scholarly journals A Stepwise Approach for High-Dimensional Gaussian Graphical Models

2021 ◽  
Vol 1 (2) ◽  
Author(s):  
Ruben Zamar ◽  
Marcelo Ruiz ◽  
Ginette Lafit ◽  
Javier Nogales
Keyword(s):  
Graphical Models ◽  
Partial Correlation ◽  
Pearson Correlation ◽  
Real Life ◽  
Correlation Coefficients ◽  
High Dimensional ◽  
Prediction Errors ◽  
Gaussian Graphical Models ◽  
Stepwise Approach ◽  
Linear Predictors

We present a stepwise approach to estimate high dimensional Gaussian graphical models. We exploit the relation between the partial correlation coefficients and the distribution of the prediction errors, and parametrize the model in terms of the Pearson correlation coefficients between the prediction errors of the nodes’ best linear predictors. We propose a novel stepwise algorithm for detecting pairs of conditionally dependent variables. We compare the proposed algorithm with existing methods including graphical lasso (Glasso), constrained `l1-minimization(CLIME) and equivalent partial correlation (EPC), via simulation studies and real life applications. In our simulation study we consider several model settings and report the results using different performance measures that look at desirable features of the recovered graph.

scholarly journals The ‘Un-Shrunk’ Partial Correlation in Gaussian Graphical Models

2020 ◽  
Author(s):  
Victor Bernal ◽  
Rainer Bischoff ◽  
Peter Horvatovich ◽  
Victor Guryev ◽  
Marco Grzegorczyk
Keyword(s):  
Graphical Models ◽  
Regulatory Networks ◽  
Partial Correlation ◽  
High Dimensional ◽  
Dimensional Problem ◽  
Gaussian Graphical Models ◽  
High Dimensional Problem ◽  
Non Linear ◽  
Partial Correlations ◽  
Molecular Profiles

Abstract Background: In systems biology, it is important to reconstruct regulatory networks from quantitative molecular profiles. Gaussian graphical models (GGMs) are one of the most popular methods to this end. A GGM consists of nodes (representing the transcripts, metabolites or proteins) inter-connected by edges (reflecting their partial correlations). Learning the edges from quantitative molecular profiles is statistically challenging, as there are usually fewer samples than nodes (‘high dimensional problem’). Shrinkage methods address this issue by learning a regularized GGM. However, it is an open question how the shrinkage affects the final result and its interpretation.Results: We show that the shrinkage biases the partial correlation in a non-linear way. This bias does not only change the magnitudes of the partial correlations but also affects their order. Furthermore, it makes networks obtained from different experiments incomparable and hinders their biological interpretation. We propose a method, referred to as the ‘un-shrunk’ partial correlation, which corrects for this non-linear bias. Unlike traditional methods, which use a fixed shrinkage value, the new approach provides partial correlations that are closer to the actual (population) values and that are easier to interpret. We apply the ‘un-shrunk’ method to two gene expression datasets from Escherichia coli and Mus musculus.Conclusions: GGMs are popular undirected graphical models based on partial correlations. The application of GGMs to reconstruct regulatory networks is commonly performed using shrinkage to overcome the “high-dimensional” problem. Besides it advantages, we have identified that the shrinkage introduces a non-linear bias in the partial correlations. Ignoring this type of effects caused by the shrinkage can obscure the interpretation of the network, and impede the validation of earlier reported results.

scholarly journals The ‘un-shrunk’ partial correlation in Gaussian graphical models

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Victor Bernal ◽  
Rainer Bischoff ◽  
Peter Horvatovich ◽  
Victor Guryev ◽  
Marco Grzegorczyk
Keyword(s):  
Graphical Models ◽  
Regulatory Networks ◽  
Partial Correlation ◽  
High Dimensional ◽  
Dimensional Problem ◽  
Gaussian Graphical Models ◽  
High Dimensional Problem ◽  
Non Linear ◽  
Partial Correlations ◽  
Molecular Profiles

Abstract Background In systems biology, it is important to reconstruct regulatory networks from quantitative molecular profiles. Gaussian graphical models (GGMs) are one of the most popular methods to this end. A GGM consists of nodes (representing the transcripts, metabolites or proteins) inter-connected by edges (reflecting their partial correlations). Learning the edges from quantitative molecular profiles is statistically challenging, as there are usually fewer samples than nodes (‘high dimensional problem’). Shrinkage methods address this issue by learning a regularized GGM. However, it remains open to study how the shrinkage affects the final result and its interpretation. Results We show that the shrinkage biases the partial correlation in a non-linear way. This bias does not only change the magnitudes of the partial correlations but also affects their order. Furthermore, it makes networks obtained from different experiments incomparable and hinders their biological interpretation. We propose a method, referred to as ‘un-shrinking’ the partial correlation, which corrects for this non-linear bias. Unlike traditional methods, which use a fixed shrinkage value, the new approach provides partial correlations that are closer to the actual (population) values and that are easier to interpret. This is demonstrated on two gene expression datasets from Escherichia coli and Mus musculus. Conclusions GGMs are popular undirected graphical models based on partial correlations. The application of GGMs to reconstruct regulatory networks is commonly performed using shrinkage to overcome the ‘high-dimensional problem’. Besides it advantages, we have identified that the shrinkage introduces a non-linear bias in the partial correlations. Ignoring this type of effects caused by the shrinkage can obscure the interpretation of the network, and impede the validation of earlier reported results.

scholarly journals The First National Survey of Cadmium in Cacao Farm Soil in Colombia

