Skip to main content
SpringerLink
Log in

Sliced Inverse Regression for Spatial Data

  • Chapter
  • First Online:
Festschrift in Honor of R. Dennis Cook

Abstract

Sliced inverse regression is one of the most popular sufficient dimension reduction methods. Originally, it was designed for independent and identically distributed data and recently extend to the case of serially and spatially dependent data. In this work we extend it to the case of spatially dependent data where the response might depend also on neighbouring covariates when the observations are taken on a grid-like structure as it is often the case in econometric spatial regression applications. We suggest guidelines on how to decide upon the dimension of the subspace of interest and also which spatial lag might be of interest when modeling the response. These guidelines are supported by a conducted simulation study.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  • P. Adragni, R.D. Cook, Sufficient dimension reduction and prediction in regression. Philos. Trans. R. Soc. A 367, 4385–4405 (2009)

    Article  MathSciNet  Google Scholar 

  • M.M.R. Affossogbe, G. Martial Nkiet, C. Ogouyandjou, Dimension reduction in spatial regression with kernel SAVE method (2019). arXiv:1909.09996

    Google Scholar 

  • F. Bachoc, M.G Genton, K. Nordhausen, A. Ruiz-Gazen, J. Virta, Spatial blind source separation. Biometrika 107, 627–646 (2020)

    Google Scholar 

  • C. Becker, R. Fried, Sliced inverse regression for high-dimensional time series, in Exploratory Data Analysis in Empirical Research, ed. by M. Schwaiger, O. Opitz. Studies in Classification, Data Analysis, and Knowledge Organization (Springer, Berlin, Heidelberg, 2003), pp. 3–11

    Google Scholar 

  • A. Belouchrani, K. Abed Meraim, J.-F. Cardoso, E. Moulines, A blind source separation technique based on second order statistics. IEEE Trans. Signal. Process. 45, 434–444 (1997)

    Article  Google Scholar 

  • E. Bura, R.D. Cook, Extending sliced inverse regression: the weighted chi-squared test. J. Am. Stat. Assoc. 96, 996–1003 (2001)

    Article  MathSciNet  Google Scholar 

  • J.-F. Cardoso, A. Souloumiac, Jacobi angles for simultaneous diagonalization. SIAM J. Math. Anal. Appl. 17, 161–164 (1996)

    Article  MathSciNet  Google Scholar 

  • R.D. Cook, SAVE: a method for dimension reduction and graphics in regression. Commun. Stat. Theory Methods 29, 2109–2121 (2000)

    Article  Google Scholar 

  • R.D. Cook, Fisher lecture: dimension reduction in regression. Stat. Sci. 22, 1–26 (2007)

    MathSciNet  MATH  Google Scholar 

  • R.D. Cook, An Introduction to Envelopes: Dimension Reduction for Efficient Estimation in Multivariate Statistics (Wiley, Hoboken, NJ, 2018)

    Book  Google Scholar 

  • R.D. Cook, Principal components, sufficient dimension reduction, and envelopes. Annu. Rev. Stat. Appl. 5, 533–559 (2018)

    Article  MathSciNet  Google Scholar 

  • R.D. Cook, S. Weisberg, Sliced inverse regression for dimension reduction: comment. J. Am. Stat. Assoc. 86, 328–332 (1991)

    MATH  Google Scholar 

  • Y. Guan, H. Wang, Sufficient dimension reduction for spatial point processes directed by gaussian random fields. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 72, 367–387 (2010)

    Article  MathSciNet  Google Scholar 

  • R.J. Hijmans, raster: Geographic Data Analysis and Modeling (2019). R package version 2.8-19

    Google Scholar 

  • K. Illner, J. Miettinen, C. Fuchs, S. Taskinen, K. Nordhausen, H. Oja, F.J. Theis, Model selection using limiting distributions of second-order blind source separation algorithms. Signal Process. 113, 95–103 (2015)

    Article  Google Scholar 

  • H. Kelejian, G. Piras, Spatial Econometrics (Academic, New York, 2017)

    Google Scholar 

  • J. LeSage, R.K. Pace, Introduction to Spatial Econometrics. Statistics: A Series of Textbooks and Monographs (Chapman & Hall/CRC, Boca Raton, FL, 2009)

