Quadratic Spline Research Articles

Multi-threshold de-noising of electrical imaging logging data based on the wavelet packet transform

A key problem of effectiveness evaluation for fractured-vuggy carbonatite reservoir is how to accurately extract fracture and vug information from electrical imaging logging data. Drill bits quaked during drilling and resulted in rugged surfaces of borehole walls and thus conductivity fluctuations in electrical imaging logging data. The occurrence of the conductivity fluctuations (formation background noise) directly affects the fracture/vug information extraction and reservoir effectiveness evaluation. We present a multi-threshold de-noising method based on wavelet packet transform to eliminate the influence of rugged borehole walls. The noise is present as fluctuations in button-electrode conductivity curves and as pockmarked responses in electrical imaging logging static images. The noise has responses in various scales and frequency ranges and has low conductivity compared with fractures or vugs. Our de-noising method is to decompose the data into coefficients with wavelet packet transform on a quadratic spline basis, then shrink high-frequency wavelet packet coefficients in different resolutions with minimax threshold and hard-threshold function, and finally reconstruct the thresholded coefficients. We use electrical imaging logging data collected from fractured-vuggy Ordovician carbonatite reservoir in Tarim Basin to verify the validity of the multi-threshold de-noising method. Segmentation results and extracted parameters are shown as well to prove the effectiveness of the de-noising procedure.

A spline collocation method for integrating a class of chemical reactor equations

. In this paper, we develop a quadratic spline collocation method for integrating the nonlinear partial differential equations (PDEs) of a plug flow reactor model. The method is proposed in order to be used for the operation of control design and/or numerical simulations. We first present the Crank-Nicolson method to temporally discretize the state variable. Then, we develop and analyze the proposed spline collocation method for the spatial discretization. The design of the collocation method is interpreted as one order error convergent. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence.

Quadratic Spline Function for the Approximate Solution of an Intermediate Space-Fractional Advection Diffusion Equation

The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.

An integro quadratic spline approach for a class of variable-order fractional initial value problems

This paper develops a technique for the approximate solution of a class of variable-order fractional differential equations useful in the area of fluid dynamics. The method adopts a piecewise integro quadratic spline interpolation and is used in the study of the variable-order fractional Bagley–Torvik and Basset equations. The accuracy of the proposed algorithm is verified by means of illustrative examples.

A quadrature-based superconvergent isogeometric frequency analysis with macro-integration cells and quadratic splines

A quadrature-based superconvergent isogeometric method is presented for efficient frequency analysis with macro-integration cells and quadratic splines. The present development particularly refers to the isogeometric analysis of scalar wave equations. The single element, two-element and three-element macro-integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy, in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix. The 1D quadrature rules with various macro-integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix. Consequently, 3-point, 5-point and 7-point quadrature rules with identical precision are established for the single element, two-element and three-element macro-integration cells, respectively. By construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1D case. The 2D and 3D superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1D integration algorithms. It is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem, which needs special treatments for the multidimensional higher order mass matrix formulation. The proposed approach is featured by its simplicity for numerical implementation and efficiency using macro-integration cells. Numerical examples confirm the efficacy of the present methodology.

Analysis of switched quantizer based on the quadratic spline functions

ABSTRACTIn this paper, an approximation of the optimal compressor function using the quadratic spline functions with 2L = 8 segments is described. Since the quadratic spline with 2L = 8 segments provides better approximation of the optimal compression function than quadratic spline with 2L = 4 segments, capitalizing on the benefits of the obtained spline approximation, quantizer designing process is firstly performed for the so assumed number of segments and the Laplacian source of a unit variance. Then, to enhance the usability of the proposed model, the switched quantization technique is applied and a beneficial analysis is derived, providing insight in the robustness of the proposed quantizer performances with respect to the mismatch in designed for and applied to variances. Reached quality has been compared to another model from the literature, and it has been shown that the proposed model outperforms the previous model by almost 1.3 dB.

