algorithms.clustering.hierarchical_clustering¶
Module: algorithms.clustering.hierarchical_clustering¶
Inheritance diagram for nipy.algorithms.clustering.hierarchical_clustering:
These routines perform some hierrachical agglomerative clustering of some input data. The following alternatives are proposed: - Distance based average-link - Similarity-based average-link - Distance based maximum-link - Ward’s algorithm under graph constraints - Ward’s algorithm without graph constraints
In this latest version, the results are returned in a ‘WeightedForest’ structure, which gives access to the clustering hierarchy, facilitates the plot of the result etc.
For back-compatibility, *_segment versions of the algorithms have been appended, with the old API (except the qmax parameter, which now represents the number of wanted clusters)
Author : Bertrand Thirion,Pamela Guevara, 2006-2009
Class¶
WeightedForest¶
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class
nipy.algorithms.clustering.hierarchical_clustering.WeightedForest(V, parents=None, height=None)[source]¶ Bases:
nipy.algorithms.graph.forest.ForestThis is a weighted Forest structure, i.e. a tree - each node has one parent and children (hierarchical structure) - some of the nodes can be viewed as leaves, other as roots - the edges within a tree are associated with a weight: +1 from child to parent -1 from parent to child - additionally, the nodes have a value, which is called ‘height’, especially useful from dendrograms
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__init__(V, parents=None, height=None)[source]¶ - Parameters
V: the number of edges of the graph
parents=None: array of shape (V)
the parents of the graph by default, the parents are set to range(V), i.e. each node is its own parent, and each node is a tree
height=None: array of shape(V)
the height of the nodes
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plot(ax=None)[source]¶ Plot the dendrogram associated with self the rank of the data in the dendogram is returned
- Parameters
ax: axis handle, optional
- Returns
ax, the axis handle
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list_of_subtrees()[source]¶ returns the list of all non-trivial subtrees in the graph Caveat: theis function assumes that the vertices are sorted in a way such that parent[i]>i forall i Only the leaves are listeed, not the subtrees themselves
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adjacency()¶ returns the adjacency matrix of the graph as a sparse coo matrix
- Returns
adj: scipy.sparse matrix instance,
that encodes the adjacency matrix of self
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all_distances(seed=None)¶ returns all the distances of the graph as a tree
- Parameters
seed=None array of shape(nbseed) with valuesin [0..self.V-1]
set of vertices from which tehe distances are computed
- Returns
dg: array of shape(nseed, self.V), the resulting distances
Notes
By convention infinite distances are given the distance np.inf
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anti_symmeterize()¶ anti-symmeterize self, i.e. produces the graph whose adjacency matrix would be the antisymmetric part of its current adjacency matrix
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cc()¶ Compte the different connected components of the graph.
- Returns
label: array of shape(self.V), labelling of the vertices
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check()¶ Check that self is indeed a forest, i.e. contains no loop
- Returns
a boolean b=0 iff there are loops, 1 otherwise
Notes
Slow implementation, might be rewritten in C or cython
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cliques()¶ Extraction of the graphe cliques these are defined using replicator dynamics equations
- Returns
cliques: array of shape (self.V), type (np.int)
labelling of the vertices according to the clique they belong to
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compact_neighb()¶ returns a compact representation of self
- Returns
idx: array of of shape(self.V + 1):
the positions where to find the neighors of each node within neighb and weights
neighb: array of shape(self.E), concatenated list of neighbors
weights: array of shape(self.E), concatenated list of weights
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compute_children()¶ Define the children of each node (stored in self.children)
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copy()¶ returns a copy of self
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cut_redundancies()¶ Returns a graph with redundant edges removed: ecah edge (ab) is present ony once in the edge matrix: the correspondng weights are added.
- Returns
the resulting WeightedGraph
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define_graph_attributes()¶ define the edge and weights array
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degrees()¶ Returns the degree of the graph vertices.
- Returns
rdegree: (array, type=int, shape=(self.V,)), the right degrees
ldegree: (array, type=int, shape=(self.V,)), the left degrees
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depth_from_leaves()¶ compute an index for each node: 0 for the leaves, 1 for their parents etc. and maximal for the roots.
