Abstract This paper aims to present the first Frequentist framework on signal region detection in high-resolution and high-order image regression problems. Image data and scalar-on-image regression are intensively studied in recent years. However, most existing studies on such topics focussed on outcome prediction, while the research on region detection is rather limited, even though the latter is often more important. In this paper, we develop a general framework named Sparse Kronecker Product Decomposition (SKPD) to tackle this issue. The SKPD framework is general in the sense that it works for both matrices and tensors represented image data. Our framework includes one-term, multi-term, and nonlinear SKPDs. We propose nonconvex optimization problems for one-term and multi-term SKPDs and develop path-following algorithms for the nonconvex optimization. Under a Restricted Isometric Property, the computed solutions of the path-following algorithm are guaranteed to converge to the truth with a particularly chosen initialization even though the optimization is nonconvex. Moreover, the region detection consistency could also be guaranteed. The nonlinear SKPD is highly connected to shallow convolutional neural networks (CNN), particularly to CNN with one convolutional layer and one fully-connected layer. Effectiveness of SKPD is validated by real brain imaging data in the UK Biobank database.
{"title":"Sparse Kronecker product decomposition: a general framework of signal region detection in image regression","authors":"Sanyou Wu, Long Feng","doi":"10.1093/jrsssb/qkad024","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad024","url":null,"abstract":"Abstract This paper aims to present the first Frequentist framework on signal region detection in high-resolution and high-order image regression problems. Image data and scalar-on-image regression are intensively studied in recent years. However, most existing studies on such topics focussed on outcome prediction, while the research on region detection is rather limited, even though the latter is often more important. In this paper, we develop a general framework named Sparse Kronecker Product Decomposition (SKPD) to tackle this issue. The SKPD framework is general in the sense that it works for both matrices and tensors represented image data. Our framework includes one-term, multi-term, and nonlinear SKPDs. We propose nonconvex optimization problems for one-term and multi-term SKPDs and develop path-following algorithms for the nonconvex optimization. Under a Restricted Isometric Property, the computed solutions of the path-following algorithm are guaranteed to converge to the truth with a particularly chosen initialization even though the optimization is nonconvex. Moreover, the region detection consistency could also be guaranteed. The nonlinear SKPD is highly connected to shallow convolutional neural networks (CNN), particularly to CNN with one convolutional layer and one fully-connected layer. Effectiveness of SKPD is validated by real brain imaging data in the UK Biobank database.","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136223251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Standard Markov chain Monte Carlo methods struggle to explore distributions that concentrate in the neighbourhood of low-dimensional submanifolds. This pathology naturally occurs in Bayesian inference settings when there is a high signal-to-noise ratio in the observational data but the model is inherently over-parametrised or nonidentifiable. In this paper, we propose a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher-dimensional space; in contrast to the original posterior this lifted distribution remains diffuse in the limit of vanishing observation noise. We employ a constrained Hamiltonian Monte Carlo method, which exploits the geometry of this lifted distribution, to perform efficient approximate inference. We demonstrate in numerical experiments that, contrarily to competing approaches, the sampling efficiency of our proposed methodology does not degenerate as the target distribution to be explored concentrates near low-dimensional submanifolds. Python code reproducing the results is available at https://doi.org/10.5281/zenodo.6551654.
{"title":"Manifold lifting: scaling Markov chain Monte Carlo to the vanishing noise regime","authors":"K. Au, Matthew M. Graham, Alexandre Hoang Thiery","doi":"10.1093/jrsssb/qkad023","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad023","url":null,"abstract":"\u0000 Standard Markov chain Monte Carlo methods struggle to explore distributions that concentrate in the neighbourhood of low-dimensional submanifolds. This pathology naturally occurs in Bayesian inference settings when there is a high signal-to-noise ratio in the observational data but the model is inherently over-parametrised or nonidentifiable. In this paper, we propose a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher-dimensional space; in contrast to the original posterior this lifted distribution remains diffuse in the limit of vanishing observation noise. We employ a constrained Hamiltonian Monte Carlo method, which exploits the geometry of this lifted distribution, to perform efficient approximate inference. We demonstrate in numerical experiments that, contrarily to competing approaches, the sampling efficiency of our proposed methodology does not degenerate as the target distribution to be explored concentrates near low-dimensional submanifolds. Python code reproducing the results is available at https://doi.org/10.5281/zenodo.6551654.","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"33 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74984921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In many practical experiments, both the level combinations of factors and the addition orders will affect the responses. However, virtually no construction methods have been provided for such experimental designs. This paper focuses on such experiments, introduces a new type of design called the ordering factorial design, and proposes the nominal main effect component-position model and interaction-main effect component-position model. To obtain efficient fractional designs, we provide some deterministic construction methods. The resulting designs are D-optimal, and the run sizes are much smaller than that of the full designs. Moreover, in some cases, some constructed designs are still D-optimal after reducing the number of components and factors.
