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Wan F, Liu W, Bretz F. Confidence sets for a level set in linear regression. Stat Med 2024; 43:1103-1118. [PMID: 38183296 DOI: 10.1002/sim.9996] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2022] [Revised: 11/14/2023] [Accepted: 12/05/2023] [Indexed: 01/08/2024]
Abstract
Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It has been realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, ie, the set of covariate values for which the regression function exceeds a predefined level, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this article, the construction of confidence sets for the level set of linear regression is considered. In particular,1 - α $$ 1-\alpha $$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding1 - α $$ 1-\alpha $$ level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore, the method proposed in this article is widely applicable. Simulation studies with both linear and generalized linear models are conducted to assess the method and real examples are used to illustrate the method.
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Affiliation(s)
- Fang Wan
- Department of Mathematics and Statistics, Lancaster University, Bilrigg lane, Lancaster, LA1 4YF, UK
| | - Wei Liu
- School of Mathematical Sciences, University of Southampton, Southampton, UK
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2
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Maullin-Sapey T, Schwartzman A, Nichols TE. Spatial confidence regions for combinations of excursion sets in image analysis. J R Stat Soc Series B Stat Methodol 2024; 86:177-193. [PMID: 38344135 PMCID: PMC10852994 DOI: 10.1093/jrsssb/qkad104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2022] [Revised: 04/03/2023] [Accepted: 07/27/2023] [Indexed: 06/02/2024]
Abstract
The analysis of excursion sets in imaging data is essential to a wide range of scientific disciplines such as neuroimaging, climatology, and cosmology. Despite growing literature, there is little published concerning the comparison of processes that have been sampled across the same spatial region but which reflect different study conditions. Given a set of asymptotically Gaussian random fields, each corresponding to a sample acquired for a different study condition, this work aims to provide confidence statements about the intersection, or union, of the excursion sets across all fields. Such spatial regions are of natural interest as they directly correspond to the questions 'Where do all random fields exceed a predetermined threshold?', or 'Where does at least one random field exceed a predetermined threshold?'. To assess the degree of spatial variability present, our method provides, with a desired confidence, subsets and supersets of spatial regions defined by logical conjunctions (i.e. set intersections) or disjunctions (i.e. set unions), without any assumption on the dependence between the different fields. The method is verified by extensive simulations and demonstrated using task-fMRI data to identify brain regions with activation common to four variants of a working memory task.
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Affiliation(s)
- Thomas Maullin-Sapey
- Nuffield Department of Population Health, Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, University of Oxford, Oxford, UK
| | - Armin Schwartzman
- Division of Biostatistics, University of California, San Diego, CA, USA
- Halicioğlu Data Science Institute, University of California, San Diego, CA, USA
| | - Thomas E Nichols
- Nuffield Department of Population Health, Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, University of Oxford, Oxford, UK
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Cui H, Liu Y, Mao G, Zhang J. Model-free conditional screening for ultrahigh-dimensional survival data via conditional distance correlation. Biom J 2023; 65:e2200089. [PMID: 36526602 DOI: 10.1002/bimj.202200089] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2022] [Revised: 09/15/2022] [Accepted: 09/20/2022] [Indexed: 12/23/2022]
Abstract
How to select the active variables that have significant impact on the event of interest is a very important and meaningful problem in the statistical analysis of ultrahigh-dimensional data. In many applications, researchers often know that a certain set of covariates are active variables from some previous investigations and experiences. With the knowledge of the important prior knowledge of active variables, we propose a model-free conditional screening procedure for ultrahigh dimensional survival data based on conditional distance correlation. The proposed procedure can effectively detect the hidden active variables that are jointly important but are weakly correlated with the response. Moreover, it performs well when covariates are strongly correlated with each other. We establish the sure screening property and the ranking consistency of the proposed method and conduct extensive simulation studies, which suggests that the proposed procedure works well for practical situations. Then, we illustrate the new approach through a real dataset from the diffuse large-B-cell lymphoma study S1.