An investigation of export–import ratios in Turkey using spline regression models

This paper examines the use of spline functions in linear, squared, and cubic spline regression models and exhibits the estimation of spline parameters from data by ordinary least squares. Determination of the number and the location of knots is central to spline regression. In this paper, we initially propose a method based on the coefficient of determination R2 related to the estimation of knots in spline regression. This proposed method as applied to export–import ratio distributions in Turkey for the years 1923–2010 determines the knots, and linear, quadratic, and cubic spline regression models are established accordingly. Results reveal that spline regression models offer better results than polynomial regression models, and that the quadratic spline regression model is the best explanatory model for export–import ratio distributions in the smoothest spline regression models.

A Hybrid Wavelet-Based Method for the Peak Detection of Photoplethysmography Signals.

The noninvasive peripheral oxygen saturation (SpO2) and the pulse rate can be extracted from photoplethysmography (PPG) signals. However, the accuracy of the extraction is directly affected by the quality of the signal obtained and the peak of the signal identified; therefore, a hybrid wavelet-based method is proposed in this study. Firstly, we suppressed the partial motion artifacts and corrected the baseline drift by using a wavelet method based on the principle of wavelet multiresolution. And then, we designed a quadratic spline wavelet modulus maximum algorithm to identify the PPG peaks automatically. To evaluate this hybrid method, a reflective pulse oximeter was used to acquire ten subjects' PPG signals under sitting, raising hand, and gently walking postures, and the peak recognition results on the raw signal and on the corrected signal were compared, respectively. The results showed that the hybrid method not only corrected the morphologies of the signal well but also optimized the peaks identification quality, subsequently elevating the measurement accuracy of SpO2 and the pulse rate. As a result, our hybrid wavelet-based method profoundly optimized the evaluation of respiratory function and heart rate variability analysis.

Viral kinetics in untreated versus treated acute HIV infection in prospective cohort studies in Thailand.

Introduction: The extent of viral replication during acute HIV infection (AHI) influences HIV disease progression. However, information comparing viral load (VL) kinetics with and without antiretroviral therapy (ART) in AHI is limited. The knowledge gained could inform preventive strategies aimed at reducing VL during AHI and therapeutic strategies to alter the viral kinetics that may enhance the likelihood of achieving HIV remission.Methods: The analysis utilized VL data captured during the first year of HIV infection from two studies in Thailand: the RV217 study (untreated AHI, 30 participants and 412 visits) and the RV254 study (treated AHI, 235 participants and 2803 visits). Fiebig stages were I/II (HIV RNA+, HIV IgM−) and Fiebig III/IV (HIV IgM+, Western blot-/indeterminate). Data were modelled utilizing spline effects within a linear mixed model, with a random intercept and slope to allow for between-subject variability and adjustment for the differences in variability between studies. The number of knots in the quadratic spline basis functions was determined by comparing models with differing numbers of knots via the Akaike Information Criterion. Models were fit using PROC GLIMMIX in SAS v9.3.Results: At enrolment, there were 24 Fiebig I/II and 6 Fiebig III/IV individuals in the untreated group and 137 Fiebig I/II and 98 Fiebig III/IV individuals in the treated group. Overall, the median age was 27.5 years old, most were male (89%), and CRF01_AE was the most common HIV clade (76%). By day 12 (4 days after ART in RV254), the untreated group had a 2.7-fold higher predicted mean VL level compared to those treated (predicted log VL 6.19 for RV217 and 5.76 for RV254, p = 0.05). These differences increased to 135-fold by day 30 (predicted log VL 4.89 for RV217 and 2.76 for RV254) and 1148-fold by day 120 (predicted log VL 4.68 for RV217 and 1.63 for RV254) (p < 0.0001 for both) until both curves were similarly flat at about day 150 (p = 0.17 between days 150 and 160). The VL trajectories were significantly different between Fiebig I/II and Fiebig III/IV participants when comparing the two groups and within the treated group (p < 0.001 for both).Conclusions: Initiating ART in AHI dramatically changed the trajectory of VL very early in the course of infection that could have implications for reducing transmission potential and enhancing responses to future HIV remission strategies. There is an urgency of initiating ART when acute infection is identified. New and inexpensive strategies to engage and test individuals at high risk for HIV as well as immediate treatment access will be needed to improve the treatment of acute infection globally.Clinical Trial Number: NCT00796146 and NCT00796263

