Quadratic Spline Function Research Articles

Splines for Trend Analysis and Continuous Confounder Control

Spline regression often represents a less biased and more efficient alternative to standard linear, curvilinear, or categorical analyses of continuous exposures and confounders. Benefits of restricted cubic and quadratic splines have been described in the epidemiologic and biomedical literature.1-2 Analogous to the SAS (SAS Institute, Inc., Cary, North Carolina) code provided by Harrell3 for estimating restricted cubic splines, we present straightforward SAS code for estimating restricted quadratic splines. Using data from the HIV clinical cohort at the University of North Carolina Center for AIDS Research,4 we illustrate use of restricted quadratic splines in regression modeling for trend analysis and control of a continuous confounder. Details regarding the functional form of the restricted quadratic splines as well as SAS code for estimating restricted quadratic spline functions are provided in the eAppendix (http://links.lww.com). The data and SAS code used to generate the results included in this paper are also in the eAppendix. First, we illustrate the use of a restricted quadratic splines when estimating the association between log10 HIV-1 viral load centered at 2.301 log10 copies/ml and mortality. Figures S1 A-1C show the unadjusted association between centered log10 HIV-1 viral load at therapy initiation and the relative hazard of death estimated from several Cox proportional hazards models that (A) assume a log-linear relationship, (B) use indicators corresponding to quartiles of centered log10 HIV-1 viral load, or (C) include restricted quadratic splines with 4 equal knots based on the case distribution. Figure 1 Unadjusted associations between centered log10 HIV-1 viral load at therapy initiation and relative hazard of death among 557 male participants in the University of North Carolina Center for AIDS Research HIV clinical cohort, 1999-2010. HIV-1 viral load ... Based on the Akaike information criterion (AIC),5 presented in Figure 1, the restricted quadratic splines model provides the best fit to the data. The P- value for a joint Wald test of the three restricted quadratic splines basis functions included in the model was 0.010. The restricted quadratic splines model suggests a non-log-linear relationship between centered log10 HIV-1 viral load at therapy initiation and the relative hazard of death. Second, we illustrate the use of restricted quadratic splines when controlling for centered log10 HIV-1 viral load as a confounder using a Cox model. The table shows the hazard ratios for the association between an indicator of CD4 cell count ≤350 cells/mm3 at therapy initiation and hazard of death, both unadjusted and adjusted for confounding by viral load at therapy initiation. Adjusting for viral load using a log-linear term attenuated the point estimate corresponding to the CD4 cell count indicator by 26%. Adjustment using restricted quadratic splines with 4 equal knots based on the case distribution attenuated the point estimate by 30%. Attenuation upon control for viral load is expected given that higher viral load was associated with lower CD4 cell count (http://links.lww.com), and an elevated risk of subsequent mortality. Similar results were observed when restricted cubic splines was used instead of a restricted quadratic splines with the same degrees of freedom and comparable knot locations (http://links.lww.com). Table Hazard ratio for association between CD4 cell count less than or equal to 350 cells/mm3 versus greater than 350 cells/mm3 at therapy initiation and death among 557 male participants in the University of North Carolina Center for AIDS Research HIV clinical ... For the first example, use of restricted quadratic splines rather than linear terms or indicators provided a better fit, revealing non-linear relationships that otherwise may have not been apparent. In the second example, use of a restricted quadratic spline resulted in stronger attenuation of a crude association, which likely represents better control of confounding by viral load. The macro presented here offers users a straightforward SAS option for implementing restricted quadratic splines regression. This code is intended to aid in model selection as well as assessing robustness of inferences when comparing various modeling strategies.3,6-7 Furthermore, we hope the examples and code will facilitate the use of splines among researchers hesitant to employ less intuitive but largely equivalent modeling strategies,3,7 and in turn broaden the use of splines in applied epidemiologic research.

Quadratic spline solution for boundary value problem of fractional order

Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.

Shape Preserving Surfaces for the Visualization of Positive and Convex Data using Rational Biquadratic Splines

A smooth surface interpolation scheme for positive and convex data has been developed. This scheme has been extended from the rational quadratic spline function of Sarfraz [11] to a rational bi-quadratic spline function. Simple data dependent constraints are derived on the free parameters in the description of rational bi-quadratic spline function to preserve the shape of 3D positive and convex data. The rational spline scheme has a unique representation. The developed scheme is computationally economical and visually pleasant.

