In the past decades, the growing amount of network data has lead to many novel statistical models. In this paper we consider so called geometric networks. Typical examples are road networks or other infrastructure networks. But also the neurons or the blood vessels in a human body can be interpreted as a geometric network embedded in a three-dimensional space. In all these applications a network specific metric rather than the Euclidean metric is usually used, which makes the analyses on network data challenging. We consider network based point processes and our task is to estimate the intensity (or density) of the process which allows to detect high- and low- intensity regions of the underlying stochastic processes. Available routines that tackle this problem are commonly based on kernel smoothing methods. However, kernel based estimation in general exhibits some drawbacks such as suffering from boundary effects and the locality of the smoother. In an Euclidean space, the disadvantages of kernel methods can be overcome by using penalized spline smoothing. We here extend penalized spline smoothing towards smooth intensity estimation on geometric networks and apply the approach to both, simulated and real world data. The results show that penalized spline based intensity estimation is numerically efficient and outperforms kernel based methods. Furthermore, our approach easily allows to incorporate covariates, which allows to respect the network geometry in a regression model framework.
For multivariate nonparametric regression, functional analysis-of-variance (ANOVA) modeling aims to capture the relationship between a response and covariates by decomposing the unknown function into various components, representing main effects, two-way interactions, etc. Such an approach has been pursued explicitly in smoothing spline ANOVA modeling and implicitly in various greedy methods such as MARS. We develop a new method for functional ANOVA modeling, based on doubly penalized estimation using total-variation and empirical-norm penalties, to achieve sparse selection of component functi…
Penalized spline smoothing is a popular and flexible method of obtaining estimates in nonparametric regression but the classical least-squares criterion is highly susceptible to model deviations and atypical observations. Penalized spline estimation with a resistant loss function is a natural remedy, yet to this day the asymptotic properties of M-type penalized spline estimators have not been studied. We show in this paper that M-type penalized spline estimators achieve the same rates of convergence as their least-squares counterparts, even with auxiliary scale estimation. We further find theo…
Data analysed here derive from experiments conducted to study neurons' activity in the visual cortex of behaving monkeys. We consider a spatio-temporal adaptive penalized spline (P-spline) approach for modelling the firing rate of visual neurons. To the best of our knowledge, this is the first attempt in the statistical literature for locally adaptive smoothing in three dimensions. Estimation is based on the Separation of Overlapping Penalties (SOP) algorithm, which provides the stability and speed we look for.
Bayesian P-splines assume an intrinsic Gaussian Markov random field prior on the spline coefficients, conditional on a precision hyper-parameter $τ$. Prior elicitation of $τ$ is difficult. To overcome this issue we aim to building priors on an interpretable property of the model, indicating the complexity of the smooth function to be estimated. Following this idea, we propose Penalized Complexity (PC) priors for the number of effective degrees of freedom. We present the general ideas behind the construction of these new PC priors, describe their properties and show how to implement them in P-s…
Penalized spline estimation with discrete difference penalties (P-splines) is a popular estimation method for semiparametric models, but the classical least-squares estimator is highly sensitive to deviations from its ideal model assumptions. To remedy this deficiency, a broad class of P-spline estimators based on general loss functions is introduced and studied. Robust estimators are obtained by well-chosen loss functions, such as the Huber or Tukey loss function. A preliminary scale estimator can also be included in the loss function. It is shown that this class of P-spline estimators enjoys…
Penalized spline regression is a popular method for scatterplot smoothing, but there has long been a debate on how to construct confidence intervals for penalized spline fits. Due to the penalty, the fitted smooth curve is a biased estimate of the target function. Many methods, including Bayesian intervals and the simple-shift bias-reduction, have been proposed to upgrade the coverage of the confidence intervals, but these methods usually fail to adequately improve the situation at predictor values where the function is sharply curved. In this paper, we develop a novel approach to improving th…