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Affiliation(s)
- Hengjian Cui
- School of Mathematical Sciences, Capital Normal University, Beijing, China
| | - Yanyan Liu
- School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, China
| | - Guangcai Mao
- School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei, China
| | - Jing Zhang
- School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, China
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4
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Muñoz A, Martos G, Gonzalez J. Level Sets Semimetrics for Probability Measures with Applications in Hypothesis Testing. Methodol Comput Appl Probab 2023. [DOI: 10.1007/s11009-023-09990-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/13/2023]
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5
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Merlo L, Petrella L, Salvati N, Tzavidis N. Marginal M-quantile regression for multivariate dependent data. Comput Stat Data Anal 2022. [DOI: 10.1016/j.csda.2022.107500] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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6
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Berthet P, Einmahl JHJ. Cube root weak convergence of empirical estimators of a density level set. Ann Stat 2022. [DOI: 10.1214/21-aos2157] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Philippe Berthet
- Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse; CNRS
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Rodríguez-Casal A, Saavedra-Nieves P. A data-adaptive method for estimating density level sets under shape conditions. Ann Stat 2022. [DOI: 10.1214/21-aos2168] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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8
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Affiliation(s)
- Yen-Chi Chen
- Department of Statistics, University of Washington, Seattle
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Mao G, Liu Y, Wu Y, Yang Q. The hazard level set. J Nonparametr Stat 2021. [DOI: 10.1080/10485252.2021.2015767] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Guangcai Mao
- School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei, People's Republic of China
| | - Yanyan Liu
- School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, People's Republic of China
| | - Yuanshan Wu
- School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, People's Republic of China
| | - Qinglong Yang
- School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, People's Republic of China
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11
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Pan Y, Liu Z, Song G, Wei S. Case-cohort and inference for the proportional hazards model with covariate adjustment. COMMUN STAT-THEOR M 2021. [DOI: 10.1080/03610926.2021.1996607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Yingli Pan
- Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China
| | - Zhan Liu
- Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China
| | - Guangyu Song
- Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China
| | - Sha Wei
- Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China
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Chazal F, Michel B. An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists. Front Artif Intell 2021; 4:667963. [PMID: 34661095 PMCID: PMC8511823 DOI: 10.3389/frai.2021.667963] [Citation(s) in RCA: 55] [Impact Index Per Article: 18.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2021] [Accepted: 07/16/2021] [Indexed: 11/30/2022] Open
Abstract
With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for data analysis and statistical learning. Topological data analysis (tda) is a recent and fast-growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. It proposes new well-founded mathematical theories and computational tools that can be used independently or in combination with other data analysis and statistical learning techniques. This article is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for nonexperts.
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Affiliation(s)
- Frédéric Chazal
- Inria Saclay - Île-de-France Research Centre, Palaiseau, France
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Saavedra-Nieves P, Crujeiras RM. Nonparametric estimation of directional highest density regions. ADV DATA ANAL CLASSI 2021. [DOI: 10.1007/s11634-021-00457-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Abstract
AbstractHighest density regions (HDRs) are defined as level sets containing sample points of relatively high density. Although Euclidean HDR estimation from a random sample, generated from the underlying density, has been widely considered in the statistical literature, this problem has not been contemplated for directional data yet. In this work, directional HDRs are formally defined and plug-in estimators based on kernel smoothing and associated confidence regions are proposed. We also provide a new suitable bootstrap bandwidth selector for plug-in HDRs estimation based on the minimization of an error criteria that involves the Hausdorff distance between the boundaries of the theoretical and estimated HDRs. An extensive simulation study shows the performance of the resulting estimator for the circle and for the sphere. The methodology is applied to analyze two real data sets in animal orientation and seismology.
