Robust and sparse Gaussian graphical modelling under cell‐wise contamination

Katayama, Shota; Fujisawa, Hironori; Drton, Mathias

doi:10.1002/sta4.181

Cited by 9 publications

(6 citation statements)

References 34 publications

(61 reference statements)

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“…For example, we are interested in investigating gene expression variation under different tissue types. In the similar notation as in (Katayama et al, 2018), consider the response variable y coming from a mixture model, where I ∈ ℝ n×n is the identity matrix and B 1 is a diagonal matrix where the diagonal elements are Bernoulli variables with probability π 1 ∈ [0, 0.5) to be 1, and y 0 is the clean part forming the population, and y 1 is the outlier. The 0-1 elements in the diagonal of B 1 indicate the samples coming from the clean part y 0 or the outlier y 1 .…”

Section: Methodsmentioning

confidence: 99%

“…Fujisawa and Eguchi (2008) revisited these old works and constructed a γ -cross entropy robust criterion, assuming under a proper γ (≥ 0), the outliers go to the tails of density power and thus do not contribute much in the population estimation. Recently, the γ -cross entropy criterion has gained much attention and there are a series of variant works including robust estimation using an unnormalized model (Kanamori and Fujisawa, 2015), robust clustering (Chen et al, 2014), Gaussian graphical modeling (Katayama et al, 2018; Miyamura and Kano, 2006), and others.…”

Section: Introductionmentioning

confidence: 99%

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AdaReg: Data Adaptive Robust Estimation in Linear Regression with Application in GTEx Gene Expressions

Wang

Jiang

Snyder

2019

Preprint

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With the development of high-throughput RNA sequencing (RNA-seq) technology, the Genotype Tissue-Expression (GTEx) project (Consortium et al., 2015) generated a valuable resource of gene expression data from more than 11,000 samples. The large-scale data set is a powerful resource for understanding the human transcriptome. However, the technical variation, sequencing background noise and unknown factors make the statistical analysis challenging. To eliminate the possibility that outliers might affect the estimation of population distribution, we need a more robust estimation method, a method that will adapt to heterogeneous genes and further optimize the estimate for each gene. We followed the approach of the robust estimation based on γ-density-power-weight (Fujisawa and Eguchi, 2008; Windham, 1995), where γ is the exponent of density weight which controls the balance between bias and variance. As far as we know, our work is the first to propose a procedure to tune the parameter γ to balance the bias-variance trade-off under the mixture distributions. We constructed a robust likelihood criterion based on weighted densities in the mixture model of Gaussian population distribution mixed with unknown outlier distribution, and developed a data-adaptive γ-selection procedure embedded into the robust estimation. We provided a heuristic analysis on the selection criterion and found that our practical selection trend under various γ’s in average performance has similar capability to capture minimizer γ as the inestimable Mean Squared Error (MSE) trend from our simulation studies under a series of settings. Our data-adaptive robustifying procedure in the linear regression problem (AdaReg) shows a significant advantage in both simulation studies and real data application of heart samples from the GTEx project compared to the fixed γ procedure and other robust methods. This paper discusses some limitations of this method, and future work.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

AdaReg: Data Adaptive Robust Estimation in Linear Regression with Application in GTEx Gene Expressions

Wang

Jiang

Snyder

2019

Preprint

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“…where d measures the distance between two matrices. For instance, the Frobenius norm ( [46,21]) and the element-wise maximum norm ( [27]) are used pre-viously. Then, the nearest matrix Σ psd would be put into the subsequent multivariate analyses (e.g.…”

Section: More General Solution: Matrix Approximationmentioning

confidence: 99%

“…+ λI is positive definite. [21] also point out that the solution of (23) may not exist, unless an input matrix Σ plug is guaranteed to be PSD. We conjecture this irregularity is due to the initialization.…”

Section: Algorithm 2 the Clime Algorithmmentioning

confidence: 99%

Estimating high-dimensional covariance and precision matrices under general missing dependence

Park

Wang

Lim

2021

Electron. J. Statist.

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A sample covariance matrix S of completely observed data is the key statistic in a large variety of multivariate statistical procedures, such as structured covariance/precision matrix estimation, principal component analysis, and testing of equality of mean vectors. However, when the data are partially observed, the sample covariance matrix from the available data is biased and does not provide valid multivariate procedures. To correct the bias, a simple adjustment method called inverse probability weighting (IPW) has been used in previous research, yielding the IPW estimator. The estimator can play the role of S in the missing data context, thus replacing S in off-the-shelf multivariate procedures such as the graphical lasso algorithm. However, theoretical properties (e.g. concentration) of the IPW estimator have been only established in earlier work under very simple missing structures; every variable of each sample is independently subject to missingness with equal probability. We investigate the deviation of the IPW estimator when observations are partially observed under general missing dependency. We prove the optimal convergence rate Op( log p/n) of the IPW estimator based on the element-wise maximum norm, even when two unrealistic assumptions (known mean and/or missing probabilities) frequently assumed to be known in the past work are relaxed. The optimal rate is especially crucial in estimating a precision matrix, because of the "meta-theorem" [26] that claims the rate of the IPW estimator governs that of the resulting precision matrix estimator. In

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“…where d measures the distance between two matrices. For instance, the Frobenius norm (Katayama et al 2018;Wang et al 2014) and the element-wise maximum norm (Loh and Tan 2018) are used previously. Then, the nearest matrix Σ psd would be put into the subsequent multivariate analyses (e.g.…”

mentioning

confidence: 99%

Estimating High-dimensional Covariance and Precision Matrices under General Missing Dependence

Park¹,

Wang²,

Lim³

2020

Preprint

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Robust and sparse Gaussian graphical modelling under cell‐wise contamination

Cited by 9 publications

References 34 publications

AdaReg: Data Adaptive Robust Estimation in Linear Regression with Application in GTEx Gene Expressions

AdaReg: Data Adaptive Robust Estimation in Linear Regression with Application in GTEx Gene Expressions

Estimating high-dimensional covariance and precision matrices under general missing dependence

Estimating High-dimensional Covariance and Precision Matrices under General Missing Dependence

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