Estimating spatial covariance using penalised likelihood with weighted L1 penalty

Journal of Nonparametric Statistics

Volumn 21, Issue 7, 2009, Pages 925-942

  Zhua, Zhengyuan ^a   Liu, Yufeng ^b  

^a Iowa State University   (United States)

^b University of North Carolina at Chapel Hill   (United States)

Author keywords

Cholesky decomposition; Gaussian Markov random fields; LASSO; Maximum likelihood; Nonstationarity

Indexed keywords

EID: 70449359522     PISSN: 10485252     EISSN: 10290311     Source Type: Journal    
DOI: 10.1080/10485250903023632     Document Type: Article

References (35)

1. O. Banerjee, L.E. Ghaoui, and A. d'Aspremont, Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data, J. Mach. Learn. Res. 9 (2008), pp. 485-516.
2. N. Cressie, Statistics for Spatial Data, Wiley, NewYork, 1993.
3. M. Brewer and A. Nolan, Variable smoothing in Bayesian intrinsic autoregressions, Environmetrics, 18 (2007), pp. 841-857.
4. H. Lu, C.S. Reilly, S. Banerjee, and B.P. Carlin, Bayesian areal wombling via adjacency modelling, Environ. Ecol. Stat. 14 (2007), pp. 433-452.
5. J.Z. Huang, N. Liu, M. Pourahmadi, and L. Liu, Covariance selection and estimation via penalised normal likelihood, Biometrika 93 (2006), pp. 85-98.
6. L. Levina, A. Rothman, and J. Zhu, Sparse estimation of large covariance matrices via a hierarchical lasso penalty, Ann. Appl. Stat. 2 (2008), pp. 245-263.
7. S. Banerjee, B.P. Carlin, and A.E. Gelfand, Hierarchical Modeling and Analysis for Spatial Data, Chapman and Hall/CRC, Boca Raton, FL, 2004.
8. Meinshusen, and P. Buhlmann, High-dimensional graphs and variable selection with the Lasso, Ann. Stat. 34 (2006), pp. 1436-1462.
9. A.J. Rothman, P.J. Bickel, E. Levina, and J. Zhu, Sparse permutation invariant covariance estimation, Electron. J. Stat. 2 (2008), pp. 494-515.
10. M. Yuan and Y. Lin, Model selection and estimation in the Gaussian graphical model, Biometrika 94 (2007), pp. 19-35.
11. J. Fan, and R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Am. Statist. Assoc. 96 (2001), pp. 1348-1360.
12. R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B, 58 (1996), pp. 267-288.
13. H.H. Zhang and W. Lu, Adaptive lasso for Cox's proportional hazards model, Biometrika 94 (2007), pp. 691-703.
14. H. Zou, The adaptive Lasso and its oracle properties, J. Am. Stat. Assoc. 101 (2006), pp. 1418-1429.
15. M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation, Biometrika, 86 (1999), pp. 677-690.
16. B.D. Ripley, Spatial Statistics, John Wiley and Sons, NewYork, 1981.
17. P. Whittle, On stationary processes in the plane, Biometrika 41 (1954), pp. 434-449.
18. L.F. Lee, Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models, Econometrica, 72 (2004), pp. 1899-1925.
19. K. Ord, Estimation methods for models of spatial interaction, J. Am. Stat. Assoc. 70 (1975), pp. 120-126.
20. O. Smirnov and L. Anselin, Fast maximum likelihood estimation of very large spatial autoregressive models: A characteristic polynomial approach, Comput. Stat. Data Anal. 35 (2001), pp. 301-319.
21. I.S. Duff, A.M. Erisman, and J.K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, New York, 1989.
22. A. George and J.W.H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice Haal Inc., Englewood Cliffs, 1981.
23. P.J. Green and R. Sibson, Computing Dirichlet Tessellations in the Plane, Comput. J. 21 (1978), pp. 168-173.
24. E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 24th National Conference ACM, CM: NewYork, 1969, pp. 157-172.
25. C.H. Papadimitriou, The NP-Completeness of the Bandwidth Minimization Problem, Computing 16 (1976), pp. 263-270.
26. M. Yannakakis, Computing the minimum fill-in is NP-complete, SIAM J. Algebr. Discrete Methods 2 (1981), pp. 77-79.
27. S. Rosset and J. Zhu, Piecewise linear regularized solution paths, Ann. Stat. 35 (2007), pp. 1012-1030.
28. B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, Least angle regression (with discussion), Ann. Stat. 32 (2004), pp. 407-499.
29. K. Knight and W.J. Fu, Asymptotics for lasso-type estimators, Ann. Stat. 28 (2000), pp. 1356-1378.
30. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer-Verlag, NewYork, 2001.
31. P. Groisman, Data Documentation for TD-3721: Gridded US Daily Precipitation and SnowfallTime Series, National Climatic Data Center, Asheville, NC, 2000.
32. H. Rue, and H. Tjelmeland, Fitting Gaussian Markov random fields to Gaussian fields, Scand. J. Stat. 29 (2002), pp. 31-49.
33. D.J. Nott and W.T.M. Dunsmuir, Estimation of nonstationary spatial covariance structure, Biometrika 89 (2002), pp. 819-829.
34. P.Z. Chinn, J. Chvatalova, A.K. Dewdney, and N.E. Gibbs, The bandwidth problem for graphs and matrices - a survey, J. Graph Theory 16 (1982), pp. 223-254.
35. J. Friedman, T. Hastie, and R. Tibshirani, Sparse inverse covariance estimation with the graphical lasso, Biostatistics 9 (2008), pp. 432-441.