Thresholding Methods: Difference between revisions
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===Global Thresholding=== | ===Global Thresholding=== | ||
A simple filtering technique is to apply a continuously variable threshold <math>\tau</math> to the association matrix <math>A</math>, so that | A simple filtering technique is to apply a continuously variable threshold,<math> \tau</math>, to the association matrix <math>A</math>, so that <math>A_{i,j} = 1</math> if <math> \rho_{i,j} > \tau</math>, and <math>A_{i,j} = 0</math> otherwise. | ||
<math>A_{i,j} = 1 | |||
and <math>A_{i,j} = 0</math> otherwise. | |||
===Fixed Cost=== | ===Fixed Cost=== | ||
As <math>\tau</math> is continuously variable, it is possible to use this and related filtering techniques to construct binary graphs of arbitrary connection density or topological cost, <math>0 < \kappa < 1</math>, where <math>\kappa</math> is the number of edges in the graph (or non-zero elements in the adjacency matrix) divided by the maximum possible number of edges, <math>N | As <math>\tau</math> is continuously variable, it is possible to use this and related filtering techniques to construct binary graphs of arbitrary connection density or topological cost, <math>0 < \kappa < 1</math>, where <math>\kappa</math> is the number of edges in the graph (or non-zero elements in the adjacency matrix) divided by the maximum possible number of edges, <math>N . (N - 1)</math>. | ||
The advantage of this approach is that the resulting networks can now be compared across subjects or groups since they contain the same number of edges and nodes. Indeed, all network measures are sensitive to the number of nodes and edges and comparing networks of different size and connectivity is therefore tricky: [[http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0013701]] | The advantage of this approach is that the resulting networks can now be compared across subjects or groups since they contain the same number of edges and nodes. Indeed, all network measures are sensitive to the number of nodes and edges and comparing networks of different size and connectivity is therefore tricky: [[http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0013701]] | ||
===MST-based methods=== | ===MST-based methods=== | ||
The Minimum Spanning Tree (MST) is a simply connected acyclic graph that connects all <math>n</math> nodes with <math>n | The Minimum Spanning Tree (MST) is a simply connected acyclic graph that connects all <math>n</math> nodes with <math>n − 1</math> edges such that the sum of the `distance' of included edges <math>d_{ij}</math> is minimum. Clearly, in the case of brain networks one would like to include edges so as to maximise the total correlation strength, the distance <math>d_{ij}</math> is therefore defined as: <math> d_{ij} =</math><math> (2(1-\rho_{ij}))^{1/2} </math> | ||
====MST + Global thresholding==== | ====MST + Global thresholding==== | ||
Latest revision as of 23:40, 18 October 2022
Global Thresholding
A simple filtering technique is to apply a continuously variable threshold,, to the association matrix , so that if , and otherwise.
Fixed Cost
As is continuously variable, it is possible to use this and related filtering techniques to construct binary graphs of arbitrary connection density or topological cost, , where is the number of edges in the graph (or non-zero elements in the adjacency matrix) divided by the maximum possible number of edges, . The advantage of this approach is that the resulting networks can now be compared across subjects or groups since they contain the same number of edges and nodes. Indeed, all network measures are sensitive to the number of nodes and edges and comparing networks of different size and connectivity is therefore tricky: [[1]]
MST-based methods
The Minimum Spanning Tree (MST) is a simply connected acyclic graph that connects all nodes with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n − 1} edges such that the sum of the `distance' of included edges is minimum. Clearly, in the case of brain networks one would like to include edges so as to maximise the total correlation strength, the distance is therefore defined as:
MST + Global thresholding
The MST ensures that the network is fully connected and contains mostly (but not only) links with high correlation strength that would have survived global thresholding anyway. Using the MST as a backbone, we can now include further edges in the network by adding links in order of decreasing correlation strength up to the desired cost .
