Quadratic Spline Research Articles

Mapping the Three‐Dimensional Distribution of Soil Organic Matter across a Subtropical Hilly Landscape

There is a serious lack of detailed and accurate three‐dimensional soil distribution information worldwide. This study examined the effectiveness of combining radial basis function (RBF) neural networks and profile depth functions to map the three‐dimensional distribution of soil organic matter (SOM) in a subtropical hilly landscape in southern Anhui Province, China. The RBF networks were used to predict the lateral distribution of SOM based on its relations with terrain attributes and land uses, while the depth functions were used to fit its vertical distribution based on sparse measurements of SOM in soil genetic horizons. Compared with power and logarithmic functions, the equal‐area quadratic splines had smaller bias, higher accuracy, and more stable performance in fitting the vertical SOM distribution. The prediction accuracy of the whole three‐dimensional mapping method decreased with depth within the upper 60 cm, while the best accuracy occurred below 60 cm. In the upper 30 cm, areas with high elevation tended to have high predicted SOM content and vice versa. There were local deviations from this pattern in areas where toeslopes and ravines had higher predicted SOM content than backslopes, even though the latter are at higher elevations. Multiple regressions with dummy variables showed that the influence of terrain conditions on SOM content was strong in the upper 60 cm and weak below 60 cm, while that of land use was strong in the upper 30 cm and weak below 30 cm. Both influences were the strongest in the upper 15‐cm soil layer. Under the same terrain conditions, agricultural cultivation is associated with SOM accumulation in the upper 30 cm.

Computational method for solving space fractional Fisher's nonlinear equation

This paper aims to present a general framework of the quadratic spline functions to develop a numerical method for solving the nonlinear space fractional Fisher's equation. Using Von Neumann method, the proposed method is shown to be conditionally stable. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm. The results reveal that the proposed approach is very effective, convenient, and quite accurate to such considered problems. Copyright © 2013 John Wiley & Sons, Ltd.

Electricity Consumption Analysis Using Spline Regression Models: The Case of a Turkish Province

Energy is one of the indispensible elements of human life and electrical energy is adopted as the most frequently used energy type. As this type of energy can not be stored at the present time, it has to be instantly consumed. In other words, the demand of the consumers has to be compensated, immediately. This paper employs to model the electrical consumption of Erzurum province in 2011 by spline regression and to decide whether a statistically seasonal variation exists for this consumption. The one-year data set of the investigation was obtained from Turkish Electricity Transmission Company Provincial Directorate of Erzurum and was analyzed by the agency of continuous partial polynomial spline regressions. This analysis determined three knots and fits linear, quadratic and cubic spline regression models.

Local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation

In this paper, we describe the construction of a suitable normalized B-spline representation for special multivariate quadratic spline space S21,0(Δ) over a refined quadrangulation. We then develop as an application, a general theory of quasi-interpolants based on this representation.

Monotone and convex interpolation by weighted quadratic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted C 1 quadratic splines in such a way that the monotonicity and convexity of the data are preserved...

Blood 25-Hydroxy Vitamin D Levels and Incident Type 2 Diabetes

OBJECTIVETo quantitatively assess the strength and shape of the association between blood 25-hydroxy vitamin D [25(OH)D] levels and incident risk of type 2 diabetes.RESEARCH DESIGN AND METHODSA systematic search of the MEDLINE and Embase databases and a hand search of references from original reports were conducted up to 31 October 2012. Prospective observational studies that assessed the association between blood levels of 25(OH)D and risk of incident type 2 diabetes were included for meta-analysis. DerSimonian and Laird’s random-effects model was used. A quadratic spline regression analysis was used to examine the shape of the association with a generalized least-squares trend test performed for the dose-response relation.RESULTSA total of 21 prospective studies involving 76,220 participants and 4,996 incident type 2 diabetes cases were included for meta-analysis. Comparing the highest to the lowest category of 25(OH)D levels, the summary relative risk for type 2 diabetes was 0.62 (95% CI 0.54–0.70). A spline regression model showed that higher 25(OH)D levels were monotonically associated with a lower diabetes risk. This inverse association did not differ by sex, duration of follow-up, study sample size, diabetes diagnostic criteria, or 25(OH)D assay method. A linear trend analysis showed that each 10 nmol/L increment in 25(OH)D levels was associated with a 4% lower risk of type 2 diabetes (95% CI 3–6; P for linear trend < 0.0001).CONCLUSIONSOur meta-analysis showed an inverse and significant association between circulating 25(OH)D levels and risk of type 2 diabetes across a broad range of blood 25(OH)D levels in diverse populations.

High‐Resolution 3‐D Mapping of Soil Texture in Denmark

Soil texture which is spatially variable in nature, is an important soil physical property that governs most physical, chemical, biological, and hydrological processes in soils. Detailed information on soil texture variability both in vertical and lateral dimensions is crucial for proper crop and land management and environmental studies, especially in Denmark where mechanized agriculture covers two thirds of the land area. We modeled the continuous depth function of texture distribution from 1958 Danish soil profiles (up to a 2‐m depth) using equal‐area quadratic splines and predicted clay, silt, fine sand, and coarse sand content at six standard soil depths of GlobalSoilMap project (0–5, 5–15, 15–30, 30–60, 60–100, and 100–200 cm) via regression rules using the Cubist data mining tool. Seventeen environmental variables were used as predictors and their strength of prediction was also calculated. For example, in the prediction of silt content at 0 to 5 cm depth, factors that registered a higher level of importance included the soil map scored (90%), landscape types (54%), and landuse (27%), while factors with lower scores were direct insolation (17%) and slope aspect (14%). Model validation (20% of the data selected randomly) showed a higher prediction performance in the upper depth intervals but increasing prediction error in the lower depth intervals (e.g.,R2= 0.54, RMSE = 33.7 g kg−1for silt 0–5 cm andR2= 0.29, RMSE = 38.8 g kg−1from 100–200 cm). Danish soils have a high sand content (mean values for clay, silt, fine sand, and coarse sand content for 0‐ to 5‐cm depth were 79, 84, 324, and 316 g kg−1, respectively). Northern parts of the country have a higher content of fine sand compared to the rest of the study area, whereas in the western part of the country there was little clay but a high coarse sand content at all soil depths. The eastern and central parts of the country are rich in clay, but due to leaching, surface soils are clay eluviated with subsequent accumulation at lower depths. We found equal‐area quadratic splines and regression rules to be promising tools for soil profile harmonization and spatial prediction of texture properties at national extentacross Denmark.

Recovering Gradients from Sparsely Observed Functional Data

The recovery of gradients of sparsely observed functional data is a challenging ill-posed inverse problem. Given observations of smooth curves (e.g., growth curves) at isolated time points, the aim is to provide estimates of the underlying gradients (or growth velocities). To address this problem, we develop a Bayesian inversion approach that models the gradient in the gaps between the observation times by a tied-down Brownian motion, conditionally on its values at the observation times. The posterior mean and covariance kernel of the growth velocities are then found to have explicit and computationally tractable representations in terms of quadratic splines. The hyperparameters in the prior are specified via nonparametric empirical Bayes, with the prior precision matrix at the observation times estimated by constrained ℓ₁ minimization. The infinitessimal variance of the Brownian motion prior is selected by cross-validation. The approach is illustrated using both simulated and real data examples.

DELO-BEZIER FORMAL SOLUTIONS OF THE POLARIZED RADIATIVE TRANSFER EQUATION

We present two new accurate and efficient method to compute the formal solution of the polarized radiative transfer equation. In this work, the source function and the absorption matrix are approximated using quadratic and cubic Bezier spline interpolants. These schemes provide 2nd and 3rd order approximation respectively and don't suffer from erratic behavior of the polynomial approximation (overshooting). The accuracy and the convergence of the new method are studied along with other popular solutions of the radiative transfer equation, using stellar atmospheres with strong gradients in the line-of-sight velocity and in the magnetic-field vector.

Numerical Determination of the Optimal Value of Quantizer’s Segment Threshold Using Quadratic Spline Functions

In this paper, an approximation of the optimal compressor function using the quadratic spline functions has been presented. The coefficients of the quadratic spline functions are determined by minimizing the mean-square error (MSE). Based on the obtained approximative quadratic spline functions, the design for companding quantizer for Gaussian source is done. The support region of proposed companding quantizer is divided on segments of unequal size, where the optimal value of segment threshold is numerically determined depending on maximal value of the signal to quantization noise ratio (SQNR). It is shown that by the companding quantizer proposed in this paper, the SQNR that is very close to SQNR of nonlinear optimal companding quantizer is achieved.

An Arc Fault Detection Method Based on Wavelet Feature Extraction and The Design & Realization by LabWindows/CVI

Arc fault is one of the primary reasons that cause electrical fire. When arc fault occurs in power supply line, the current can not make protective equipments act and the arc fault can not be found and cut off easily, so electrical fire comes into being. In the comparisons, the current signal is used for the detection physical parameter and the wavelet transform is used to study the arc fault detection. First according to the select principle of wavelet singularity detection, select and construct orthogonal quadratic spline wavelet as wavelet function, and use the porous algorithm dyadic wavelet transform to realize wavelet transform fast algorithm. Then carried out the arc fault's wavelet singularity detection through modulus maxima detection method, finally analyzed the current signal by the method of wavelet approximation. LabWindows/CVI is used as development platform to design and implement the above analysis. Based on LabWindows/CVI, the upper computer program uses multithreading technology to detect and analyze the current signal. By arc fault detection algorithm, the system judges whether there has arc fault or not. The algorithm judges whether there has arc fault or not through detect if there has periodicity singularity points or not. The experiments show that this detection method of arc fault can detect arc fault in power supply circuits exactly and efficaciously.

Numeric Solution of the Fokker-Planck-Kolmogorov Equation

The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested; 2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.

Iterative methods for solving nonlinear equations by using quadratic spline function

In this paper, iterative methods of order three and four constructed based on quadratic spline function for solving nonlinear equations. Several numerical examples are given to illustrate the efficiency and performance of the iterative methods; the methods are also compared with some other known iterative methods.

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Healthcare electronics count on the effectiveness of the on-patient signal preprocessing unit to moderate the wireless data transfer for better power efficiency. In order to reduce the system power in long-time ECG acquisition, this work describes an on-patient QRS detection processor for arrhythmia monitoring. It extracts the concerned ECG part, i.e., the RR-interval between the QRS complex for evaluating the heart rate variability. The processor is structured by a scale-3 quadratic spline wavelet transform followed by a maxima modulus recognition stage. The former is implemented via a symmetric FIR filter, whereas the latter includes a number of feature extraction steps: zero-crossing detection, peak (zero-derivative) detection, threshold adjustment and two finite state machines for executing the decision rules. Fabricated in 0.35-μm CMOS the 300-Hz processor draws only 0.83 μW, which is favorably comparable with the prior arts. In the system tests, the input data is placed via an on-chip 10-bit SAR analog-to-digital converter, while the output data is emitted via an off-the-shelf wireless transmitter (TI CC2500) that is configurable by the processor for different data transmission modes: 1) QRS detection result, 2) raw ECG data or 3) both. Validated with all recordings from the MIT-BIH arrhythmia database, 99.31% sensitivity and 99.70% predictivity are achieved. Mode 1 with solely the result of QRS detection exhibits 6× reduction of system power over modes 2 and 3.