Using multi-scale product spectrum for single and multi-pitch estimation

The authors present an algorithm for pitch estimation including voiced/unvoiced decision in the case of a noisy speech and when two speakers are talking simultaneously. The approach is based on the spectral multi-scale product (SMP) analysis of the sound mixture. SMP is the spectrum of the product of three successive wavelet transform coefficients of the speech. The wavelet used for SMP analysis is the quadratic spline function. The proposed method is compared with other state-of-the-art algorithms. It is robust in the presence of a noise and permits the pitch estimation of the dominant speech and the concurrent one from the sound mixture with high accuracy.

A two-dimensional minimum-derivative spline

We consider the construction of a C (1,1) interpolation parabolic spline function of two variables on a uniform rectangular grid, i.e., a function continuous in a given region together with its first partial derivatives which on every partial grid rectangle is a polynomial of second degree in x and second degree in y. The spline function is constructed as a minimum-derivative one-dimensional quadratic spline in one of the variables, and the spline coefficients themselves are minimum-derivative quadratic spline functions of the other variable.

MPM Method and DYPSA Algorithm Evaluation for GCI Detection in a Noisy Speech Signal

This paper deals with the evaluation and the comparison of two methods for determining glottal closure instants (GCIs) from the speech signal. These methods are the Multi-scale Product Method (MPM) and the Dynamic Programming Phase Slope Algorithm (DYPSA). The MPM consists of determining abscissa of peaks appeared on the product of wavelet transform coefficients at three adjacent dyadic scales. The wavelet used to compute the product is the quadratic spline function. However, the DYPSA algorithm is based on the group delay function to generate the GCI candidates and the dynamic programming technique to select the true GCIs. The comparison of the two algorithms is operated without noise and with additive Gaussian noise at various SNR levels. The evaluation is operated on the Keele University database. Results show that the MPM is more robust to noise than the DYPSA with a good detection rate nearby 90% at an SNR level of 0 dB and a small false alarm rate. Besides, the DYPSA is more sensitive to noise with a good detection rate of 20% at 0 dB and a high false alarm rate surpassing 70%.

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.

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.

Electroglottographic Measures Based on GCI and GOI Detection Using Multiscale Product

This paper deals with glottal parameter estimation such as local pitch and open quotient from electroglottographic signal (EGG). This estimation is based on glottal closing instants and glottal opening instants determined by a multi-scale product of this signal. Wavelet transform of EGG signal is made with a quadratic spline function. Wavelet coefficients calculated on different dyadic scales, show modulus maxima at localized discontinuities of EGG signal. The detected maxima and minima correspond to the glottal opening and closing instants called GOIs and GCIs. To improve the estimate precision, we operate the multi-scale product of wavelet transform coefficients of three successive dyadic scales. This processing enhances edge detection. A Multi-scale product is a nonlinear combination of successive scales; it reduces noise and spurious peaks. We apply cubic root amplitude on the product to improve the representation of weak amplitudes. The method has a good representation of GCI and a best detection of GOI. The method was tested on the Keele University database; it is effective and robust in multiple cases even for a typical signal showing undetermined GOIs and multiple peaks at GCIs. Finally precise measurement of these instants allows accurate estimation of prosodic parameters as local pitch and open quotient.

A quantitative model for aviation safety risk assessment

The objective of this research is to develop an analytic method that uses data on both accident and safety indicators to quantify the aviation risk which are caused by human errors. A specified proportional hazard model considering the baseline hazard function as a quadratic spline function has investigated and demonstrated its applicability in aviation risk assessment. The use of the proposed model allows investigation of non-linear effects of aviation safety factors and flexible assessment of aviation risk. A subset of data gathered from the Fight Safety Management Information System (FSMIS) developed by the office of the Taiwan Civil Aeronautics Administration (CAA) was applied to accomplish this study. The results demonstrate that the proposed model is a more promising approach with the potential of becoming very useful in practice and leading to further generalization of aviation risk analysis.

Prognostic Value of Admission Laboratory Parameters in Traumatic Brain Injury: Results from The IMPACT Study