    Google Scholar 

  • K.-C. Li, Sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86, 316–327 (1991)

    Article  MathSciNet  Google Scholar 

  • B. Li, Sufficient Dimension Reduction Methods and Applications with R (CRC Press, Boca Raton, 2018)

    Book  Google Scholar 

  • E. Liski, K. Nordhausen, H. Oja, A. Ruiz-Gazen, Combining linear dimension reduction subspaces, in Recent Advances in Robust Statistics: Theory and Applications, ed. by C. Agostinelli, A. Basu, P. Filzmoser, D. Mukherjee (Springer, New Delhi, 2016), pp. 131–149

    Chapter  Google Scholar 

  • E. Liski, K. Nordhausen, H. Oja, A. Ruiz-Gazen, LDRTools: Tools for Linear Dimension Reduction (2018). R package version 0.2-1

    Google Scholar 

  • J.-M. Loubes, A.-F. Yao, Kernel Inverse Regression for spatial random fields. Int. J. Appl. Math. Stat. 32, 1–26 (2013)

    MathSciNet  Google Scholar 

  • W. Luo, B. Li, Combining eigenvalues and variation of eigenvectors for order determination. Biometrika 103, 875–887 (2016)

    Article  MathSciNet  Google Scholar 

  • Y. Ma, L. Zhu, A review on dimension reduction. Int. Stat. Rev. 81, 134–150 (2013)

    Article  MathSciNet  Google Scholar 

  • M. Matilainen, C. Croux, K. Nordhausen, H. Oja, Supervised dimension reduction for multivariate time series. Econ. Stat. 4, 57–69 (2017)

    MathSciNet  MATH  Google Scholar 

  • M. Matilainen, C. Croux, K. Nordhausen, H. Oja, Sliced average variance estimation for multivariate time series. Statistics 53, 630–655 (2019)

    Article  MathSciNet  Google Scholar 

  • J. Miettinen, K. Illner, K. Nordhausen, H. Oja, S. Taskinen, F. Theis, Separation of uncorrelated stationary time series using autocovariance matrices. J. Time Ser. Anal. 37(3), 337–354 (2016)

    Article  MathSciNet  Google Scholar 

  • J. Miettinen, K. Nordhausen, S. Taskinen, Blind source separation based on joint diagonalization in R: the packages JADE and BSSasymp. J. Stat. Softw. 76, 1–31 (2017)

    Article  Google Scholar 

  • K. Nordhausen, H. Oja, P. Filzmoser, C. Reimann, Blind source separation for spatial compositional data. Math. Geosci. 47, 753–770 (2015)

    Article  Google Scholar 

  • K. Nordhausen, H. Oja, D.E. Tyler, Asymptotic and bootstrap tests for subspace dimension (2016). arXiv:1611.04908

    Google Scholar 

  • R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, 2018)

    Google Scholar 

  • M. Schlather, A. Malinowski, P.J. Menck, M. Oesting, K. Strokorb, Analysis, simulation and prediction of multivariate random fields with package RandomFields. J. Stat. Softw. 63, 1–25 (2015)

    Article  Google Scholar 

  • M. van Lieshout, Theory of Spatial Statistics (Chapman & Hall/CRC, New York, 2019)

    Book  Google Scholar 

Download references

Acknowledgements

The work of CM and KN was supported by the Austrian Science Fund (FWF) Grant number P31881-N32, and we are grateful for the comments from the referees and from the editors which helped to improve the paper.

Author information

Authors and Affiliations

  1. Institute of Statistics & Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria

    Christoph Muehlmann & Klaus Nordhausen

  2. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Hannu Oja

Authors
  1. Christoph Muehlmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hannu Oja
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Nordhausen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Nordhausen .

Editor information

Editors and Affiliations

  1. Applied Statistics, Vienna University of Technology, Vienna, Wien, Austria

    Efstathia Bura

  2. Department of Statistics, The Pennsylvania State University, University Park, PA, USA

    Bing Li

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Muehlmann, C., Oja, H., Nordhausen, K. (2021). Sliced Inverse Regression for Spatial Data. In: Bura, E., Li, B. (eds) Festschrift in Honor of R. Dennis Cook. Springer, Cham. https://doi.org/10.1007/978-3-030-69009-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation