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% DBSCAN DBSCAN clustering algorithm |
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% |
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% Usage: [C, ptsC, centres] = dbscan(P, E, minPts) |
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% |
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% Arguments: |
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% P - dim x Npts array of points. |
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% E - Distance threshold. |
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% minPts - Minimum number of points required to form a cluster. |
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% |
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% Returns: |
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% C - Cell array of length Nc listing indices of points associated with |
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% each cluster. |
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% ptsC - Array of length Npts listing the cluster number associated with |
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% each point. If a point is denoted as noise (not enough nearby |
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% elements to form a cluster) its cluster number is 0. |
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% centres - dim x Nc array of the average centre of each cluster. |
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% Reference: |
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% Martin Ester, Hans-Peter Kriegel, Jörg Sander, Xiaowei Xu (1996). "A |
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% density-based algorithm for discovering clusters in large spatial databases |
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% with noise". Proceedings of the Second International Conference on Knowledge |
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% Discovery and Data Mining (KDD-96). AAAI Press. pp. 226-231. |
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% Also see: http://en.wikipedia.org/wiki/DBSCAN |
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% Copyright (c) 2013 Peter Kovesi |
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% Centre for Exploration Targeting |
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% The University of Western Australia |
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% peter.kovesi at uwa edu au |
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% |
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% Permission is hereby granted, free of charge, to any person obtaining a copy |
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% of this software and associated documentation files (the "Software"), to deal |
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% in the Software without restriction, subject to the following conditions: |
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% |
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% The above copyright notice and this permission notice shall be included in |
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% all copies or substantial portions of the Software. |
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% |
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% The Software is provided "as is", without warranty of any kind. |
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% PK January 2013 |
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function [C, ptsC, centres] = dbscan(P, E, minPts) |
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[dim, Npts] = size(P); |
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ptsC = zeros(Npts,1); |
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C = {}; |
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Nc = 0; % Cluster counter. |
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Pvisit = zeros(Npts,1); % Array to keep track of points that have been visited. |
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for n = 1:Npts |
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if ~Pvisit(n) % If this point not visited yet |
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Pvisit(n) = 1; % mark as visited |
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neighbourPts = regionQuery(P, n, E); % and find its neighbours |
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if length(neighbourPts) < minPts-1 % Not enough points to form a cluster |
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ptsC(n) = 0; % Mark point n as noise. |
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else % Form a cluster... |
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Nc = Nc + 1; % Increment number of clusters and process |
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% neighbourhood. |
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C{Nc} = [n]; % Initialise cluster Nc with point n |
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ptsC(n) = Nc; % and mark point n as being a member of cluster Nc. |
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ind = 1; % Initialise index into neighbourPts array. |
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% For each point P' in neighbourPts ... |
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while ind <= length(neighbourPts) |
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nb = neighbourPts(ind); |
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if ~Pvisit(nb) % If this neighbour has not been visited |
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Pvisit(nb) = 1; % mark it as visited. |
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% Find the neighbours of this neighbour and if it has |
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% enough neighbours add them to the neighbourPts list |
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neighbourPtsP = regionQuery(P, nb, E); |
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if length(neighbourPtsP) >= minPts |
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neighbourPts = [neighbourPts neighbourPtsP]; |
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end |
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end |
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% If this neighbour nb not yet a member of any cluster add it |
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% to this cluster. |
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if ~ptsC(nb) |
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C{Nc} = [C{Nc} nb]; |
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ptsC(nb) = Nc; |
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end |
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ind = ind + 1; % Increment neighbour point index and process |
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% next neighbour |
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end |
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end |
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end |
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end |
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% Find centres of each cluster |
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centres = zeros(dim,length(C)); |
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for n = 1:length(C) |
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for k = 1:length(C{n}) |
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centres(:,n) = centres(:,n) + P(:,C{n}(k)); |
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end |
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centres(:,n) = centres(:,n)/length(C{n}); |
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end |
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end % of dbscan |
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%------------------------------------------------------------------------ |
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% Find indices of all points within distance E of point with index n |
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% This function could make use of a precomputed distance table to avoid |
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% repeated distance calculations, however this would require N^2 storage. |
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% Not a big problem either way if the number of points being clustered is |
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% small. For large datasets this function will need to be optimised. |
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% Arguments: |
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% P - the dim x Npts array of data points |
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% n - Index of point of interest |
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% E - Distance threshold |
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function neighbours = regionQuery(P, n, E) |
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E2 = E^2; |
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[dim, Npts] = size(P); |
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neighbours = []; |
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for i = 1:Npts |
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if i ~= n |
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% Test if distance^2 < E^2 |
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v = P(:,i)-P(:,n); |
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dist2 = v'*v; |
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if dist2 < E2 |
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neighbours = [neighbours i]; |
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end |
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end |
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end |
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end % of regionQuery |
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