Locally Adaptive Bayes Nonparametric Regression via Nested Gaussian Processes

REFERENCES

• Abramovich , F. and Steinberg , D. M. 1996 . Improved Inference in Nonparametric Regression Using L-Smoothing Splines . Journal of Statistical Planning and Inference , 49 : 327 – 341 .
   Web of Science ®Google Scholar
• Aronszajn , N. 1950 . Theory of Reproducing Kernels . Transactions of the American Mathematical Society , 68 : 337 – 404 .
   Web of Science ®Google Scholar
• Berlinet , A. and Thomas-Agnan , C. 2004 . Reproducing Kernel Hilbert Spaces in Probability and Statistics , Netherlands : Springer .
   Google Scholar
• Bhattacharya , A. , Pati , D. and Dunson , D. B. 2011 . “Adaptive Dimension Reduction With a Gaussian Process Prior,” . Arxiv preprint arXiv:1111.1044
   Google Scholar
• Choi , T. and Schervish , M. J. 2007 . On Posterior Consistency in Nonparametric Regression Problems . Journal of Multivariate Analysis , 98 : 1969 – 1987 .
   Web of Science ®Google Scholar
• Constantine , W. and Percival , D. 2010 . wmtsa: Insightful Wavelet Methods for Time Series Analysis . Available at http://CRAN.R-project.org/package=wmtsa
   Google Scholar
• Coombes , K. , Koomen , J. , Baggerly , K. , Morris , J. and Kobayashi , R. 2005a . Understanding the Characteristics of Mass Spectrometry Data Through the Use of Simulation . Cancer Informatics , 1 : 41
   PubMedGoogle Scholar
• Coombes , K. R. , Tsavachidis , S. , Morris , J. S. , Baggerly , K. A. , Hung , M. C. and Kuerer , H. M. 2005b . Improved Peak Detection and Quantification of Mass Spectrometry Data Acquired From Surface-Enhanced Laser Desorption and Ionization by Denoising Spectra With the Undecimated Discrete Wavelet Transform . Proteomics , 5 : 4107 – 4117 .
   PubMed Web of Science ®Google Scholar
• Cottrell , J. and London , U. 1999 . Probability-Based Protein Identification by Searching Sequence Databases Using Mass Spectrometry Data . Electrophoresis , 20 : 3551 – 3567 .
   PubMed Web of Science ®Google Scholar
• Crainiceanu , C. M. , Ruppert , D. , Carroll , R. J. , Joshi , A. and Goodner , B. 2007 . Spatially Adaptive Bayesian Penalized Splines With Heteroscedastic Errors . Journal of Computational and Graphical Statistics , 16 : 265 – 288 .
   Web of Science ®Google Scholar
• Denison , D. G. T. , Mallick , B. K. and Smith , A. F. M. 1998 . Automatic Bayesian Curve Fitting . Journal of the Royal Statistical Society, Series B , 60 : 333 – 350 .
   Google Scholar
• Dimatteo , I. , Genovese , C. R. and Kass , R. E. 2001 . Bayesian Curve-Fitting With Free-Knot Splines . Biometrika , 88 : 1055
   Web of Science ®Google Scholar
• Domon , B. and Aebersold , R. 2006 . Mass Spectrometry and Protein Analysis . Science , 312 : 212
   PubMed Web of Science ®Google Scholar
• Donoho , D. L. and Johnstone , I. M. 1995 . “Adapting to Unknown Smoothness via Wavelet Shrinkage,” . Journal of the American Statistical Association , : 1200 – 1224 .
   Web of Science ®Google Scholar
• Donoho , D. L. and Johnstone , I. M. 1994 . Ideal Spatial Adaptation by Wavelet Shrinkage . Biometrika , 81 : 425 – 455 .
   Web of Science ®Google Scholar
• Durbin , J. and Koopman , S. 2002 . A Simple and Efficient Simulation Smoother for State Space Time Series Analysis . Biometrika , 89 : 603
   Web of Science ®Google Scholar
• Durbin , J. and Koopman , S. 2001 . Time Series Analysis by State Space Methods (vol. 24) , Oxford : Oxford University Press .
   Google Scholar
• Fan , J. and Gijbels , I. 1995 . Data-Driven Bandwidth Selection in Local Polynomial Fitting: Variable Bandwidth and Spatial Adaptation . Journal of the Royal Statistical Society, Series B , 57 : 371 – 394 .