Abnormalities in laboratory parameters are frequent following traumatic brain injury (TBI), but few studies have investigated their predictive value. We aimed to describe and quantify the relation between laboratory parameters that are routinely determined on admission and final outcome following TBI. Individual patient data were available in the IMPACT database from six Phase III randomized controlled trials and one observational study in TBI. We studied glucose (N = 4834), sodium ( N = 5270), pH ( N = 3398), hemoglobin (Hb, N = 3875), platelet count ( N = 1629), and prothrombin time (PT; N = 840) for their associations with outcome at 6 months (Glasgow Outcome Scale [GOS]). We used logistic regression models with linear, quadratic, and restricted cubic spline functions. The strength of the associations was expressed as an unadjusted odds ratio, calculated over the shift in outcome between the 25th and 75th percentiles. Proportional odds methodology was further applied to quantify the strength of the associations across the full range of the GOS. All parameters were consistently associated with outcome in a continuous relationship: glucose and prothrombin time showed a positive linear relation to outcome (i.e., increasing values associated with poorer outcome) and Hb, platelets, and pH an inverse linear relation (i.e., low values associated with poorer outcome). Sodium demonstrated a U-shaped relation to outcome, with low levels being more strongly related to poorer outcome. Effects were strongest for increasing levels of glucose (odds ratio 1.7; 95% CI 1.54-1.83) and decreasing levels of Hb (odds ratio 0.7; CI 0.60-0.78). Higher glucose values were associated with increasing age, but on adjusted analysis, the strength of the association with outcome remained. Whether treatment of abnormal values may improve outcome needs further rigorous study.

Quadratic non-polynomial spline approach to the solution of a system of second-order boundary-value problems

A quadratic non-polynomial spline functions based method is developed to find approximations solution to a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. The present approach has less computational cost and gives better approximations than those produced by other collocation, finite-difference and spline methods. Convergence analysis of the method is discussed. A numerical example is given to illustrate practical usefulness of the new method.

Study on the Modeling Method Suitable for Pavement Rutting

A paved road surface is in contact with a vehicle directly through its tire therefore its condition is closely associated with vehicle running 1) . It also greatly affects road user’s ride quality, safety and the living environment of roadside resident. Therefore, it is important to keeping track of the conditions and evaluate the conditions of the paved road surface, then make the suitable maintenance at the appropriate time according to the evaluation 2) . A major distress mode of the paved road is caused by road surface characteristics, such as rutting (pavement rutting), roughness and crack. Rutting and roughness are a special concern because these will affect road user and roadside resident directly. A longitudinal profile such as roughness has associated with elements which are related to the road user’s rating 3) . On the other hand, for the problems of a transverse profile such as the rutting, although there are the attempts being made to evaluate the rutting which affects the vehicle stability and controllability 3) , it has not yet reached to establish the evaluation index with the road user’s point of view. In Japan today, rut depth as an index for maintenance of the rutting has some disadvantages 2) . For instance, the index has problems the measurement method and definition of the rut depth differs from one administrator to another and it is does not reflect the road user’s point of view. There are two experimental approaches to estimate the road profile which affects vehicle behavior and road users. One approach is to use the actual vehicle with real road conditions. And the other is to use the simulation based on the vehicle dynamics (vehicle dynamics simulation). Recently vehicle dynamics simulation have developed and its achievements have become a center of attraction because it can do experiments to diagnose the road profile safely, rapidly and efficiently with same driving situation 3)-6) . The experiment for the rutting evaluation with the simulation has studied to concentrate on the vehicle dynamics 3) . However, the rut shape which is the object of investigation, especially the rutting caused by flowing which depends on heavy traffic, did not take into account the actual shape at the existing road, although it approximated only depression in the wheel path. Against this background, for the purpose of quantitative grasp of the rut shape, we developed the geometrical models of the rutting (the pavement rutting models) which simplified and approximated by use of comparatively easy functions and which have a moderate flexibility. The developed models were verified in respect of the adaptability of these shapes for actual road profile data and the applicability for the simulation method of the vehicle dynamics. The actual road profile is using part of the data which is obtained from the second Permanent International Association of Road Congress – World Road Association (PIARC) International Experiment to Harmonize Longitudinal and Transverse Profile Measure and Reporting procedure (EVEN Project) 1) . While there are various types of vehicle dynamics models or simulation Abstract: In case of fitting geometric model of pavement rutting to the real data, the model of a high order function has the well advantage of fitness. However, it also has the disadvantage of versatility. In this study, we developed the pavement rutting models with quadratic function and spline function for quantification of pavement rutting characteristics on road surface. The two developed models are verified in respect of the adaptability of these shapes and the accommodation for the simulation method of the vehicle dynamics using the road profile data of the PIARC EVEN experiment in 1998. As a result, it is indicated that the spline function is applicable for representing pavement rutting characteristics.

舗装わだち掘れのモデリング方法に関する研究

In case of fitting geometric model of pavement rutting to the real data, the model of a high order function has the advantage of goodness offitness. However, it also has the disadvantage of versatility. In this study, we developed the pavement rutting models with quadratic function and spline function for quantification of pavement rutting characteristics on road surface. The two developed models are verified in respect of the adaptability of these shapes and the accommodation for the simulation method of the vehicle dynamics using the road profile data of the PIARC EVEN experiment in 1998. As a result, it is indicated that the spline function is applicable for representing pavement rutting characteristics.

