From 8fd80522f0d7dcbe5b2a58588a4000dd286e9f3b Mon Sep 17 00:00:00 2001 From: daniel Date: Thu, 28 Mar 2024 14:19:24 +0100 Subject: [PATCH] effie changes (abstract / intro) --- LaTeX/main.bib | 2891 ++++++++++++++++++++++++++++++++++------------- LaTeX/paper.tex | 30 +- 2 files changed, 2119 insertions(+), 802 deletions(-) diff --git a/LaTeX/main.bib b/LaTeX/main.bib index 08a6478..feb3fa7 100644 --- a/LaTeX/main.bib +++ b/LaTeX/main.bib @@ -1,1033 +1,2338 @@ @book{AbadirMagnus2005, - title = {Matrix Algebra}, - author = {Abadir, Karim M. and Magnus, Jan R.}, - year = {2005}, - publisher = {Cambridge University Press}, - series = {Econometric Exercises}, - collection = {Econometric Exercises}, - place = {Cambridge}, - doi = {10.1017/CBO9780511810800} + title = {Matrix Algebra}, + author = {Abadir, Karim M. and Magnus, Jan R.}, + year = {2005}, + publisher = {Cambridge University Press}, + collection = {Econometric Exercises}, + doi = {10.1017/CBO9780511810800}, + place = {Cambridge}, + series = {Econometric Exercises} } -@book{AbsilEtAt2007, - title = {{Optimization Algorithms on Matrix Manifolds}}, - author = {Absil, P.-A. and Mahony, R. and Sepulchre, R.}, - year = {2007}, - publisher = {Princeton University Press}, - isbn = {9780691132983}, - note = {Full Online Text \url{https://press.princeton.edu/absil}} +@book{AbsilEtAl2007, + title = {{Optimization Algorithms on Matrix Manifolds}}, + author = {Absil, P.-A. and Mahony, R. and Sepulchre, R.}, + year = {2008}, + pages = {xvi+224}, + publisher = {Princeton University Press, Princeton, NJ}, + doi = {10.1515/9781400830244}, + isbn = {978-0-691-13298-3}, + mrclass = {90-02 (58E17 90C30 90C52)}, + mrnumber = {2364186}, + note = {Full Online Text \url{https://press.princeton.edu/absil}}, + url = {https://doi.org/10.1515/9781400830244} +} + +@article{AdragniCook2009, + title = {Sufficient dimension reduction and prediction in regression}, + author = {Adragni, Kofi P. and Cook, R. 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Math.}, + fjournal = {Annals of the Institute of Statistical Mathematics}, + volume = {54}, + number = {4}, + pages = {768--795}, + doi = {10.1023/A:1022411301790}, + issn = {0020-3157,1572-9052}, + mrclass = {62J05 (62H99)}, + mrnumber = {1954046}, + url = {https://doi.org/10.1023/A:1022411301790} +} + +@article{ClevelandDevlin1988, + title = {{Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting}}, + author = {William S. Cleveland and Susan J. Devlin}, + year = {1988}, + journal = {J. Amer. Statist. 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Dennis}, - journal = {Statistical Science}, - month = {02}, - number = {1}, - pages = {1--26}, - publisher = {The Institute of Mathematical Statistics}, - title = {{Fisher Lecture: Dimension Reduction in Regression}}, - volume = {22}, - year = {2007}, - doi = {10.1214/088342306000000682} + title = {{Fisher Lecture: Dimension Reduction in Regression}}, + author = {Cook, R. Dennis}, + year = {2007}, + journal = {Statistical Science}, + volume = {22}, + number = {1}, + pages = {1--26}, + publisher = {The Institute of Mathematical Statistics}, + doi = {10.1214/088342306000000682}, + month = {02} } -@article{Dai2012, - author = {B. Dai}, - title = {Multivariate bernoulli distribution models}, - year = {2012} +@article{Cook2018, + title = {Principal Components, Sufficient Dimension Reduction, and Envelopes}, + author = {Cook, R. Dennis}, + year = {2018}, + journal = {Annual Review of Statistics and Its Application}, + volume = {5}, + number = {1}, + pages = {533-559}, + doi = {10.1146/annurev-statistics-031017-100257} +} + +@article{CookForzani2008, + title = {Principal fitted components for dimension reduction in regression}, + author = {Cook, R. D. and Forzani, L.}, + year = {2008}, + journal = {Statistical Science}, + volume = {23}, + number = {4}, + pages = {485-501} +} + +@article{CookForzani2009, + title = {Likelihood-based sufficient dimension reduction}, + author = {R. Dennis Cook and Liliana Forzani}, + year = {2009}, + journal = {Journal of the American Statistical Association}, + volume = {104}, + number = {485}, + pages = {197--208}, + publisher = {Taylor and Francis Ltd.}, + doi = {10.1198/jasa.2009.0106}, + issn = {0162-1459}, + month = {3} +} + +@article{CookLi2002, + title = {Dimension reduction for conditional mean in regression}, + author = {Cook, R.D. and Li, B.}, + year = {2002}, + journal = {The Annals of Statistics}, + fjournal = {The Annals of Statistics}, + volume = {30}, + number = {2}, + pages = {455--474}, + publisher = {The Institute of Mathematical Statistics}, + doi = {10.1214/aos/1021379861} +} + +@article{CookLi2004, + title = {Determining the dimension of iterative {H}essian transformation}, + author = {Cook, R. Dennis and Li, Bing}, + year = {2004}, + journal = {Ann. 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Mota} +@article{DeAlmeidaEtAl2007, + title = {PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization}, + author = {André L.F. {de Almeida} and Gérard Favier and João Cesar M. Mota}, + year = {2007}, + journal = {Signal Processing}, + volume = {87}, + number = {2}, + pages = {337-351}, + doi = {https://doi.org/10.1016/j.sigpro.2005.12.014}, + issn = {0165-1684}, + note = {Tensor Signal Processing}, + url = {https://www.sciencedirect.com/science/article/pii/S0165168406001757} } @article{DeesMandic2019, - title = {A Statistically Identifiable Model for Tensor-Valued Gaussian Random Variables}, - author = {Bruno Scalzo Dees and Danilo P. Mandic}, - journal = {ArXiv}, - year = {2019}, - volume = {abs/1911.02915}, - url = {https://api.semanticscholar.org/CorpusID:207847615} + title = {A Statistically Identifiable Model for Tensor-Valued Gaussian Random Variables}, + author = {Bruno Scalzo Dees and Danilo P. Mandic}, + year = {2019}, + journal = {ArXiv}, + volume = {abs/1911.02915}, + url = {https://api.semanticscholar.org/CorpusID:207847615} } @article{DeLathauwerCastaing2007, - title = {Tensor-based techniques for the blind separation of DS-CDMA signals}, - journal = {Signal Processing}, - volume = {87}, - number = {2}, - pages = {322-336}, - year = {2007}, - note = {Tensor Signal Processing}, - issn = {0165-1684}, - doi = {10.1016/j.sigpro.2005.12.015}, - url = {https://www.sciencedirect.com/science/article/pii/S0165168406001745}, - author = {Lieven {De Lathauwer} and Joséphine Castaing} + title = {Tensor-based techniques for the blind separation of DS-CDMA signals}, + author = {Lieven {De Lathauwer} and Joséphine Castaing}, + year = {2007}, + journal = {Signal Processing}, + volume = {87}, + number = {2}, + pages = {322-336}, + doi = {10.1016/j.sigpro.2005.12.015}, + issn = {0165-1684}, + note = {Tensor Signal Processing}, + url = {https://www.sciencedirect.com/science/article/pii/S0165168406001745} +} + +@article{deLeeuwMichailidis2000, + title = {Discussion article on the paper by Lange, Hunter \& Yang (2000)}, + author = {J. de Leeuw and G. 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Hoff}, - title = {{Separable covariance arrays via the Tucker product, with applications to multivariate relational data}}, - volume = {6}, - journal = {Bayesian Analysis}, - number = {2}, - publisher = {International Society for Bayesian Analysis}, - pages = {179 -- 196}, - keywords = {Gaussian, matrix normal, multiway data, network, tensor, Tucker decomposition}, - year = {2011}, - doi = {10.1214/11-BA606} + title = {{Separable covariance arrays via the Tucker product, with applications to multivariate relational data}}, + author = {Peter D. Hoff}, + year = {2011}, + journal = {Bayesian Analysis}, + volume = {6}, + number = {2}, + pages = {179 -- 196}, + publisher = {International Society for Bayesian Analysis}, + doi = {10.1214/11-BA606}, + keywords = {Gaussian, matrix normal, multiway data, network, tensor, Tucker decomposition} } @article{Hoff2015, - author = {Peter D. Hoff}, - title = {{Multilinear tensor regression for longitudinal relational data}}, - volume = {9}, - journal = {The Annals of Applied Statistics}, - number = {3}, - publisher = {Institute of Mathematical Statistics}, - pages = {1169 -- 1193}, - keywords = {Array normal, Bayesian inference, event data, international relations, network, Tucker product, vector autoregression}, - year = {2015}, - doi = {10.1214/15-AOAS839} + title = {{Multilinear tensor regression for longitudinal relational data}}, + author = {Peter D. Hoff}, + year = {2015}, + journal = {The Annals of Applied Statistics}, + volume = {9}, + number = {3}, + pages = {1169 -- 1193}, + publisher = {Institute of Mathematical Statistics}, + doi = {10.1214/15-AOAS839}, + keywords = {Array normal, Bayesian inference, event data, international relations, network, Tucker product, vector autoregression} +} + +@article{Hornik1991, + title = {Approximation capabilities of multilayer feedforward networks}, + author = {Hornik, Kurt}, + year = {1991}, + journal = {Neural Networks}, + volume = {4}, + number = {2}, + pages = {251-257}, + issn = {0893-6080}, + note = {\url{https://doi.org/10.1016/0893-6080(91)90009-T}} } @article{HuLeeWang2022, - author = {Hu, Jiaxin and Lee, Chanwoo and Wang, Miaoyan}, - title = {Generalized Tensor Decomposition With Features on Multiple Modes}, - journal = {Journal of Computational and Graphical Statistics}, - volume = {31}, - number = {1}, - pages = {204-218}, - year = {2022}, - publisher = {Taylor \& Francis}, - doi = {10.1080/10618600.2021.1978471}, + title = {Generalized Tensor Decomposition With Features on Multiple Modes}, + author = {Hu, Jiaxin and Lee, Chanwoo and Wang, Miaoyan}, + year = {2022}, + journal = {Journal of Computational and Graphical Statistics}, + volume = {31}, + number = {1}, + pages = {204-218}, + publisher = {Taylor \& Francis}, + doi = {10.1080/10618600.2021.1978471} } -@article{Ising1924, - author = {Ising, Ernst}, - title = {{Beitrag zur Theorie des Ferromagnetismus}}, - journal = {Zeitschrift f\"ur Physik}, - pages = {253-258}, - volume = {31}, - number = {1}, - year = {1924}, - month = {2}, - issn = {0044-3328}, - doi = {10.1007/BF02980577} +@article{Ising1925, + title = {{Beitrag zur Theorie des Ferromagnetismus}}, + author = {Ising, Ernst}, + year = {1925}, + journal = {Zeitschrift f\"ur Physik}, + volume = {31}, + number = {1}, + pages = {253-258}, + doi = {10.1007/BF02980577}, + issn = {0044-3328}, + month = {2} } -@article{JungEtAt2019, - author = {Sungkyu Jung and Jeongyoun Ahn and Yongho Jeon}, - title = {Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem}, - journal = {Journal of Computational and Graphical Statistics}, - volume = {28}, - number = {3}, - pages = {710-721}, - year = {2019}, - publisher = {Taylor & Francis}, - doi = {10.1080/10618600.2019.1568014} +@book{JamesEtAl2021, + title = {An introduction to statistical learning---with applications in {R}}, + author = {James, Gareth and Witten, Daniela and Hastie, Trevor and Tibshirani, Robert}, + year = {2021}, + pages = {xv+607}, + publisher = {Springer, New York}, + doi = {10.1007/978-1-0716-1418-1}, + edition = {Second}, + isbn = {978-1-0716-1418-1}, + mrclass = {62-01 (62-04 62H30 62Jxx 62M45 62N01)}, + mrnumber = {4309209}, + series = {Springer Texts in Statistics}, + url = {https://doi.org/10.1007/978-1-0716-1418-1} +} + +@incollection{JennyHaselmayerKapla2021, + title = {Measuring incivility in parliamentary debates : validating a sentiment analysis procedure with calls to order in the Austrian Parliament}, + author = {Jenny, Marcelo and Haselmayer, Martin and Kapla, Daniel}, + year = {2021}, + pages = {1--11}, + publisher = {Routledge}, + address = {London}, + booktitle = {Political Incivility in the Parliamentary, Electoral and Media Arena : Crossing Boundaries}, + editor = {Walter, Annemarie S.}, + isbn = {978-0-367-46273-4}, + series = {Routledge studies on political parties and party systems} +} + +@book{JohnsonEtAl1997, + title = {{Discrete Multivariate Distributions}}, + author = {Johnson, Norman L. and Kotz, Samuel and Balakrishnan, N.}, + year = {1997}, + pages = {xxii+299}, + publisher = {John Wiley \& Sons, Inc., New York}, + isbn = {0-471-12844-9}, + mrclass = {62E15 (60C05 60E05 62H05)}, + mrnumber = {1429617}, + note = {A Wiley-Interscience Publication}, + series = {Wiley Series in Probability and Statistics: Applied Probability and Statistics} +} + +@article{Jolliffe1982, + title = {A Note on the Use of Principal Components in Regression}, + author = {Ian T. Jolliffe}, + year = {1982}, + journal = {Journal of the Royal Statistical Society. Series C (Applied Statistics)}, + volume = {31}, + number = {3}, + pages = {300--303}, + publisher = {[Wiley, Royal Statistical Society]}, + issn = {00359254, 14679876}, + url = {http://www.jstor.org/stable/2348005} +} + +@article{JungEtAl2019, + title = {Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem}, + author = {Sungkyu Jung and Jeongyoun Ahn and Yongho Jeon}, + year = {2019}, + journal = {Journal of Computational and Graphical Statistics}, + volume = {28}, + number = {3}, + pages = {710-721}, + publisher = {Taylor & Francis}, + doi = {10.1080/10618600.2019.1568014} } @book{Kaltenbaeck2021, - title = {Aufbau Analysis}, - author = {Kaltenb\"ack, Michael}, - isbn = {978-3-88538-127-3}, - series = {Berliner Studienreihe zur Mathematik}, - edition = {27}, - year = {2021}, - publisher = {Heldermann Verlag} + title = {Aufbau Analysis}, + author = {Kaltenb\"ack, Michael}, + year = {2021}, + publisher = {Heldermann Verlag}, + edition = {27}, + isbn = {978-3-88538-127-3}, + series = {Berliner Studienreihe zur Mathematik} } @article{Kapla2019, - title = {Comparison of Different Word Embeddings and Neural Network Types for Sentiment Analysis of German Political Speeches}, - author = {Kapla, Daniel}, - year = {2019} + title = {Comparison of Different Word Embeddings and Neural Network Types for Sentiment Analysis of German Political Speeches}, + author = {Kapla, Daniel}, + year = {2019} +} + +@article{KaplaFertlBura2022, + title = {Fusing sufficient dimension reduction with neural networks}, + author = {Kapla, Daniel and Fertl, Lukas and Bura, Efstathia}, + year = {2022}, + journal = {Comput. Statist. Data Anal.}, + fjournal = {Computational Statistics \& Data Analysis}, + volume = {168}, + pages = {Paper No. 107390, 20}, + doi = {10.1016/j.csda.2021.107390}, + issn = {0167-9473,1872-7352}, + mrclass = {99-01}, + mrnumber = {4343643}, + url = {https://doi.org/10.1016/j.csda.2021.107390} +} + +@misc{KingmaWelling2019, + title = {An {I}ntroduction to {V}ariational {A}utoencoders}, + author = {Kingma, Diederik P. and Welling, Max}, + year = 2019, + howpublished = {arXiv:1906.02691 [cs.LG]}, + note = {\url{http://arxiv.org/abs/1906.02691}} } @inproceedings{KofidisRegalia2005, - title = {Tensor Approximation and Signal Processing Applications}, - author = {Eleftherios Kofidis and Phillip A. Regalia}, - year = {2005}, - url = {https://api.semanticscholar.org/CorpusID:13667742} + title = {Tensor Approximation and Signal Processing Applications}, + author = {Eleftherios Kofidis and Phillip A. Regalia}, + year = {2005}, + url = {https://api.semanticscholar.org/CorpusID:13667742} } @article{Kolda2006, - title = {Multilinear operators for higher-order decompositions.}, - author = {Kolda, Tamara Gibson}, - doi = {10.2172/923081}, - url = {https://www.osti.gov/biblio/923081}, - place = {United States}, - year = {2006}, - month = {4}, - type = {Technical Report} + title = {Multilinear operators for higher-order decompositions.}, + author = {Kolda, Tamara Gibson}, + year = {2006}, + doi = {10.2172/923081}, + month = {4}, + place = {United States}, + type = {Technical Report}, + url = {https://www.osti.gov/biblio/923081} } @article{KoldaBader2009, - author = {Kolda, Tamara G. and Bader, Brett W.}, - title = {Tensor Decompositions and Applications}, - journal = {SIAM Review}, - volume = {51}, - number = {3}, - pages = {455-500}, - year = {2009}, - doi = {10.1137/07070111X} + title = {Tensor Decompositions and Applications}, + author = {Kolda, Tamara G. and Bader, Brett W.}, + year = {2009}, + journal = {SIAM Review}, + volume = {51}, + number = {3}, + pages = {455-500}, + doi = {10.1137/07070111X} } @book{KolloVonRosen2005, - title = {Advanced Multivariate Statistics with Matrices}, - isbn = {978-1-4020-3419-0}, - doi = {10.1007/1-4020-3419-9}, - publisher = {Springer Dordrecht}, - author = {Kollo, T\~onu and von Rosen, Dietrich}, - editor = {Hazewinkel, M.}, - year = {2005} + title = {Advanced Multivariate Statistics with Matrices}, + author = {Kollo, T\~onu and von Rosen, Dietrich}, + year = {2005}, + publisher = {Springer Dordrecht}, + doi = {10.1007/1-4020-3419-9}, + editor = {Hazewinkel, M.}, + isbn = {978-1-4020-3419-0} +} + +@inproceedings{KongEtAl2005, + title = {Generalized 2D principal component analysis}, + author = {Hui Kong and Xuchun Li and Lei Wang and Earn Khwang Teoh and Jian-Gang Wang and R. Venkateswarlu}, + year = {2005}, + volume = {1}, + number = {}, + pages = {108-113}, + booktitle = {Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005.}, + doi = {10.1109/IJCNN.2005.1555814}, + issn = {2161-4393} +} + +@article{Kramer1991, + title = {Nonlinear principal component analysis using autoassociative neural networks}, + author = {Kramer, Mark A.}, + year = {1991}, + journal = {AIChE Journal}, + volume = {37}, + number = {2}, + pages = {233-243}, + note = {\url{https://doi.org/10.1002/aic.690370209}} } @book{Kroonenberg2008, - title = {Applied Multiway Data Analysis}, - author = {Kroonenberg, Pieter M.}, - year = {2008}, - publisher = {John Wiley \& Sons, Ltd}, - address = {New York}, - isbn = {9780470238004}, - doi = {10.1002/9780470238004}, + title = {Applied Multiway Data Analysis}, + author = {Kroonenberg, Pieter M.