In the presence of unmeasured spatial confounding, spatial models may actually increase (rather than decrease) bias, leading to uncertainty as to how they should be applied in practice. We evaluated spatial modeling approaches through simulation and application to a big data electronic health record study. Whereas the risk of bias was high for purely spatial exposures (e.g., built environment), we found very limited potential for increased bias for individual-level exposures that cluster spatially (e.g., smoking status). We also proposed a novel exposure-penalized spline approach that selects …
Regression spline is a useful tool in nonparametric regression. However, finding the optimal knot locations is a known difficult problem. In this article, we introduce the Non-concave Penalized Regression Spline. This proposal method not only produces smoothing spline with optimal convergence rate, but also can adaptively select optimal knots simultaneously. It is insensitive to the number of origin knots. The method's performance in a simulation has been studied to compare the other methods. The problem of how to choose smoothing parameters, i.e. penalty parameters in the non-concave regressi…
Nonparametric Density Estimation for Data Scattered on Irregular Spatial Domains: A Likelihood-Based Approach Using Bivariate Penalized Spline Smoothing
Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first non-parametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse …
Generalized additive models (GAMs) are a well-established statistical tool for modeling complex nonlinear relationships between covariates and a response assumed to have a conditional distribution in the exponential family. In this article, P-splines and the Laplace approximation are coupled for flexible and fast approximate Bayesian inference in GAMs. The proposed Laplace-P-spline model contributes to the development of a new methodology to explore the posterior penalty space by considering a deterministic grid-based strategy or a Markov chain sampler, depending on the number of smooth additi…
The paper motivates high dimensional smoothing with penalized splines and its numerical calculation in an efficient way. If smoothing is carried out over three or more covariates the classical tensor product spline bases explode in their dimension bringing the estimation to its numerical limits. A recent approach by Siebenborn and Wagner(2019) circumvents storage expensive implementations by proposing matrix-free calculations which allows to smooth over several covariates. We extend their approach here by linking penalized smoothing and its Bayesian formulation as mixed model which provides a …
P-splines are penalized B-splines, in which finite order differences in coefficients are typically penalized with an $\ell_2$ norm. P-splines can be used for semiparametric regression and can include random effects to account for within-subject variability. In addition to $\ell_2$ penalties, $\ell_1$-type penalties have been used in nonparametric and semiparametric regression to achieve greater flexibility, such as in locally adaptive regression splines, $\ell_1$ trend filtering, and the fused lasso additive model. However, there has been less focus on using $\ell_1$ penalties in P-splines, pa…
Stochastic volatility (SV) models mimic many of the stylized facts attributed to time series of asset returns, while maintaining conceptual simplicity. The commonly made assumption of conditionally normally distributed or Student-t-distributed returns, given the volatility, has however been questioned. In this manuscript, we introduce a novel maximum penalized likelihood approach for estimating the conditional distribution in an SV model in a nonparametric way, thus avoiding any potentially critical assumptions on the shape. The considered framework exploits the strengths both of the powerful …
Despite their widespread use in practice, the asymptotic properties of Bayesian penalized splines have not been investigated so far. We close this gap and study posterior concentration rates for Bayesian penalized splines in a Gaussian nonparametric regression model. A key feature of the approach is the hyperprior on the smoothing variance, which allows for adaptive smoothing in practice but complicates the theoretical analysis considerably as it destroys conjugacy and precludes analytic expressions for the posterior moments. To derive our theoretical results, we rely on several new concepts i…
Density estimation plays a fundamental role in many areas of statistics and machine learning. Parametric, nonparametric and semiparametric density estimation methods have been proposed in the literature. Semiparametric density models are flexible in incorporating domain knowledge and uncertainty regarding the shape of the density function. Existing literature on semiparametric density models is scattered and lacks a systematic framework. In this paper, we consider a unified framework based on the reproducing kernel Hilbert space for modeling, estimation, computation and theory. We propose gene…
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - μ(t_j))^2 + λ\int_a^b [μ"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may…
