Advances in Applied Probability

Sliced inverse regression and independence in random marked sets with covariates

Ondřej} Šedivý, Jakub Stanek, Blažena Kratochvílová, and Viktor Beneš

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Dimension reduction of multivariate data was developed by Y. Guan for point processes with Gaussian random fields as covariates. The generalization to fibre and surface processes is straightforward. In inverse regression methods, we suggest slicing based on geometrical marks. An investigation of the properties of this method is presented in simulation studies of random marked sets. In a refined model for dimension reduction, the second-order central subspace is analyzed in detail. A real data pattern is tested for independence of a covariate.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 3 (2013), 626-644.

Dates
First available in Project Euclid: 30 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1377868532

Digital Object Identifier
doi:10.1239/aap/1377868532

Mathematical Reviews number (MathSciNet)
MR3102465

Zentralblatt MATH identifier
1292.60015

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 62M30: Spatial processes

Keywords
Dimension reduction central subspace random marked set Gaussian random field covariate

Citation

Šedivý, Ondřej}; Stanek, Jakub; Kratochvílová, Blažena; Beneš, Viktor. Sliced inverse regression and independence in random marked sets with covariates. Adv. in Appl. Probab. 45 (2013), no. 3, 626--644. doi:10.1239/aap/1377868532. https://projecteuclid.org/euclid.aap/1377868532


Export citation

References

  • Ballani, F., Kabluchko, Z. and Schlather, M. (2012). Random marked sets. Adv. Appl. Prob. 44, 603–616.
    Mathematical Reviews (MathSciNet): MR3024601
    Zentralblatt MATH: 1266.60091
    Digital Object Identifier: doi:10.1239/aap/1346955256
    Project Euclid: euclid.aap/1346955256
  • Beneš, V. and Rataj, J. (2004). Stochastic Geometry: Selected Topics. Kluwer Academic, Boston, MA.
    Mathematical Reviews (MathSciNet): MR2068607
  • Brillinger, D. R. (2010). Modeling spatial trajectories. In Handbook of Spatial Statistics, eds A. E. Gelfand et al., Chapman& Hall/CRC, Boca Raton, FL, pp. 463–476.
    Mathematical Reviews (MathSciNet): MR2730961
    Digital Object Identifier: doi:10.1201/9781420072884-c26
  • Cook, R. D. (1998). Regression Graphics. John Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1645673
  • Cook, R. D. and Weisberg, S. (1991). Sliced inverse regression for dimension reduction: comment. J. Amer. Statist. Assoc. 86, 328–332.
    Mathematical Reviews (MathSciNet): MR1137117
    Zentralblatt MATH: 0742.62044
    Digital Object Identifier: doi:10.1080/01621459.1991.10475035
  • Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edn. Arnold, London.
    Mathematical Reviews (MathSciNet): MR743593
  • Frcalová, B. and Beneš, V. (2009). Spatio-temporal modelling of a Cox point process sampled by a curve, filtering and inference. Kybernetika 45, 912–930.
    Mathematical Reviews (MathSciNet): MR2650073
    Zentralblatt MATH: 1192.60073
  • Frcalová, B., Beneš, V. and Klement, D. (2010). Spatio-temporal point process filtering methods with an application. Environmetrics 21, 240–252.
    Mathematical Reviews (MathSciNet): MR2842241
  • Guan, Y. (2008). On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. J. Amer. Statist. Assoc. 103, 1238–1247.
    Mathematical Reviews (MathSciNet): MR2528839
    Zentralblatt MATH: 1205.62139
    Digital Object Identifier: doi:10.1198/016214508000000526
  • Guan, Y. and Wang, H. (2010). Sufficient dimension reduction for spatial point processes directed by Gaussian random fields. J. R. Statist. Soc. B 72, 367–387.
    Mathematical Reviews (MathSciNet): MR2758117
    Digital Object Identifier: doi:10.1111/j.1467-9868.2010.00738.x
  • Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley, Chichester.
    Mathematical Reviews (MathSciNet): MR2384630
    Zentralblatt MATH: 1197.62135
  • Lavancier, F., Møller, J. and Rubak, E. (2012). Statistical aspects of determinantal point processes. Preprint. Available at http://arxiv.org/abs/.1205.4818v2.
  • Li, K.-C. (1991). Sliced inverse regression for dimension reduction. J. Amer. Statist. Assoc. 86, 316–327.
    Mathematical Reviews (MathSciNet): MR1137117
    Zentralblatt MATH: 0742.62044
    Digital Object Identifier: doi:10.1080/01621459.1991.10475035
  • Li, K.-C. (2000). High dimensional data analysis via the SIR/PHD approach. Lecture Notes, UCLA.
  • Li, B. and Wang, S. (2007). On directional regression for dimension reduction. J. Amer. Statist. Assoc. 102, 997–1008.
    Mathematical Reviews (MathSciNet): MR2354409
    Zentralblatt MATH: 05564427
    Digital Object Identifier: doi:10.1198/016214507000000536
  • Molchanov, I. (1983). Labelled random sets (Russian). Teor. Veroyat. Matem. Statist. 29, 93–98 (in Russian). English translation: Theory Prob. Math. Statist. 29 (1984), 113–119.
    Mathematical Reviews (MathSciNet): MR727111
  • Møller, J. and Waagepetersen, R. (2004). Statistics and Simulations of Spatial Point Processes. World Scientific, Singapore.
  • Nguyen, X. and Zessin, H. (1979). Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.
    Mathematical Reviews (MathSciNet): MR534841
  • Pawlas, Z. (2003). Central limit theorem for random measures generated by stationary processes of compact sets. Kybernetika 39, 719–729.
    Mathematical Reviews (MathSciNet): MR2035646
    Zentralblatt MATH: 1249.60015
  • Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge University Press.
    Mathematical Reviews (MathSciNet): MR1780932
    Zentralblatt MATH: 0977.60005
  • Schlather, M., Ribeiro, P. J. and Diggle, P. J. (2004). Detecting dependence between marks and locations of marked point processes. J. R. Statist. Soc. B 66, 79–93.
    Mathematical Reviews (MathSciNet): MR2035760
    Zentralblatt MATH: 1061.62151
    Digital Object Identifier: doi:10.1046/j.1369-7412.2003.05343.x
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2455326
    Zentralblatt MATH: 1175.60003
  • Staněk, J. and Štěpán, J. (2010). Diffusion with a reflecting and absorbing level set boundary–-a simulation study. Acta Univ. Carol. Math. Phys. 51, 73–86.
    Mathematical Reviews (MathSciNet): MR2789277
    Zentralblatt MATH: 1217.60048
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. John Wiley, New York.
    Mathematical Reviews (MathSciNet): MR895588
  • Zähle, M. (1982). Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 49–72. \endharvreferences
    Mathematical Reviews (MathSciNet): MR695116
    Digital Object Identifier: doi:10.1002/mana.19821080105

Applied Probability Trust

Advances in Applied Probability
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more