}, + year = {2008}, + publisher = {John Wiley \& Sons, Ltd}, + address = {New York}, + doi = {10.1002/9780470238004}, + isbn = {9780470238004} } @book{Kusolitsch2011, - title = {{M}a\ss{}- und {W}ahrscheinlichkeitstheorie}, - subtitle = {{E}ine {E}inf{\"u}hrung}, - author = {Kusolitsch, Norbert}, - series = {Springer-Lehrbuch}, - year = {2011}, - publisher = {Springer Vienna}, - isbn = {978-3-7091-0684-6}, - doi = {10.1007/978-3-7091-0685-3} + title = {{M}a\ss{}- und {W}ahrscheinlichkeitstheorie}, + author = {Kusolitsch, Norbert}, + year = {2011}, + publisher = {Springer Vienna}, + doi = {10.1007/978-3-7091-0685-3}, + isbn = {978-3-7091-0684-6}, + series = {Springer-Lehrbuch}, + subtitle = {{E}ine {E}inf{\"u}hrung} } @article{LandgrafLee2020, - title = {Dimensionality reduction for binary data through the projection of natural parameters}, - journal = {Journal of Multivariate Analysis}, - volume = {180}, - pages = {104668}, - year = {2020}, - issn = {0047-259X}, - doi = {10.1016/j.jmva.2020.104668}, - author = {Andrew J. Landgraf and Yoonkyung Lee} + title = {Dimensionality reduction for binary data through the projection of natural parameters}, + author = {Andrew J. Landgraf and Yoonkyung Lee}, + year = {2020}, + journal = {Journal of Multivariate Analysis}, + volume = {180}, + pages = {104668}, + doi = {10.1016/j.jmva.2020.104668}, + issn = {0047-259X} } -@article{LeBihanEtAt2001, - title = {Diffusion tensor imaging: Concepts and applications}, - author = {Le Bihan, Denis and Mangin, Jean-François and Poupon, Cyril and Clark, Chris A. and Pappata, Sabina and Molko, Nicolas and Chabriat, Hughes}, - year = {2001}, - journal = {Journal of Magnetic Resonance Imaging}, - volume = {13}, - number = {4}, - pages = {534-546}, - doi = {https://doi.org/10.1002/jmri.1076}, - url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/jmri.1076} +@book{Lauritzen1996, + title = {{Graphical Models}}, + author = {Lauritzen, Steffen L}, + year = {1996}, + publisher = {Oxford University Press}, + doi = {10.1093/oso/9780198522195.001.0001}, + isbn = {9780198522195}, + month = {05} +} + +@article{LauritzenRichardson2002, + title = {Chain Graph Models and Their Causal Interpretations}, + author = {Steffen L. Lauritzen and Thomas S. Richardson}, + year = {2002}, + journal = {Journal of the Royal Statistical Society. Series B (Statistical Methodology)}, + volume = {64}, + number = {3}, + pages = {321--361}, + publisher = {[Royal Statistical Society, Wiley]}, + issn = {13697412, 14679868}, + url = {http://www.jstor.org/stable/3088778}, + urldate = {2024-01-20} +} + +@article{LeBihanEtAl2001, + title = {Diffusion tensor imaging: Concepts and applications}, + author = {Le Bihan, Denis and Mangin, Jean-Fran\c{c}ois and Poupon, Cyril and Clark, Chris A. and Pappata, Sabina and Molko, Nicolas and Chabriat, Hughes}, + year = {2001}, + journal = {Journal of Magnetic Resonance Imaging}, + volume = {13}, + number = {4}, + pages = {534-546}, + doi = {https://doi.org/10.1002/jmri.1076}, + url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/jmri.1076} } @book{Lee2012, - title = {Introduction to Smooth Manifolds}, - author = {Lee, John M.}, - year = {2012}, - journal = {Graduate Texts in Mathematics}, - publisher = {Springer New York}, - doi = {10.1007/978-1-4419-9982-5} + title = {Introduction to Smooth Manifolds}, + author = {Lee, John M.}, + year = {2012}, + journal = {Graduate Texts in Mathematics}, + publisher = {Springer New York}, + doi = {10.1007/978-1-4419-9982-5} } @book{Lee2018, - title = {Introduction to Riemannian Manifolds}, - author = {Lee, John M.}, - year = {2018}, - journal = {Graduate Texts in Mathematics}, - publisher = {Springer International Publishing}, - doi = {10.1007/978-3-319-91755-9} + title = {Introduction to Riemannian Manifolds}, + author = {Lee, John M.}, + year = {2018}, + journal = {Graduate Texts in Mathematics}, + publisher = {Springer International Publishing}, + doi = {10.1007/978-3-319-91755-9} } @article{LengPan2018, - author = {Leng, Chenlei and Pan, Guangming}, - title = {{Covariance estimation via sparse Kronecker structures}}, - volume = {24}, - journal = {Bernoulli}, - number = {4B}, - publisher = {Bernoulli Society for Mathematical Statistics and Probability}, - pages = {3833 -- 3863}, - year = {2018}, - doi = {10.3150/17-BEJ980} + title = {{Covariance estimation via sparse Kronecker structures}}, + author = {Leng, Chenlei and Pan, Guangming}, + year = {2018}, + journal = {Bernoulli}, + volume = {24}, + number = {4B}, + pages = {3833 -- 3863}, + publisher = {Bernoulli Society for Mathematical Statistics and Probability}, + doi = {10.3150/17-BEJ980} } -@article{LeporeEtAt2008, - author = {Lepore, Natasha and Brun, Caroline and Chou, Yi-Yu and Chiang, Ming-Chang and Dutton, Rebecca A. and Hayashi, Kiralee M. and Luders, Eileen and Lopez, Oscar L. and Aizenstein, Howard J. and Toga, Arthur W. and Becker, James T. and Thompson, Paul M.}, - journal = {IEEE Transactions on Medical Imaging}, - title = {Generalized Tensor-Based Morphometry of HIV/AIDS Using Multivariate Statistics on Deformation Tensors}, - year = {2008}, - volume = {27}, - number = {1}, - pages = {129-141}, - doi = {10.1109/TMI.2007.906091} +@article{Lenz1920, + title = {Beitrag zum Verst{\"a}ndnis der magnetischen Erscheinungen in festen K{\"o}rpern}, + author = {W. Lenz}, + year = {1920}, + journal = {European Physical Journal A}, + volume = {21}, + pages = {613--615}, + url = {https://cds.cern.ch/record/460663} +} + +@article{LeporeEtAl2008, + title = {Generalized Tensor-Based Morphometry of HIV/AIDS Using Multivariate Statistics on Deformation Tensors}, + author = {Lepore, Natasha and Brun, Caroline and Chou, Yi-Yu and Chiang, Ming-Chang and Dutton, Rebecca A. and Hayashi, Kiralee M. and Luders, Eileen and Lopez, Oscar L. and Aizenstein, Howard J. and Toga, Arthur W. and Becker, James T. and Thompson, Paul M.}, + year = {2008}, + journal = {IEEE Transactions on Medical Imaging}, + volume = {27}, + number = {1}, + pages = {129-141}, + doi = {10.1109/TMI.2007.906091} } @article{LeurgansRoss1992, - title = {{Multilinear Models: Applications in Spectroscopy}}, - author = {Sue Leurgans and Robert T. Ross}, - year = {1992}, - journal = {Statistical Science}, - volume = {7}, - number = {3}, - publisher = {Institute of Mathematical Statistics}, - pages = {289 -- 310}, - keywords = {Multi-mode factor analysis, nonlinear least-squares, PARAFAC, three-way arrays}, - doi = {10.1214/ss/1177011225} + title = {{Multilinear Models: Applications in Spectroscopy}}, + author = {Sue Leurgans and Robert T. Ross}, + year = {1992}, + journal = {Statistical Science}, + volume = {7}, + number = {3}, + pages = {289 -- 310}, + publisher = {Institute of Mathematical Statistics}, + doi = {10.1214/ss/1177011225}, + keywords = {Multi-mode factor analysis, nonlinear least-squares, PARAFAC, three-way arrays} } -@article{LengPan2018, - author = {Leng, Chenlei and Pan, Guangming}, - title = {{Covariance estimation via sparse Kronecker structures}}, - volume = {24}, - journal = {Bernoulli}, - number = {4B}, - publisher = {Bernoulli Society for Mathematical Statistics and Probability}, - pages = {3833 -- 3863}, - year = {2018}, - doi = {10.3150/17-BEJ980} -} - -@article{LeporeEtAt2008, - author = {Lepore, Natasha and Brun, Caroline and Chou, Yi-Yu and Chiang, Ming-Chang and Dutton, Rebecca A. and Hayashi, Kiralee M. and Luders, Eileen and Lopez, Oscar L. and Aizenstein, Howard J. and Toga, Arthur W. and Becker, James T. and Thompson, Paul M.}, - journal = {IEEE Transactions on Medical Imaging}, - title = {Generalized Tensor-Based Morphometry of HIV/AIDS Using Multivariate Statistics on Deformation Tensors}, - year = {2008}, - volume = {27}, - number = {1}, - pages = {129-141}, - doi = {10.1109/TMI.2007.906091} -} - -@article{LeurgansRoss1992, - title = {{Multilinear Models: Applications in Spectroscopy}}, - author = {Sue Leurgans and Robert T. Ross}, - year = {1992}, - journal = {Statistical Science}, - volume = {7}, - number = {3}, - publisher = {Institute of Mathematical Statistics}, - pages = {289 -- 310}, - keywords = {Multi-mode factor analysis, nonlinear least-squares, PARAFAC, three-way arrays}, - doi = {10.1214/ss/1177011225} +@article{LezonEtAl2006, + title = {Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns}, + author = {Timothy R. Lezon and Jayanth R. Banavar and Marek Cieplak and Amos Maritan and Nina V. Fedoroff}, + year = {2006}, + journal = {Proceedings of the National Academy of Sciences}, + volume = {103}, + number = {50}, + pages = {19033-19038}, + doi = {10.1073/pnas.0609152103} } @article{Li1991, - title = {Sliced Inverse Regression for Dimension Reduction}, - author = {Li, K. C.}, - journal = {Journal of the American Statistical Association}, - volume = {86}, - number = {414}, - pages = {316-327}, - year = {1991}, + title = {{Sliced Inverse Regression for Dimension Reduction}}, + author = {Li, Ker-Chau}, + year = {1991}, + journal = {J. Amer. Statist. Assoc.