Nonlinear Splines Research Articles

Assessing the safety effects of multiple roadside treatments using parametric and nonparametric approaches

This study evaluates the safety effectiveness of multiple roadside elements on roadway segments by estimating crash modification factors (CMFs) using the cross-sectional method. To consider the nonlinearity in crash predictors, the study develops generalized nonlinear models (GNMs) and multivariate adaptive regression splines (MARS) models. The MARS is one of the promising data mining techniques due to its ability to consider the interaction impact of more than one variables and nonlinearity of predictors simultaneously. The CMFs were developed for four roadside elements (driveway density, poles density, distance to poles, and distance to trees) and combined safety effects of multiple treatments were interpreted by the interaction terms from the MARS models. Five years of crash data from 2008 to 2012 were collected for rural undivided four-lane roadways in Florida for different crash types and severity levels. The results show that the safety effects decrease as density of driveways and roadside poles increase. The estimated CMFs also indicate that increasing distance to roadside poles and trees reduces crashes. The study demonstrates that the GNMs show slightly better model fitness than negative binomial (NB) models. Moreover, the MARS models outperformed NB and GNM models due to its strength to reflect the nonlinearity of crash predictors and interaction impacts among variables under different ranges. Therefore, it can be recommended that the CMFs are estimated using MARS when there are nonlinear relationships between crash rate and roadway characteristics, and interaction impacts among multiple treatments.

Abstract 1274: Alcohol and lung cancer risk: a pooled analysis using International Lung Cancer Consortium studies

Abstract Background: Alcohol consumption is known to be associated with risk of developing several cancers. It is unclear, however, whether alcohol consumption is a risk factor for lung cancer. The relationship between lung cancer and alcohol consumption is likely to be confounded by smoking. To minimize potential confounding by tobacco consumption, we conducted a pooled analysis to examine the association of alcohol consumption with lung cancer risk in a large sample of never-smokers. Methods: We pooled data from 22 case-control and cohort studies from North America, Europe and Asia within the International Lung Cancer Consortium (ILCCO) and SYNERGY Consortium. We examined the association of average lifetime alcohol consumption (expressed as average grams per day intake) with lung cancer risk in never smokers using logistic regression to model categories of alcohol consumption (0<5g per day, 5<10g per day, 10<20g per day, 20<30g per day, 30<45g per day, 45+ g per day). To investigate the shape of the dose response relationship, we applied restricted cubic spline models to examine the association for lung cancer risk overall and by histological subtype. Additional analyses examined wine, beer and liquor consumption in relation to risk, with mutual adjustment for each alcoholic beverage. All analyses were adjusted for age, sex, education, ethnicity and study. Results: A total of 2548 never-smoking cases and 9362 never-smoking controls were included in the analysis. The results showed lower risk among consumers of alcohol with strongest evidence found for moderate drinkers relative to non-drinkers with ORs of 0.80 (95% CI 0.70-0.90) and 0.82 (95% CI 0.69-0.99) for <5grams and 5-10 grams of alcohol per day respectively. Non-linear restricted cubic splines showed reduced lung cancer risk among moderate drinkers relative to non-drinkers with risk increasing towards the null as consumption increased. Similar results were seen for adenocarcinoma and squamous cell carcinoma. Associations with lung cancer differed for wine and beer consumption. Reduced risk was observed for wine drinking particularly at low levels of drinking, OR of 0.80 (95% CI=0.69-0.94) for <5g per day. Risk for beer consumption increased from close to null among occasional drinkers to 1.54 (95% CI 0.90-2.65) among consumers of 20-30g of alcohol per day (test for trend P=0.09). Conclusions: These results indicate an inverse association between moderate drinking and lung cancer risk relative to never drinkers. However, the inverse association was restricted to wine consumption, not consumption of beer. Lifestyle differences between consumers of beer and wine may play a role in differing patterns of risk found by alcohol type. Citation Format: Gordon Fehringer, Darren Brenner, Zuo-Feng Zhang, Yuan-Chin Amy Lee, Keitaro Matsuo, Isabelle Stucker, Paolo Vineis, Paolo Boffetta, Paul Brennan, Maria T. Landi, Hal Morgenstern, Curtis C. Harris, Qing Lan, Yun-Chul Hong, Jack Siemiatycki, John R. McLaughlin, Philip Lazarus, Joshua Muscat, Ann G. Schwartz, Juan M. Barros Dios, Alberto R. Raviña, Gad Rennert, David C. Christiani, Adonina Tardon, Loic Le Marchand, Irene Orlow, Eric J. Duell, Angeline S. Andrew, Hermann Brenner, Dario Consonni, Ann Olsson, Kurt Straif, Rayjean J. Hung. Alcohol and lung cancer risk: a pooled analysis using International Lung Cancer Consortium studies. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 1274. doi:10.1158/1538-7445.AM2014-1274

Forecasting Stock Prices With Linear And Nonlinear Settings: A Comparison

This paper is devoted to the application and comparison of linear (VAR) and nonlinear Multiple Adaptive Regression Splines (MARS) forecasting models, in estimating, evaluating, and selecting among linear and non-linear forecasting models for economic and financial time series. We argue that although the evidence in favor of constructing forecasts using non-linear models is rather sparse, there is reason to be optimistic. Nonlinear models reduce nonlinearity and Gaussianity in the residuals of the linear models. Linear models, however, demonstrate better forecasts than nonlinear. However, much remains to be done. Finally, we outline a variety of topics for future research, and discuss a number of areas which have received considerable attention in the recent literature, but where many questions remain.

Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines

This paper presents a new method for zonal wavefront reconstruction (WFR) with application to adaptive optics systems. This new method, indicated as Spline based ABerration REconstruction (SABRE), uses bivariate simplex B-spline basis functions to reconstruct the wavefront using local wavefront slope measurements. The SABRE enables WFR on nonrectangular and partly obscured sensor grids and is not subject to the waffle mode. The performance of SABRE is compared to that of the finite difference (FD) method in numerical experiments using data from a simulated Shack-Hartmann lenslet array. The results show that SABRE offers superior reconstruction accuracy and noise rejection capabilities compared to the FD method.

ELASTIC FLOW WITH JUNCTIONS: VARIATIONAL APPROXIMATION AND APPLICATIONS TO NONLINEAR SPLINES

We consider stable semidiscrete approximations of parametrized curve networks for gradient flows of elastic type functionals. Here meaningful and relevant conditions at junction points, such as double and triple junctions, need to be derived and suitably discretized. Examples for double junction types are C0 attachment and C1 continuity. We develop strong and weak formulations for the elastic flow for curve networks with such junction points. For junctions with three or more curves, the conditions at the junctions are derived here for the first time. Possible applications include a simplified one-dimensional model of geometric biomembranes, as well as nonlinear splines in two and higher dimensions. The numerical results presented in this paper demonstrate the practicality of the introduced finite element approximations.

Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function

We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss–Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G2, and G3 non-linear superspiral splines.

Blood pressure and risk of cardiovascular diseases in type 2 diabetes

Estimate risks of coronary heart disease (CHD), stroke and cardiovascular disease (CVD) with updated mean systolic (SBP) and diastolic (DBP) blood pressure in an observational study of patients with type 2 diabetes. Thirty-five thousand and forty-one patients treated with antihypertensive drugs, and 18 512 untreated patients, aged 30-75 years, without previous heart failure, followed for 6 years until 2009. In treated patients, nonlinear splines for 6-year risk of fatal/nonfatal CHD, stroke and CVD by BP as a continuous variable showed a progressive increase with higher SBP from 140 mmHg and higher, and with DBP from 80 mmHg, with a J-shaped risk curve at lowest SBP levels, but not obviously at lowest DBP levels. Analysing intervals of SBP with 130-134 mmHg as reference at Cox regression, adjusted hazard ratios (HR) for fatal/nonfatal CHD, stroke and CVD with at least 140 mmHg were 1.22 [95% confidence interval (CI): 1.08-1.39], 1.43 (1.18-1.72), 1.26 (1.13-1.41), all P < 0.001. HR with 115-129 and 135-139 mmHg were nonsignificant, whereas increased with 100-114 mmHg, 1.96 (P < 0.001), 1.75 (P = 0.02), 2.08 (P < 0.001), respectively. With DBP 75-79 mmHg as reference, adjusted HR for fatal/nonfatal CHD, stroke and CVD with DBP 80-84 mmHg were 1.42 (1.26-1.59), 1.46 (1.24-1.72), 1.39 (1.26-1.53), all P < 0.001. Corresponding HR with DBP at least 85 mmHg were 1.70 (1.50-1.92), 2.35 (1.99-2.77), 1.87 (1.69-2.07), all P < 0.001. Corresponding HR with DBP 60-69 and 70-74 mmHg were nonsignificant. The picture was similar in 7059 patients with previous CVD and in untreated patients. BP around 130-135/75-79 mmHg showed lower risks of cardiovascular diseases in patients with type 2 diabetes.

Integrative analysis of transcriptomic and proteomic data of Shewanella oneidensis: missing value imputation using temporal datasets

Despite significant improvements in recent years, proteomic datasets currently available still suffer from large number of missing values. Integrative analyses based upon incomplete proteomic and transcriptomic datasets could seriously bias the biological interpretation. In this study, we applied a non-linear data-driven stochastic gradient boosted trees (GBT) model to impute missing proteomic values using a temporal transcriptomic and proteomic dataset of Shewanella oneidensis. In this dataset, genes' expression was measured after the cells were exposed to 1 mM potassium chromate for 5, 30, 60, and 90 min, while protein abundance was measured for 45 and 90 min. With the ultimate objective to impute protein values for experimentally undetected samples at 45 and 90 min, we applied a serial set of algorithms to capture relationships between temporal gene and protein expression. This work follows four main steps: (1) a quality control step for gene expression reliability, (2) mRNA imputation, (3) protein prediction, and (4) validation. Initially, an S control chart approach is performed on gene expression replicates to remove unwanted variability. Then, we focused on the missing measurements of gene expression through a nonlinear Smoothing Splines Curve Fitting. This method identifies temporal relationships among transcriptomic data at different time points and enables imputation of mRNA abundance at 45 min. After mRNA imputation was validated by biological constrains (i.e. operons), we used a data-driven GBT model to impute protein abundance for the proteins experimentally undetected in the 45 and 90 min samples, based on relevant predictors such as temporal mRNA gene expression data and cellular functional roles. The imputed protein values were validated using biological constraints such as operon and pathway information through a permutation test to investigate whether dispersion measures are indeed smaller for known biological groups than for any set of random genes. Finally, we demonstrated that such missing value imputation improved characterization of the temporal response of S. oneidensis to chromate.

Customized sequential designs for random simulation experiments: Kriging metamodeling and bootstrapping

This paper proposes a novel method to select an experimental design for interpolation in random simulation, especially discrete event simulation. (Though the paper focuses on Kriging, this design approach may also apply to other types of metamodels such as non-linear regression models and splines.) Assuming that simulation requires much computer time, it is important to select a design with a small number of observations (or simulation runs). The proposed method is therefore sequential. Its novelty is that it accounts for the specific input/output behavior (or response function) of the particular simulation at hand; i.e., the method is customized or application-driven. A tool for this customization is bootstrapping, which enables the estimation of the variances of predictions for inputs not yet simulated. The method is tested through two classic simulation models, namely the expected steady-state waiting time of the M/M/1 queuing model, and the mean costs of a terminating ( s, S) inventory simulation. For these two simulation models the novel design indeed gives better results than a popular alternative design, namely Latin Hypercube Sampling (LHS) with a prefixed sample.

Evaluation of chromatographic descriptors for the prediction of gastro-intestinal absorption of drugs

The use of chromatographic descriptors in QSAR was evaluated. Therefore, retentions were measured on an immobilized artificial membrane system, 2 micellar liquid chromatography systems and 17 orthogonal or disimilar reversed-phase liquid chromatographic systems. It was investigated whether it was possible to model gastro-intestinal absorption as a function of chromatographic retentions applying two linear and one non-linear multivariate modeling technique. In a second step it was evaluated if models built with theoretical descriptors could be improved by adding the measured retention factors to the data set of descriptive variables. It was seen that gastro-intestinal absorption could be modelled in function of chromatographic retention using the non-linear modeling technique multivariate adaptive regression splines (MARS). The best models were obtained using a combination of theoretical and chromatographic descriptors with MARS as modeling technique.

