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105
Bayesian analysis for penalized spline regression using WinBUGS
- J. Statist. Soft
, 2005
"... Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys t ..."
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Cited by 56 (7 self)
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Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in WinBUGS. Good mixing properties of the MCMC chains are obtained by using low-rank thin-plate splines, while simulation times per iteration are reduced employing WinBUGS specific computational tricks.
Smoothing with Mixed Model Software
- Journal of Statistical Software
, 2004
"... Smoothing with mixed model software Smoothing methods that use basis functions with penalization can be formulated as fits in a mixed model framework. One of the major benefits is that software for mixed model analysis can be used for smoothing. We illustrate this for several smoothing models such a ..."
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Cited by 36 (2 self)
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Smoothing with mixed model software Smoothing methods that use basis functions with penalization can be formulated as fits in a mixed model framework. One of the major benefits is that software for mixed model analysis can be used for smoothing. We illustrate this for several smoothing models such as additive and varying coefficient models for both S-PLUS and SAS software. Code for each of the illustrations is available on the Internet.
On the asymptotics of penalized splines
, 2007
"... The asymptotic behaviour of penalized spline estimators is studied in the univari-ate case. We use B-splines and a penalty is placed on mth-order differences of the coefficients. The number of knots is assumed to converge to infinity as the sample size increases. We show that penalized splines behav ..."
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Cited by 31 (3 self)
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The asymptotic behaviour of penalized spline estimators is studied in the univari-ate case. We use B-splines and a penalty is placed on mth-order differences of the coefficients. The number of knots is assumed to converge to infinity as the sample size increases. We show that penalized splines behave similarly to Nadaraya-Watson kernel estimators with ‘equivalent ’ kernels depending upon m. The equivalent kernels we obtain for penalized splines are the same as those found by Silverman for smooth-ing splines. The asymptotic distribution of the penalized spline estimator is Gaussian and we give simple expressions for the asymptotic mean and variance. Provided that it is fast enough, the rate at which the number of knots converges to infinity does not affect the asymptotic distribution. The optimal rate of convergence of the penalty parameter is given. Penalized splines are not design-adaptive.
Splines, knots, and penalties
, 2004
"... Abstract Penalized splines have gained much popularity as a flexible tool for smoothing and semi-parametric models. Two approaches have been advocated: 1) use a B-spline basis, equally-spaced knots and difference penalties ..."
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Cited by 30 (1 self)
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Abstract Penalized splines have gained much popularity as a flexible tool for smoothing and semi-parametric models. Two approaches have been advocated: 1) use a B-spline basis, equally-spaced knots and difference penalties
Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis
- Biometrics
, 2008
"... Summary. In this article, we present new methods to analyze data from an experiment using rodent models to investigate the role of p27, an important cell-cycle mediator, in early colon carcinogenesis. The responses modeled here are essentially functions nested within a two-stage hierarchy. Standard ..."
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Cited by 26 (4 self)
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Summary. In this article, we present new methods to analyze data from an experiment using rodent models to investigate the role of p27, an important cell-cycle mediator, in early colon carcinogenesis. The responses modeled here are essentially functions nested within a two-stage hierarchy. Standard functional data analysis literature focuses on a single stage of hierarchy and conditionally independent functions with near white noise. However, in our experiment, there is substantial biological motivation for the existence of spatial correlation among the functions, which arise from the locations of biological structures called colonic crypts: this possible functional correlation is a phenomenon we term crypt signaling. Thus, as a point of general methodology, we require an analysis that allows for functions to be correlated at the deepest level of the hierarchy. Our approach is fully Bayesian and uses Markov chain Monte Carlo methods for inference and estimation. Analysis of this data set gives new insights into the structure of p27 expression in early colon carcinogenesis and suggests the existence of significant crypt signaling. Our methodology uses regression splines, and because of the hierarchical nature of the data, dimension reduction of the covariance matrix of the spline coefficients is important: we suggest simple methods for overcoming this problem.
2007), ”A Note on Penalized Spline Smoothing with Correlated Errors”, mimeo
"... This note investigates the behavior of data driven smoothing parameters for penalized spline regression in the presence of correlated data. It has been shown for other smoothing methods before, that mean squared error minimiz-ers, such as (generalized) cross validation or Akaike criterion, are extre ..."
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Cited by 25 (5 self)
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This note investigates the behavior of data driven smoothing parameters for penalized spline regression in the presence of correlated data. It has been shown for other smoothing methods before, that mean squared error minimiz-ers, such as (generalized) cross validation or Akaike criterion, are extremely sensitive to misspecifications of the correlation structure over or (under) fit-ting the data. In contrast to this, we show that a maximum likelihood based choice of the smoothing parameter is more robust and for moderately mis-specified correlation structure over or (under) fitting does not occur. This is demonstrated in simulations, data examples and supported by theoretical investigations.
On semiparametric regression with O’Sullivan penalized splines
- Australian and New Zealand Journal of Statistics
, 2008
"... An exposition on the use of O’Sullivan penalized splines in contemporary semiparamet-ric regression, including mixed model and Bayesian formulations, is presented. O’Sullivan penalized splines are similar to P-splines, but have the advantage of being a direct generaliza-tion of smoothing splines. Ex ..."
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Cited by 24 (7 self)
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An exposition on the use of O’Sullivan penalized splines in contemporary semiparamet-ric regression, including mixed model and Bayesian formulations, is presented. O’Sullivan penalized splines are similar to P-splines, but have the advantage of being a direct generaliza-tion of smoothing splines. Exact expressions for the O’Sullivan penalty matrix are obtained. Comparisons between the two types of splines reveal that O’Sullivan penalized splines more closely mimic the natural boundary behaviour of smoothing splines. Implementation in modern computing environments such as MATLAB, R and BUGS is discussed.
Some asymptotic results on generalized penalized spline smoothing
- B
"... The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be a ..."
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Cited by 24 (7 self)
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The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be a priori normally distributed links the smoothing framework to generalized linear mixed models (GLMM). We consider the asymptotic rates such that Laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates. Secondly, we make use of a fully Bayesian viewpoint by imposing a priori distribution on all parameters and coefficients. We argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coefficients is approximately normal. The validity of this result is investigated in finite samples by comparing Markov Chain Monte Carlo (MCMC) results with their asymptotic approximation in a simulation study. 1 1
Agus Sudjianto. Analysis of computer experiments using penalized likelihood in gaussian kriging models
- Journal of the American Statistical Association
"... Kriging is a popular analysis approach for computer experiments for the purpose of creating a cheap-to-compute “meta-model ” as a surrogate to a computationally expensive engineering simulation model. The maximum likelihood approach is used to estimate the parameters in the kriging model. However, t ..."
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Cited by 23 (2 self)
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Kriging is a popular analysis approach for computer experiments for the purpose of creating a cheap-to-compute “meta-model ” as a surrogate to a computationally expensive engineering simulation model. The maximum likelihood approach is used to estimate the parameters in the kriging model. However, the likelihood function near the optimum may be flat in some situations, which leads to maximum likelihood estimates for the parameters in the covariance matrix that have very large variance. To overcome this difficulty, a penalized likelihood approach is proposed for the kriging model. Both theoretical analysis and empirical experience using real world data suggest that the proposed method is particularly important in the context of a computationally intensive simulation model where the number of simulation runs must be kept small because collection of a large sample set is prohibitive. The proposed approach is applied to the reduction of piston slap, an unwanted engine noise due to piston secondary motion. Issues related to practical implementation of the proposed approach are discussed.
