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This article introduces a new R package (pspatreg) for the estimation of semiparametric spatial autoregressive models. pspatreg fits penalized spline semiparametric spatial autoregressive models via Restricted Maximum Likelihood or Maximum Likelihood. These models are very flexible since they make it possible to simultaneously control for spatial dependence, nonlinearities in the functional form, and spatio-temporal heterogeneity. The package also allows to estimate parametric spatial autoregressive models for both cross sectional and panel data (with fixed effects), thus avoiding the use of different libraries. The official demos, vignettes, and tutorials of the package are distributed either in CRAN or GitHub. This article illustrates the potential of the package by using an application to cross-sectional data.

The methodological contribution in this paper is motivated by biomechanical studies where data characterizing human movement are waveform curves representing joint measures such as flexion angles, velocity, acceleration, and so on. In many cases the aim consists of detecting differences in gait patterns when several independent samples of subjects walk or run under different conditions (repeated measures). Classic kinematic studies often analyse discrete summaries of the sample curves discarding important information and providing biased results. As the sample data are obviously curves, a Functional Data Analysis approach is proposed to solve the problem of testing the equality of the mean curves of a functional variable observed on several independent groups under different treatments or time periods. A novel approach for Functional Analysis of Variance (FANOVA) for repeated measures that takes into account the complete curves is introduced. By assuming a basis expansion for each sample curve, two-way FANOVA problem is reduced to Multivariate ANOVA for the multivariate response of basis coefficients. Then, two different approaches for MANOVA with repeated measures are considered. Besides, an extensive simulation study is developed to check their performance. Finally, two applications with gait data are developed.

In many applications, it is useful to use piecewise polynomials that satisfy certain regularity conditions at the joint points. Cubic spline functions emerge as good candidates having C2 regularity. On the other hand, if the data points present discontinuities, the classical spline approximations produce Gibbs oscillations. In a recent paper, we have introduced a new nonlinear spline approximation avoiding the presence of these oscillations. Unfortunately, this new reconstruction loses the C2 regularity. This paper introduces a new nonlinear spline that preserves the regularity at all the joint points except at the end points of an interval containing a discontinuity, and that avoids the Gibbs oscillations. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was funded by the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MMTM2015-64382-P and PID2019-108336GB-I00 (MINECO/FEDER), by grant MTM2017-83942 funded by Spanish MINECO and by grant PID2020-117211GB-I00 funded by MCIN/AEI/10.13039/501100011033.

In this paper we study the pricing of exchange options under a dynamic described by stochastic correlation with random jumps. In particular, we consider a Ornstein-Uhlenbeck covariance model with Levy Background Noise Process driven by Inverse Gaussian subordinators. We use expansion in terms of Taylor polynomials and cubic splines to approximately compute the price of the derivative contract. Our findings show that this approach provides an efficient way to compute the price when compared with a Monte Carlo method while maintaining an equivalent degree of accuracy with the latter., Comment: 27 pages, 7 figures, 2 tables

In computer graphics, the efficient rendering of a clear sky may greatly enhance the realism of synthesised interactive virtual environments. However, light atmospheric scattering, lying behind a reliable sky synthesis, is a computationally demanding process. Obtaining the highest possible accuracy and quality of the synthesised image in an efficient manner is not trivial as even minor integration errors may result in huge image artefacts. In this paper we propose two new approaches, namely the first based on the splines and the second on the Taylor expansion, in order to obtain the efficient computing of the transmittance (optical depth) component of the physically based single scattering model. We used Bruneton’s framework to compare the proposed solution with state-of-the-art methods in a single, scattering-based clear sky synthesis, in terms of image quality and performance (time). We managed to improve the quality of the single scattering effect and time performance and we removed the necessity of pre-calculating values to look-up tables for the transmittance part of the scattering integral, reducing exploited memory usage.

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper. We would like to thank the anonymous referees for their valuable comments, which have helped to significantly improve this work. This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia), by the national research project MTM2015- 64382-P (MINECO/FEDER) and by NSF grant DMS-1719410.

Understanding the learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. Partially, this is due to symmetries inherent within the NN parameterization, allowing multiple different parameter settings to result in an identical output function, resulting in both an unclear relationship and redundant degrees of freedom. The NN parameterization is invariant under two symmetries: permutation of the neurons and a continuous family of transformations of the scale of weight and bias parameters. We propose taking a quotient with respect to the second symmetry group and reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that standard weight initializations yield very flat initial functions, and that this flatness, together with overparametrization and the initial weight scale, is responsible for the strength and type of implicit regularization, consistent with previous work. Our implicit regularization results are complementary to recent work, showing that initialization scale critically controls implicit regularization via a kernel-based argument. Overall, removing the weight scale symmetry enables us to prove these results more simply and enables us to prove new results and gain new insights while offering a far more transparent and intuitive picture. Looking forward, our quotiented spline-based approach will extend naturally to the multivariate and deep settings, and alongside the kernel-based view, we believe it will play a foundational role in efforts to understand neural networks. Videos of learning dynamics using a spline-based visualization are available athttp://shorturl.at/tFWZ2.