}, + fjournal = {Journal of the American Statistical Association}, + volume = {86}, + number = {414}, + pages = {316--327}, + doi = {10.1080/01621459.1991.10475035} } @article{Li1992, - author = {Ker-Chau Li}, - title = {On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma}, - journal = {Journal of the American Statistical Association}, - volume = {87}, - number = {420}, - pages = {1025-1039}, - year = {1992}, - publisher = {Taylor \& Francis}, - doi = {10.1080/01621459.1992.10476258}, + title = {On principal {H}essian directions for data visualization and dimension reduction: another application of {S}tein's lemma}, + author = {Li, Ker-Chau}, + year = {1992}, + journal = {J. Amer. Statist. Assoc.}, + fjournal = {Journal of the American Statistical Association}, + volume = {87}, + number = {420}, + pages = {1025--1039}, + publisher = {Taylor \& Francis}, + doi = {10.1080/01621459.1992.10476258}, + issn = {0162-1459,1537-274X} +} + +@book{Li2018, + title = {Sufficient dimension reduction}, + author = {Li, Bing}, + year = {2018}, + volume = {161}, + pages = {xxi+283}, + publisher = {CRC Press, Boca Raton, FL}, + doi = {10.1201/9781315119427}, + isbn = {978-1-4987-0447-2}, + mrclass = {62-02 (62G08 62H12 62H20 62L10)}, + mrnumber = {3838449}, + note = {Methods and applications with R}, + series = {Monographs on Statistics and Applied Probability}, + url = {https://doi.org/10.1201/9781315119427} +} + +@article{LiArtemiouLi2011, + title = {Principal support vector machines for linear and nonlinear sufficient dimension reduction}, + author = {Li, Bing and Artemiou, Andreas and Li, Lexin}, + year = {2011}, + journal = {Ann. Statist.}, + fjournal = {The Annals of Statistics}, + volume = {39}, + number = {6}, + pages = {3182--3210}, + publisher = {The Institute of Mathematical Statistics}, + doi = {10.1214/11-AOS932}, + month = {12}, + url = {https://doi.org/10.1214/11-AOS932} } @article{LiKimAltman2010, - author = {Bing Li and Min Kyung Kim and Naomi Altman}, - title = {{On dimension folding of matrix- or array-valued statistical objects}}, - volume = {38}, - journal = {The Annals of Statistics}, - number = {2}, - publisher = {Institute of Mathematical Statistics}, - pages = {1094 -- 1121}, - keywords = {directional regression, electroencephalography, Kronecker envelope, sliced average variance estimate, sliced inverse regression}, - year = {2010}, - doi = {10.1214/09-AOS737} + title = {{On dimension folding of matrix- or array-valued statistical objects}}, + author = {Bing Li and Min Kyung Kim and Naomi Altman}, + year = {2010}, + journal = {The Annals of Statistics}, + volume = {38}, + number = {2}, + pages = {1094 -- 1121}, + publisher = {Institute of Mathematical Statistics}, + doi = {10.1214/09-AOS737}, + keywords = {directional regression, electroencephalography, Kronecker envelope, sliced average variance estimate, sliced inverse regression} +} + +@article{Lin2019, + title = {Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition}, + author = {Lin, Zhenhua}, + year = {2019}, + journal = {SIAM Journal on Matrix Analysis and Applications}, + volume = {40}, + number = {4}, + pages = {1353--1370}, + doi = {10.1137/18M1221084} } -# TODO: check this! @inbook{LiuKoike2007, - title = {Extending Multivariate Space-Time Geostatistics for Environmental Data Analysis}, - author = {Chunxue Liu and Katsuaki Koike}, - year = {2007}, - journal = {Mathematical Geology}, - publisher = {International Association for Mathematical Geology}, - pages = {289--305}, - doi = {10.1007/s11004-007-9085-9} + title = {Extending Multivariate Space-Time Geostatistics for Environmental Data Analysis}, + author = {Chunxue Liu and Katsuaki Koike}, + year = {2007}, + journal = {Mathematical Geology}, + pages = {289--305}, + publisher = {International Association for Mathematical Geology}, + doi = {10.1007/s11004-007-9085-9} +} + +@article{LiWang2007, + title = {On Directional Regression for Dimension Reduction}, + author = {Bing Li and Shaoli Wang}, + year = {2007}, + journal = {Journal of the American Statistical Association}, + volume = {102}, + number = {479}, + pages = {997-1008}, + publisher = {Taylor \& Francis}, + doi = {10.1198/016214507000000536} +} + +@article{LiZhaChiaromonte2005, + title = {Contour regression: A general approach to dimension reduction}, + author = {Li, Bing and Zha, Hongyuan and Chiaromonte, Francesca}, + year = {2005}, + journal = {Ann. Statist.}, + fjournal = {The Annals of Statistics}, + volume = {33}, + number = {4}, + pages = {1580--1616}, + publisher = {The Institute of Mathematical Statistics}, + doi = {10.1214/009053605000000192}, + url = {https://doi.org/10.1214/009053605000000192} +} + +@article{LuoLi2016, + title = {Combining eigenvalues and variation of eigenvectors for order determination}, + author = {Luo, Wei and Li, Bing}, + year = {2016}, + journal = {Biometrika}, + volume = {103}, + number = {4}, + pages = {875--887}, + doi = {10.1093/biomet/asw051}, + issn = {0006-3444, 1464-3510}, + month = {12}, + url = {https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/asw051}, + urldate = {2021-10-06} +} + +@article{LuoLi2021, + title = {On order determination by predictor augmentation}, + author = {Luo, Wei and Li, Bing}, + year = {2021}, + journal = {Biometrika}, + volume = {108}, + number = {3}, + pages = {557--574}, + doi = {10.1093/biomet/asaa077}, + issn = {0006-3444, 1464-3510}, + month = {08}, + url = {https://academic.oup.com/biomet/article/108/3/557/5917626}, + urldate = {2021-10-06} } @article{LuZimmerman2005, - title = {The likelihood ratio test for a separable covariance matrix}, - author = {Nelson Lu and Dale L. Zimmerman}, - year = {2005}, - journal = {Statistics \& Probability Letters}, - volume = {73}, - number = {4}, - pages = {449-457}, - issn = {0167-7152}, - doi = {10.1016/j.spl.2005.04.020}, - url = {https://www.sciencedirect.com/science/article/pii/S0167715205001495} + title = {The likelihood ratio test for a separable covariance matrix}, + author = {Nelson Lu and Dale L. Zimmerman}, + year = {2005}, + journal = {Statistics \& Probability Letters}, + volume = {73}, + number = {4}, + pages = {449-457}, + doi = {10.1016/j.spl.2005.04.020}, + issn = {0167-7152}, + url = {https://www.sciencedirect.com/science/article/pii/S0167715205001495} } @article{MagnusNeudecker1986, - title = {Symmetry, 0-1 Matrices and Jacobians: A Review}, - author = {Magnus, Jan R. and Neudecker, Heinz}, - ISSN = {02664666, 14694360}, - URL = {http://www.jstor.org/stable/3532421}, - journal = {Econometric Theory}, - number = {2}, - pages = {157--190}, - publisher = {Cambridge University Press}, - urldate = {2023-10-03}, - volume = {2}, - year = {1986} + title = {Symmetry, 0-1 Matrices and Jacobians: A Review}, + author = {Magnus, Jan R. and Neudecker, Heinz}, + year = {1986}, + journal = {Econometric Theory}, + volume = {2}, + number = {2}, + pages = {157--190}, + publisher = {Cambridge University Press}, + issn = {02664666, 14694360}, + url = {http://www.jstor.org/stable/3532421}, + urldate = {2023-10-03} } @book{MagnusNeudecker1999, - title = {Matrix Differential Calculus with Applications in Statistics and Econometrics (Revised Edition)}, - author = {Magnus, Jan R. and Neudecker, Heinz}, - year = {1999}, - publisher = {John Wiley \& Sons Ltd}, - isbn = {0-471-98632-1} + title = {Matrix differential calculus with applications in statistics and econometrics}, + author = {Magnus, Jan R. and Neudecker, Heinz}, + year = {1999}, + pages = {xviii+395}, + publisher = {John Wiley \& Sons, Ltd., Chichester}, + isbn = {0-471-98633-X}, + mrclass = {15-01 (26-01 62-01)}, + mrnumber = {1698873}, + note = {Revised reprint of the 1988 original}, + series = {Wiley Series in Probability and Statistics} } @article{ManceurDutilleul2013, - title = {Maximum likelihood estimation for the tensor normal distribution: Algorithm, minimum sample size, and empirical bias and dispersion}, - author = {Ameur M. Manceur and Pierre Dutilleul}, - journal = {Journal of Computational and Applied Mathematics}, - volume = {239}, - pages = {37-49}, - year = {2013}, - issn = {0377-0427}, - doi = {10.1016/j.cam.2012.09.017}, - url = {https://www.sciencedirect.com/science/article/pii/S0377042712003810} + title = {Maximum likelihood estimation for the tensor normal distribution: Algorithm, minimum sample size, and empirical bias and dispersion}, + author = {Ameur M. Manceur and Pierre Dutilleul}, + year = {2013}, + journal = {Journal of Computational and Applied Mathematics}, + volume = {239}, + pages = {37-49}, + doi = {10.1016/j.cam.2012.09.017}, + issn = {0377-0427}, + url = {https://www.sciencedirect.com/science/article/pii/S0377042712003810} } -@article{MardiaGoodall1993, - title = {Spatial-Temporal Analysis of Multivariate Environmental Monitoring Data}, - author = {Mardia, Kanti and Goodall, Colin}, - year = {1993}, - month = {01}, - pages = {}, - volume = {6}, - publisher = {Elsevier Science Publisher B.V.