TY - JOUR

T1 - Identification of unknown parameters and prediction with hierarchical matrices

AU - Litvinenko, A.

AU - Kriemann, R.

AU - Berikov, V.

N1 - Funding Information: Acknowledgment The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no 0314-2019-0015). The work was partly supported by RFBR grant 19-29-01175. A. Litvinenko was supported by funding from the Alexander von Humboldt Foundation. Publisher Copyright: © 2021 UNCECOMP Proceedings. All rights reserved.

PY - 2021

Y1 - 2021

N2 - Statistical analysis of massive datasets very often implies expensive linear algebra operations with large dense matrices. Typical tasks are an estimation of unknown parameters of the underlying statistical model and prediction of missing values. We developed the H-MLE procedure, which solves these typical tasks. The unknown parameters can be estimated by maximizing the joint Gaussian log-likelihood function, which depends on a covariance matrix. To decrease the high computational cost, we approximate the covariance matrix in the hierarchical (H-) matrix format, which has only a log-linear computational cost. The H-matrix technique allows inhomogeneous covariance matrices and almost arbitrary locations. Especially, H-matrices can be applied in cases when the matrices under consideration are dense and unstructured. For validation purposes, we implemented three machine learning methods: the kNN, random forest, and deep neural network. The best results (for the given datasets) were obtained by the kNN method with three or seven neighbors depending on the dataset. The results computed with the H-MLE method were compared with the results obtained by the kNN method. The developed H-matrix code and all datasets are freely available online.

AB - Statistical analysis of massive datasets very often implies expensive linear algebra operations with large dense matrices. Typical tasks are an estimation of unknown parameters of the underlying statistical model and prediction of missing values. We developed the H-MLE procedure, which solves these typical tasks. The unknown parameters can be estimated by maximizing the joint Gaussian log-likelihood function, which depends on a covariance matrix. To decrease the high computational cost, we approximate the covariance matrix in the hierarchical (H-) matrix format, which has only a log-linear computational cost. The H-matrix technique allows inhomogeneous covariance matrices and almost arbitrary locations. Especially, H-matrices can be applied in cases when the matrices under consideration are dense and unstructured. For validation purposes, we implemented three machine learning methods: the kNN, random forest, and deep neural network. The best results (for the given datasets) were obtained by the kNN method with three or seven neighbors depending on the dataset. The results computed with the H-MLE method were compared with the results obtained by the kNN method. The developed H-matrix code and all datasets are freely available online.

KW - Computational statistics

KW - Data analysis

KW - Hierarchical matrix

KW - Matérn covariance

KW - Parameter inference

KW - Prediction

KW - Random field

KW - Spatial statistics

UR - http://www.scopus.com/inward/record.url?scp=85121119835&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:85121119835

VL - 2021-June

JO - UNCECOMP Proceedings

JF - UNCECOMP Proceedings

SN - 2623-3339

T2 - 4th International Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2021

Y2 - 28 June 2021 through 30 June 2021

ER -