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Register a new userData Distillery: Effective Dimension Estimation via Penalized Probabilistic PCA
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Added by
Wei Deng
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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The paper tackles the unsupervised estimation of the effective dimension of a sample of dependent random vectors. The proposed method uses the principal components (PC) decomposition of sample covariance to establish a low-rank approximation that helps uncover the hidden structure. The number of PCs to be included in the decomposition is determined via a Probabilistic Principal Components Analysis (PPCA) embedded in a penalized profile likelihood criterion. The choice of penalty parameter is guided by a data-driven procedure that is justified via analytical derivations and extensive finite sample simulations. Application of the proposed penalized PPCA is illustrated with three gene expression datasets in which the number of cancer subtypes is estimated from all expression measurements. The analyses point towards hidden structures in the data, e.g. additional subgroups, that could be of scientific interest.
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Quadratic regression goes beyond the linear model by simultaneously including main effects and interactions between the covariates. The problem of interaction estimation in high dimensional quadratic regression has received extensive attention in the past decade. In this article we introduce a novel method which allows us to estimate the main effects and interactions separately. Unlike existing methods for ultrahigh dimensional quadratic regressions, our proposal does not require the widely used heredity assumption. In addition, our proposed estimates have explicit formulas and obey the invariance principle at the population level. We estimate the interactions of matrix form under penalized convex loss function. The resulting estimates are shown to be consistent even when the covariate dimension is an exponential order of the sample size. We develop an efficient ADMM algorithm to implement the penalized estimation. This ADMM algorithm fully explores the cheap computational cost of matrix multiplication and is much more efficient than existing penalized methods such as all pairs LASSO. We demonstrate the promising performance of our proposal through extensive numerical studies.
In a Gaussian graphical model, the conditional independence between two variables are characterized by the corresponding zero entries in the inverse covariance matrix. Maximum likelihood method using the smoothly clipped absolute deviation (SCAD) penalty (Fan and Li, 2001) and the adaptive LASSO penalty (Zou, 2006) have been proposed in literature. In this article, we establish the result that using Bayesian information criterion (BIC) to select the tuning parameter in penalized likelihood estimation with both types of penalties can lead to consistent graphical model selection. We compare the empirical performance of BIC with cross validation method and demonstrate the advantageous performance of BIC criterion for tuning parameter selection through simulation studies.
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 consider the problem of estimating parameters of stochastic differential equations (SDEs) with discrete-time observations that are either completely or partially observed. The transition density between two observations is generally unknown. We propose an importance sampling approach with an auxiliary parameter when the transition density is unknown. We embed the auxiliary importance sampler in a penalized maximum likelihood framework which produces more accurate and computationally efficient parameter estimates. Simulation studies in three different models illustrate promising improvements of the new penalized simulated maximum likelihood method. The new procedure is designed for the challenging case when some state variables are unobserved and moreover, observed states are sparse over time, which commonly arises in ecological studies. We apply this new approach to two epidemics of chronic wasting disease in mule deer.
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