} +@incollection{MardiaGoodall1993, + title = {Spatial-temporal analysis of multivariate environmental monitoring data}, + author = {Mardia, Kanti V. and Goodall, Colin R.}, + year = {1993}, + volume = {6}, + pages = {347--386}, + publisher = {North-Holland, Amsterdam}, + booktitle = {Multivariate environmental statistics}, + isbn = {0-444-89804-2}, + mrclass = {62H11}, + mrnumber = {1268443}, + series = {North-Holland Ser. Statist. Probab.} } -@InProceedings{MartinFernandez2004, - author = {Mart{\'i}n-Fern{\'a}ndez, Marcos and Westin, Carl-Fredrik and Alberola-L{\'o}pez, Carlos}, - editor = {Barillot, Christian and Haynor, David R. and Hellier, Pierre}, - title = {3D Bayesian Regularization of Diffusion Tensor MRI Using Multivariate Gaussian Markov Random Fields}, - booktitle = {Medical Image Computing and Computer-Assisted Intervention -- MICCAI 2004}, - year = {2004}, - publisher = {Springer Berlin Heidelberg}, - address = {Berlin, Heidelberg}, - pages = {351--359}, - abstract = {3D Bayesian regularization applied to diffusion tensor MRI is presented here. The approach uses Markov Random Field ideas and is based upon the definition of a 3D neighborhood system in which the spatial interactions of the tensors are modeled. As for the prior, we model the behavior of the tensor fields by means of a 6D multivariate Gaussian local characteristic. As for the likelihood, we model the noise process by means of conditionally independent 6D multivariate Gaussian variables. Those models include inter-tensor correlations, intra-tensor correlations and colored noise. The solution tensor field is obtained by using the simulated annealing algorithm to achieve the maximum a posteriori estimation. Several experiments both on synthetic and real data are presented, and performance is assessed with mean square error measure.}, - isbn = {978-3-540-30135-6} +@inproceedings{MartinFernandez2004, + title = {3D Bayesian Regularization of Diffusion Tensor MRI Using Multivariate Gaussian Markov Random Fields}, + author = {Mart{\'i}n-Fern{\'a}ndez, Marcos and Westin, Carl-Fredrik and Alberola-L{\'o}pez, Carlos}, + year = {2004}, + pages = {351--359}, + publisher = {Springer Berlin Heidelberg}, + address = {Berlin, Heidelberg}, + booktitle = {Medical Image Computing and Computer-Assisted Intervention -- MICCAI 2004}, + editor = {Barillot, Christian and Haynor, David R. and Hellier, Pierre}, + isbn = {978-3-540-30135-6} +} + +@article{MaZhu2013, + title = {A review on dimension reduction}, + author = {Ma, Yanyuan and Zhu, Liping}, + year = {2013}, + journal = {Int. Stat. Rev.}, + fjournal = {International Statistical Review. Revue Internationale de Statistique}, + volume = {81}, + number = {1}, + pages = {134--150}, + doi = {10.1111/j.1751-5823.2012.00182.x}, + issn = {0306-7734,1751-5823}, + mrclass = {62G08 (62-02 62H12)}, + mrnumber = {3047506}, + url = {https://doi.org/10.1111/j.1751-5823.2012.00182.x} } @book{McCullagh1987, - title = {Tensor Methods in Statistics}, - subtitle = {Monographs on Statistics and Applied Probability}, - author = {McCullagh, Peter}, - year = {1987}, - publisher = {Chapman and Hall/CRC}, - doi = {10.1201/9781351077118} + title = {Tensor Methods in Statistics}, + author = {McCullagh, Peter}, + year = {1987}, + publisher = {Chapman and Hall/CRC}, + doi = {10.1201/9781351077118}, + subtitle = {Monographs on Statistics and Applied Probability} +} + +@article{McCullochPitts1943, + title = {A Logical Calculus of the Ideas Immanent in Nervous Activity}, + author = {Mc{C}ulloch, Warren S and Pitts, Walter}, + year = {1943}, + journal = {Bulletin of Mathematical Biophysics}, + volume = {5}, + pages = {115--133}, + publisher = {Springer} +} + +@article{Nadarajah2005, + title = {A generalized normal distribution}, + author = {Saralees Nadarajah}, + year = {2005}, + journal = {Journal of Applied Statistics}, + volume = {32}, + number = {7}, + pages = {685--694}, + publisher = {Taylor \& Francis}, + doi = {10.1080/02664760500079464} } @inproceedings{Nesterov1983, - title = {A method of solving a convex programming problem with convergence rate $O(1/k^2)$}, - author = {Nesterov, Yurii Evgen'evich}, - booktitle = {Doklady Akademii Nauk}, - volume = {269}, - number = {3}, - pages = {543--547}, - year = {1983}, - organization= {Russian Academy of Sciences} + title = {A method of solving a convex programming problem with convergence rate $O(1/k^2)$}, + author = {Nesterov, Yurii Evgen'evich}, + year = {1983}, + volume = {269}, + number = {3}, + pages = {543--547}, + booktitle = {Doklady Akademii Nauk}, + organization = {Russian Academy of Sciences} } -@article{OhlsonEtAt2013, - title = {The multilinear normal distribution: Introduction and some basic properties}, - author = {Ohlson, Martin and Ahmad, Mumtaz Rauf and von Rosen, Dietrich}, - year = {2013}, - journal = {Journal of Multivariate Analysis}, - volume = {113}, - pages = {37-47}, - issn = {0047-259X}, - doi = {10.1016/j.jmva.2011.05.015}, - url = {https://www.sciencedirect.com/science/article/pii/S0047259X11001047} +@article{NguyenEtAl2017, + title = {Inverse statistical problems: from the inverse {I}sing problem to data science}, + author = {H. Chau Nguyen and Riccardo Zecchina and Johannes Berg}, + year = {2017}, + journal = {Advances in Physics}, + volume = {66}, + number = {3}, + pages = {197--261}, + publisher = {Taylor \& Francis}, + doi = {10.1080/00018732.2017.1341604} } -@article{PfeifferForzaniBura, - author = {Pfeiffer, Ruth and Forzani, Liliana and Bura, Efstathia}, - year = {2012}, - month = {09}, - pages = {2414-27}, - title = {Sufficient dimension reduction for longitudinally measured predictors}, - volume = {31}, - journal = {Statistics in medicine}, - doi = {10.1002/sim.4437} +@article{Niss2005, + title = {{History of the Lenz-Ising Model 1920--1950: From Ferromagnetic to Cooperative Phenomena}}, + author = {Niss, Martin}, + year = {2005}, + journal = {Arch. Hist. Exact Sci.}, + fjournal = {Archive for History of Exact Sciences}, + volume = {59}, + number = {3}, + pages = {267--318}, + doi = {10.1007/s00407-004-0088-3}, + issn = {1432-0657} +} + +@article{OhlsonEtAl2013, + title = {The multilinear normal distribution: Introduction and some basic properties}, + author = {Ohlson, Martin and Ahmad, Mumtaz Rauf and von Rosen, Dietrich}, + year = {2013}, + journal = {Journal of Multivariate Analysis}, + volume = {113}, + pages = {37-47}, + doi = {10.1016/j.jmva.2011.05.015}, + issn = {0047-259X}, + url = {https://www.sciencedirect.com/science/article/pii/S0047259X11001047} +} + +@article{Oseledets2011, + title = {Tensor-Train Decomposition}, + author = {Oseledets, I. V.}, + year = {2011}, + journal = {SIAM Journal on Scientific Computing}, + volume = {33}, + number = {5}, + pages = {2295-2317}, + doi = {10.1137/090752286} +} + +@article{PanMaiZhang2018, + title = {Covariate-Adjusted Tensor Classification in High dimensions}, + author = {Yuqing Pan and Qing Mai and Xin Zhang}, + year = {2018}, + journal = {Journal of the American Statistical Association}, + volume = {0}, + number = {ja}, + pages = {1-41}, + publisher = {Taylor & Francis}, + doi = {10.1080/01621459.2018.1497500}, + eprint = {https://doi.org/10.1080/01621459.2018.1497500}, + url = {https://doi.org/10.1080/01621459.2018.1497500} +} + +@book{Pepe03, + title = {The Statistical Evaluation of Medical Tests for Classification and Prediction}, + author = {Pepe, M.S.}, + year = {2003}, + publisher = {Oxford University Press}, + address = {New York} +} + +@article{PfeifferForzaniBura2012, + title = {Sufficient dimension reduction for longitudinally measured predictors}, + author = {Pfeiffer, Ruth and Forzani, Liliana and Bura, Efstathia}, + year = {2012}, + journal = {Statistics in medicine}, + volume = {31}, + pages = {2414-27}, + doi = {10.1002/sim.4437}, + month = {09} } @article{PfeifferKaplaBura2021, - author = {Pfeiffer, Ruth and Kapla, Daniel and Bura, Efstathia}, - title = {{Least squares and maximum likelihood estimation of sufficient reductions in regressions with matrix-valued predictors}}, - volume = {11}, - year = {2021}, - journal = {International Journal of Data Science and Analytics}, - doi = {10.1007/s41060-020-00228-y} + title = {{Least squares and maximum likelihood estimation of sufficient reductions in regressions with matrix-valued predictors}}, + author = {Pfeiffer, Ruth and Kapla, Daniel and Bura, Efstathia}, + year = {2021}, + journal = {International Journal of Data Science and Analytics}, + volume = {11}, + doi = {10.1007/s41060-020-00228-y} +} + +@article{LiZhang2017, +author = {Lexin Li and Xin Zhang}, +title = {Parsimonious Tensor Response Regression}, +journal = {Journal of the American Statistical Association}, +volume = {112}, +number = {519}, +pages = {1131-1146}, +year = {2017}, +publisher = {Taylor & Francis}, +doi = {10.1080/01621459.2016.1193022}, + + +URL = { + + https://doi.org/10.1080/01621459.2016.1193022 + + + +}, +eprint = { + https://doi.org/10.1080/01621459.2016.1193022} +} + + +@inproceedings{RabusseauKadri2016, + author = {Rabusseau, Guillaume and Kadri, Hachem}, + booktitle = {Advances in Neural Information Processing Systems}, + editor = {D. Lee and M. Sugiyama and U. Luxburg and I. Guyon and R. Garnett}, + pages = {}, + publisher = {Curran Associates, Inc.