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Affiliation(s)
- Wanli Qiao
- Department of Statistics, George Mason University, Fairfax, VA 22030, USA
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Mao G, Zhang J. Statistical inference for nonparametric censored regression. Stat (Int Stat Inst) 2021. [DOI: 10.1002/sta4.333] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Guangcai Mao
- School of Mathematics and Statistics Central China Normal University Wuhan Hubei 430079 China
| | - Jing Zhang
- School of Statistics and Mathematics Zhongnan University of Economics and Law Wuhan Hubei 430073 China
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Liu Y, Wu Y, Zhang J, Zhou H. Cox regression analysis for distorted covariates with an unknown distortion function. Biom J 2021; 63:968-983. [PMID: 33687092 DOI: 10.1002/bimj.202000209] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Revised: 11/07/2020] [Accepted: 01/21/2021] [Indexed: 11/09/2022]
Abstract
We study inference for censored survival data where some covariates are distorted by some unknown functions of an observable confounding variable in a multiplicative form. An example of this kind of data in medical studies is normalizing some important observed exposure variables by patients' body mass index , weight, or age. Such a phenomenon also appears frequently in environmental studies where an ambient measure is used for normalization and in genomic studies where the library size needs to be normalized for the next generation sequencing of data. We propose a new covariate-adjusted Cox proportional hazards regression model and utilize the kernel smoothing method to estimate the distorting function, then employ an estimated maximum likelihood method to derive the estimator for the regression parameters. We establish the large sample properties of the proposed estimator. Extensive simulation studies demonstrate that the proposed estimator performs well in correcting the bias arising from distortion. A real dataset from the National Wilms' Tumor Study is used to illustrate the proposed approach.
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Affiliation(s)
- Yanyan Liu
- School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, P. R. China
| | - Yuanshan Wu
- School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, P. R. China
| | - Jing Zhang
- School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, P. R. China
| | - Haibo Zhou
- Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA
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Affiliation(s)
- Wei Li
- Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, U.S.A
| | - Subhashis Ghosal
- Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203, U.S.A
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20
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Deng H, Zhang CH. Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors. Ann Stat 2020. [DOI: 10.1214/20-aos1946] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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21
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Baíllo A, Chacón JE. A new selection criterion for statistical home range estimation. J Appl Stat 2020; 49:722-737. [DOI: 10.1080/02664763.2020.1822302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- A. Baíllo
- Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
| | - J. E. Chacón
- Departamento de Matemáticas, Universidad de Extremadura, Badajoz, Spain
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22
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23
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Kolaczyk ED, Lin L, Rosenberg S, Walters J, Xu J. Averages of unlabeled networks: Geometric characterization and asymptotic behavior. Ann Stat 2020. [DOI: 10.1214/19-aos1820] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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24
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Qiao W. Asymptotics and optimal bandwidth for nonparametric estimation of density level sets. Electron J Stat 2020. [DOI: 10.1214/19-ejs1668] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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25
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Cholaquidis A, Cuevas A. Set estimation under biconvexity restrictions. ESAIM-PROBAB STAT 2020. [DOI: 10.1051/ps/2020019] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
A set in the Euclidean plane is said to be biconvex if, for some angle θ ∈ [0, π∕2), all its sections along straight lines with inclination angles θ and θ + π∕2 are convex sets (i.e., empty sets or segments). Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of “rectilinear convexity”, in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set S from a random sample of points drawn on S. By analogy with the classical convex case, one would like to define the “biconvex hull” of the sample points as a natural estimator for S. However, as previously pointed out by several authors, the notion of “hull” for a given set A (understood as the “minimal” set including A and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of “biconvex hull” (often called “rectilinear convex hull”): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation of a biconvex set S and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate, from a sample of points on S, the biconvexity angle θ is also given.
