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1. On Functional Processes with Multiple Discontinuities

   https://doi.org/10.1111/rssb.12493  

   Li, Jialiang; Li, Yaguang; Hsing, Tailen (March 2022, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract We consider the problem of estimating multiple change points for a functional data process. There are numerous examples in science and finance in which the process of interest may be subject to some sudden changes in the mean. The process data that are not in a close vicinity of any change point can be analysed by the usual nonparametric smoothing methods. However, the data close to change points and contain the most pertinent information of structural breaks need to be handled with special care. This paper considers a half-kernel approach that addresses the inference of the total number, locations and jump sizes of the changes. Convergence rates and asymptotic distributional results for the proposed procedures are thoroughly investigated. Simulations are conducted to examine the performance of the approach, and a number of real data sets are analysed to provide an illustration. 

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2. Inference for change points in high-dimensional mean shift models

   Kaul, Abhishek; Michailidis, George (May 2025, Statistica sinica)

   We consider the problem of constructing confidence intervals for the locations of change points in a high-dimensional mean shift model. We develop a locally refitted least squares estimator and obtain component-wise and simultaneous rates of estimation of change points. The simultaneous rate is the sharpest available by at least a factor of log p, while the component-wise one is optimal. These results enable existence of limiting distributions for the locations of the change points. Subsequently, component-wise distributions are characterized under both vanishing and non-vanishing jump size regimes, while joint distributions of change point estimates are characterized under the latter regime, which also yields asymptotic independence of these estimates. We provide the relationship between these distributions, which allows construction of regime adaptive confidence intervals. All results are established under a high dimensional scaling, in the presence of diverging number of change points. They are illustrated on synthetic data and on sensor measurements from smartphones for activity recognition. 

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3. Simultaneous Change Point Inference and Structure Recovery for High Dimensional Gaussian Graphical Models

   Liu, B.; Zhang, X.; Liu, Y. (October 2021, Journal of machine learning research)

   In this article, we investigate the problem of simultaneous change point inference and structure recovery in the context of high dimensional Gaussian graphical models with possible abrupt changes. In particular, motivated by neighborhood selection, we incorporate a threshold variable and an unknown threshold parameter into a joint sparse regression model which combines p l1-regularized node-wise regression problems together. The change point estimator and the corresponding estimated coefficients of precision matrices are obtained together. Based on that, a classifier is introduced to distinguish whether a change point exists. To recover the graphical structure correctly, a data-driven thresholding procedure is proposed. In theory, under some sparsity conditions and regularity assumptions, our method can correctly choose a homogeneous or heterogeneous model with high accuracy. Furthermore, in the latter case with a change point, we establish estimation consistency of the change point estimator, by allowing the number of nodes being much larger than the sample size. Moreover, it is shown that, in terms of structure recovery of Gaussian graphical models, the proposed thresholding procedure achieves model selection consistency and controls the number of false positives. The validity of our proposed method is justified via extensive numerical studies. Finally, we apply our proposed method to the S&P 500 dataset to show its empirical usefulness. 

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4. PROLONG: penalized regression for outcome guided longitudinal omics analysis with network and group constraints

   https://doi.org/10.1093/bioinformatics/btaf099  

   Broll, Steven; Basu, Sumanta; Lee, Myung Hee; Wells, Martin T (March 2025, Bioinformatics)

   Martelli, Pier Luigi (Ed.)

   Abstract MotivationThere is a growing interest in longitudinal omics data paired with some longitudinal clinical outcome. Given a large set of continuous omics variables and some continuous clinical outcome, each measured for a few subjects at only a few time points, we seek to identify those variables that co-vary over time with the outcome. To motivate this problem we study a dataset with hundreds of urinary metabolites along with Tuberculosis mycobacterial load as our clinical outcome, with the objective of identifying potential biomarkers for disease progression. For such data clinicians usually apply simple linear mixed effects models which often lack power given the low number of replicates and time points. We propose a penalized regression approach on the first differences of the data that extends the lasso + Laplacian method [Li and Li (Network-constrained regularization and variable selection for analysis of genomic data. Bioinformatics 2008;24:1175–82.)] to a longitudinal group lasso + Laplacian approach. Our method, PROLONG, leverages the first differences of the data to increase power by pairing the consecutive time points. The Laplacian penalty incorporates the dependence structure of the variables, and the group lasso penalty induces sparsity while grouping together all contemporaneous and lag terms for each omic variable in the model. ResultsWith an automated selection of model hyper-parameters, PROLONG correctly selects target metabolites with high specificity and sensitivity across a wide range of scenarios. PROLONG selects a set of metabolites from the real data that includes interesting targets identified during EDA. Availability and implementationAn R package implementing described methods called “prolong” is available at https://github.com/stevebroll/prolong. Code snapshot available at 10.5281/zenodo.14804245. 

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5. Finite Sample Change Point Inference and Identification for High-Dimensional Mean Vectors

   https://doi.org/10.1111/rssb.12406  

   Yu, Mengjia; Chen, Xiaohui (December 2020, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Cumulative sum (CUSUM) statistics are widely used in the change point inference and identification. For the problem of testing for existence of a change point in an independent sample generated from the mean-shift model, we introduce a Gaussian multiplier bootstrap to calibrate critical values of the CUSUM test statistics in high dimensions. The proposed bootstrap CUSUM test is fully data dependent and it has strong theoretical guarantees under arbitrary dependence structures and mild moment conditions. Specifically, we show that with a boundary removal parameter the bootstrap CUSUM test enjoys the uniform validity in size under the null and it achieves the minimax separation rate under the sparse alternatives when the dimension p can be larger than the sample size n. Once a change point is detected, we estimate the change point location by maximising the ℓ∞-norm of the generalised CUSUM statistics at two different weighting scales corresponding to covariance stationary and non-stationary CUSUM statistics. For both estimators, we derive their rates of convergence and show that dimension impacts the rates only through logarithmic factors, which implies that consistency of the CUSUM estimators is possible when p is much larger than n. In the presence of multiple change points, we propose a principled bootstrap-assisted binary segmentation (BABS) algorithm to dynamically adjust the change point detection rule and recursively estimate their locations. We derive its rate of convergence under suitable signal separation and strength conditions. The results derived in this paper are non-asymptotic and we provide extensive simulation studies to assess the finite sample performance. The empirical evidence shows an encouraging agreement with our theoretical results. 

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