}, + title = {Low-Rank Regression with Tensor Responses}, + url = {https://proceedings.neurips.cc/paper_files/paper/2016/file/3806734b256c27e41ec2c6bffa26d9e7-Paper.pdf}, + volume = {29}, + year = {2016} +} + + +@article{HaoEtAl2021, + author = {Botao Hao and Boxiang Wang and Pengyuan Wang and Jingfei Zhang and Jian Yang and Will Wei Sun}, + title = {Sparse Tensor Additive Regression}, + journal = {Journal of Machine Learning Research}, + year = {2021}, + volume = {22}, + number = {64}, + pages = {1--43}, + url = {http://jmlr.org/papers/v22/19-769.html} +} + +@article{Rosenblatt1958, + title = {The perceptron: A probabilistic model for information storage and organization in the brain}, + author = {Frank Rosenblatt}, + year = {1958}, + journal = {Psychological Review}, + volume = {65}, + number = {6}, + pages = {386--408}, + doi = {10.1037/h0042519} +} + +@inproceedings{Rumelhart1986, + title = {Learning internal representations by error propagation}, + author = {David E. Rumelhart and Geoffrey E. Hinton and Ronald J. Williams}, + year = {1986}, + url = {https://api.semanticscholar.org/CorpusID:62245742} +} + +@article{RuppertWand1994, + title = {Multivariate Locally Weighted Least Squares Regression}, + author = {D. Ruppert and M. P. Wand}, + year = {1994}, + journal = {The Annals of Statistics}, + volume = {22}, + number = {3}, + pages = {1346--1370}, + publisher = {Institute of Mathematical Statistics}, + issn = {00905364}, + url = {http://www.jstor.org/stable/2242229}, + urldate = {2024-01-25} +} + +@inproceedings{ShanEtAl2008, + title = {Unified Principal Component Analysis with generalized Covariance Matrix for face recognition}, + author = {Shiguang Shan and Bo Cao and Yu Su and Laiyun Qing and Xilin Chen and Wen Gao}, + year = {2008}, + volume = {}, + number = {}, + pages = {1-7}, + booktitle = {2008 IEEE Conference on Computer Vision and Pattern Recognition}, + doi = {10.1109/CVPR.2008.4587375}, + issn = {1063-6919} } @inproceedings{ShashuaHazan2005, - author = {Shashua, Amnon and Hazan, Tamir}, - title = {Non-Negative Tensor Factorization with Applications to Statistics and Computer Vision}, - year = {2005}, - isbn = {1595931805}, - publisher = {Association for Computing Machinery}, - address = {New York, NY, USA}, - doi = {10.1145/1102351.1102451}, - booktitle = {Proceedings of the 22nd International Conference on Machine Learning}, - pages = {792--799}, - numpages = {8}, - location = {Bonn, Germany}, - series = {ICML '05} + title = {Non-Negative Tensor Factorization with Applications to Statistics and Computer Vision}, + author = {Shashua, Amnon and Hazan, Tamir}, + year = {2005}, + pages = {792--799}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA}, + booktitle = {Proceedings of the 22nd International Conference on Machine Learning}, + doi = {10.1145/1102351.1102451}, + isbn = {1595931805}, + location = {Bonn, Germany}, + numpages = {8}, + series = {ICML '05} } @article{Soize2008, - title = {Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size}, - author = {C. Soize}, - year = {2008}, - journal = {Probabilistic Engineering Mechanics}, - volume = {23}, - number = {2}, - pages = {307-323}, - note = {5th International Conference on Computational Stochastic Mechanics}, - issn = {0266-8920}, - doi = {10.1016/j.probengmech.2007.12.019}, - url = {https://www.sciencedirect.com/science/article/pii/S0266892007000562} + title = {Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size}, + author = {C. Soize}, + year = {2008}, + journal = {Probabilistic Engineering Mechanics}, + volume = {23}, + number = {2}, + pages = {307-323}, + doi = {10.1016/j.probengmech.2007.12.019}, + issn = {0266-8920}, + note = {5th International Conference on Computational Stochastic Mechanics}, + url = {https://www.sciencedirect.com/science/article/pii/S0266892007000562} } @article{SoloveychikTrushin2016, - title = {Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness}, - author = {I. Soloveychik and D. Trushin}, - year = {2016}, - journal = {Journal of Multivariate Analysis}, - volume = {149}, - pages = {92-113}, - issn = {0047-259X}, - doi = {10.1016/j.jmva.2016.04.001}, - url = {https://www.sciencedirect.com/science/article/pii/S0047259X16300070} + title = {Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness}, + author = {I. Soloveychik and D. Trushin}, + year = {2016}, + journal = {Journal of Multivariate Analysis}, + volume = {149}, + pages = {92-113}, + doi = {10.1016/j.jmva.2016.04.001}, + issn = {0047-259X}, + url = {https://www.sciencedirect.com/science/article/pii/S0047259X16300070} } @misc{SongHero2023, - title = {On Separability of Covariance in Multiway Data Analysis}, - author = {Dogyoon Song and Alfred O. Hero}, - year = {2023}, - eprint = {2302.02415}, - archivePrefix = {arXiv}, - primaryClass = {math.ST}, - doi = {10.48550/arXiv.2302.02415} + title = {On Separability of Covariance in Multiway Data Analysis}, + author = {Dogyoon Song and Alfred O. 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URL: \url{https://stockfishchess.org}}, + shorthand = {SF08}, + sortyear = 2008 } diff --git a/LaTeX/paper.tex b/LaTeX/paper.tex index 48cad61..c0ec5de 100644 --- a/LaTeX/paper.tex +++ b/LaTeX/paper.tex @@ -272,10 +272,9 @@ \maketitle \begin{abstract} - We consider regression or classification problems where the independent variable is matrix- or tensor valued. Modeling the inverse regression as a member of the quadratic exponential family, we derive a multilinear sufficient reduction for the regression or classification problem. Using manifold theory, we prove the consistency and asymptotic normality of the sufficient reduction. For continuous - tensor-valued predictors, we develop a computationally efficient estimation procedure of - their sufficient reductions, which is also applicable to situations where the dimension of - the reduction exceeds the available sample size. An estimation procedure for binary tensor-valued data is also provided. We conclude with simulations and real world data examples for both continuous and binary tensor-valued predictors. + We consider regression or classification problems where the independent variable is matrix- or tensor-valued. We derive a multi-linear sufficient reduction for the regression or classification problem modeling the conditional distribution of the predictors given the response as a member of the quadratic exponential family. Using manifold theory, we prove the consistency and asymptotic normality of the sufficient reduction. We develop estimation procedures of + sufficient reductions for both continuous and binary tensor-valued predictors. For continuous predictors, the algorithm is highly computationally efficient and is also applicable to situations where the dimension of + the reduction exceeds the sample size. We demonstrate the superior performance of our approach in simulations and real-world data examples for both continuous and binary tensor-valued predictors. The \textit{Chess data} analysis results agree with a human player's understanding of the game and confirm the relevance of our approach. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -286,6 +285,16 @@ In Statistics, tensors are a mathematical tool to represent data of complex stru Complex data are collected at different times and/or under several conditions often involving a large number of multi-indexed variables represented as tensor-valued data \parencite{KoldaBader2009}. They occur in large-scale longitudinal studies \parencite[e.g.][]{Hoff2015}, in agricultural experiments and chemometrics and spectroscopy \parencite[e.g.][]{LeurgansRoss1992,Burdick1995}. Also, in signal and video processing where sensors produce multi-indexed data, e.g. over spatial, frequency, and temporal dimensions \parencite[e.g.][]{DeLathauwerCastaing2007,KofidisRegalia2005}, in telecommunications \parencite[e.g.][]{DeAlmeidaEtAl2007}. Other examples of multiway data include 3D images of the brain, where the modes are the 3 spatial dimensions, and spatio-temporal weather imaging data, a set of image sequences represented as 2 spatial modes and 1 temporal mode. +\begin{itemize} + \item Review \cite{ZhouLiZhu2013} and see how you compare with them. They focus on the forward regression model with a scalar response but they claim that "Exploit- ing the array structure in imaging data, the new method substantially reduces the dimensionality of imaging data, which leads to efficient estimation and prediction." + \item Read \cite{ZhouEtAl2023} to figure out the distribution they use for the tensor-valued predictors and briefly describe what they do. + \item Read \cite{RabusseauKadri2016} to