Agronomy ◽  
2021 ◽  
Vol 11 (4) ◽  
pp. 761
Author(s):  
Daniel Bravo ◽  
Clara Leon-Moreno ◽  
Carlos Alberto Martínez ◽  
Viviana Marcela Varón-Ramírez ◽  
Gustavo Alfonso Araujo-Carrillo ◽  
...  
Keyword(s):  
Graphical Models ◽  
Partial Correlation ◽  
National Level ◽  
Prediction Method ◽  
Correlation Coefficients ◽  
Predictive Performance ◽  
Nationwide Survey ◽  
Critical Discussion ◽  
Cadmium Content ◽  
Soil Variables

This study represents the first nationwide survey regarding the distribution of Cd content in cacao-growing soils in Colombia. The soil Cd distribution was analyzed using a cold/hotspots model. Moreover, both descriptive and predictive analytical tools were used to assess the key factors regulating the Cd concentration, considering Cd content and eight soil variables in the cacao systems. A critical discussion was performed in four main cacao-growing districts. Our results suggest that the performance of a model using all the variables will always be superior to the one using Zn alone. The analyzed variables featured an appropriate predictive performance, nonetheless, that performance has to be improved to develop a prediction method that might be used nationwide. Results from the fitted graphical models showed that the largest associations (as measured by the partial correlation coefficients) were those between Cd and Zn. Ca had the second-largest partial correlation with Cd and its predictive performance ranked second. Interestingly, it was found that there was a high variability in the factors correlated with Cd in cacao growing soils at a national level. Therefore, this study constitutes a baseline for the forthcoming studies in the country and should be reinforced with an analysis of cadmium content in cacao beans.

scholarly journals A Partial Correlation Screening Approach for Controlling the False Positive Rate in Sparse Gaussian Graphical Models

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Ginette Lafit ◽  
Francis Tuerlinckx ◽  
Inez Myin-Germeys ◽  
Eva Ceulemans
Keyword(s):  
Graphical Models ◽  
False Positive ◽  
Partial Correlation ◽  
State Of The Art ◽  
False Positive Rate ◽  
False Positives ◽  
Gaussian Graphical Models ◽  
Undirected Network ◽  
Partial Correlations ◽  
Positive Rate

AbstractGaussian Graphical Models (GGMs) are extensively used in many research areas, such as genomics, proteomics, neuroimaging, and psychology, to study the partial correlation structure of a set of variables. This structure is visualized by drawing an undirected network, in which the variables constitute the nodes and the partial correlations the edges. In many applications, it makes sense to impose sparsity (i.e., some of the partial correlations are forced to zero) as sparsity is theoretically meaningful and/or because it improves the predictive accuracy of the fitted model. However, as we will show by means of extensive simulations, state-of-the-art estimation approaches for imposing sparsity on GGMs, such as the Graphical lasso, ℓ1 regularized nodewise regression, and joint sparse regression, fall short because they often yield too many false positives (i.e., partial correlations that are not properly set to zero). In this paper we present a new estimation approach that allows to control the false positive rate better. Our approach consists of two steps: First, we estimate an undirected network using one of the three state-of-the-art estimation approaches. Second, we try to detect the false positives, by flagging the partial correlations that are smaller in absolute value than a given threshold, which is determined through cross-validation; the flagged correlations are set to zero. Applying this new approach to the same simulated data, shows that it indeed performs better. We also illustrate our approach by using it to estimate (1) a gene regulatory network for breast cancer data, (2) a symptom network of patients with a diagnosis within the nonaffective psychotic spectrum and (3) a symptom network of patients with PTSD.

scholarly journals Corruption of the Pearson correlation coefficient by measurement error: estimation, bias, and correction under different error models

2019 ◽  
Author(s):  
Edoardo Saccenti ◽  
Margriet H. W. B. Hendriks ◽  
Age K. Smilde
Keyword(s):  
Pearson Correlation ◽  
Measurement Techniques ◽  
Life Sciences ◽  
Real Life ◽  
Correlation Coefficients ◽  
Gene Expressions ◽  
Science Data ◽  
Real Life Data ◽  
Comprehensive Measurement

ABSTRACTCorrelation coefficients are abundantly used in the life sciences. Their use can be limited to simple exploratory analysis or to construct association networks for visualization but they are also basic ingredients for sophisticated multivariate data analysis methods. It is therefore important to have reliable estimates for correlation coefficients. In modern life sciences, comprehensive measurement techniques are used to measure metabolites, proteins, gene-expressions and other types of data. All these measurement techniques have errors. Whereas in the old days, with simple measurements, the errors were also simple, that is not the case anymore. Errors are heterogeneous, non-constant and not independent. This hampers the quality of the estimated correlation coefficients seriously. We will discuss the different types of errors as present in modern comprehensive life science data and show with theory, simulations and real-life data how these affect the correlation coefficients. We will briefly discuss ways to improve the estimation of such coefficients.

A simplified approach to bias estimation for correlations

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Xiaofeng Steven Liu
Keyword(s):  
Partial Correlation ◽  
Pearson Correlation ◽  
Correlation Coefficients ◽  
Small Sample ◽  
Bias Estimation ◽  
Simplified Approach ◽  
Parametric Bootstrapping ◽  
Sample Data ◽  
Distributional Assumptions ◽  
Non Parametric

Abstract Objectives We introduce a simple and unified methodology to estimate the bias of Pearson correlation coefficients, partial correlation coefficients, and semi-partial correlation coefficients. Methods Our methodology features non-parametric bootstrapping and can accommodate small sample data without making any distributional assumptions. Results Two examples with R code are provided to illustrate the computation. Conclusions The computation strategy is easy to implement and remains the same, be it Pearson correlation or partial or semi-partial correlation.

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