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Non-negative matrix factorization with fixed row and column sums

LINEAR ALGEBRA AND ITS APPLICATIONS 2008年第5-6期429卷 1020-1025页

作者： Ho, Ngoc-Diep Van Dooren, Paul Catholic Univ Louvain CESAME B-1348 Louvain Belgium

In this short note, we focus on the use of the generalized Kullback-Leibler (KL) divergence in the problem of non-negative matrix factorization (NM[F). We will show that when using the generalized KL divergence as cost function for NIVIF, the row sums and the column sums of the original matrix are preserved in the approximation. We will use this special characteristic in several approximation problems. (c) 2007 Elsevier Inc. All rights reserved.

关键词： non-negative matrix factorization generalized Kullback-Leibler divergence stochastic matrix approximation row sums column sums

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Modified linear projection for large spatial datasets

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COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2017年第2期46卷 870-889页

作者： Hirano, Toshihiro Univ Tokyo Grad Sch Econ Bunkyo Ku Tokyo Japan NEC Corp Ltd Nakahara Ku 1753 Shimonumabe Kawasaki Kanagawa 2118666 Japan

Recent developments in engineering techniques for spatial data collection such as geographic information systems have resulted in an increasing need for methods to analyze large spatial datasets. These sorts of datasets can be found in various fields of the natural and social sciences. However, model fitting and spatial prediction using these large spatial datasets are impractically time-consuming, because of the necessary matrix inversions. Various methods have been developed to deal with this problem, including a reduced rank approach and a sparse matrix approximation. In this article, we propose a modification to an existing reduced rank approach to capture both the large- and small-scale spatial variations effectively. We have used simulated examples and an empirical data analysis to demonstrate that our proposed approach consistently performs well when compared with other methods. In particular, the performance of our new method does not depend on the dependence properties of the spatial covariance functions.

关键词： Covariance tapering Gaussian process Geostatistics Markov chain Monte Carlo Reduced rank approximation stochastic matrix approximation

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A multi-resolution approximation via linear projection for large spatial datasets

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JAPANESE JOURNAL OF STATISTICS AND DATA SCIENCE 2021年第1期4卷 215-256页

作者： Hirano, Toshihiro Kanto Gakuin Univ Coll Econ Kanazawa Ku 1-50-1 Mutsuura Higashi Yokohama Kanagawa 2368501 Japan

Recent technical advances in collecting spatial data have been increasing the demand for methods to analyze large spatial datasets. The statistical analysis for these types of datasets can provide useful knowledge in various fields. However, conventional spatial statistical methods, such as maximum-likelihood estimation and kriging, are impractically time-consuming for large spatial datasets due to the necessary matrix inversions. To cope with this problem, we propose a multi-resolution approximation via linear projection (M-RA-lp). The M-RA-lp conducts a linear projection approach on each subregion whenever a spatial domain is subdivided, which leads to an approximated covariance function capturing both the large- and small-scale spatial variations. Moreover, we elicit the algorithms for fast computation of the log-likelihood function and predictive distribution with the approximated covariance function obtained by the M-RA-lp. Simulation studies and a real data analysis for air dose rates demonstrate that our proposed M-RA-lp works well relative to the related existing methods.

关键词： Covariance tapering Gaussian process Geostatistics Large spatial datasets Multi-resolution approximation stochastic matrix approximation

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