figure out what they do. They seem to draw both the response and the predictors from tensor-normal with iid N(0,1) entries: "In order to leverage the tensor structure of the output data, we formulate the problem as the minimization of a least squares criterion subject to a multilinear rank constraint on the regression tensor. The rank constraint enforces the model to capture low-rank structure in the outputs and to explain dependencies between inputs and outputs in a low-dimensional multilinear subspace." + \item +\end{itemize} + + +Tensor regression models have been proposed to exploit the special structure of tensor covariates, e.g. \cite{HaoEtAl2021,ZhouLiZhu2013}, or tensor responses \cite{RabusseauKadri2016,LiZhang2017,ZhouEtAl2023} \cite{HaoEtAl2021} modeled a scalar response as a flexible nonparametric function of tensor covariates. \cite{ZhouLiZhu2013} assume the scalar response has a distribution in the exponential family given the tensor-valued predictors and model the link function as a multilinear function of the predictors. \cite{LiZhang2017} model the tensor-valued response as tensor normal. Rather than using $L_1$ type penalty functions to induce sparsity, they employ the envelope method (Cook, Li, and Chiaromonte Citation2010) to estimate the unknown regression coefficient. Moreover, the envelope method essentially identifies and uses the material information jointly. They develop an estimation algorithm and study the asymptotic properties of the estimator. the scalar response as These models try to utilize the sparse and low-rank structures in the tensors – either in the regression coefficient tensor or the response tensor – to boost performance on the regression task by reducing the number of free parameters. + Multilinear tensor normal models have been used in various applications, including medical imaging \parencite{BasserPajevic2007,DrydenEtAl2009}, spatio-temporal data analysis \parencite{GreenewaldHero2014}, regression analysis for longitudinal relational data \parencite{Hoff2015}. One of the most important uses of the multilinear normal (MLN) distribution, and hence tensor analysis, is perhaps in magnetic resonance imaging (MRI) \parencite{OhlsonEtAl2013}. A recent survey \parencite{WangEtAl2022} and references therein contain more information and potential applications of multilinear tensor normal models. In this paper we present a model-based \emph{Sufficient Dimension Reduction} (SDR) method for tensor-valued data with distribution in the quadratic exponential family assuming a Kronecker product structure of the first and second moment. By generalizing the parameter space to embedded manifolds we obtain consistency and asymtotic normality results while allowing great modeling flexibility in the linear sufficient dimension reduction. @@ -885,7 +894,7 @@ for every non-empty compact $K\subset\Xi$. Then, there exists a strong M-est %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Simulations}\label{sec:simulations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -In this section we report simulation results for the tensor normal and the Ising model where different aspects of the GMLM model are compared against other methods. The comparison methods are Tensor Sliced Inverse Regression (TSIR) \parencite{DingCook2015}, Multiway Generalized Canonical Correlation Analysis (MGCCA) \parencite{ChenEtAl2021,GirkaEtAl2024} and the Tucker decomposition that is a higher-order form of principal component analysis (HOPCA) \textcite{KoldaBader2009}, for both continuous and binary data. For the latter, the binary values are treated as continuous. As a base line we also include classic PCA on vectorized observations. In case of the Ising model, we also compare with LPCA (Logistic PCA) and CLPCA (Convex Logistic PCA), both introduced in \textcite{LandgrafLee2020}. All experiments are performed at sample size $n = 100, 200, 300, 500$ and $750$. Every experiment is repeated $100$ times. +In this section we report simulation results for the tensor normal and the Ising model where different aspects of the GMLM model are compared against other methods. The comparison methods are Tensor Sliced Inverse Regression (TSIR) \parencite{DingCook2015}, Multiway Generalized Canonical Correlation Analysis (MGCCA) \parencite{ChenEtAl2021,GirkaEtAl2024} and the Tucker decomposition that is a higher-order form of principal component analysis (HOPCA) \textcite{KoldaBader2009}, for both continuous and binary data. For the latter, the binary values are treated as continuous. As part of our baseline analysis, we also incorporate traditional Principal Component Analysis (PCA) on vectorized observations. In the case of the Ising model, we also compare with LPCA (Logistic PCA) and CLPCA (Convex Logistic PCA), both introduced in \textcite{LandgrafLee2020}. All experiments are performed at sample sizes $n = 100, 200, 300, 500$ and $750$. Every experiment is repeated $100$ times. We are interested in the quality of the estimate of the true sufficient reduction $\ten{R}(\ten{X})$ from \cref{thm:sdr}. Therefore, we compare with the true vectorized reduction matrix $\mat{B} = \bigkron_{k = r}^{1}\mat{\beta}_k$, as it is compatible with any linear reduction method. The distance $d(\mat{B}, \hat{\mat{B}})$ between $\mat{B}\in\mathbb{R}^{p\times q}$ and an estimate $\hat{\mat{B}}\in\mathbb{R}^{p\times \tilde{q}}$ is the \emph{subspace distance} which is proportional to \begin{displaymath} @@ -930,7 +939,7 @@ The final tensor normal experiment 1e) is a misspecified model to explore the ro \end{figure} -The results are visualized in \cref{fig:sim-normal}. Simulation 1a), given a 1D linear relation between the response $Y$ and $\ten{X}$, TSIR and GMLM are equivalent. This is expected as \textcite{DingCook2015} already established that TSIR gives the MLE estimate under a tensor (matrix) normal distributed setting. For the other methods, MGCCA is only a bit better than PCA which, unexpectedly, beats HOPCA. But none of them are close to the performance of TSIR or GMLM. Continuing with 1b), where we introduced a qubic relation between $Y$ and $\ten{X}$, we observe a bigger deviation in the performance of GMLM and TSIR. This is caused mainly because we are estimating an $8$ dimensional subspace now, which amplifies the small performance boost, in the subspace distance, we gain by avoiding slicing. The results of 1c) are surprising. The GMLM model behaves as expected, clearly being the best. The first surprise is that PCA, HOPCA and MGCCA are visually indistinguishable. This is explained by a high signal to noise ration in this particular example. But the biggest surprise is the failure of TSIR. Even more surprising is that the conditional distribution $\ten{X}\mid Y$ is tensor normal distributed which in conjunction with $\cov(\vec\ten{X})$ having a Kronecker structure, should give the MLE estimate. The low-rank assumption is also not an issue, this simply relates to TSIR estimating a 1D linear reduction which fulfills all the requirements. Finally, a common known issue of slicing, used in TSIR, is that conditional multi-modal distributions can cause estimation problems due to the different distribution modes leading to vanishing slice means. Again, this is not the case in simulation 1c). +The results are visualized in \cref{fig:sim-normal}. Simulation 1a), given a 1D linear relation between the response $Y$ and $\ten{X}$, TSIR and GMLM are equivalent. This is expected as \textcite{DingCook2015} already established that TSIR gives the MLE estimate under a tensor (matrix) normal distributed setting. For the other methods, MGCCA is only a bit better than PCA which, unexpectedly, beats HOPCA. But none of them are close to the performance of TSIR or GMLM. Continuing with 1b), where we introduced a cubic relation between $Y$ and $\ten{X}$, we observe a bigger deviation in the performance of GMLM and TSIR. This is caused mainly because we are estimating an $8$ dimensional subspace now, which amplifies the small performance boost, in the subspace distance, we gain by avoiding slicing. The GMLM model in 1c) behaves as expected, clearly being the best. The other results are surprising. First, PCA, HOPCA and MGCCA are visually indistinguishable. This is explained by a high signal-to-noise ratio in this particular example. But the biggest surprise is the failure of TSIR. Even more surprising is that the conditional distribution $\ten{X}\mid Y$ is tensor normal distributed which, in conjunction with $\cov(\vec\ten{X})$ having a Kronecker structure, should give the MLE estimate. The low-rank assumption is also not an issue, this simply relates to TSIR estimating a 1D linear reduction which fulfills all the requirements. Finally, a common known issue of slicing, used in TSIR, is that conditional multi-modal distributions can cause estimation problems due to the different distribution modes leading to vanishing slice