GAM analysis of the relationship between DRV PK and pharmacodynamics following DRV/r 800/100 mg qd in the phase III trials ARTEMIS and ODIN

BackgroundOnce‐daily (qd) darunavir/ritonavir (DRV/r) 800/100mg was evaluated in two randomised and controlled Phase III trials, showing high response rates in treatment‐naïve (ARTEMIS; TMC114‐C211) and treatment‐experienced patients with no DRV resistance‐associated mutations (ODIN; TMC114‐C229). Treatment interruptions, medical co‐morbidities, drug‐drug interactions, and challenges to adherence can decrease exposures and result in suboptimal DRV coverage. Consequences of suboptimal dosing could be detrimental to patients, namely the selection of drug‐resistant HIV strains. No relevant relationship between DRV area under the plasma concentration‐time curve over 24 hours (AUC24h) or plasma trough concentrations (C0h) and efficacy or safety were observed in ARTEMIS or ODIN [1,2]. The PK/PD analysis for efficacy has been extended using a generalized additive model (GAM).MethodsA univariate GAM analysis was undertaken for each trial, exploring the relationship between DRV C0h and virological response at Week 48. A quadratic spline (2 degrees of freedom) was selected, based on the Akaike information criterion value. Additionally, a linear logistic regression model was applied to model virologic response as a function of DRV C0h.ResultsIn ARTEMIS, there was no relationship between DRV C0h and the observed virological response or virological response predicated in the GAM analysis, with no indication that lower values within the observed DRV C0h range are associated with lower predicted virological response (Figure A). The mean predicted virological response was 92.8% in the total population (N = 304). A 50% reduction in DRV C0h would result in a similar predicted mean virological response (93.9%). In ODIN, for values of DRV C0h up to 3000ng/mL, there was no clear relationship between DRV C0h and observed virological response. The GAM analysis showed a counter‐intuitive inverse relationship between DRV C0h and predicted virological response (Figure B). The mean predicted virological response was 80.6% in the total population (N = 252). For each trial, similar results to GAM were obtained when a linear logistic regression model was applied.ConclusionIn patients who received 800/100mg qd DRV/r in the ARTEMIS and ODIN trials, a reduction in DRV C0h of up to 50% is not associated with a reduced virologic response at Week 48, as predicted by both the GAM and logistic regression model. DRV/r doses below 800/100mg qd should not be administered.image

Research and Realization of Highly Accurate Data Generation Based on Channel Simulator

With high accuracy, the channel simulator plays an important role in the docking experiment between the ground and the responder beacon. To begin with, this paper introduces the data generation algorithm including the data generation based on simulation technology, the principle of the linear least squares algorithm and then proposes the least squares quadratic spline method to generate highly accurate data in this channel simulator. Secondly, this paper introduces the system design to realize the data generation. Finally, a case which studies the approximation and an error analysis of the data generation algorithm is realized. The algorithm is considered to be accurate and easy to get the source data. The core of the algorithm is using data from Satellite Tool Kit to generate distance and speed sequence, and using the least squares to approximate real data and quadratic spline to fit for obtaining highly accurate data.

Multivariate normalized Powell–Sabin B-splines and quasi-interpolants

We present the construction of a multivariate normalized B-spline basis for the quadratic C1-continuous spline space defined over a triangulation in Rs (s⩾1) with a generalized Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction can be interpreted geometrically as the determination of a set of s-simplices that must contain a specific set of points. We also propose a family of quasi-interpolants based on this multivariate Powell–Sabin B-spline representation. Their spline coefficients only depend on a set of local function values. The multivariate quasi-interpolants reproduce quadratic polynomials and have an optimal approximation order.

Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ℝ2. By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.

Error bounds on the approximation of functions and partial derivatives by quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of a rectangular domain

Given a non-uniform criss-cross triangulation of a rectangular domain Ω, we consider the approximation of a function f and its partial derivatives, by general C1 quadratic spline quasi-interpolants and their derivatives. We give error bounds in terms of the smoothness of f and the characteristics of the triangulation. Then, the preceding theoretical results are compared with similar results in the literature. Finally, several examples are proposed for illustrating various applications of the quasi-interpolants studied in the paper.