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1. More efficient approximation of smoothing splines via space-filling basis selection

   https://doi.org/10.1093/biomet/asaa019  

   Meng, Cheng; Zhang, Xinlian; Zhang, Jingyi; Zhong, Wenxuan; Ma, Ping (May 2020, Biometrika)

   null (Ed.)

   Summary We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $$n$$, the smoothing spline estimator can be expressed as a linear combination of $$n$$ basis functions, requiring $O(n^3)$ computational time when the number $$d$$ of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on $$q$$ randomly selected basis functions, resulting in a computational cost of $O(nq^2)$. It is known that these two estimators converge at the same rate when $$q$$ is of order $$O\{n^{2/(pr+1)}\}$$, where $$p\in [1,2]$$ depends on the true function and $r > 1$ depends on the type of spline. Such a $$q$$ is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease $$q$$ to around $$O\{n^{1/(pr+1)}\}$$ when $$d\leq pr+1$$. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods. 

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2. Scanner: Simultaneously temporal trend and spatial cluster detection for spatial‐temporal data

   https://doi.org/10.1002/env.2849  

   Wang, Xin; Zhang, Xin (April 2024, Environmetrics)

   Abstract Identifying the underlying trajectory pattern in the spatial‐temporal data analysis is a fundamental but challenging task. In this paper, we study the problem of simultaneously identifying temporal trends and spatial clusters of spatial‐temporal trajectories. To achieve this goal, we propose a novel method named spatial clustered and sparse nonparametric regression (). Our method leverages the B‐spline model to fit the temporal data and penalty terms on spline coefficients to reveal the underlying spatial‐temporal patterns. In particular, our method estimates the model by solving a doubly‐penalized least square problem, in which we use a group sparse penalty for trend detection and a spanning tree‐based fusion penalty for spatial cluster recovery. We also develop an algorithm based on the alternating direction method of multipliers (ADMM) algorithm to efficiently minimize the penalized least square loss. The statistical consistency properties of estimator are established in our work. In the end, we conduct thorough numerical experiments to verify our theoretical findings and validate that our method outperforms the existing competitive approaches. 

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3. An asymptotic and empirical smoothing parameters selection method for smoothing spline ANOVA models in large samples

   https://doi.org/10.1093/biomet/asaa047  

   Sun, Xiaoxiao; Zhong, Wenxuan; Ma, Ping (August 2020, Biometrika)

   null (Ed.)

   Summary Large samples are generated routinely from various sources. Classic statistical models, such as smoothing spline ANOVA models, are not well equipped to analyse such large samples because of high computational costs. In particular, the daunting computational cost of selecting smoothing parameters renders smoothing spline ANOVA models impractical. In this article, we develop an asympirical, i.e., asymptotic and empirical, smoothing parameters selection method for smoothing spline ANOVA models in large samples. The idea of our approach is to use asymptotic analysis to show that the optimal smoothing parameter is a polynomial function of the sample size and an unknown constant. The unknown constant is then estimated through empirical subsample extrapolation. The proposed method significantly reduces the computational burden of selecting smoothing parameters in high-dimensional and large samples. We show that smoothing parameters chosen by the proposed method tend to the optimal smoothing parameters that minimize a specific risk function. In addition, the estimator based on the proposed smoothing parameters achieves the optimal convergence rate. Extensive simulation studies demonstrate the numerical advantage of the proposed method over competing methods in terms of relative efficacy and running time. In an application to molecular dynamics data containing nearly one million observations, the proposed method has the best prediction performance. 

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4. Layer-Wise Learning Strategy for Nonparametric Tensor Product Smoothing Spline Regression and Graphical Models

   Tan, Kean Ming (October 2019, Journal of machine learning research)

   Nonparametric estimation of multivariate functions is an important problem in statisti- cal machine learning with many applications, ranging from nonparametric regression to nonparametric graphical models. Several authors have proposed to estimate multivariate functions under the smoothing spline analysis of variance (SSANOVA) framework, which assumes that the multivariate function can be decomposed into the summation of main effects, two-way interaction effects, and higher order interaction effects. However, existing methods are not scalable to the dimension of the random variables and the order of inter- actions. We propose a LAyer-wiSE leaRning strategy (LASER) to estimate multivariate functions under the SSANOVA framework. The main idea is to approximate the multivari- ate function sequentially starting from a model with only the main effects. Conditioned on the support of the estimated main effects, we estimate the two-way interaction effects only when the corresponding main effects are estimated to be non-zero. This process is con- tinued until no more higher order interaction effects are identified. The proposed strategy provides a data-driven approach for estimating multivariate functions under the SSANOVA framework. Our proposal yields a sequence of estimators. To study the theoretical prop- erties of the sequence of estimators, we establish the notion of post-selection persistency. Extensive numerical studies are performed to evaluate the performance of LASER. 

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5. A Kernel Multiple Change-point Algorithm via Model Selection

   Arlot, Sylvain; Celisse, Alain; Harchaoui, Zaid (January 2019, Journal of machine learning research)

   We consider a general formulation of the multiple change-point problem, in which the data is assumed to belong to a set equipped with a positive semidefinite kernel. We propose a model-selection penalty allowing to select the number of change points in Harchaoui and Cappe's kernel-based change-point detection method. The model-selection penalty generalizes non-asymptotic model-selection penalties for the change-in-mean problem with univariate data. We prove a non-asymptotic oracle inequality for the resulting kernel-based change-point detection method, whatever the unknown number of change points, thanks to a concentration result for Hilbert-space valued random variables which may be of independent interest. Experiments on synthetic and real data illustrate the proposed method, demonstrating its ability to detect subtle changes in the distribution of data. 

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