A Study on the Stiffness Characteristics according to the Body Pressure on the Seat Cushion for Vehicle

Objectives: Since the riding comfort upon seating is determined according to the driving environment or the driver’s preference, objective evaluations are accompanied by many constraints. Methods/Statistical Analysis: Variations in stiffness distribution of the seat cushion were analyzed by using body pressure distribution. First, to identify loaddeflection characteristics of the seat cushion, static load tests were conducted and stiffness characteristics as a function of load were affirmed by application of quadratic spline interpolation method. And, body pressure distribution of the male subject produced upon seating in the cushion was measured under actual vehicle conditions. Findings: Variations in stiffness distribution of the seat cushion produced when a driver comes aboard a vehicle, and the following conclusions could be obtained. First, by conducting static load tests for the seat cushion to obtain load-deflection curves, deflection characteristics per load point were identified. Second, through quadratic function for the load-deflection curves derived from static load tests, stiffness distributions were derived. Stiffness characteristics were continuously varied as a function of load distribution applied to the seat cushion, which was analyzed to have an effect on stiffness distribution as a function of body pressure distribution for the driver. Third, based on body pressure distribution data for the male subject and load-deflection curves for the seat cushion, variations in stiffness of the seat cushion were analyzed as a function of body pressure distribution. As a result, the extent of hardness felt by the human body was greater for the right side even if body pressure of the male subject was shown to be larger in the left-side region. Application/Improvements: Characteristics of seat stiffness distribution as a function of body pressure distribution which were derived in the present study are considered to be utilized for pad development of car seat.

Optimal designs for spline wavelet regression models

In this article we investigate the optimal design problem for some wavelet regression models. Wavelets are very flexible in modeling complex relations, and optimal designs are appealing as a means of increasing the experimental precision. In contrast to the designs for the Haar wavelet regression model (Herzberg and Traves 1994; Oyet and Wiens 2000), the I-optimal designs we construct are different from the D-optimal designs. We also obtain c-optimal designs. Optimal (D- and I-) quadratic spline wavelet designs are constructed, both analytically and numerically. A case study shows that a significant saving of resources may be realized by employing an optimal design. We also construct model robust designs, to address response misspecification arising from fitting an incomplete set of wavelets.

Implementation of a sigmoid depth function to describe change of soil pH with depth

Soil pH controls the availability of the majority of plant nutrients, if not all, and determines the growth environment for plant roots. Profile depth functions have been used to represent the vertical distribution of soil attributes and to predict them at continuous depths. This paper proposes a new model to predict pH for a whole soil profile. Soil properties including pH are often similar within the plow layer from mixing during tillage and other agricultural operations. Similarly, soil pH below the root zone tends to be uniform due to low disturbance, leaving a transition zone from the bottom of the tillage layer to the bottom of the root zone due to depth-dependent root density and related soil processes. Keeping this physical description of agricultural field soil profile in mind, a closed form equation (model) was developed similar to a sigmoid curve. The model has 4 parameters including 1) soil pH at the top of a soil profile, 2) soil pH at the bottom of a soil profile, 3) hillslope parameter representing steepness of the curve that is determined by the length of the root zone, and 4) inflection point representing almost the midpoint of the transition zone or root zone. A total of 32 soil cores down to about 1.1m depths were collected from an agricultural field of Macdonald farm, McGill University. The sub-samples were taken at every 10cm and analyzed for soil pH in soil: water suspension in the laboratory. The measured pH was used to test the fitting performance of the sigmoid model. Additionally, a global dataset with 432 profiles with various soil classes, drainage types, land use, and altitude was also used to test the generality of the new model. The performance of this model was compared with the results of the commonly used 3rd order polynomial regression function and the equal-area quadratic spline function. Good performance of the sigmoid model with explicit physical explanation showed promise in predicting soil pH at depths. The spline function had the highest accuracy but lacked a general trend in its shape and parameters. The polynomial function had good accuracy and displayed a non-monotonic trend, which can also be used as a substitute for some profiles with complex variability.

A Fast and Stable Algorithm for Linear Parabolic Partial Differential Equations

We report a new algorithm for solving linear parabolic partial differential equations (PDEs) in one space dimension. The algorithm employs optimal quadratic spline collocation (QSC) for the space discretization and deferred correction (DC) for the time discretization. We can obtain a numerical solution with theoretical convergence order \(\mathcal {O}({\Delta } x^{4}+{\Delta } t^{\min [k_{0}+1,N_{m}]})\) by solving k0+1 tridiagonal linear systems at each time step, where k0 is the number of deferred corrections per time step and Nm is usually an integer which is not bigger than 8. The stability properties of the new algorithm are analyzed, and compared with the QSC-CN0 algorithm [5], which is a very efficient algorithm. Numerical experiments are attached to demonstrate the effectiveness of the new algorithm.

Associations of Breast Cancer Risk Factors with Premenopausal Sex Hormones in Women with Very Low Breast Cancer Risk.

Breast cancer incidence rates are low but rising in urban Mongolia. We collected reproductive and lifestyle factor information and measured anthropometrics and serum sex steroid concentrations among 314 premenopausal women living in Ulaanbaatar, Mongolia. Mean differences in hormone concentrations by these factors were calculated using age-adjusted quadratic regression splines. Estrone and estradiol in college-educated women were, respectively, 18.2% (p = 0.03) and 23.6% (p = 0.03) lower than in high-school-educated women. Progesterone concentrations appeared 55.8% lower (p = 0.10) in women residing in modern housing compared with women living in traditional housing (gers), although this finding was not statistically significant. Testosterone concentrations were positively associated with adiposity and central fat distribution; 17.1% difference (p = 0.001) for highest vs. lowest quarter for body mass index and 15.1% difference (p = 0.005) for waist-to-height ratio. Estrogens were higher in the follicular phase of women who breastfed each child for shorter durations. A distinct hormonal profile was associated with an urban lifestyle in premenopausal, Mongol women. In particular, heavier, more-educated women living in urban dwellings had higher testosterone and lower estrogen and progesterone levels. Higher breast cancer incidence in urban compared with rural women suggest that the hormonal profile associated with a more traditional lifestyle may be protective among Mongol women.

Quadratic spline collocation method and efficient preconditioner for the Helmholtz equation with the Sommerfeld boundary conditions

In this paper, using the quadratic spline collocation method (QSC), we numerically solve the Helmholtz equations with the Sommerfeld boundary conditions. By reordering the unknowns, we obtain a $$3\times 3$$ block linear system. Then, we introduce a two-step preconditioner based on the approximate inverse block polynomial preconditioner. Theoretical analysis show this preconditioner can largely gather the eigenvalues around 1. Numerical examples are presented to test the error of QSC method and check the efficiency of the presented preconditioner.

Analytic integration of singular kernels for boundary element analysis of plane orthotropic media

Abstract Both composite materials and stress concentration are common issues in modern structural engineering. In this paper, analytical integration of the singular kernels is performed for boundary element analysis (BEM) of elastic, plane orthotropic media with stress concentrations. Analytical integration leads to accuracy and efficiency improvements over FEM. Furthermore, high continuity (HC), quadratic spline interpolation on the boundary is used to further improve accuracy at low computational cost when compared to FEM. The advantages of BEM for calculation of displacements and stresses near stress raisers in orthotropic plates are shown. Particular attention is paid to efficient interpolation for approximating boundary quantities and to precision of computation for evaluating boundary integrals. Such improvements lead to accurate computation of both displacements and stresses in both the boundary and the domain. Thus, the advantages of the proposed method are accuracy and low computational cost.

Detailed analysis of the impact of age on the QT interval.

ObjectiveTo analyze the effect of age on the ECG QT interval, an important predictor of cardiovascular mortality and drug-induced cardiac arrhythmias, and determine whether QT-heart rate correction formulae (QTc) have differential relationships with age and sex.MethodsData were examined from the US National Health and Nutrition Examination Survey (NHANES) II and III, civilian population aged 25 to 90 years. QT weighted means and standard deviations were calculated for all ages. The QTc were evaluated for six QTc: proposed by Bazett (QTcBZT), Fridericia (QTcFRD), Hodges (QTcHDG), Dmitrienko (QTcDMT), Rautaharju (QTcRTHa) and Framingham (QTcFRM).ResultsQTc was strongly related to age and gender, for all formulae except for QTcBZT for women. The relationship between QTc and age was significant regardless of whether the relationship was approximated by a linear or non-linear (quadratic or cubic spline) model. QTc increased more dramatically with age in men. There was a significant (P < 0.001) positive relationship between QTc variance and age for each QTc formula for both men and women. There were a greater proportion of individuals with longer QTc with older ages especially age 80 years and above.ConclusionQTc and its variance increase with age. Prolonged QTc is more prevalent in older individuals, especially men.

Strain Echocardiography and Functional Capacity in Asymptomatic Primary Mitral Regurgitation With Preserved Ejection Fraction

BackgroundThe potential additive utility of baseline resting left ventricular global longitudinal strain (LV-GLS) and exercise stress testing in risk stratification of patients with significant mitral regurgitation (MR) has not been studied. ObjectivesThe goal of this study was to determine whether resting LV-GLS and exercise testing provide incremental prognostic utility in asymptomatic patients with ≥3+ primary MR and preserved left ventricular ejection fraction. MethodsBetween 2000 and 2011, resting and exercise echocardiography data, Society of Thoracic Surgeons (STS) scores, and death were recorded in 737 patients (mean age 58 ± 13 years; 68% men). ResultsCoronary artery disease and flail leaflet were seen in 10% and 28% of patients, respectively. STS score, resting left ventricular ejection fraction, mitral effective regurgitant orifice, resting right ventricular systolic pressure (RVSP), exercise metabolic equivalents (METs), and percentage of age-/sex-predicted METs were 1.5 ± 1%, 62 ± 2%, 0.45 ± 0.2 cm2, 31 ± 12 mm Hg, 9.8 ± 3, and 115 ± 27, respectively. Median LV-GLS was –21.7%. Within 3 months (interquartile range: 1 to 15 months), 65% underwent mitral valve surgery. At 8.3 ± 3 years, 64 (9%) patients died (0% 30-day post-operative deaths). On multivariable Cox survival analysis, higher STS score (hazard ratio [HR]: 1.14), more abnormal resting LV-GLS (HR: 1.60), higher baseline RVSP (HR: 1.35), and lower percentage of age-/sex-predicted METs (HR: 1.13) were associated with higher mortality, whereas mitral valve surgery (HR: 0.82) was associated with improved survival (all p < 0.01). Addition of predicted METs and resting LV-GLS to STS, resting RVSP, left ventricular end-systolic dimension, and mitral effective regurgitant orifice increased the C-statistic for longer-term mortality from 0.61 to 0.69 and 0.78, respectively (all p < 0.01). On quadratic spline analysis, the risk of death progressively increased as resting LV-GLS worsened below –21%. ConclusionsReduced exercise capacity and worsening resting LV-GLS were associated with mortality, providing additive prognostic utility.

Numerical benchmarking of fluid–structure interaction: An isogeometric finite element approach

In this paper we describe and evaluate an isogeometric finite element program, IFEM-FSI, for doing coupled fluid–structure interaction simulations. We investigate the role played by employing higher polynomial orders and higher regularity for solving a well known benchmark problem for flow past a circular cylinder with an attached flexible bar at Reynolds number Re=100. Furthermore, we investigate the sensitivity to resolution in the fluid mesh as well as stiffness distribution in the mesh movement algorithm. Mesh quality is also assessed. Our simulations indicate that quadratic and cubic spline elements give better estimation of lift, drag and displacements than linear spline elements.

Numerical solution of fractional bioheat equation by quadratic spline collocation method

Numerical solution of fractional bioheat equation by quadratic spline collocation method