Implied winning probabilities are usually derived from betting odds by the normalization: inverse odds are divided by the booksum (sum of the inverse odds) to ensure that the implied probabilities add up to 1. Another, less frequently used method, is Shin’s model, which endogenously accounts for a possible favourite-longshot bias. In this paper, we compare these two methods in two betting markets on soccer games. The method we use for the comparison is new and has two advantages. Unlike the binning method that is used predominantly, it is based on match-level data. The method allows for residual favourite-longshot bias, and also allows for incorporation of match specific variables that may determine the relation between the actual probability of the outcome and the implied winning probabilities. The method can be applied to any probabilistic classification problem. In our application, we find that Shin’s model yields unbiased estimates for the actual probability of outcome in the English Premier League. In the Spanish La Liga, implied probabilities derived from the betting odds using either the method of normalization or Shin’s model suffer from favourite bias: favourites tend to win their matches more frequently than the implied probabilities suggest.

11 Pág. Instituto de Ciencias Forestales (ICIFOR), Many oak stands (Quercus spp.) have been managed as coppices for firewood production for centuries in the Mediterranean area. After the abandonment of firewood production during the 1980s, current management practices attempt to convert coppices into coppices-with-standards through thinning and promoting forest regeneration via sexual reproduction. In this work, we used long-term data from repeated forest inventories and dendrometers in a thinning trial to assess the effects of thinning and climate on the intra- and inter-annual growth dynamics of Quercus pyrenaica Willd. coppices. Our results revealed that thinning favored the growth of Q. pyrenaica trees, especially when the stand density reduction was high (ca. 50% of the basal area extracted). Unthinned plots displayed more natural mortality i.e., self-thinning. Growth was enhanced with low vapor pressure deficit. Intense thinning treatments displayed higher intra-annual growth rates and interacted positively with rainfall to induce higher growth. We conclude that thinning, especially intensive thinning, may alleviate the negative effects of dry years and thus could provide a potential measure to adapt these stands to the different climatic scenarios with higher temperatures and less precipitation within the framework of sustainable forest management., This study has been funded through the Agreement EG17-042 between the Ministry of Agriculture and INIA. DMF was supported by Juan de la Cierva Formación Post-Doctoral program (FJC2018-037870-I) and DMB was funded by projects AGL2015-73190-JIN and RYC-2017-23389 from the Spanish Ministry of Science and Innovation. We also thank two anonymous reviewers for their helpful comments, which improved the quality of the work.

Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an $$\ell _1$$ ℓ 1 -norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones.

WOS: 000560864700001 In this paper, we propose a Pade-type approximation based on truncated total least squares (P - TTLS) and compare it with three commonly used smoothing methods: Penalized spline, Kernel smoothing and smoothing spline methods that have become very powerful smoothing techniques in the non-parametric regression setting. We consider the nonparametric regression model, y(i) = g(x(i)) + epsilon(i), and discuss how to estimate smooth regression function g where we are unsure of the underlying functional form of g. The Pade approximation provides a linear model with multi-collinearities and errors in all its variables. The P - TTLS method is primarily designed to address these issues, especially for solving error-contaminated systems and ill-conditioned problems. To demonstrate the ability of the method, we conduct Monte Carlo simulations under different conditions and employ a real data example. The outcomes of the experiments show that the fitted curve solved by P - TTLS is superior to and more stable than the benchmarked penalized spline (B - PS), Kernel smoothing (KS) and smoothing spline (SS) techniques.

A test score on a psychological test is usually expressed as a normed score, representing its position relative to test scores in a reference population. These typically depend on predictor(s) such as age. The test score distribution conditional on predictors is estimated using regression, which may need large normative samples to estimate the relationships between the predictor(s) and the distribution characteristics properly. In this study, we examine to what extent this burden can be alleviated by using prior information in the estimation of new norms with Bayesian Gaussian distributional regression. In a simulation study, we investigate to what extent this norm estimation is more efficient and how robust it is to prior model deviations. We varied the prior type, prior misspecification and sample size. In our simulated conditions, using a fixed effects prior resulted in more efficient norm estimation than a weakly informative prior as long as the prior misspecification was not age dependent. With the proposed method and reasonable prior information, the same norm precision can be achieved with a smaller normative sample, at least in empirical problems similar to our simulated conditions. This may help test developers to achieve cost-efficient high-quality norms. The method is illustrated using empirical normative data from the IDS-2 intelligence test.

