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Register a new userUltrahigh dimensional variable selection: beyond the linear model
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Added by
Yichao Wu
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
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Variable selection in high-dimensional space characterizes many contemporary problems in scientific discovery and decision making. Many frequently-used techniques are based on independence screening; examples include correlation ranking (Fan and Lv, 2008) or feature selection using a two-sample t-test in high-dimensional classification (Tibshirani et al., 2003). Within the context of the linear model, Fan and Lv (2008)showed that this simple correlation ranking possesses a sure independence screening property under certain conditions and that its revision, called iteratively sure independent screening (ISIS), is needed when the features are marginally unrelated but jointly related to the response variable. In this paper, we extend ISIS, without explicit definition of residuals, to a general pseudo-likelihood framework, which includes generalized linear models as a special case. Even in the least-squares setting, the new method improves ISIS by allowing variable deletion in the iterative process. Our technique allows us to select important features in high-dimensional classification where the popularly used two-sample t-method fails. A new technique is introduced to reduce the false discovery rate in the feature screening stage. Several simulated and two real data examples are presented to illustrate the methodology.
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High-dimensional variable selection is an important issue in many scientific fields, such as genomics. In this paper, we develop a sure independence feature screening pro- cedure based on kernel canonical correlation analysis (KCCA-SIS, for short). KCCA- SIS is easy to be implemented and applied. Compared to the sure independence screen- ing procedure based on the Pearson correlation (SIS, for short) developed by Fan and Lv [2008], KCCA-SIS can handle nonlinear dependencies among variables. Compared to the sure independence screening procedure based on the distance correlation (DC- SIS, for short) proposed by Li et al. [2012], KCCA-SIS is scale free, distribution free and has better approximation results based on the universal characteristic of Gaussian Kernel (Micchelli et al. [2006]). KCCA-SIS is more general than SIS and DC-SIS in the sense that SIS and DC-SIS correspond to certain choice of kernels. Compared to supremum of Hilbert Schmidt independence criterion-Sure independence screening (sup-HSIC-SIS, for short) developed by Balasubramanian et al. [2013], KCCA-SIS is scale free removing the marginal variation of features and response variables. No model assumption is needed between response and predictors to apply KCCA-SIS and it can be used in ultrahigh dimensional data analysis. Similar to DC-SIS and sup- HSIC-SIS, KCCA-SIS can also be used directly to screen grouped predictors and for multivariate response variables. We show that KCCA-SIS has the sure screening prop- erty, and has better performance through simulation studies. We applied KCCA-SIS to study Autism genes in a spatiotemporal gene expression dataset for human brain development, and obtained better results based on gene ontology enrichment analysis comparing to the other existing methods.
We develop a Bayesian variable selection method, called SVEN, based on a hierarchical Gaussian linear model with priors placed on the regression coefficients as well as on the model space. Sparsity is achieved by using degenerate spike priors on inactive variables, whereas Gaussian slab priors are placed on the coefficients for the important predictors making the posterior probability of a model available in explicit form (up to a normalizing constant). The strong model selection consistency is shown to be attained when the number of predictors grows nearly exponentially with the sample size and even when the norm of mean effects solely due to the unimportant variables diverge, which is a novel attractive feature. An appealing byproduct of SVEN is the construction of novel model weight adjusted prediction intervals. Embedding a unique model based screening and using fast Cholesky updates, SVEN produces a highly scalable computational framework to explore gigantic model spaces, rapidly identify the regions of high posterior probabilities and make fast inference and prediction. A temperature schedule guided by our model selection consistency derivations is used to further mitigate multimodal posterior distributions. The performance of SVEN is demonstrated through a number of simulation experiments and a real data example from a genome wide association study with over half a million markers.
In this paper, we introduce a new methodology for Bayesian variable selection in linear regression that is independent of the traditional indicator method. A diagonal matrix $mathbf{G}$ is introduced to the prior of the coefficient vector $boldsymbol{beta}$, with each of the $g_j$s, bounded between $0$ and $1$, on the diagonal serves as a stabilizer of the corresponding $beta_j$. Mathematically, a promising variable has a $g_j$ value that is close to $0$, whereas the value of $g_j$ corresponding to an unpromising variable is close to $1$. This property is proven in this paper under orthogonality together with other asymptotic properties. Computationally, the sample path of each $g_j$ is obtained through Metropolis-within-Gibbs sampling method. Also, in this paper we give two simulations to verify the capability of this methodology in variable selection.
Most of the consistency analyses of Bayesian procedures for variable selection in regression refer to pairwise consistency, that is, consistency of Bayes factors. However, variable selection in regression is carried out in a given class of regression models where a natural variable selector is the posterior probability of the models. In this paper we analyze the consistency of the posterior model probabilities when the number of potential regressors grows as the sample size grows. The novelty in the posterior model consistency is that it depends not only on the priors for the model parameters through the Bayes factor, but also on the model priors, so that it is a useful tool for choosing priors for both models and model parameters. We have found that some classes of priors typically used in variable selection yield posterior model inconsistency, while mixtures of these priors improve this undesirable behavior. For moderate sample sizes, we evaluate Bayesian pairwise variable selection procedures by comparing their frequentist Type I and II error probabilities. This provides valuable information to discriminate between the priors for the model parameters commonly used for variable selection.
Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called two-stage drift condition to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.
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