of the finite element method (FEM). Good agreements have been achieved. The law of dynamic stress intensity factors of the three-point bending specimen under impact loadings varing with crack depths and loading rates is studied.

Online Support Vector Regression With Varying Parameters for Time-Dependent Data

Support vector regression (SVR) is a machine learning technique that continues to receive interest in several domains, including manufacturing, engineering, and medicine. In order to extend its application to problems in which data sets arrive constantly and in which batch processing of the data sets is infeasible or expensive, an accurate online SVR (AOSVR) technique was proposed. The AOSVR technique efficiently updates a trained SVR function whenever a sample is added to or removed from the training set without retraining the entire training data. However, the AOSVR technique assumes that the new samples and the training samples are of the same characteristics; hence, the same value of SVR parameters is used for training and prediction. This assumption is not applicable to data samples that are inherently noisy and nonstationary, such as sensor data. As a result, we propose AOSVR with varying parameters that uses varying SVR parameters rather than fixed SVR parameters and hence accounts for the variability that may exist in the samples. To accomplish this objective, we also propose a generalized weight function to automatically update the weights of SVR parameters in online monitoring applications. The proposed function allows for lower and upper bounds for SVR parameters. We tested our proposed approach and compared results with the conventional AOSVR approach using two benchmark time-series data and sensor data from a nuclear power plant. The results show that using varying SVR parameters is more applicable to time-dependent data.