{"title":"Ordering factorial experiments","authors":"Liuqing Yang, Yongdao Zhou, Min-Qian Liu","doi":"10.1093/jrsssb/qkad027","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad027","url":null,"abstract":"\u0000 In many practical experiments, both the level combinations of factors and the addition orders will affect the responses. However, virtually no construction methods have been provided for such experimental designs. This paper focuses on such experiments, introduces a new type of design called the ordering factorial design, and proposes the nominal main effect component-position model and interaction-main effect component-position model. To obtain efficient fractional designs, we provide some deterministic construction methods. The resulting designs are D-optimal, and the run sizes are much smaller than that of the full designs. Moreover, in some cases, some constructed designs are still D-optimal after reducing the number of components and factors.","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"30 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81753929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Peter J Bickel, Derek Bean, Aiyou Chen and Purnamrita Sarkar’s contribution to the Discussion of “Vintage Factor Analysis with Varimax Performs Statistical Inference” by Rohe & Zeng","authors":"","doi":"10.1093/jrsssb/qkad037","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad037","url":null,"abstract":"","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"40 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73761918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-06eCollection Date: 2023-07-01DOI: 10.1093/jrsssb/qkad017
Shu Yang, Chenyin Gao, Donglin Zeng, Xiaofei Wang
We propose a test-based elastic integrative analysis of the randomised trial and real-world data to estimate treatment effect heterogeneity with a vector of known effect modifiers. When the real-world data are not subject to bias, our approach combines the trial and real-world data for efficient estimation. Utilising the trial design, we construct a test to decide whether or not to use real-world data. We characterise the asymptotic distribution of the test-based estimator under local alternatives. We provide a data-adaptive procedure to select the test threshold that promises the smallest mean square error and an elastic confidence interval with a good finite-sample coverage property.
{"title":"Elastic integrative analysis of randomised trial and real-world data for treatment heterogeneity estimation.","authors":"Shu Yang, Chenyin Gao, Donglin Zeng, Xiaofei Wang","doi":"10.1093/jrsssb/qkad017","DOIUrl":"10.1093/jrsssb/qkad017","url":null,"abstract":"<p><p>We propose a test-based elastic integrative analysis of the randomised trial and real-world data to estimate treatment effect heterogeneity with a vector of known effect modifiers. When the real-world data are not subject to bias, our approach combines the trial and real-world data for efficient estimation. Utilising the trial design, we construct a test to decide whether or not to use real-world data. We characterise the asymptotic distribution of the test-based estimator under local alternatives. We provide a data-adaptive procedure to select the test threshold that promises the smallest mean square error and an elastic confidence interval with a good finite-sample coverage property.</p>","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"85 3","pages":"575-596"},"PeriodicalIF":3.1,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10376438/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10035092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Seconder of the vote of thanks to Rohe & Zeng and contribution to the Discussion of “Vintage Factor Analysis with Varimax Performs Statistical Inference”","authors":"M. Pensky","doi":"10.1093/jrsssb/qkad031","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad031","url":null,"abstract":"","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"13 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90038425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tao Wang’s contribution to the Discussion of “Vintage Factor Analysis with Varimax Performs Statistical Inference” by Rohe & Zeng","authors":"T. VanderWeele","doi":"10.1093/jrsssb/qkad046","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad046","url":null,"abstract":"","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"10 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84755440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Yang Liu’s contribution to the Discussion of “Vintage Factor Analysis with Varimax Performs Statistical Inference” by Rohe & Zeng","authors":"Y. Liu","doi":"10.1093/jrsssb/qkad042","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad042","url":null,"abstract":"","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"439 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91081087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mark Pilling's contribution to the Discussion of “Vintage Factor Analysis with Varimax Performs Statistical Inference” by Rohe & Zeng","authors":"M. Pilling","doi":"10.1093/jrsssb/qkad044","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad044","url":null,"abstract":"","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"36 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73894761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rungang Han and Anru R Zhang’s contribution to the Discussion of “Vintage Factor Analysis with Varimax Performs Statistical Inference” by Rohe & Zeng","authors":"Anru R. Zhang","doi":"10.1093/jrsssb/qkad034","DOIUrl":"https://doi.org/10.1093/jrsssb/qkad034","url":null,"abstract":"","PeriodicalId":49982,"journal":{"name":"Journal of the Royal Statistical Society Series B-Statistical Methodology","volume":"25 1","pages":""},"PeriodicalIF":5.8,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90079238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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