Surface subsidence is a dominant component of the displacement vector triggered by underground mining. Over the last few decades, Differential Interferometry Synthetic Aperture Radar (DInSAR) has been used to efficiently monitor this phenomenon with great spatial and temporal coverage. More advanced multi-temporal DInSAR (MTInSAR) algorithms have been proposed to overcome some of the limitations of conventional DInSAR. However, advanced MTInSAR approaches are also not perfect in terms of measuring mining subsidence (e.g., temporal decorrelation, ambiguity, nonlinearity). For this reason, we propose a fusion of the Persistent Scatterer Interferometry (PSInSAR) and DInSAR results. By combining these complementary techniques, the atmospheric errors in PSInSAR data are reduced and larger deformation rates could have been detected more accurately (thanks to DInSAR) than by an approach solely based on PS-InSAR. This allows to measure areas with fast-moving subsidence (1 m/year) due to ongoing underground coal exploitation. Data from ascending and descending orbits of Sentinel-1A\B were used to obtain the vertical deformation component. The resulting integrated vertical deformation map was compared with the results from levelling benchmarks. The Root Mean Square Error (RMSE) calculated based on this comparison was 22 mm. Moreover, the maximal vertical cumulative subsidence detected in the study area was 1.05 m/year.

International audience; This paper presents a general survey of digital predistortion (DPD) techniques with segmentation. A comparison of global DPD with two segmented approaches namely Vector-Switched DPD and Decomposed Vector Rotation DPD is presented with the support of experimentation on a strongly non-linear 3 ways Doherty PA. It shows the interest of both segmented approaches in terms of linearization performance, complexity and ease of implementation compared to the global DPD. The paper starts with some mathematical generalities on interpolation and splines. It focuses on segmented models derived from Volterra series even if the presented principles can also be applied to neural networks.

Modelling the effects of solar irradiation plays an important role in various applications. This paper describes a semi-parametric (combined grey-box and spline-based), data-driven technique that can be used to model systems in which the solar gain depends on the sun position. The solar gain factor is introduced, i.e. the absorbed fraction of measured solar irradiation, and estimated as a continuous non-parametric function of the sun position. The implementation of the spline-based solar gain factor in a grey-box model framework is described. The method is tested in two case studies—in a model of the internal temperature of a dwelling in Aalborg, Denmark, and a model of the return temperature of a solar collector field in Solrod, Denmark. It is shown that the solar gain factor as a function of sun position is able to account for structural variations in solar gain that may occur due to factors such as shading obstacles and window or absorber orientation. In both test cases, the spline-based solar gain function improved the model accuracy significantly, and largely reduced structural errors in prediction residuals. In addition, the shape of the estimated function provided insight into the dynamics of the system and the local solar input characteristics. Accurate representation of such site characteristics was not possible with any data-driven method found in the literature. Besides the grey-box models used in this study, the solar gain factor can be used in a variety of data-driven models, for example in linear regression models.

We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output relation. To that end, we first establish a global bound for the Lipschitz constant of neural networks. Based on the obtained bound, we then formulate a variational problem for learning activation functions. Our variational problem is infinite-dimensional and is not computationally tractable. However, we prove that there always exists a solution that has continuous and piecewise-linear (linear-spline) activations. This reduces the original problem to a finite-dimensional minimization where an $\ell _1$ penalty on the parameters of the activations favors the learning of sparse nonlinearities. We numerically compare our scheme with standard ReLU network and its variations, PReLU and LeakyReLU and we empirically demonstrate the practical aspects of our framework.

We focus on the generalized-interpolation problem. There, one reconstructs continuous-domain signals that honor discrete data constraints. This problem is infinite-dimensional and ill-posed. We make it well-posed by imposing that the solution balances data fidelity and some $L_{p}$ -norm regularization. More specifically, we consider $p\geq 1$ and the multi-order derivative regularization operator ${\mathrm{L}}={\mathrm{D}}^{N_{0}}$ . We reformulate the regularized problem exactly as a finite-dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid. Our splines are represented in a B-spline basis, which results in a well-conditioned discretization. For a sufficiently fine grid, our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted. This remarkable property is due to the approximation power of splines. We use the alternating-direction method of multipliers along with a multiresolution strategy to compute our solution. We present numerical results for spatial and Fourier interpolation. Through our experiments, we investigate features induced by the $L_{p}$ -norm regularization, namely, sparsity, regularity, and oscillatory behavior.

