Sufficient dimension reduction via random‐partitions for the large‐p‐small‐n problem

Summary Sufficient dimension reduction (SDR) continues to be an active field of research. When estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem, which can be problematic under the large‐p‐small‐n situation. In recent years, n...

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Veröffentlicht in:Biometrics 2019-03, Vol.75 (1), p.245-255
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description Summary Sufficient dimension reduction (SDR) continues to be an active field of research. When estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem, which can be problematic under the large‐p‐small‐n situation. In recent years, new techniques have emerged in numerical linear algebra, called randomized algorithms or random sketching, for high‐dimensional and large scale problems. To overcome the large‐p‐small‐n SDR problem, we combine the idea of statistical inference with random sketching to propose a new SDR method, called integrated random‐partition SDR (iRP‐SDR). Our method consists of the following three steps: (i) Randomly partition the covariates into subsets to construct an envelope subspace with low dimension. (ii) Obtain a sketch of the CS by applying a conventional SDR method within the constructed envelope subspace. (iii) Repeat the above two steps many times and integrate these multiple sketches to form the final estimate of the CS. After describing the details of these steps, the asymptotic properties of iRP‐SDR are established. Unlike existing methods, iRP‐SDR does not involve the determination of the structural dimension until the last stage, which makes it more adaptive to a high‐dimensional setting. The advantageous performance of iRP‐SDR is demonstrated via simulation studies and a practical example analyzing EEG data.
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When estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem, which can be problematic under the large‐p‐small‐n situation. In recent years, new techniques have emerged in numerical linear algebra, called randomized algorithms or random sketching, for high‐dimensional and large scale problems. To overcome the large‐p‐small‐n SDR problem, we combine the idea of statistical inference with random sketching to propose a new SDR method, called integrated random‐partition SDR (iRP‐SDR). Our method consists of the following three steps: (i) Randomly partition the covariates into subsets to construct an envelope subspace with low dimension. (ii) Obtain a sketch of the CS by applying a conventional SDR method within the constructed envelope subspace. (iii) Repeat the above two steps many times and integrate these multiple sketches to form the final estimate of the CS. After describing the details of these steps, the asymptotic properties of iRP‐SDR are established. Unlike existing methods, iRP‐SDR does not involve the determination of the structural dimension until the last stage, which makes it more adaptive to a high‐dimensional setting. 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When estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem, which can be problematic under the large‐p‐small‐n situation. In recent years, new techniques have emerged in numerical linear algebra, called randomized algorithms or random sketching, for high‐dimensional and large scale problems. To overcome the large‐p‐small‐n SDR problem, we combine the idea of statistical inference with random sketching to propose a new SDR method, called integrated random‐partition SDR (iRP‐SDR). Our method consists of the following three steps: (i) Randomly partition the covariates into subsets to construct an envelope subspace with low dimension. (ii) Obtain a sketch of the CS by applying a conventional SDR method within the constructed envelope subspace. (iii) Repeat the above two steps many times and integrate these multiple sketches to form the final estimate of the CS. After describing the details of these steps, the asymptotic properties of iRP‐SDR are established. Unlike existing methods, iRP‐SDR does not involve the determination of the structural dimension until the last stage, which makes it more adaptive to a high‐dimensional setting. 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When estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem, which can be problematic under the large‐p‐small‐n situation. In recent years, new techniques have emerged in numerical linear algebra, called randomized algorithms or random sketching, for high‐dimensional and large scale problems. To overcome the large‐p‐small‐n SDR problem, we combine the idea of statistical inference with random sketching to propose a new SDR method, called integrated random‐partition SDR (iRP‐SDR). Our method consists of the following three steps: (i) Randomly partition the covariates into subsets to construct an envelope subspace with low dimension. (ii) Obtain a sketch of the CS by applying a conventional SDR method within the constructed envelope subspace. (iii) Repeat the above two steps many times and integrate these multiple sketches to form the final estimate of the CS. After describing the details of these steps, the asymptotic properties of iRP‐SDR are established. Unlike existing methods, iRP‐SDR does not involve the determination of the structural dimension until the last stage, which makes it more adaptive to a high‐dimensional setting. The advantageous performance of iRP‐SDR is demonstrated via simulation studies and a practical example analyzing EEG data.</abstract><cop>United States</cop><pub>Blackwell Publishing Ltd</pub><pmid>30052272</pmid><doi>10.1111/biom.12926</doi><tpages>11</tpages><orcidid>https://orcid.org/0000-0003-4022-2081</orcidid><oa>free_for_read</oa></addata></record>
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source MEDLINE; Access via Wiley Online Library; Oxford University Press Journals All Titles (1996-Current)
subjects Alcoholism - pathology
Algorithms
Asymptotic methods
Asymptotic properties
Brain - drug effects
Computer Simulation
Distance correlation screening
Eigenvalues
Electroencephalography - statistics & numerical data
Humans
Linear algebra
Machine Learning
Models, Theoretical
Partitions (mathematics)
Random sketching
Randomized algorithm
Random‐partition
Reduction
Regression analysis
Sketches
Statistical analysis
Statistical inference
Subspaces
Sufficient dimension reduction
Sure screening property
title Sufficient dimension reduction via random‐partitions for the large‐p‐small‐n problem
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