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Register a new userRobust approach for variable selection with high dimensional Logitudinal data analysis
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Added by
Jiaqi Li
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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This paper proposes a new robust smooth-threshold estimating equation to select important variables and automatically estimate parameters for high dimensional longitudinal data. A novel working correlation matrix is proposed to capture correlations within the same subject. The proposed procedure works well when the number of covariates p increases as the number of subjects n increases. The proposed estimates are competitive with the estimates obtained with the true correlation structure, especially when the data are contaminated. Moreover, the proposed method is robust against outliers in the response variables and/or covariates. Furthermore, the oracle properties for robust smooth-threshold estimating equations under large n, diverging p are established under some regularity conditions. Extensive simulation studies and a yeast cell cycle data are used to evaluate the performance of the proposed method, and results show that our proposed method is competitive with existing robust variable selection procedures.
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When fitting statistical models, some predictors are often found to be correlated with each other, and functioning together. Many group variable selection methods are developed to select the groups of predictors that are closely related to the continuous or categorical response. These existing methods usually assume the group structures are well known. For example, variables with similar practical meaning, or dummy variables created by categorical data. However, in practice, it is impractical to know the exact group structure, especially when the variable dimensional is large. As a result, the group variable selection results may be selected. To solve the challenge, we propose a two-stage approach that combines a variable clustering stage and a group variable stage for the group variable selection problem. The variable clustering stage uses information from the data to find a group structure, which improves the performance of the existing group variable selection methods. For ultrahigh dimensional data, where the predictors are much larger than observations, we incorporated a variable screening method in the first stage and shows the advantages of such an approach. In this article, we compared and discussed the performance of four existing group variable selection methods under different simulation models, with and without the variable clustering stage. The two-stage method shows a better performance, in terms of the prediction accuracy, as well as in the accuracy to select active predictors. An athletes data is also used to show the advantages of the proposed method.
High-dimensional, low sample-size (HDLSS) data problems have been a topic of immense importance for the last couple of decades. There is a vast literature that proposed a wide variety of approaches to deal with this situation, among which variable selection was a compelling idea. On the other hand, a deep neural network has been used to model complicated relationships and interactions among responses and features, which is hard to capture using a linear or an additive model. In this paper, we discuss the current status of variable selection techniques with the neural network models. We show that the stage-wise algorithm with neural network suffers from disadvantages such as the variables entering into the model later may not be consistent. We then propose an ensemble method to achieve better variable selection and prove that it has probability tending to zero that a false variable is selected. Then, we discuss additional regularization to deal with over-fitting and make better regression and classification. We study various statistical properties of our proposed method. Extensive simulations and real data examples are provided to support the theory and methodology.
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.
As an effective nonparametric method, empirical likelihood (EL) is appealing in combining estimating equations flexibly and adaptively for incorporating data information. To select important variables and estimating equations in the sparse high-dimensional model, we consider a penalized EL method based on robust estimating functions by applying two penalty functions for regularizing the regression parameters and the associated Lagrange multipliers simultaneously, which allows the dimensionalities of both regression parameters and estimating equations to grow exponentially with the sample size. A first inspection on the robustness of estimating equations contributing to the estimating equations selection and variable selection is discussed from both theoretical perspective and intuitive simulation results in this paper. The proposed method can improve the robustness and effectiveness when the data have underlying outliers or heavy tails in the response variables and/or covariates. The robustness of the estimator is measured via the bounded influence function, and the oracle properties are also established under some regularity conditions. Extensive simulation studies and a yeast cell data are used to evaluate the performance of the proposed method. The numerical results reveal that the robustness of sparse estimating equations selection fundamentally enhances variable selection accuracy when the data have heavy tails and/or include underlying outliers.
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.
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