This contribution investigates the connection between Isogeometric Analysis (IgA) and the Partial Element Equivalent Circuit (PEEC) method for electrostatic problems. We demonstrate that using the spline-based geometry concepts from IgA allows for extracting circuit elements without a meshing step. Moreover, the proposed IgA-PEEC method converges for complex geometries up to three times faster than the conventional PEEC approach and, in turn, it requires a significantly lower number of degrees of freedom to solve a problem with comparable accuracy. The resulting method is closely related to the isogeometric boundary element method. However, it uses lowest-order basis functions to allow for straightforward physical and circuit interpretations. The findings are validated by an analytical example with complex geometry, i.e., significant curvature, and by a realistic model of a surge arrester.

Locally refined B-spline (LRB) surfaces provide a representation that is well suited to scattered data approximation. When a data set has local details in some areas and is largely smooth elsewhere, LR B-splines allow the spatial distribution of degrees of freedom to follow the variations of the data set. An LRB surface approximating a data set is refined in areas where the accuracy does not meet a required tolerance. In this paper we address, in a systematic study, different LRB refinement strategies and polynomial degrees for surface approximation. We study their influence on the resulting data volume and accuracy when applied to geospatial data sets with different structural behaviour. The relative performance of the refinement strategies is reasonably coherent for the different data sets and this paper concludes with some recommendations. An overall evaluation indicates that bi-quadratic LRB are preferable for the use cases tested, and that the strategies we denote as “full span" have the overall best performance.

Experimental and computational data and field data obtained from measurements are often sparse and noisy. Consequently, interpolating unknown functions under these restrictions to provide accurate predictions is very challenging. This study compares machine-learning methods and cubic splines on the sparsity of training data they can handle, especially when training samples are noisy. We compare deviation from a true function f using the mean square error, signal-to-noise ratio and the Pearson R2 coefficient. We show that, given very sparse data, cubic splines constitute a more precise interpolation method than deep neural networks and multivariate adaptive regression splines. In contrast, machine-learning models are robust to noise and can outperform splines after a training data threshold is met. Our study aims to provide a general framework for interpolating one-dimensional signals, often the result of complex scientific simulations or laboratory experiments.

Spline functions are a useful tool for modelling the shape of shell structures. They have curvature continuity that allows good approximation accuracy for various objects, including hyperboloid cooling towers, spherical domes, paraboloid bowls of radio telescopes, or many other types of smooth free surfaces. Spline models can be used to determine the displacement of structures based on point clouds from laser scanning or photogrammetry. The curvature continuity of splines may, however, cause local distortions in models that have edges. Edges may appear in point clouds where surface patches are joined, on surfaces equipped with additional technical infrastructure or with cracks and shifts in the structure. Taking the properties of spline functions into account, several characteristic types of edge configurations can be distinguished, which may, to a different extent, affect the values of modelling errors. The research conducted below was aimed at identifying such configurations based on theoretical considerations and then assessing their effect on the accuracy of modelling shell structures measured by laser scanning. It turned out to be possible to distinguish between edge configurations, based on the deviation values.

In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization., 29 pages, 8 figures

Artificial beach nourishment is one of the most important environmentally friendly coastal protection methods since it protects the aesthetic and recreational values of the beach and increases its protective properties. Therefore, the main aim of the current study is to assess the accuracy of multivariate adaptive regression splines (MARS) in predicting the parameters, namely sediment transport coefficients (K) and the diffusion rate (omega), which affect beach nourishment performance. The performance of the MARS was determined by comparison of the models using exponential, linear, and power regression equations trained by conventional regression analyses (CRA) and the teaching-learning based optimization (TLBO) algorithm. In all models, two different input data obtained from the experimental study were used, one dimensional and one non-dimensional. The results presented that the MARS models gave lower error values than the CRA and TLBO models according to the root mean square error, mean absolute error, and scattering index criteria. When the models were evaluated, it was revealed that dimensional and non-dimensional models gave approximate results. We proved that the dimensional and non-dimensional MARS models can be used to estimate the (K) and (omega) values.

International audience; Interpolation is a vital piece of many image processing pipelines. In this document we present how to optimize the implementation of image B-spline interpolation for GPU architectures. The implementation is based on the one proposed by Briand and Monasse in 2018 and works for orders up to 11. The two main optimizations consist in: (1) transposing the B-spline coefficients before the prefiltering of the rows and (2) dividing columns into subregions in order to use more threads. We assess the impact of the floating point precision and of using high B-spline orders.

This paper describes a method for planning energy-efficient trajectories for industrial robots driven by brushless or DC motors with regenerative braking. The optimization problem is defined upon spline interpolation methods, using piecewise polynomial functions to produce a trajectory passing through a sequence of via-points, and on the electromechanical model of the robot. The formulation introduced in this work is aimed at estimating and optimizing the energy consumption using closed-form expressions and therefore without the need for any numerical integration of the robot dynamics. The method accounts for kinematic constraints on speed, acceleration, and jerk, as well as constraints due to the limitations of the power supply and of the regenerated energy storage system.

