Studies have shown that in many practical applications, data interpolation by splines leads to better approximation and higher computational efficiency as compared to data interpolation by a single polynomial. Data interpolation by splines can be significantly improved if knots are allowed to be free rather than at a priori fixed locations such as data points. In practical applications, the smallest possible curvature is often desired. Therefore, optimal splines are determined by minimizing a derivative of continuously differentiable functions comprising the spline of the required order. The problem of obtaining an optimal spline is tantamount to minimizing derivatives of a nonlinear differentiable function over a Banach space on a compact set. While the problem of data interpolation by quadratic splines has been accomplished analytically, interpolation by splines of higher orders or in higher dimensions is challenging. In this paper, to overcome difficulties associated with the complexity of the interpolation problem, the interval over which data points are defined, is discretized and continuous derivatives are substituted by their discrete counterparts. It is shown that as the mesh of the discretization approaches zero, a resulting near-optimal spline approaches an optimal spline. Splines with the desired accuracy can be obtained by choosing an appropriate mesh of the discretization. By using cubic splines as an example, numerical results demonstrate that the linear programming (LP) formulation, resulting from the discretization of the interpolation problem, can be solved by linear solvers with high computational efficiency and resulting splines provide a good approximation to the optimal splines.

Fitting parameters on spline curves produces more parsimonious models, maintaining fit quality. Smoothing the splines reduces predictive variance. Individual splines are often fit by type of variable, e.g., in age-period-cohort models. Linear and cubic splines are most common. Several smoothing criteria have been used for parameter curves, with cubic splines fit by constraining the integral of splines’ squared-second derivatives popular recently. Constraining the sum of squared second differences for linear splines is analogous. Generally the degree of smoothing is selected using cross-validation. Known spline dummy-variable matrices allow regression estimation of splines, with smoothing done via constrained regression. Smoothing criteria based on sums of squares or absolute values of parameters, as in ridge regression or LASSO, improves predictive accuracy and produces splines similar to smoothing by second-derivative constraints. Variables with very low t-statistics represent points where curve-shapes barely change. Eliminating those variables leaves knots concentrated where spline shapes change. A Bayesian version of this puts shrinkage priors on spline parameters. This yields realistic joint parameter distributions, avoids problems associated with using cross-validation for parameter estimation, and readily expands to non-linear modeling, such as interactions among variable types. Regularized regression and Bayesian spline methods are compared for two example datasets.

The class of A-stable symmetric one-step Hermite-Obrechkoff (HO) methods introduced in [1] for dealing with Initial Value Problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed in [2] for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss Runge-Kutta schemes and Euler-Maclaurin formulas of the same order.

Accurate geometric modeling of blood vessels lumen given their 3D images is a basis for vessel quantification in support of cardiovascular diseases diagnosis, treatment and monitoring. Unlike other approaches, which assume circular or elliptical vessel cross-section, the proposed method employs parametric B-splines combined with the image formation system equations to accurately localize the highly curved lumen boundaries. No image segmentation is involved which might reduce the localization accuracy in effect of spatial discretization. It is shown that the model parameters can be reliably identified by a feedforward neural network which, driven by the cross-section images, predicts the parameter values many times faster than a reference least-squares (LS) model fitting algorithm. Two example applications: modeling the lower extremities artery-vein complexes visualized in steady-state contrast-enhanced magnetic resonance images (MRI) and the coronary arteries pictured in computed tomography angiograms (CTA), are presented and discussed to demonstrate the method potential. Besides application to medical diagnosis, blood-flow simulation and vessel-phantom design, it can serve as a tool for automated annotation of image datasets required to train machine-learning algorithms.

This paper focuses on a joint model to analyse longitudinal proportion and survival data. After transforming the longitudinal proportional data by a logit function, we adopt a partially linear mixed-effect model for it, in which nonlinear function of time is fitted via using B-splines technique and a centered Dirichlet Process Mixture Model (CDPMM) is specified for a general distribution of random effects. The survival data is assumed to a Cox proportional hazard model, the sharing random effects joint model is developed for the two types of data. Combining the Gibbs sampler and the Metropolis-Hastings algorithm, we propose a Bayesian Lasso (BLasso) method to simultaneously estimate unknown parameters and select important covariates. Simulation studies are conducted to investigate the finite sample performance of the proposed methods. An example from the MA.5 research experiment is used to illustrate the proposed methodologies.

In January and February 2001, El Salvador was hit by two strong earthquakes that triggered thousands of landslides, causing 1,259 fatalities and extensive damage. The analysis of aerial and SPOT-4 satellite images taken a few days after the events allowed us to map 6,491 coesismic landslides, which occurred in 14 study areas extending for about 400 km2. Four different Multivariate Adaptive Regression Splines (MARS) models were produced by using different covariate sets and landslide inventories, the latter containing the slope failures triggered by an extreme rainfall event of November 2009 and those induced by the earthquakes of 2001. Moreover, two validation scenarios were employed, including the 25% and 95% of the mapped landslides, respectively. The results of our experiment revealed that: (i) MARS algorithm provides reliable predictions of coesismic landslides; (ii) models calibrated with rainfall-induced landslides predict with acceptable accuracy landslides caused by deep earthquakes and distributed over vast areas; (iii) the best accuracy is achieved by models trained with both preparatory and trigger variables; (iv) a small portion of the landslides produced by an earthquake can be used to calibrate MARS predictive models that help to identify slopes where yet unreported landslides may have occurred.