Exploration of linear modelling techniques and their combination with multivariate adaptive regression splines to predict gastro-intestinal absorption of drugs

In general, linear modelling techniques such as multiple linear regression (MLR), principal component regression (PCR) and partial least squares (PLS), are used to model QSAR data. This type of data can be very complex and linear modelling techniques often model only a limited part of the information captured in the data. In this study, it was tried to combine linear techniques with the flexible non-linear technique multivariate adaptive regression splines (MARS). Models were built using an MLR model, combined with either a stepwise procedure or a genetic algorithm for variable selection, a PCR model or a PLS model as starting points for the MARS algorithm. The descriptive and predictive power of the models was evaluated in a QSAR context and compared to the performances of the individual linear models and the single MARS model. In general, the combined methods resulted in significant improvements compared to the linear models and can be considered valuable techniques in modelling complex QSAR data. For the used data set the best model was obtained using a combination of PLS and MARS. This combination resulted in a model with a Pearson correlation coefficient of 0.90 and a cross-validation error, evaluated with 10-fold cross-validation of 9.9%, pointing at good descriptive and high predictive properties.

A Sobolev gradient method for construction of elastic curves in regular surfaces

Given a sequence of control points and tangent vectors in a regular surface, we treat the problem of constructing a curve that lies in the surface and has minimum total curvature subject to the constraint that it interpolates the control points and tangent vectors. Geodesics and parametric planar curves (nonlinear splines) are included as special cases. The surface is defined implicitly by a smooth function, and the curve is approximated by a discrete set of vertices along with first and second derivative vectors. The nonlinear optimization problem of minimizing curvature subject to the constraints is solved by a variable metric gradient descent method based on Neuberger's Sobolev gradient theory. We compare different methods for treating the nonlinear constraints.

Scotch pine adaptation to climate changes.

tree rings were approximated by linear or nonlinear splines. The responses of trees induced by climatic factors were calculated from age curves using the corresponding index [4]. The coefficients of correlation, coefficients of synchronicity, and coefficients of sensitivity of tree ring indices to climatic factors were used for a comparative analysis of provenance growth and the effect of changes in monthly mean temperatures and atmospheric precipitations on the provenances [3]. Standard climatic response functions were also calculated. These functions show the relative contribution of each climatic factor to the variability of the tree-ring parameter studied [8]. Cluster analysis was based on the calculation of normalized Euclidean distances over a set of independent climatically determined parameters of the tree-ring structure. The results of the cluster analysis were used to obtain quantitative integrated estimates of the provenance closeness [9]. The provenances were divided into three groups within each provenance trial. The latitudes of the sites where the seeds of the provenances of the first group were collected almost coincided with the latitudes of the origin of the local provenance. The second group included provenances whose origins were remote from the geographical locations of the provenance trials. The third group comprised provenances from the far north (forest-tundra for southern taiga provenance trials and near-tundra forests for the forest-steppe zone). The climatic response functions of these groups slightly differed from one another. In the provenances of the first group growing in southern taiga, the April temperature rise causes an increase in the cell production rate and induces formation of a broader zone of late wood. Warm spring facilitates the growth of provenances of the second group. The enhanced precipitation in July accelerates the growth of the provenances from areas with insufficient humidity. Under the growth conditions of forest-steppe, the April temperature rise has a positive effect on the radial growth of all provenances. The enhanced precipitation in July causes an increase in the fraction of late wood in provenances of the second group. A statistically significant effect of May temperature on the maximum density of tree ring was observed only for the northernmost provenance.

Hamilton equations for elasticae in the Euclidean 3-space

The variational problem on spatial curves defined by the integral of the squared curvature, whose solutions are the elasticae or nonlinear splines, is analyzed from the Hamiltonian point of view, using a procedure developed by Muñoz Masquéand Pozo Coronado [J. Muñoz Masqué, L.M. Pozo Coronado, J. Phys. A 31 (1998) 6225–6242]. The symmetry of the problem under rigid motions is then used to reduce the Euler–Lagrange equations to a first-order dynamical system.

