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[DBSCAN](https://en.wikipedia.org/wiki/DBSCAN) is a clustering algorithm that can be seen as a way of rejecting points that are positioned in low dense regions of a point cloud. This introduces holes and may result in a larger segment, that would otherwise be connected via a non dense link to become disconnected and form two segments, or clusters, instead. The rejection criterion is simple. The central concern is to evaluate a distance matrix <img src="https://render.githubusercontent.com/render/math?math=A_{ij}"> with an applied cutoff <img src="https://render.githubusercontent.com/render/math?math=\epsilon"> this turns the distances into true or false values depending on if a pair distance between point i and j is within the distance cutoff. This new binary Neighbour matrix <img src="https://render.githubusercontent.com/render/math?math=N_{ij}=A_{ij}<\epsilon"> tells you wether or not two points are neighbours (including itself). The DBSCAN criterion states that a point is not part of any cluster if it has fewer than `minPts` neighbors. Once you've calculated the distance matrix you can immediately evaluate the number of neighbors each point has and the rejection criterion, via <img src="https://render.githubusercontent.com/render/math?math=R_i=(\sum_{j} A_{ij}<\epsilon)-1 < minPts">. If the rejection vector R value of a point is True then all the pairwise distances in the distance matrix of that point is set to value larger than epsilon. This ensures that a distance matrix search will reject these as Neighbours of any other for the choosen epsilon. By tracing out all points that are neighbors and assessing the [connectivity](https://github.com/richardtjornhammar/impetuous/blob/master/src/impetuous/clustering.py) (search for connectivity) you can find all the clusters.
In this [example](https://gist.githubusercontent.com/richardtjornhammar/e84056e0b10f8d550258a1e8944ee375/raw/45fb8322487ff3a384e7f56eb06ac1073aee4da1/example9.py) we do exactly this for two gaussian point clouds. The dbscan search is just a single line `dbscan ( data_frame = point_cloud_df , eps=0.45 , minPts=4 )`, while the last lines are there to plot the [results](https://bl.ocks.org/richardtjornhammar/raw/0cc0ff037e88c76a9d65387155674fd1/?raw=true) with [graph revision dates](https://gist.github.com/richardtjornhammar/0cc0ff037e88c76a9d65387155674fd1/revisions)
In this [example](https://gist.githubusercontent.com/richardtjornhammar/e84056e0b10f8d550258a1e8944ee375/raw/e44e7226b6cb8ca486ff539ccfa775be981a549c/example9.py) we do exactly this for two gaussian point clouds. The dbscan search is just a single line `dbscan ( data_frame = point_cloud_df , eps=0.45 , minPts=4 )`, while the last lines are there to plot the [results](https://bl.ocks.org/richardtjornhammar/raw/0cc0ff037e88c76a9d65387155674fd1/?raw=true) ( has [graph revision dates](https://gist.github.com/richardtjornhammar/0cc0ff037e88c76a9d65387155674fd1/revisions) )
# Notes
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