The selection of smoothing parameter is central to the estimation of penalized splines. The best value of the smoothing parameter is often the one that optimizes a smoothness selection criterion, such as generalized cross-validation error (GCV) and restricted likelihood (REML). To correctly identify the global optimum rather than being trapped in an undesired local optimum, grid search is recommended for optimization. Unfortunately, the grid search method requires a pre-specified search interval that contains the unknown global optimum, yet no guideline is available for providing this interval…
We propose a fast penalized spline method for bivariate smoothing. Univariate P-spline smoothers (Eilers and Marx, 1996) are applied simultaneously along both coordinates. The new smoother has a sandwich form which suggested the name "sandwich smoother" to a referee. The sandwich smoother has a tensor product structure that simplifies an asymptotic analysis and it can be fast computed. We derive a local central limit theorem for the sandwich smoother, with simple expressions for the asymptotic bias and variance, by showing that the sandwich smoother is asymptotically equivalent to a bivariate …
Spatial Capture-Recapture With Penalized Regression Splines to Flexibly Model Wildlife Density and Distribution
Utrecht University（乌得勒支大学）；University of Glasgow（格拉斯哥大学）；University of St Andrews（圣安德鲁斯大学）；World Wide Fund for Nature（世界自然基金会）；University of Auckland（奥克兰大学）
Spatial data collected worldwide at a huge number of locations are frequently used in environmental and climate studies. Spatial modelling for this type of data presents both methodological and computational challenges. In this work we illustrate a computationally efficient non parametric framework to model and estimate the spatial field while accounting for geodesic distances between locations. The spatial field is modelled via penalized splines (P-splines) using intrinsic Gaussian Markov Random Field (GMRF) priors for the spline coefficients. The key idea is to use the sphere as a surrogate …
Handling Overlapping Asymmetric Datasets -- A Twice Penalized P-Spline Approach
Newcastle University（纽卡斯尔大学）；School of Computing（计算机学院）；School of Mathematics, Statistics and Physics（数学、统计与物理学院）；Translational & Clinical Research Institute（转化与临床研究院）
We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the parametric and the nonparametric components given independent and identically distributed (i.i.d.)~data. We prove the consistency and find the rates of convergence of the estimators. We establish asymptotic normality under under mild assumption and prove asympt…
In many psychometric applications, the relationship between the mean of an outcome and a quantitative covariate is too complex to be described by simple parametric functions; instead, flexible nonlinear relationships can be incorporated using penalized splines. Penalized splines can be conveniently represented as a linear mixed effects model (LMM), where the coefficients of the spline basis functions are random effects. The LMM representation of penalized splines makes the extension to multivariate outcomes relatively straightforward. In the LMM, no effect of the quantitative covariate on the …
Despite the fundamental nature of the inhomogeneous Poisson process in the theory and application of stochastic processes, and its attractive generalizations (e.g. Cox process), few tractable nonparametric modeling approaches of intensity functions exist, especially when observed points lie in a high-dimensional space. In this paper we develop a new, computationally tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the inhomogeneous Poisson process. We model the square root of the intensity as an RKHS function. Whereas RKHS models used in supervised learning rely on the so-call…
In this paper we introduce a new methodology to determine an optimal coefficient of penalized functional regression. We assume the dependent, independent variables and the regression coefficients are functions of time and error dynamics follow a stochastic differential equation. First we construct our objective function as a time dependent residual sum of square and then minimize it with respect to regression coefficients subject to different error dynamics such as LASSO, group LASSO, fused LASSO and cubic smoothing spline. Then we use Feynman-type path integral approach to determine a Schrödi…
A theorem of McCann shows that for any two absolutely continuous probability measures on R^d there exists a monotone transformation sending one probability measure to the other. A consequence of this theorem, relevant to statistics, is that density estimation can be recast in terms of transformations. In particular, one can fix any absolutely continuous probability measure, call it P, and then reparameterize the whole class of absolutely continuous probability measures as monotone transformations from P. In this paper we utilize this reparameterization of densities, as monotone transformations…
Smoothing of noisy sample covariances is an important component in functional data analysis. We propose a novel covariance smoothing method based on penalized splines and associated software. The proposed method is a bivariate spline smoother that is designed for covariance smoothing and can be used for sparse functional or longitudinal data. We propose a fast algorithm for covariance smoothing using leave-one-subject-out cross validation. Our simulations show that the proposed method compares favorably against several commonly used methods. The method is applied to a study of child growth led…
