Quadratic Spline Research Articles

Polar forms and quadratic spline quasi-interpolants on Powell--Sabin partitions

In this paper we introduce the Powell-Sabin B-spline representation of quadratic polynomials or splines in terms of their polar forms. We use this B-representation for constructing several differen...

Humanoid robot Lola: Design and walking control

In this paper we present the humanoid robot LOLA, its mechatronic hardware design, simulation and real-time walking control. The goal of the LOLA-project is to build a machine capable of stable, autonomous, fast and human-like walking. LOLA is characterized by a redundant kinematic configuration with 7-DoF legs, an extremely lightweight design, joint actuators with brushless motors and an electronics architecture using decentralized joint control. Special emphasis was put on an improved mass distribution of the legs to achieve good dynamic performance. Trajectory generation and control aim at faster, more flexible and robust walking. Center of mass trajectories are calculated in real-time from footstep locations using quadratic programming and spline collocation methods. Stabilizing control uses hybrid position/force control in task space with an inner joint position control loop. Inertial stabilization is achieved by modifying the contact force trajectories.

Quadratic spline quasi-interpolants and collocation methods

Univariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocation methods are deduced for the solution of ordinary or partial differential equations with boundary conditions. Their convergence properties are illustrated and compared with finite difference methods on some numerical examples of elliptic boundary value problems.

Solitary waves for the MRLW equation

This work presents a computational comparison study of quadratic, cubic, quartic and quintic splines for solving the modified regularized long wave (MRLW) equation. Collocation schemes with quadratic and cubic splines are found to be unconditionally stable. The fourth-order Runge–Kutta method has been used to solve the collocation schemes when quartic and quintic B-splines are used. The three invariants of motion have been evaluated to determine the conservation properties of the suggested algorithms. Comparisons of results due to different schemes with the exact values shows the accuracy and efficiency of the proposed schemes. Results corresponding to higher order splines are more accurate than those corresponding to lower order splines.

The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem

We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the discrete minimum principle. Error bounds for the numerical approximations are established. Numerical results give justification of the parameter-uniform convergence of the numerical approximations.

How to Obtain Long Term Projections for Smoking Behaviour: A Case Study in the Dutch Population

A state‐transition model is presented to project the prevalence rates of never, current and former smokers within the Dutch population. Changes are determined by the transition rates among these three classes: start rates, quit rates and relapse rates. Both the initial prevalence rates and transition rates are calculated from cross‐sectional data using a restricted quadratic multinomial regression spline and a restricted quadratic logistic regression spline, respectively. Through a Monte Carlo experiment an uncertainty analysis was performed to assess the level of reliability of the results, while a sensitivity analysis was carried out in order to detect the input variables that mostly contribute to the output variability. The results obtained through this model show a considerable but rather slow decrease of smokers up to year 2050.

FMF-14 OPTIMUM WINDING TENSION AND NIP-LOAD INTO WOUND WEBS FOR PROTECTING WRINKLES AND SLIPPAGE(Flexible Media Feeding and Handling Machines IV,Technical Program of Oral Presentations)

This paper presented about the optimization technique (SQP) for wound roll in order to protect the problem of wrinkles and the slippage phenomena into wound roll. Regarding the optimum design, the objective function which includes tangential stresses and friction forces would be presented in this paper. Tangential stresses and radial stresses could be predicted by Hakiels nonlinear model. The friction coefficients could be calculated from air entrainment which was approximated by Hamrock and Dowson model as focusing on within nip-load. The design variables in this paper were winding tensions and were updated by the quadratic spline interpolation method. The optimum winding tension and nip-load condition from numerical results would be used for experimental. The study shows the results of the minimum air films between web layers, friction forces and the in-roll stress distributions in case of constant winding tensions, taper winding tension, and optimum winding tensions.

Minimal-norm-derivative spline function in interpolation and approximation

The paper is concerned with applications of quadratic splines with minimal derivative to approximation of functions in approximation and interpolation problems. A smooth spline is constructed on a uniform mesh so as the norm of the spline derivative is minimal; the nodes of the spline and the nodes of interpolations coincide. This approach allows construction of a spline from given values of the function on the mesh without additional assignment of the value of the function derivative at the initial point, because the derivative can be determined from the minimality condition for the norm of the spline derivative in L2.

A robust layer-resolving spline collocation method for a convection–diffusion problem

We consider finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. The problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified Shishkin mesh. The position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle. We prove pointwise convergence of order O ( N - 2 ln 2 N ) inside the boundary layer and second order convergence elsewhere. The uniform convergence of the approximate continual solution is also given. Further, we approximate normalized flux and give estimates of the error at the mesh points and between them. The numerical experiments presented in the paper confirm our theoretical results.

A Shape Representation with Elastic Quadratic Polynomials—Preservation of High Curvature Points under Noisy Conditions

We present a new shape representation technique for a planar shape using overlapping quadratic splines. The technique refines the shape by iteratively updating each spline to reduce the cost associated with C 1 discontinuity. It consists of a series of affine commutative linear operators, producing a smooth bandpass frequency response. The primary purpose of the technique is to remove minute high-curvature points while preserving salient ones. We compare the performance of the technique against those that are based on either linear splines, cubic splines, or wavelets. We consider three criteria: sensitivity of detecting salient high-curvature points, Hausdorff distance between the representation and the original shape, and computation time. The results show that the proposed technique is highly effective in preserving salient high curvature points with a relatively small Hausdorff distance and a computational cost.