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Bello R, Miao D, Falcon R, Nakata M, Rosete A, Ciucci D. Rough Sets Meet Statistics - A New View on Rough Set Reasoning About Numerical Data. ROUGH SETS 2020. [PMCID: PMC7338170 DOI: 10.1007/978-3-030-52705-1_6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
Affiliation(s)
- Rafael Bello
- Department of Computer Science, Universidad Central de Las Villas, Santa Clara, Cuba
| | - Duoqian Miao
- Department of Computer Science and Technology, Tongji University, Shanghai, China
| | - Rafael Falcon
- School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON Canada
| | - Michinori Nakata
- Department of Management and Information Science, Josai International University, Togane, Chiba Japan
| | - Alejandro Rosete
- Departamento de Inteligencia Artificial e Infraestructura de Sistemas Informáticos, Universidad Tecnológica de La Habana “José Antonio Echeverría” (CUJAE), Havana, Cuba
| | - Davide Ciucci
- Department of Informatics, Systems and Communication, Università degli Studi di Milano-Bicocca, Milan, Italy
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28
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Qiao W, Polonik W. Nonparametric confidence regions for level sets: Statistical properties and geometry. Electron J Stat 2019. [DOI: 10.1214/19-ejs1543] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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29
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Kamiya H. A note on estimation of the shape of density level sets of star-shaped distributions. COMMUN STAT-THEOR M 2018. [DOI: 10.1080/03610926.2018.1473606] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Hidehiko Kamiya
- Faculty of Economics, Osaka University of Economics, Osaka, Japan
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30
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Ma B, Entezari A. An Interactive Framework for Visualization of Weather Forecast Ensembles. IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 2018; 25:1091-1101. [PMID: 30130213 DOI: 10.1109/tvcg.2018.2864815] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Numerical Weather Prediction (NWP) ensembles are commonly used to assess the uncertainty and confidence in weather forecasts. Spaghetti plots are conventional tools for meteorologists to directly examine the uncertainty exhibited by ensembles, where they simultaneously visualize isocontours of all ensemble members. To avoid visual clutter in practical usages, one needs to select a small number of informative isovalues for visual analysis. Moreover, due to the complex topology and variation of ensemble isocontours, it is often a challenging task to interpret the spaghetti plot for even a single isovalue in large ensembles. In this paper, we propose an interactive framework for uncertainty visualization of weather forecast ensembles that significantly improves and expands the utility of spaghetti plots in ensemble analysis. Complementary to state-of-the-art methods, our approach provides a complete framework for visual exploration of ensemble isocontours, including isovalue selection, interactive isocontour variability exploration, and interactive sub-region selection and re-analysis. Our framework is built upon the high-density clustering paradigm, where the mode structure of the density function is represented as a hierarchy of nested subsets of the data. We generalize the high-density clustering for isocontours and propose a bandwidth selection method for estimating the density function of ensemble isocontours. We present novel visualizations based on high-density clustering results, called the mode plot and the simplified spaghetti plot. The proposed mode plot visually encodes the structure provided by the high-density clustering result and summarizes the distribution of ensemble isocontours. It also enables the selection of subsets of interesting isocontours, which are interactively highlighted in a linked spaghetti plot for providing spatial context. To provide an interpretable overview of the positional variability of isocontours, our system allows for selection of informative isovalues from the simplified spaghetti plot. Due to the spatial variability of ensemble isocontours, the system allows for interactive selection and focus on sub-regions for local uncertainty and clustering re-analysis. We examine a number of ensemble datasets to establish the utility of our approach and discuss its advantages over state-of-the-art visual analysis tools for ensemble data.
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Abstract
Multivariate quantiles are of increasing importance in applications of hydrology. This calls for reliable methods to evaluate the precision of the estimated quantile sets. Therefore, we focus on two recently developed approaches to estimate confidence regions for level sets and extend them to provide confidence regions for multivariate quantiles based on copulas. In a simulation study, we check coverage probabilities of the employed approaches. In particular, we focus on small sample sizes. One approach shows reasonable coverage probabilities and the second one obtains mixed results. Not only the bounded copula domain but also the additional estimation of the quantile level pose some problems. A small sample application gives further insight into the employed techniques.
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Affiliation(s)
- Yen‐Chi Chen
- Department of Statistics University of Washington Seattle Washington
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33
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Statistical, Spatial and Temporal Mapping of 911 Emergencies in Ecuador. APPLIED SCIENCES-BASEL 2018. [DOI: 10.3390/app8020199] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/14/2023]
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34
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Doss CR, Weng G. Bandwidth selection for kernel density estimators of multivariate level sets and highest density regions. Electron J Stat 2018. [DOI: 10.1214/18-ejs1501] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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