means. Again, this is not the case in simulation 1c). An investigation into this behaviour revealed the problem in the estimation of the mode covariance matrices $\mat{O}_k = \E[(\ten{X} - \E\ten{X})_{(k)}\t{(\ten{X} - \E\ten{X})_{(k)}}]$. The mode wise reductions provided by TSIR are computed as $\hat{\mat{O}}_k^{-1}\hat{\mat{\Gamma}}_k$ where the poor estimation of $\hat{\mat{O}}_k$ causes the failure of TSIR. The poor estimate of $\mat{O}_k$ is rooted in the high signal to noise ratio in this particular simulation. GMLM does not have degenerate behaviour for high signal to noise ratios but it is less robust in low signal to noise ratio setting where TSIR performs better in this specific example. Simulation 1d), incorporating information about the covariance structure behaves similar to 1b), except that GMLM gains a statistically significant lead in estimation performance. The last simulation, 1e), where the model was misspecified for GMLM. GMLM, TSIR as well as MGCCA are on par where GMLM has a sligh lead in the small sample size setting and MGCCA overtakes in higher sample scenarios. The PCA and HOPCA methods both still outperformed. A wrong assumption about the relation to the response is still better than no relation at all. @@ -1056,7 +1065,7 @@ In \cref{tab:eeg} we provide the AUC and its standard deviation. For all applied %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Chess}\label{sec:chess} -The data set is provided by the \citetitle{lichess-database}\footnote{\fullcite{lichess-database}}. We downloaded November of 2023 consisting of more than $92$ million games. We removed all games without position evaluations. The evaluations, also denoted scores, are from Stockfish\footnote{\fullcite{stockfish}}, a free and strong chess engine. The scores take the role of the response $Y$ and correspond to a winning probability from whites point of few. Positive scores are good for white and negative scores indicate an advantage for black. We ignore all highly unbalanced positions, which we set to be positions with absolute score above $5$. We also remove all positions with a mate score (one side can force check mate). Furthermore, we only consider positions after $10$ half-moves to avoid oversampling the beginning of the most common openings including the start position which is in every game. Finally, we only consider positions with white to move. This leads to a final data set of roughly $64$ million positions, including duplicates. +The data set is provided by the \citetitle{lichess-database}\footnote{\fullcite{lichess-database}}. We randomly selected the November of 2023 data that consist of more than $92$ million games. We removed all games without position evaluations. The evaluations, also denoted as scores, are from Stockfish\footnote{\fullcite{stockfish}}, a free and strong chess engine. The scores take the role of the response $Y$ and correspond to a winning probability from the white pieces point of view. Positive scores are good for white and negative scores indicate an advantage for black pieces. We ignore all highly unbalanced positions, which we set to be positions with absolute score above $5$. We also remove all positions with a mate score (one side can force checkmate). Furthermore, we only consider positions after $10$ half-moves to avoid oversampling the beginning of the most common openings including the start position which is in every game. Finally, we only consider positions with white to move. This leads to a final data set of roughly $64$ million positions, including duplicates. A chess position is encoded as a set of $12$ binary matrices $\ten{X}_{\mathrm{piece}}$ of dimensions $8\times 8$. Every binary matrix encodes the positioning of a particular piece by containing a $1$ if the piece is present at the corresponding board position. The $12$ pieces derive from the $6$ types of pieces, namely pawns (\pawn), knights (\knight), bishops (\bishop), queens (\queen) and kings (\king) of two colors, black and white. See \cref{fig:fen2tensor} for a visualization. @@ -1119,10 +1128,13 @@ If for every piece type ($6$ types, \emph{not} distinguishing between color) hol \caption{\label{fig:psqt}Extracted PSQTs (piece square tables) from the chess example GMLM reduction.} \end{figure} -The first visual effect in \cref{fig:psqt} is the dark blue PSQT of the Queen followed by a not so dark Rook PSQT. This indicated that the Queen, followed by the Rook, are the most value pieces (after the king, but a king piece value makes no sense). The next two are the Knight and Bishop which have higher value than the Pawns, ignoring the $6$th and $7$th rank as this makes the pawns a potential queen. This is the classic piece value order known in chess. +The first visual effect in \cref{fig:psqt} is the dark blue PSQT of the Queen followed by a not-so-dark Rook PSQT. This indicates that the Queen, followed by the Rook, are the most valuable pieces (after the king which is the most valuable, which also implies that assigning value to the king makes no sense). The next two are the Knight and Bishop which have higher value than the Pawns, ignoring the $6$th and $7$th rank as this makes the pawns potential queens. This is the classic piece value order known in chess. -Next, goint one by one through the PSQTs, a few words about the prefered positions for every piece type. The pawn positions are specifically good on the $6$th and especially on the $7$th rank as this threatens a promotion to a Queen (or Knight, Bishop, Rook). The Knight PSQT is a bit surprising, the most likely explanation for the knight being good in the enemy territory is that it got there by capturing an enemy piece for free. A common occurency in low rated games which is a big chunk of the training data, ranging over all levels. The Bishops sem to have no specific prefered placement, only a slight higher overall value than pawns, excluding pawns iminent of a promotion. Continuing with the rooks, we see that the rook is a good attacking piece, indicated by a save rook infiltration. The Queen is powerfull almost everywhere, only the outer back rank squares (lower left and right) tend to reduce her value. This is rooted in the queen being there is a sign for being pushed by enemy pieces. Leading to a lot of squares being controled by the enemy hindering one own movement. Finally, the king, given the goal of the game is to checkmate the king, a save position for the king is very valuable. This is seen by the back rank (rank $1$) being the only non-penalized squares. Furthermore, the most save squares are the castling target squares ($g1$ and $c1$) as well as the $b1$ square. Shifting the king over to $b1$ is quite common protecting the $a2$ pawn providing a complete protected pawn shield infront of the king. +Next, going over the PSQTs one by one, a few words about the preferred positions for every piece type. The pawn positions are specifically good on the $6$th and especially on the $7$th rank as this threatens a promotion to a Queen (or Knight, Bishop, Rook). The Knight PSQT is a bit surprising, the most likely explanation for the knight being good in the enemy territory is that it got there by capturing an enemy piece for free. A common occurrence in low-rated games is a big chunk of the training data, ranging over all levels. The Bishops seem to have no specific preferred placement, only a slightly higher overall value than pawns, excluding pawns imminent for a promotion. Continuing with the rooks, we see that the rook is a good attacking piece, indicated by a save rook infiltration. \footnote{Rook infiltration is a strategic concept in chess that involves skillfully maneuvering your rook to penetrate deep into your opponent’s territory.} The Queen is powerful almost everywhere, only the outer back rank squares (lower left and right) tend to reduce her value. This is rooted in the queen's presence there being a sign for being chased by enemy pieces. Leading to a lot of squares being controlled by the enemy hindering one own movement. Finally, given the goal of the game is to checkmate the king, a safe position for the king is very valuable. This is seen by the back rank (rank $1$) being the only non-penalized squares. Furthermore, the safest squares are the castling \footnote{Castling is a maneuver that combines king safety with rook activation.} target squares ($g1$ and $c1$) as well as the $b1$ square. Shifting the king over to $b1$ is quite common to protect the $a2$ pawn so that the entire pawn shield in front of the king is protected. +The results of our analysis in the previous paragraph agree with the configuration of the chess board most associated with observed chess game outcomes. This arrangement also aligns with the understanding of human chess players of an average configuration at any moment during the game. + +\section{Discussion} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%