On the Characterization of Quadratic Splines

A quadratic spline is a differentiable piecewise quadratic function. Many problems in the numerical analysis and optimization literature can be reformulated as unconstrained minimizations of quadratic splines. However, only special cases of quadratic splines have been studied in the existing literature and algorithms have been developed on a case-by-case basis. There lacks an analytical representation of a general or even convex quadratic spline. The current paper fills this gap by providing an analytical representation of a general quadratic spline. Furthermore, for a convex quadratic spline, it is shown that the representation can be refined in the neighborhood of a nondegenerate point and a set of nondegenerate minimizers. Based on these characterizations, many existing algorithms for specific convex quadratic splines are also finitely convergent for a general convex quadratic spline. Finally, we study the relationship between the convexity of a quadratic spline function and the monotonicity of the corresponding linear complementarity problem. It is shown that, although both conditions lead to easy solvability of the problem, they are different in general.

Noise reduction in helioseismic power spectra by non-orthogonal wavelets

We present a method to reduce noise in helioseismic power spectra using a non-orthogonal wavelet transform based on quadratic spline functions. The quality of our method is tested by applying it to artificially generated time-series approximating solar p-modes. The mode frequencies and line widths obtained from least-squares fits to the smoothed spectra are compared with the corresponding parameters deduced from maximum likelihood fits to the original spectra. The results from both approaches are very similar and suggest that there is no major bias in either of these rather independent approaches. As a practical example we denoise parts of the power spectrum obtained from the two first years of operation of the GOLF instrument onboard SOHO.

Application of adaptive matching pursuit to adaptive control of nonlinear dynamic systems

A common problem in the adaptive control of nonlinear systems is the selection of an appropriate trade-off between performance and computational complexity. A reduced-complexity recursive algorithm is proposed for the identification of nonlinear auto-regressive with exogenous inputs (NARX) models of a nonlinear system. This algorithm, which follows the matching pursuit algorithm, is used for the adaptive control of nonlinear dynamic systems, where quadratic spline functions are used as the basic building blocks. The adaptive controller is applied to several nonlinear systems, previously solved by other nonlinear control methods.

Quadratic spline interpolator

A simple system is proposed which generates a quadratic spline function interpolating a given discrete-time signal. It works as a D-A converter which is free from phase distortions and whose output is smooth. The output is constructed as two-fold integration of a staircase signal. A kind of sampled data feedback is employed to stabilize the system. Its circuit is mainly composed of a pair of D-A and A-D converter chips and two analogue integrators. This hardware complexity is about a quarter of current systems.

Knot positions for the smoothest periodic quadratic spline interpolation of equispaced data

The knot positions of a periodic quadratic spline function which furnishes the smoothest interpolation of equispaced data are clarified. It is proved that the knots are to be at the midpoints of every two adjacent data points for the smoothest interpolation. This gives a partial answer to the problem where to locate the knots to obtain the smoothest spline interpolation of even degree.

PAH extraction and estimation of plasma flow in diseased human kidneys

We have analyzed the efficiency with which p-amino-hippuric acid (PAH) is extracted (EPAH) by patients with healthy kidneys (n = 13) or kidneys damaged by chronic cyclosporin nephropathy (n = 21) or primary glomerulopathy (n = 12); respective values (mean +/- SE) for EPAH were 0.87 +/- 0.03, 0.77 +/- 0.03, and 0.69 +/- 0.04. Judged by a 131I-hippuran-to-PAH clearance ratio of 0.75 +/- 0.05, extraction ratio of hippuran was less efficient than EPAH in three glomerulopathic patients. A direct relationship was defined between EPAH and glomerular filtration rate (GFR) (r = 0.54) or calculated efferent oncotic pressure (IIE; r = 0.41, P less than 0.01). Curve fitting by means of quadratic spline functions revealed GFR and IIE to be additive in predicting EPAH (R2 = 0.45). Linear model prediction methods and a sample reuse technique failed to predict EPAH reliably from GFR and preglomerular oncotic pressure (IIA); however, 95% prediction intervals exceed 0.30 EPAH units in width. We conclude that oncotic pressure (presumably reflecting albumin concentration) along with GFR is predictive of EPAH depression in humans with chronic renal disease. However, even sophisticated curve-fitting techniques are too imprecise for accurate prediction of EPAH in a given individual. We submit that renal venous sampling to determine EPAH continues to be necessary for the accurate determination of the rate of plasma flow in the injured human kidney.