Statistical Corrections of Invalid Correlation Matrices

Løland, Anders; Huseby, Ragnar Bang; Hjort, Nils Lid; Frigessi, Arnoldo

doi:10.1111/sjos.12035

Cited by 5 publications

(4 citation statements)

References 50 publications

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“…While the estimated correlation matrix Ĉ can fail to be positive definite using this procedure (although this did not occur in our analyses), methods to adjust this can be easily implemented e.g. (Løland et al 2013). Note that by using the approximate posterior sample from…”

Section: Gaussian Copula Abcmentioning

confidence: 99%

High-dimensional ABC

Nott¹,

Ong²,

Fan³

et al. 2018

Preprint

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approximation of (1) involves a kernel density estimation of the intractable likelihood based on s − s obs , the distance between simulated and observed summary statistics. As a result, the quality of the approximation decreases rapidly as the dimension of the summary statistic q increases, as the distance between s and s obs necessarily increases with their dimension, even setting aside the approximations involved in the choice of an informative S(y).Several authors (e.g. Blum 2010, Barber et al. 2015 have given results which illuminate the way that the dimension of the summary statistic q impacts the performance of standard ABC methods. For example, Blum ( 2010) obtains the result that the minimal mean squared error of certain kernel ABC density estimators is of the order of N −4/(q+5) , where N is the number of Monte Carlo samples in the kernel approximation. Barber et al. (2015) consider a simple rejection ABC algorithm where the kernel K h is uniform, and obtain a similar result concerned with optimal estimation of posterior expectations. Biau et al. (2015) extend the analysis of Blum (2010) using a nearest neighbour perspective, which accounts for the common ABC practice of choosing h adaptively based on a large pool of samples (e.g. Blum et al. 2013).Regression adjustments (e.g. Blum 2010; Beaumont et al. 2002;Blum and François 2010;Blum et al. 2013) are extremely valuable in practice for extending the applicability of ABC approximations to higher dimensions, since the regression model has some ability to compensate for the mismatch between the simulated summary statistics s and the observed value s obs . However, except when the true relationship between θ and s is known precisely (allowing for a perfect adjustment), these approaches may only extend ABC applicability to moderately higher dimensions. For example, Nott et al. ( 2014) demonstrated a rough doubling of the number of acceptably estimated parameters for a fixed computational cost when using regression adjustment compared to just rejection sampling, for a simple toy model. Nonparametric regression approaches are also subject to the curse of dimensionality, and the results of Blum (2010) also apply to certain density estimators which include nonparametric regression adjustments. Nevertheless, it has been observed that these theoretical results may

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Section: Gaussian Copula Abcmentioning

confidence: 99%

High-dimensional ABC

Nott¹,

Ong²,

Fan³

et al. 2018

Preprint

View full text Add to dashboard Cite

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“…Finally, we note that the estimate Λ obtained by combining the Λi,j is not guaranteed to be positive definite (although in all our later analyses it was). If this occurs then alternative procedures for constructing Λ can be adopted, such as the methods considered in Løland et al (2013). We also note that the use of a plug-in estimator for Λ ignores the possibility of large estimation errors.…”

Section: Gaussian Copula Abcmentioning

confidence: 99%

Extending approximate Bayesian computation methods to high dimensions via a Gaussian copula model

Nott

Fan

et al. 2017

Computational Statistics & Data Analysis

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Approximate Bayesian computation (ABC) refers to a family of inference methods used in the Bayesian analysis of complex models where evaluation of the likelihood is difficult. Conventional ABC methods often suffer from the curse of dimensionality, and a marginal adjustment strategy was recently introduced in the literature to improve the performance of ABC algorithms in high-dimensional problems. The marginal adjustment approach is extended using a Gaussian copula approximation. The method first estimates the bivariate posterior for each pair of parameters separately using a 2-dimensional Gaussian copula, and then combines these estimates together to estimate the joint posterior. The approximation works well in large sample settings when the posterior is approximately normal, but also works well in many cases which are far from that situation due to the nonparametric estimation of the marginal posterior distributions. If each bivariate posterior distribution can be well estimated with a lowdimensional ABC analysis then this Gaussian copula method can extend ABC methods to problems of high dimension. The method also results in an analytic expression for the approximate posterior which is useful for many purposes such as approximation of the likelihood itself. This method is illustrated with several examples.

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“…The resulting covariance matrix is not guaranteed to be PD because it has been built from inconsistent data sets. Motivated by the same problem, Løland et al (2013) propose both a pseudo-likelihood and a Bayesian approach to find PD estimates of pairwise correlation matrices. However, their approach relies on expert knowledge to formulate priors for the pairwise covariances.…”

Section: Nearest Positive Definite Matrix Proceduresmentioning

confidence: 99%

Robust estimation of precision matrices under cellwise contamination

Tarr

Müller

Weber

2016

Computational Statistics & Data Analysis

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There is a great need for robust techniques in data mining and machine learning contexts where many standard techniques such as principal component analysis and linear discriminant analysis are inherently susceptible to outliers. Furthermore, standard robust procedures assume that less than half the observation rows of a data matrix are contaminated, which may not be a realistic assumption when the number of observed features is large. This work looks at the problem of estimating covariance and precision matrices under cellwise contamination. We consider using a robust pairwise covariance matrix as an input to various regularisation routines, such as the graphical lasso, QUIC and CLIME. To ensure the input covariance matrix is positive semidefinite, we use a method that transforms a symmetric matrix of pairwise covariances to the nearest covariance matrix. The result is a potentially sparse precision matrix that is resilient to moderate levels of cellwise contamination. Since this procedure is not based on subsampling it scales well as the number of variables increases.

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Statistical Corrections of Invalid Correlation Matrices

Cited by 5 publications

References 50 publications

High-dimensional ABC

High-dimensional ABC

Extending approximate Bayesian computation methods to high dimensions via a Gaussian copula model

Robust estimation of precision matrices under cellwise contamination

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