DESIGN AND EVALUATION OF MATCHED WAVELETS WITH MAXIMUM CODING GAIN AND MINIMUM APPROXIMATION ERROR CRITERIA FOR R PEAK DETECTION IN ECG

Recently, several wavelet-based algorithms have been proposed for feature extraction in non-stationary signals such as ECG. These methods, however, have mainly used general purpose (unmatched) wavelet bases such as Daubechies and Quadratic Spline. In this paper, five new matched wavelet bases, with minimum approximation error and maximum coding gain criteria, are designed and applied to ECG signal analysis. To study the effect of using different wavelet bases for this application, two different wavelet-based R peak detection algorithms are implemented: (1) a conventional wavelet-based method; and (2) a modified wavelet-based R peak detection algorithm. Both algorithms are evaluated using the MIT-BIH Arrhythmia database. Experimental results show lower computational complexity (up to 76%) of the proposed R peak detection method compared to the conventional method. They also show considerable decrease in the number of failed detections (up to 55%) for both the conventional and the proposed algorithms when using matched wavelets instead of Quadratic Spline wavelet which, according to the literature, has generated the best detection results among all conventional wavelet bases studied previously for ECG signal analysis.

A [formula omitted] quadratic trivariate macro-element space defined over arbitrary tetrahedral partitions

In 1988, Worsey and Piper constructed a trivariate macro-element based on C 1 quadratic splines defined over a split of a tetrahedron into 24 subtetrahedra. However, this local element can only be used to construct a corresponding macro-element spline space over tetrahedral partitions that satisfy some very restrictive geometric constraints. We show that by further refining their split, it is possible to construct a macro-element also based on C 1 quadratic splines that can be used with arbitrary tetrahedral partitions. The resulting macro-element space is stable and provides full approximation power.

Voice source parameter measurement based on multi-scale analysis of electroglottographic signal

This paper deals with glottal parameter measurement from electroglottographic signal (EGG). The proposed approach is based on GCI and GOI determined by the multi-scale analysis of the EGG signal. Wavelet transform of EGG signal is done with a quadratic spline function. Wavelet coefficients calculated on different dyadic scales, show modulus maxima at localized discontinuities of the EGG signal. The detected maxima and minima correspond to the so-called GOIs and GCIs. To improve the GCI and GOI localization precision, the product of wavelet transform coefficients of three successive dyadic scales, called multi-scale product (MP), is operated. This process enhances edges and reduces noise and spurious peaks. Applying the cubic root amplitude on the multi-scale product improves the detection of weak GOI maximum and avoids the GCI misses. Applied on the Keele University database, the method brings about a good detection of GCI and GOI. Based on the GCI and GOI, voicing classification, pitch frequency and open quotient measurements are processed. The proposed voicing classification approach is evaluated with additive noise. For clean signal the performance is of 96.4%, and at SNR level of 5 dB, the performance is of 93%. For the fundamental frequency and the open quotient measurement, the comparison of the MP with the DEGG, Howard (3/7), the threshold (35% and 50%), and the DECOM methods show that this new proposed approach is similar to the major methods with an improvement displayed by its lowest deviation.

Special multivariate quadratic spline space

In this paper, we present a special multivariate quadratic spline space S 2 1 , 0 ( ♦ ) over a refined quadrangulation. We get its dimension, and construct its basis splines. At the same time, we obtain the explicit representations of the basis splines by the smoothing cofactor-conformality method. The approximation properties of two constructed quasi-interpolation operators are discussed, some supporting numerical results are presented. We also compare our spline with some traditional splines in this paper. The results can be applied to many fields such as CAGD, function approximation, numerical analysis, finite element method, and so on.

Refined-mesh interpolation method for functions with a boundary-layer component

Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented.

Optimal superconvergent one step quadratic spline collocation methods

We formulate new optimal quadratic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructed so that the collocation equations can be solved using a matrix decomposition algorithm. The results of numerical experiments exhibit the expected optimal global accuracy as well as superconvergence phenomena.