POLYGONAL CURVE EVOLUTIONS FOR PLANAR SHAPE MODELING AND ANALYSIS

The paper develops a numerical method to analyze and model planar shapes via polygonal curve evolutions. Consider a smooth curve each point of which moves in the normal direction with speed equal to a function of the curvature and curvature derivatives at the point. Chosen the speed function properly, the evolving curve converges to a desired shape. Smooth curve evolutions are approximated by evolutions of polygonal curves. Discrete analogs of the curvature and its derivatives are considered and analyzed. We apply our approach to simplify curves for purposes of multiscale shape analysis, to model geodesics on a plane equipped with a Riemannian metric, to design of nonlinear splines satisfying various boundary conditions, and to detect objects in grey-scale images.

Unrolling and rolling of curves in non-convex surfaces

The notion of unrolling of a spherical curve is proved to coincide with its development into the tangent plane. The development of a curve in an arbitrary surface in the Euclidean 3-space is then studied from the point of view of unrolling. The inverse operation, called the rolling of a curve onto a surface, is also analysed and the relationship of such notions with the functional defined by the square of curvature is stated. An application to the construction of nonlinear splines on Riemannian surfaces is suggested.

Unified Representations of Nonlinear Splines

Golomb and Jerome's framework is modified and extended. The new framework is more general since it also handles interpolants which are not allowed to “slide” at the nodes. The space of interpolants of variable length is shown to be a smooth manifold. If the length is fixed, and there are no nodes, then the space of interpolants is a manifold. When there is at least one node, and at least one node is not on the line segment between the endpoints, then the space of interpolants of fixed length is a smooth manifold. Sufficient conditions are given which ensure the space of interpolants continues to be a smooth manifold in the presence of additional constraints such as clamping and pinning. A new fundamental finite-dimensional equation is derived. When it is solved it yields all nonlinear splines, and every nonlinear spline appears in this way. An important feature is that the same symbolic equation is used for all possible combinations of the constraints considered. It is shown how to take the solutions of the fundamental equation and use them to express the corresponding nonlinear splines in terms of a pair of elliptic functions. An inequality is derived that specifies which elliptic function appears along each section of the spline. The nonlinear splines are in a unified way shown to beC2for all possible combinations of the constraints considered.

Strong unicity in nonlinear approximation and free knot splines

We give a necessary alternation condition for unique local best approximation from Sm,k, the set of splines of degree m with k free knots. This result is related to a conjecture of L.L. Schumaker. Moreover, we give a characterization of functions from the interior of the strong unicity set for S1m,k, the set of splines of degree m with k free simple knots, and show that this set is dense in the unicity set. Then we give a general characterization of suns for strong unicity and show that S1m,k is a set of this type, although it is not a sun.

Curve interpolation with nonlinear spiral splines

Interpolating spiral splines are derived as an approximation to the curve of least energy. The defining equations, although nonlinear, are easily solved because the Jacobian matrix has banded structure. A simple but effective iterative scheme for the solution of these equations is described together with a useful scheme for determining initial approximations for nonlinear splines. The resulting curve is invariant with respect to translation and rotation of axes and is usually much smoother than is possible with polynomial splines because the curvature of the spiral spline varies linearly with respect to arc length

Exact equations of the nonlinear spline

We define the spline interpolating function, and obtain in directly computable form the elementary set of nonlinear equations describing nonlinear spline curves. Using Newton's and Newton-like methods, we solve typical spline configurations, and hence infer that the procedure will reliably yield precise extremum-energy solutions to nonlinear splines of arbitrary (but presumably reasonable) size and complexity. In order to distinguish between stable and unstable states of spline equilibria, we evaluate the energy change resulting from a perturbation, and we briefly discuss aspects of spline existence and uniqueness in relation to the solved examples. We demonstrate the abrupt transition which occurs at the threshold between spline existence and nonexistence, and conclude that proof of a spline's existence is implicit in the solution set of constants yielded by the method. The procedure may be regarded on the one hand as a precise and efficient research instrument for investigating the properties of true splines and elastica, and on the other as an everyday method for obtaining “the smoothest interpolating curve of all. Contact is always maintained with the physical analogue to the curve, the thin uniform elastic beam, since the four assignable parameters used in each spline interval comprise the necessary and sufficient three angles and one length dimension of the actual physical spline. On an historical note, the method may be seen to offer progress in the search, begun in the late 17th century, for a definitive solution to the elastica problem.