We present a novel method for the estimation of variance parameters in generalised linear mixed models. The method has its roots in Harville (1977)'s work, but it is able to deal with models that have a precision matrix for the random-effect vector that is linear in the inverse of the variance parameters (i.e., the precision parameters). We call the method SOP (Separation of Overlapping Precision matrices). SOP is based on applying the method of successive approximations to easy-to-compute estimate updates of the variance parameters. These estimate updates have an appealing form: they are the ratio of a (weighted) sum of squares to a quantity related to effective degrees of freedom. We provide the sufficient and necessary conditions for these estimates to be strictly positive. An important application field of SOP is penalised regression estimation of models where multiple quadratic penalties act on the same regression coefficients. We discuss in detail two of those models: penalised splines for locally adaptive smoothness and for hierarchical curve data. Several data examples in these settings are presented., MTM2014-55966-P MTM2014-52184-P

In this paper, the problem of vehicle guidance by means of an external leader is described. The objective is to navigate a four-wheeled vehicle through unstructured environments, characterized by the lack of availability of typical guidance infrastructure like lane markings or HD maps. The trajectory-following approach is based on an estimate of the leader’s path. For that, position measurements are stored over time with respect to an inertial frame. A new strategy is proposed to rate the significance of position measurements and ensure that a certain threshold of stored samples is not exceeded. Having an estimate of the leader path is essential to prevent the cutting-corner phenomenon and for exact path following in general. A spline-approximation technique is applied to obtain a smooth reference path for the underlying lateral and longitudinal motion controllers. For longitudinal tracking, a constant time-headway policy was implemented, to follow the leader with a constant time gap along the estimated path. The algorithm was first developed and tested in a simulation framework and then deployed in a demonstrator vehicle for validation under real operating conditions. The presented experimental results were achieved using only on-board sensors of the demonstrator vehicle, while high-accuracy differential GPS-based position measurements serve as the ground truth data for visualization.

Copula models are flexible tools to represent complex structures of dependence for multivariate random variables. According to Sklar's theorem (Sklar, 1959), any d-dimensional absolutely continuous density can be uniquely represented as the product of the marginal distributions and a copula function which captures the dependence structure among the vector components. In real data applications, the interest of the analyses often lies on specific functionals of the dependence, which quantify aspects of it in a few numerical values. A broad literature exists on such functionals, however extensions to include covariates are still limited. This is mainly due to the lack of unbiased estimators of the copula function, especially when one does not have enough information to select the copula model. Recent advances in computational methodologies and algorithms have allowed inference in the presence of complicated likelihood functions, especially in the Bayesian approach, whose methods, despite being computationally intensive, allow us to better evaluate the uncertainty of the estimates. In this work, we present several Bayesian methods to approximate the posterior distribution of functionals of the dependence, using nonparametric models which avoid the selection of the copula function. These methods are compared in simulation studies and in two realistic applications, from civil engineering and astrophysics.

Functional data analysis is typically performed in two steps: first, functionally representing discrete observations, and then applying functional methods, such as the functional principal component analysis, to the so-represented data. While the initial choice of a functional representation may have a significant impact on the second phase of the analysis, this issue has not gained much attention in the past. Typically, a rather ad hoc choice of some standard basis such as Fourier, wavelets, splines, etc. is used for the data transforming purpose. To address this important problem, we present its mathematical formulation, demonstrate its importance, and propose a data-driven method of functionally representing observations. The method chooses an initial functional basis by an efficient placement of the knots. A simple machine learning style algorithm is utilized for the knot selection and recently introduced orthogonal spline bases - splinets - are eventually taken to represent the data. The benefits are illustrated by examples of analyses of sparse functional data.

Based on multidisciplinary data, including seismological and geodetic observations, as well as seismic reflection profiles and gravity maps, we analysed the pattern of crustal deformation and active tectonics in the Sicily Channel, a key observation point to unravel the complex interaction between two major plates, Nubia and Eurasia, in the Mediterranean Sea. Our data highlight the presence of an active ~ 220-km-long complex lithospheric fault system (here named the Lampedusa-Sciacca Shear Zone), approximately oriented N–S, crossing the study area with left-lateral strike-slip deformations, active volcanism and high heat flow. We suggest that this shear zone represents the most active tectonic domain in the area, while the NW–SE elongated rifting pattern, considered the first order tectonic feature, appears currently inactive and sealed by undeformed recent (Lower Pleistocene?) deposits. Estimates of seismological and geodetic moment-rates, 6.58 × 1015 Nm/year and 7.24 × 1017 Nm/year, respectively, suggests that seismicity accounts only for ~ 0.9% of crustal deformation, while the anomalous thermal state and the low thickness of the crust would significantly inhibit frictional sliding in favour of creeping and aseismic deformation. We therefore conclude that a significant amount of the estimated crustal deformation-rate occurs aseismically, opening new scenarios for seismic risk assessments in the region.

Scanning transmission X-ray microscopy (STXM) produces images in which each pixel value is related to the measured attenuation of an X-ray beam. In practice, the location of the illuminated region does not exactly match the desired uniform pixel grid. This error can be measured using an interferometer. In this paper, we propose a spline-based reconstruction method for STXM which takes these position errors into account. We achieve this by formulating the reconstruction problem as a continuous-domain inverse problem in a spline basis, and by using Hessian nuclear-norm regularization. We solve this problem using the standard ADMM algorithm, and we demonstrate the pertinence of our approach on both simulated and real STXM data.

Introduction The frequently used Cox regression applies two critical assumptions, which might not hold for all predictors. In this study, the results from a Cox regression model (CM) and a generalized Cox regression model (GCM) are compared. Methods Data are from the Survey of Health, Ageing and Retirement in Europe (SHARE), which includes approximately 140,000 individuals aged 50 or older followed over seven waves. CMs and GCMs are used to estimate dementia risk. The results are internally and externally validated. Results None of the predictors included in the analyses fulfilled the assumptions of Cox regression. Both models predict dementia moderately well (10‐year risk: 0.737; 95% confidence interval [CI]: 0.699, 0.773; CM and 0.746; 95% CI: 0.710, 0.785; GCM). Discussion The GCM performs significantly better than the CM when comparing pseudo‐R2 and the log‐likelihood. GCMs enable researcher to test the assumptions used by Cox regression independently and relax these assumptions if necessary.

Along this work we study an indefinite abstract smoothing problem. After establishing necessary and sufficient conditions for the existence of solutions to this problem, the set of admissible parameters is discussed in detail. Then, its relationship with a linearly constrained interpolation problem is analyzed., Comment: 16 pages. Submitted

We present a non-parametric method to estimate the discount curve from market quotes based on the Moore-Penrose pseudoinverse. The discount curve reproduces the market quotes perfectly, has maximal smoothness, and is given in closed-form. The method is easy to implement and requires only basic linear algebra operations. We provide a full theoretical framework as well as several practical applications., Forthcoming in SIAM Journal on Financial Mathematics

The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial PH curve, we consider the Euler-Rodrigues (ER) frame. Moreover G 1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of G k continuity of ER frames implies G k + 1 continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results.

We present a generic method to construct orthogonal projectors for two-dimensional landmark-based parametric spline curves. We construct vector spaces that define a geometric transformation (e.g., affine, similarity, and scaling) that is applied to a reference curve. These vector spaces contain all parametric curves up to the chosen transformation. We define the vector spaces implicitly through an orthogonal projection operator and present a theorem that characterizes the projector for landmark-based spline curves, which are popular for the user-interactive analysis of biomedical images. Finally, we show how shape priors are constructed with the spline projector and provide an example of application for the segmentation of microscopy images in biology.

Planar curvature continuous path generation with obstacle avoidance is considered by dealing with environments described either in terms of a structure directly specifying the boundaries of the obstacles (path planning) or by an adjacency matrix associated with uniform cells covering the whole scene (path finding). The second representation can be obtained for example from an image segmentation method when only an image of the scene is available. The method is composed by two main steps: the first is devoted to produce an intermediate admissible polyline, while the second defines a final curvature continuous path. Since the shape of the intermediate piecewise linear path remarkably influences the smoothness of the final path, different approaches are proposed to solve the path planning/finding problem in the first step. Then the vertices of the previously produced polyline are interpolated by using a C1G2 interpolation scheme with tension based on Pythagoreanhodograph (PH) splines which have attractive features for applications. A strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. The results of several numerical experiments confirm the effectiveness of the method. An automatic twostage curvature continuous path planning approach is presented.The minimal cumulative angle polyline in the visibility graph is identified.Interpolation with C1G2 PH splines with tension is adopted.A new strategy for the automatic selection of the tension parameters is introduced.Complex environments possibly described by images are also addressed.

This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 5–18, 2018

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for p large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree k -refinement in isogeometric analysis.

CD4-based multi-state back-calculation methods are key for monitoring the HIV epidemic, providing estimates of HIV incidence and diagnosis rates by disentangling their inter-related contribution to the observed surveillance data. This paper, extends existing approaches to age-specific settings, permitting the joint estimation of age- and time-specific incidence and diagnosis rates and the derivation of other epidemiological quantities of interest. This allows the identification of specific age-groups at higher risk of infection, which is crucial in directing public health interventions. We investigate, through simulation studies, the suitability of various bivariate splines for the non-parametric modelling of the latent age- and time-specific incidence and illustrate our method on routinely collected data from the HIV epidemic among gay and bisexual men in England and Wales. Electronic supplementary material The online version of this article (10.1007/s10985-019-09465-1) contains supplementary material, which is available to authorized users.

Background. Parametric modeling of survival data is important, and reimbursement decisions may depend on the selected distribution. Accurate predictions require sufficiently flexible models to describe adequately the temporal evolution of the hazard function. A rich class of models is available among the framework of generalized linear models (GLMs) and its extensions, but these models are rarely applied to survival data. This article describes the theoretical properties of these more flexible models and compares their performance to standard survival models in a reproducible case study. Methods. We describe how survival data may be analyzed with GLMs and their extensions: fractional polynomials, spline models, generalized additive models, generalized linear mixed (frailty) models, and dynamic survival models. For each, we provide a comparison of the strengths and limitations of these approaches. For the case study, we compare within-sample fit, the plausibility of extrapolations, and extrapolation performance based on data splitting. Results. Viewing standard survival models as GLMs shows that many impose a restrictive assumption of linearity. For the case study, GLMs provided better within-sample fit and more plausible extrapolations. However, they did not improve extrapolation performance. We also provide guidance to aid in choosing between the different approaches based on GLMs and their extensions. Conclusions. The use of GLMs for parametric survival analysis can outperform standard parametric survival models, although the improvements were modest in our case study. This approach is currently seldom used. We provide guidance on both implementing these models and choosing between them. The reproducible case study will help to increase uptake of these models.

A novel surface interrogation technique is proposed to compute the intersection of curves with spline surfaces in isogeometric analysis. The intersection points are determined in one-shot without resorting to a Newton-Raphson iteration or successive refinement. Surface-curve intersection is required in a wide range of applications, including contact, immersed boundary methods and lattice-skin structures, and requires usually the solution of a system of nonlinear equations. It is assumed that the surface is given in form of a spline, such as a NURBS, T-spline or Catmull-Clark subdivision surface, and is convertible into a collection of B\'ezier patches. First, a hierarchical bounding volume tree is used to efficiently identify the B\'ezier patches with a convex-hull intersecting the convex-hull of a given curve segment. For ease of implementation convex-hulls are approximated with k-dops (discrete orientation polytopes). Subsequently, the intersections of the identified B\'ezier patches with the curve segment are determined with a matrix-based implicit representation leading to the computation of a sequence of small singular value decompositions (SVDs). As an application of the developed interrogation technique the isogeometric design and analysis of lattice-skin structures is investigated. The skin is a spline surface that is usually created in a computer-aided design (CAD) system and the periodic lattice to be fitted consists of unit cells, each containing a small number of struts. The lattice-skin structure is generated by projecting selected lattice nodes onto the surface after determining the intersection of unit cell edges with the surface. For mechanical analysis, the skin is modelled as a Kirchhoff-Love thin-shell and the lattice as a pin-jointed truss. The two types of structures are coupled with a standard Lagrange multiplier approach., Comment: 23 pages, 15 figures

In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical and non-hierarchical T-meshes. For hierarchical T-meshes it is shown that mixed smoothness spline spaces that contain the space of PHT-splines ( Deng et al., 2008 ) always have stable dimension.

Motivated by mapping adverse artifactual events caused by body movements in electroencephalographic (EEG) signals, we present a functional independent component analysis based on the spectral decomposition of the kurtosis operator of a smoothed principal component expansion. A discrete roughness penalty is introduced in the orthonormality constraint of the covariance eigenfunctions in order to obtain the smoothed basis for the proposed independent component model. To select the tuning parameters, a cross-validation method that incorporates shrinkage is used to enhance the performance on functional representations with a large basis dimension. This method provides an estimation strategy to determine the penalty parameter and the optimal number of components. Our independent component approach is applied to real EEG data to estimate genuine brain potentials from a contaminated signal. As a result, it is possible to control high-frequency remnants of neural origin overlapping artifactual sources to optimize their removal from the signal. An R package implementing our methods is available at CRAN., Methusalem funding from the Flemish Government, Project MTM2017-88708-P of the Spanish Ministry of Science, Innovation and Universities, Project FQM-307 of the Government of Andalusia (Spain)

We propose a real-time approximation of R0 in an SIR-type model that applies to the COVID-19 epidemic outbreak. A very useful direct formula expressing R0 is found. Then, various type of models are considered, namely, finite differences, cubic splines, Piecewise Cubic Hermite interpolation and linear least squares approximation. Preserving the monotonicity of the formula under consideration proves to be of crucial importance. This latter property is preferred over accuracy, since it maintains positive R0. Only the Linear Least Squares technique guarantees this, and is finally proposed here. Tests on real COVID-19 data confirm the usefulness of our approach.