We consider efficient estimation of flexible transformation models with interval-censored data. To reduce the dimension of semi-parametric models, the unknown monotone transformation function is approximated via monotone splines. A penalization technique is used to provide more computationally efficient estimation of all parameters. To accomplish model fitting, a computationally efficient nested iterative expectation-maximization (EM) based algorithm is developed for estimation, and an easily implemented variance-covariance approach is proposed for inference on regression parameters. Theoretic…
Covariance estimation is essential yet underdeveloped for analyzing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor-product B-spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B-spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters…
We consider the problem of detecting a `bump' in the intensity of a Poisson process or in a density. We analyze two types of likelihood ratio based statistics which allow for exact finite sample inference and asymptotically optimal detection: The maximum of the penalized square root of log likelihood ratios (`penalized scan') evaluated over a certain sparse set of intervals, and a certain average of log likelihood ratios (`condensed average likelihood ratio'). We show that penalizing the {\sl square root} of the log likelihood ratio - rather than the log likelihood ratio itself - leads to a si…
Current status data are commonly encountered in medical and epidemiological studies in which the failure time for study units is the outcome variable of interest. Data of this form are characterized by the fact that the failure time is not directly observed but rather is known relative to an observation time; i.e., the failure times are either left- or right-censored. Due to its structure, the analysis of such data can be challenging. To circumvent these challenges and to provide for a flexible modeling construct which can be used to analyze current status data, herein, a partially linear addi…
Motivated by a hemodialysis monitoring study, we propose a logistic model with a functional predictor, called the Sparse Functional Logistic Regression (SFLR), where the corresponding coefficient function is {\it locally sparse}, that is, it is completely zero on some subregions of its domain. The coefficient function, together with the intercept parameter, are estimated through a doubly-penalized likelihood approach with a B-splines expansion. One penalty is for controlling the roughness of the coefficient function estimate and the other penalty, in the form of the $L_1$ norm, enforces the lo…
Laplacian-P-splines for shared Gamma frailty models applied to clustered right-censored time-to-event data
Department of Neurodegeneration Diagnostics, University Hospital Bialystok, Poland（比亚韦斯托克医院神经退行性疾病诊断系）；Interuniversity Institute for Biostatistics and statistical Bioinformatics (I-BioStat), Data Science Institute, Hasselt University, Belgium（哈瑟尔特大学生物统计与统计生物信息学跨大学研究所（I-BioStat））；Global Health Institute, Family Medicine and Population Health, University of Antwerp, Belgium（安特卫普大学全球健康学院）
A generalized linear point process is specified in terms of an intensity that depends upon a linear predictor process through a fixed non-linear function. We present a framework where the linear predictor is parametrized by a Banach space and give results on Gateaux differentiability of the log-likelihood. Of particular interest is when the intensity is expressed in terms of a linear filter parametrized by a Sobolev space. Using that the Sobolev spaces are reproducing kernel Hilbert spaces we derive results on the representation of the penalized maximum likelihood estimator in a special case a…
We discuss the semiparametric modeling of mark-recapture-recovery data where the temporal and/or individual variation of model parameters is explained via covariates. Typically, in such analyses a fixed (or mixed) effects parametric model is specified for the relationship between the model parameters and the covariates of interest. In this paper, we discuss the modeling of the relationship via the use of penalized splines, to allow for considerably more flexible functional forms. Corresponding models can be fitted via numerical maximum penalized likelihood estimation, employing cross-validatio…
Small area estimation under linear mixed models often assumes that the small area effect is random effect in almost all previous studies. However, in this paper a new approach is proposed explaining small area effect as the unknown function of an area-indicative variable, which is a kind of semiparametric models. The nonparametric part for area effect is represented by using penalized splines, from which the estimation and inference are done in the linear mixed model framework. The mean-squared error of empirical estimators is shown, and the testing for small area effect also considered. Addit…