   Google Scholar
• Friedman , J. H. 1991 . Multivariate Adaptive Regression Splines . The Annals of Statistics , 19 : 1 – 67 .
   Web of Science ®Google Scholar
• Friedman , J. H. and Silverman , B. W. 1989 . Flexible Parsimonious Smoothing and Additive Modeling . Technometrics , 31 : 3 – 21 .
   Web of Science ®Google Scholar
• George , E. I. and McCulloch , R. E. 1993 . Variable Selection via Gibbs Sampling . Journal of the American Statistical Association , 88 : 881 – 889 .
   Web of Science ®Google Scholar
• Ghosal , S. and Roy , A. 2006 . Posterior Consistency of Gaussian Process Prior for Nonparametric Binary Regression . The Annals of Statistics , 34 : 2413 – 2429 .
   Web of Science ®Google Scholar
• Green , P. J. 1995 . Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination . Biometrika , 82 : 711
   Web of Science ®Google Scholar
• Heckman , N. E. and Ramsay , J. O. 2000 . Penalized Regression With Model-Based Penalties . Canadian Journal of Statistics , 28 : 241 – 258 .
   Web of Science ®Google Scholar
• Kloeden , P. E. and Platen , E. 1992 . Numerical Solution of Stochastic Differential Equations , New York : Springer Verlag .
   Google Scholar
• Krivobokova , T. , Crainiceanu , C. M. and Kauermann , G. 2008 . Fast Adaptive Penalized Splines . Journal of Computational and Graphical Statistics , 17 : 1 – 20 .
   Web of Science ®Google Scholar
• Lawrence , N. D. , Seeger , M. and Herbrich , R. 2002 . Fast Sparse Gaussian Process Methods: The Informative Vector Machine . Advances in Neural Information Processing Systems , 15 : 609 – 616 .
   Google Scholar
• Luo , Z. and Wahba , G. 1997 . Hybrid Adaptive Splines . Journal of the American Statistical Association , 92 : 107 – 116 .
   Web of Science ®Google Scholar
• Morris , J. , Coombes , K. , Koomen , J. , Baggerly , K. and Kobayashi , R. 2005 . Feature Extraction and Quantification for Mass Spectrometry in Biomedical Applications Using the Mean Spectrum . Bioinformatics , 21 : 1764 – 1775 .
   PubMed Web of Science ®Google Scholar
• Morris , J. S. , Brown , P. J. , Herrick , R. C. , Baggerly , K. A. and Coombes , K. R. 2008 . Bayesian Analysis of Mass Spectrometry Proteomic Data Using Wavelet-Based Functional Mixed Models . Biometrics , 64 : 479 – 489 .
   PubMed Web of Science ®Google Scholar
• Neal , R. 1998 . Regression and Classification Using Gaussian Process Priors . Bayesian Statistics , 6 : 475 – 501 .
   Google Scholar
• Pintore , A. , Speckman , P. and Holmes , C. C. 2006 . Spatially Adaptive Smoothing Splines . Biometrika , 93 : 113
   Web of Science ®Google Scholar
• Quinonero-Candela , J. and Rasmussen , C. E. 2005 . A Unifying View of Sparse Approximate Gaussian Process Regression . The Journal of Machine Learning Research , 6 : 1939 – 1959 .
   Web of Science ®Google Scholar
• R Development Core Team . 2011 . R: A Language and Environment for Statistical Computing , R Foundation for Statistical Computing, Vienna, Austria, Available at http://www.R-project.org
   Google Scholar
• Rasmussen , C. E. and Williams , C. K. I . 2006 . Gaussian Processes for Machine Learning , Boston : MIT Press .
   Google Scholar
• Ruppert , D. and Carroll , R. J. 2000 . Spatially-Adaptive Penalties for Spline Fitting . Australian & New Zealand Journal of Statistics , 42 : 205 – 223 .
   Web of Science ®Google Scholar
• Savitsky , T. , Vannucci , M. and Sha , N. 2011 . Variable Selection for Nonparametric Gaussian Process Priors: Models and Computational Strategies . Statistical Science , 26 : 130 – 149 .
   PubMed Web of Science ®Google Scholar
• Scheipl , F. and Kneib , T. 2009 . Locally Adaptive Bayesian P-splines with a Normal-Exponential-Gamma Prior . Computational Statistics and Data Analysis , 53 : 3533 – 3552 .
   Web of Science ®Google Scholar
• Searle , S. R. 1982 . Matrix Algebra Useful for Statistics , New York : Wiley .
   Google Scholar
• Shepp , L. A. 1966 . Radon-Nikodym Derivatives of Gaussian Measures . The Annals of Mathematical Statistics , 37 : 321 – 354 .
   Google Scholar
• Shi , J. Q. and Choi , T. 2011 . Gaussian Process Regression Analysis for Functional Data , London : Chapman & Hall/CRC Press .
   Google Scholar
• Smith , M. and Kohn , R. 1996 . Nonparametric Regression Using Bayesian Variable Selection . Journal of Econometrics , 75 : 317 – 343 .
   Web of Science ®Google Scholar
• Smola , A. J. and Bartlett , P. 2001 . Sparse Greedy Gaussian Process Regression . Advances in Neural Information Processing Systems 13 , Citeseer, Boston: MIT Press
   Google Scholar
• Tibshirani , R. , Hastie , T. , Narasimhan , B. , Soltys , S. , Shi , G. , Koong , A. and Le , Q. T. 2004 . Sample Classification From Protein Mass Spectrometry, by ‘Peak Probability Contrasts’ . Bioinformatics , 20 : 3034 – 3044 .
   PubMed Web of Science ®Google Scholar
• Van der Vaart , A. W. and Van Zanten , J. H. 2008a . Rates of Contraction of Posterior Distributions Based on Gaussian Process Priors . The Annals of Statistics , 36 : 1435 – 1463 .
   Web of Science ®Google Scholar
• Van der Vaart , A. W. and Van Zanten , J. H. 2008b . Reproducing Kernel Hilbert Spaces of Gaussian Priors . Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh , 3 : 200 – 222 .
   Google Scholar
• Van der Vaart , A. W. and Wellner , J. A. 1996 . Weak Convergence and Empirical Processes , New York : Springer Verlag .
   Google Scholar
• Wahba , G. 1990 . Spline Models for Observational Data (vol. 59) , Philadelphia : Society for Industrial Mathematics .
   Google Scholar
• Wahba , G. 1995 . Discussion of “Wavelet Shrinkage: Asymptopia?” by Donoho et al. . Journal of the Royal Statistical Society, Series B , : 360 – 361 .
   Google Scholar
• West , M. and Harrison , J. 1997 . Bayesian Forecasting and Dynamic Models , New York : Springer Verlag .
   Google Scholar
• Wolpert , R. L. , Clyde , M. A. and Tu , C. 2011 . Stochastic Expansions Using Continuous Dictionaries: Lévy Adaptive Regression Kernels . The Annals of Statistics , 39 : 1916 – 1962 .
   Web of Science ®Google Scholar
• Wood , S. A. , Jiang , W. and Tanner , M. 2002 . Bayesian Mixture of Splines for Spatially Adaptive Nonparametric Regression . Biometrika , 89 : 513
   Web of Science ®Google Scholar
• Wood , S. A. , Kohn , R. , Cottet , R. , Jiang , W. and Tanner , M. 2008 . Locally Adaptive Nonparametric Binary Regression . Journal of Computational and Graphical Statistics , 17 : 352 – 372 .
   Web of Science ®Google Scholar
• Wu , J. Q. , Sklar , J. , Wang , Y. D. and Meiring , W. 2011 . bsml: Basis Selection from Multiple Libraries , Available at http://CRAN.R-project.org/package=bsml
   Google Scholar
• Zhou , S. and Shen , X. 2001 . Spatially Adaptive Regression Splines and Accurate Knot Selection Schemes . Journal of the American Statistical Association , 96 : 247 – 259 .
   Web of Science ®Google Scholar
• Zhu , B. , Song , P. X. K. and Taylor , J. M. G. 2011 . Stochastic Functional Data Analysis: A Diffusion Model-Based Approach . Biometrics , 67 : 1295 – 1304 .
   PubMed Web of Science ®Google Scholar
• Zou , F. , Huang , H. , Lee , S. and Hoeschele , I. 2010 . Nonparametric Bayesian Variable Selection With Applications to Multiple Quantitative Trait Loci Mapping With Epistasis and Gene–Environment Interaction . Genetics , 186 : 385
   PubMed Web of Science ®Google Scholar