The impact of excess heat events in Maricopa County, Arizona: 2000–2005

Exposure to excess heat is preventable yet it is the primary weather-related cause of mortality in the United States. In the Southwest United States, high temperatures are common and indoor environments often have cooling devices. In summer 2005, Maricopa County, Arizona experienced a 182% increase in reported heat-related deaths in comparison to 2000--2004. We examined at-risk populations and excess mortality. We characterized heat-related deaths using descriptive and multivariate time-series analyses of county vital record data from June-September 2000--2005. Dose-response relationships for heat-related mortality and heat index were evaluated using linear and quadratic splines. From June-September, 2000--2005, 136 heat-related deaths (0.68 per 100,000) were reported; 49 (36%) occurred in 2005. In July 2005, a 14-day heat wave resulted in 28 (57%) reported deaths--a 102% increase in comparison to the same time period in 2000--2004. Decedent demographics in 2005 did not differ from previous years. The mean age of all 136 deaths was 56 years (range: 7-92 years). Of those with discernable reported injury locations, 62 (66%) were identified outdoors. Forty-eight (77%) decedents identified outdoors were < 5 years; conversely, 26 (82%) decedents who were found indoors were > or = 65 years. A 6% (95% CI: 1.00-1.13) increase in mortality risk was observed for each degree (F) increase in heat index. Excess heat impacted a younger population in Maricopa County and many deaths occurred outdoors. Consecutive days of heat exposure--even among a heat-acclimated population--can increase mortality risk. Public health response activities guided by locally obtained data will better target those at risk.

Modeling time variant distributions of cellular lifespans: increases in circulating reticulocyte lifespans following double phlebotomies in sheep.

Many pharmacodynamic (PD) models of cellular response assume a single and time invariant lifespan of all cells, despite the existence of a true underlying distribution of cellular lifespans and known changes in the lifespan distributions with time. To account for these features of cellular populations, a time variant cellular lifespan distribution PD model was formulated and theoretical aspects of modeling cellular populations presented. The model extends prior work assuming time variant "point distributions" of cellular lifespans (Freise et al. J Pharmacokinet Pharmacodyn 34:519-547, 2007) and models assuming a time invariant lifespan distribution (Krzyzanski et al. J Pharmacokinet Pharmacodyn 33:125-166, 2006). The formulated time variant lifespan distribution model was fitted to endogenous plasma erythropoietin (EPO), reticulocyte, and red blood cell (RBC) concentrations in sheep phlebotomized on two occasions, 8 days apart. The time variant circulating reticulocyte lifespan was modeled as a truncated and scaled Weibull distribution, with the location parameter of the distribution non-parametrically represented by an end constrained quadratic spline function. The formulated time variant lifespan distribution model was compared to the identical time invariant distribution, time variant "point distribution", and time invariant "point distribution" cellular lifespan models. Parameters of the time variant lifespan distribution model were well estimated with low standard errors. The mean circulating reticulocyte lifespan was estimated at 0.304 days, which rapidly increased over 3-fold following the first phlebotomy to a maximum of 1.03 days (P = 0.009). On average, the percentage of erythrocytes being released as reticulocytes maximally increased an estimated two-fold following the phlebotomies. The primary features of immature RBC physiology were captured by the model and gave results consistent with other estimates in sheep and humans. The comparison of the four lifespan models gave similar parameter estimates of the stimulation function and fits to the RBC data. However, the time invariant models fit the reticulocyte data poorly, while the time variant "point distribution" cellular lifespan model gave physiologically unrealistic estimates of the changes in the circulating reticulocyte lifespan under stress erythropoiesis. Thus the underlying physiology must be considered when selecting the most appropriate cellular lifespan model and not just the goodness-of-fit criteria. The proposed PD model and the numerical implementation allows for a flexible framework to incorporate time variant lifespan distributions when modeling populations of cells whose production or stimulation depends on endogenous growth factors and/or exogenous drugs.

Polar forms and quadratic spline quasi-interpolants on Powell–Sabin partitions

In this paper we introduce the Powell–Sabin B-spline representation of quadratic polynomials or splines in terms of their polar forms. We use this B-representation for constructing several differential or discrete quasi-interpolants which have an optimal approximation order. This new approach is simple and provides an efficient tool for describing many schemes of approximation involving values and (or) derivatives of a given function.

Did it fall or was it pushed? The contribution of trends in established risk factors to the decline in premature coronary heart disease mortality in New Zealand

To estimate the contribution of trends in three risk factors--systolic blood pressure (SBP), total blood cholesterol (TBC) and cigarette smoking--to the decline in premature coronary heart disease (CHD) mortality in New Zealand from 1980-2004. Risk factor prevalence data by 10-year age group (35-64 years) and sex was sourced from six national or Auckland regional health surveys and three population censuses (the latter only for smoking). The data were smoothed using two-point moving averages, then further smoothed by fitting quadratic regression equations (SBP and TBC) or splines (smoking). Risk factor/CHD mortality hazard ratios estimated by expert working groups for the World Health Organization Global Burden of Disease Study 2001 were used to translate average annual changes in risk factor prevalences to the corresponding percentage changes in premature CHD mortality. The expected trends in CHD mortality were then compared with the observed trend to estimate the contribution of each risk factor to the decline. Approximately 80% (73% for males, 87% for females) of the decline in premature CHD mortality from 1980 to 2004 is estimated to have resulted from the joint trends in population SBP and TBC distributions and smoking prevalence. Overall, approximately 42%, 36% and 22% of the joint risk factor effect was contributed by trends in SBP, TBC and smoking respectively. Our estimate for the joint risk factor contribution to the CHD mortality decline of 80% exceeds those of two earlier New Zealand studies, but agrees closely with a similar Australian study. This provides an indicator of the scope that still remains for further reduction in CHD mortality through primary and secondary prevention.