Variable renewable generation demands increasing amount of flexible resources to balance the electric power system, and batteries stand out as a promising alternative. Battery models for optimization typically represent the battery with power and energy variables, while the voltage, current, charge variable space is used for simulation models. This paper proposes a non-linear battery storage optimization model in the voltage, current, charge variable space. The battery voltage is conceived as an empirical function of both state-of-charge and charge current and represented through bi-variate cubic splines. The voltage source converter losses are also approximated with a cubic spline function. Compared to energy-based storage models, the results show that this approach enables safe operation closer to the battery voltage and current limits. Furthermore, it prefers operating around high state-of-charge due to the higher efficiency in that region. © 2020 The Authors This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

There are many cases in the engineering practice when it is necessary to know the transient temperature field in a flat plate. At that, temperature distribution is studied experimentally or by calculation. However, the obtained calculation results do not always provide the required accuracy. This paper obtains the spline-method grade 3 defect 1, which allows simulating the process of the thermal conductivity for the infinitely extended plate at nonlinear initial and boundary conditions of the first kind on the plate sides. For accurate evaluation of the solution of the heat conduction equation with the help of the spline-method, the standard problem is created for a one-dimensional infinite flat plane which has the exact analytical solution. This paper shows the use of spline method for the solution of transient heat conduction problem which gives an error of no more than 1·10 -5 and thus satisfies obtaining of the sufficient accuracy to solve many practical problems.

Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the success of deep networks. In this article we take another route by comparing the expressive power of DNNs with ReLU activation function to linear spline methods. We show that MARS (multivariate adaptive regression splines) is improper learnable by DNNs in the sense that for any given function that can be expressed as a function in MARS with M parameters there exists a multilayer neural network with O ( M log ( M ∕ e ) ) parameters that approximates this function up to sup-norm error e . We show a similar result for expansions with respect to the Faber–Schauder system. Based on this, we derive risk comparison inequalities that bound the statistical risk of fitting a neural network by the statistical risk of spline-based methods. This shows that deep networks perform better or only slightly worse than the considered spline methods. We provide a constructive proof for the function approximations.

Satellite product validation is a key to ensure the delivery of quality products for climate and weather applications. To do this, a fundamental step is the comparison with other instruments, such as radiosonde. This is especially true for essential climate variables such as temperature and humidity. Thanks to a functional data representation, this paper uses a likelihood-based approach that exploits the measurement uncertainties in a natural way. In particular, the comparison of temperature and humidity radiosonde measurements collected within the network of the Universal Rawinsonde Observation Program (RAOB) and the corresponding atmospheric profiles derived from the infrared atmospheric sounding interferometer aboard MetOp-A and MetOp-B satellites is developed with the aim of understanding the vertical smoothing mismatch uncertainty. Moreover, conventional RAOB functional data representation is assessed by means of a comparison with radiosonde reference measurements given by the Global Climate Observing System (GCOS) Reference Upper-Air Network (GRUAN), which provides high-resolution fully traceable radio-sounding profiles. In this way, the uncertainty related to coarse vertical resolution, or sparseness, of the conventional RAOB is assessed. It has been found that the uncertainty of vertical smoothing mismatch averaged along the profile is 0.50 K for temperature and 0.16 g/kg for water-vapor mixing ratio. Moreover, the uncertainty related to RAOB sparseness, averaged along the profile, is 0.29 K for temperature and 0.13 g/kg for water-vapor mixing ratio.

We provide a complete pipeline for the detection of patterns of interest in an image. In our approach, the patterns are assumed to be adequately modeled by a known template, and are located at unknown positions and orientations that we aim at retrieving. We propose a continuous-domain additive image model, where the analyzed image is the sum of the patterns to localize and a background with self-similar isotropic power-spectrum. We are then able to compute the optimal filter fulfilling the SNR criterion based on one single template and background pair: it strongly responds to the template while being optimally decoupled from the background model. In addition, we constrain our filter to be steerable, which allows for a fast template detection together with orientation estimation. In practice, the implementation requires to discretize a continuous-domain formulation on polar grids, which is performed using quadratic radial B-splines. We demonstrate the practical usefulness of our method on a variety of template approximation and pattern detection experiments. We show that the detection performance drastically improves when we exploit the statistics of the background via its power-spectrum decay, which we refer to as spectral-shaping. The proposed scheme outperforms state-of-the-art steerable methods by up to 50% of absolute detection performance.