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Graph reconstruction using covariance-based methods

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ABSTRACT

Methods based on correlation and partial correlation are today employed in the reconstruction of a statistical interaction graph from high-throughput omics data. These dedicated methods work well even for the case when the number of variables exceeds the number of samples. In this study, we investigate how the graphs extracted from covariance and concentration matrix estimates are related by using Neumann series and transitive closure and through discussing concrete small examples. Considering the ideal case where the true graph is available, we also compare correlation and partial correlation methods for large realistic graphs. In particular, we perform the comparisons with optimally selected parameters based on the true underlying graph and with data-driven approaches where the parameters are directly estimated from the data.

Electronic supplementary material: The online version of this article (doi:10.1186/s13637-016-0052-y) contains supplementary material, which is available to authorized users.

No MeSH data available.


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a Transitive closure of a graph with four nodes. Solid edges indicate existing or direct edges in the graph, whereas dashed edges indicate indirect edges which are added to the graph as the result of the transitive closure effect. b Three-dimensional true graph (left), the transitive closure of the true graph (middle), and the corresponding covariance graph constructed from the covariance matrix (right). c The illustration of a star graph. d (left) The true example graph which corresponds to the concentration graph, G, and (right) the covariance graph,  constructed from the covariance matrix. The true graph is sparse, and the covariance graph is fully connected. e The covariance graph,  with edge weights given by the correlation matrix C (the graph is predicted by thresholding the correlation matrix). (left) The graph structure when the condition (A.11) holds (see Additional file 1). (right) The graph structure when (A.12) holds (see Additional file 1). Distribution of direct and indirect edges of the covariance graph (p=500), when f and g. Vertical line (blue) indicates the optimal threshold that separates two distributions (For more information about e, f, and g, see the text in the Additional file 1)
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Fig1: a Transitive closure of a graph with four nodes. Solid edges indicate existing or direct edges in the graph, whereas dashed edges indicate indirect edges which are added to the graph as the result of the transitive closure effect. b Three-dimensional true graph (left), the transitive closure of the true graph (middle), and the corresponding covariance graph constructed from the covariance matrix (right). c The illustration of a star graph. d (left) The true example graph which corresponds to the concentration graph, G, and (right) the covariance graph, constructed from the covariance matrix. The true graph is sparse, and the covariance graph is fully connected. e The covariance graph, with edge weights given by the correlation matrix C (the graph is predicted by thresholding the correlation matrix). (left) The graph structure when the condition (A.11) holds (see Additional file 1). (right) The graph structure when (A.12) holds (see Additional file 1). Distribution of direct and indirect edges of the covariance graph (p=500), when f and g. Vertical line (blue) indicates the optimal threshold that separates two distributions (For more information about e, f, and g, see the text in the Additional file 1)

Mentions: We associate to G and G∗ their weighted adjacency matrices denoted A and A∗, respectively. Observe that G∗ contains self-loops or cycles (e.g., for a node i with at least one edge, i is connected to i by a path of length two through i→j→i), and hence, A∗ will have non-zero diagonal entries. The transitive closure of the graph is depicted in Fig. 1a for illustration.Fig. 1


Graph reconstruction using covariance-based methods
a Transitive closure of a graph with four nodes. Solid edges indicate existing or direct edges in the graph, whereas dashed edges indicate indirect edges which are added to the graph as the result of the transitive closure effect. b Three-dimensional true graph (left), the transitive closure of the true graph (middle), and the corresponding covariance graph constructed from the covariance matrix (right). c The illustration of a star graph. d (left) The true example graph which corresponds to the concentration graph, G, and (right) the covariance graph,  constructed from the covariance matrix. The true graph is sparse, and the covariance graph is fully connected. e The covariance graph,  with edge weights given by the correlation matrix C (the graph is predicted by thresholding the correlation matrix). (left) The graph structure when the condition (A.11) holds (see Additional file 1). (right) The graph structure when (A.12) holds (see Additional file 1). Distribution of direct and indirect edges of the covariance graph (p=500), when f and g. Vertical line (blue) indicates the optimal threshold that separates two distributions (For more information about e, f, and g, see the text in the Additional file 1)
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Related In: Results  -  Collection

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Fig1: a Transitive closure of a graph with four nodes. Solid edges indicate existing or direct edges in the graph, whereas dashed edges indicate indirect edges which are added to the graph as the result of the transitive closure effect. b Three-dimensional true graph (left), the transitive closure of the true graph (middle), and the corresponding covariance graph constructed from the covariance matrix (right). c The illustration of a star graph. d (left) The true example graph which corresponds to the concentration graph, G, and (right) the covariance graph, constructed from the covariance matrix. The true graph is sparse, and the covariance graph is fully connected. e The covariance graph, with edge weights given by the correlation matrix C (the graph is predicted by thresholding the correlation matrix). (left) The graph structure when the condition (A.11) holds (see Additional file 1). (right) The graph structure when (A.12) holds (see Additional file 1). Distribution of direct and indirect edges of the covariance graph (p=500), when f and g. Vertical line (blue) indicates the optimal threshold that separates two distributions (For more information about e, f, and g, see the text in the Additional file 1)
Mentions: We associate to G and G∗ their weighted adjacency matrices denoted A and A∗, respectively. Observe that G∗ contains self-loops or cycles (e.g., for a node i with at least one edge, i is connected to i by a path of length two through i→j→i), and hence, A∗ will have non-zero diagonal entries. The transitive closure of the graph is depicted in Fig. 1a for illustration.Fig. 1

View Article: PubMed Central - PubMed

ABSTRACT

Methods based on correlation and partial correlation are today employed in the reconstruction of a statistical interaction graph from high-throughput omics data. These dedicated methods work well even for the case when the number of variables exceeds the number of samples. In this study, we investigate how the graphs extracted from covariance and concentration matrix estimates are related by using Neumann series and transitive closure and through discussing concrete small examples. Considering the ideal case where the true graph is available, we also compare correlation and partial correlation methods for large realistic graphs. In particular, we perform the comparisons with optimally selected parameters based on the true underlying graph and with data-driven approaches where the parameters are directly estimated from the data.

Electronic supplementary material: The online version of this article (doi:10.1186/s13637-016-0052-y) contains supplementary material, which is available to authorized users.

No MeSH data available.


Related in: MedlinePlus