- Returns
depth: array of shape (self.V): the depth values of the vertices
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dijkstra(seed=0)¶ Returns all the [graph] geodesic distances starting from seed x
- seed (int, >-1, <self.V) or array of shape(p)
edge(s) from which the distances are computed
- Returns
dg: array of shape (self.V),
the graph distance dg from ant vertex to the nearest seed
Notes
It is mandatory that the graph weights are non-negative
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floyd(seed=None)¶ Compute all the geodesic distances starting from seeds
- Parameters
seed= None: array of shape (nbseed), type np.int
vertex indexes from which the distances are computed if seed==None, then every edge is a seed point
- Returns
dg array of shape (nbseed, self.V)
the graph distance dg from each seed to any vertex
Notes
It is mandatory that the graph weights are non-negative. The algorithm proceeds by repeating Dijkstra’s algo for each seed. Floyd’s algo is not used (O(self.V)^3 complexity…)
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from_3d_grid(xyz, k=18)¶ Sets the graph to be the topological neighbours graph of the three-dimensional coordinates set xyz, in the k-connectivity scheme
- Parameters
xyz: array of shape (self.V, 3) and type np.int,
k = 18: the number of neighbours considered. (6, 18 or 26)
- Returns
E(int): the number of edges of self
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get_E()¶ To get the number of edges in the graph
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get_V()¶ To get the number of vertices in the graph
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get_children(v=-1)¶ Get the children of a node/each node
- Parameters
v: int, optional
a node index
- Returns
children: list of int the list of children of node v (if v is provided)
a list of lists of int, the children of all nodes otherwise
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get_descendants(v, exclude_self=False)¶ returns the nodes that are children of v as a list
- Parameters
v: int, a node index
- Returns
desc: list of int, the list of all descendant of the input node
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get_edges()¶ To get the graph’s edges
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get_vertices()¶ To get the graph’s vertices (as id)
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get_weights()¶
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is_connected()¶ States whether self is connected or not
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isleaf()¶ Identification of the leaves of the forest
- Returns
leaves: bool array of shape(self.V), indicator of the forest’s leaves
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isroot()¶ Returns an indicator of nodes being roots
- Returns
roots, array of shape(self.V, bool), indicator of the forest’s roots
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kruskal()¶ Creates the Minimum Spanning Tree of self using Kruskal’s algo. efficient is self is sparse
- Returns
K, WeightedGraph instance: the resulting MST
Notes
If self contains several connected components, will have the same number k of connected components
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leaves_of_a_subtree(ids, custom=False)¶ tests whether the given nodes are the leaves of a certain subtree
- Parameters
ids: array of shape (n) that takes values in [0..self.V-1]
custom == False, boolean
if custom==true the behavior of the function is more specific - the different connected components are considered as being in a same greater tree - when a node has more than two subbranches, any subset of these children is considered as a subtree
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left_incidence()¶ Return left incidence matrix
- Returns
left_incid: list
the left incidence matrix of self as a list of lists: i.e. the list[[e.0.0, .., e.0.i(0)], .., [e.V.0, E.V.i(V)]] where e.i.j is the set of edge indexes so that e.i.j[0] = i
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list_of_neighbors()¶ returns the set of neighbors of self as a list of arrays
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main_cc()¶ Returns the indexes of the vertices within the main cc
- Returns
idx: array of shape (sizeof main cc)
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merge_simple_branches()¶ Return a subforest, where chained branches are collapsed
- Returns
sf, Forest instance, same as self, without any chain
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normalize(c=0)¶ Normalize the graph according to the index c Normalization means that the sum of the edges values that go into or out each vertex must sum to 1
- Parameters
c=0 in {0, 1, 2}, optional: index that designates the way
according to which D is normalized c == 0 => for each vertex a, sum{edge[e, 0]=a} D[e]=1 c == 1 => for each vertex b, sum{edge[e, 1]=b} D[e]=1 c == 2 => symmetric (‘l2’) normalization
Notes
Note that when sum_{edge[e, .] == a } D[e] = 0, nothing is performed
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propagate_upward(label)¶ Propagation of a certain labelling from leves to roots Assuming that label is a certain positive integer field this propagates these labels to the parents whenever the children nodes have coherent properties otherwise the parent value is unchanged
- Parameters
label: array of shape(self.V)
- Returns
label: array of shape(self.V)
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propagate_upward_and(prop)¶ propagates from leaves to roots some binary property of the nodes so that prop[parents] = logical_and(prop[children])
- Parameters
prop, array of shape(self.V), the input property
- Returns
prop, array of shape(self.V), the output property field
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remove_edges(valid)¶ Removes all the edges for which valid==0
- Parameters
valid : (self.E,) array
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remove_trivial_edges()¶ Removes trivial edges, i.e. edges that are (vv)-like self.weights and self.E are corrected accordingly
- Returns
self.E (int): The number of edges
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reorder_from_leaves_to_roots()¶ reorder the tree so that the leaves come first then their parents and so on, and the roots are last.
- Returns
order: array of shape(self.V)
the order of the old vertices in the reordered graph
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right_incidence()¶ Return right incidence matrix
- Returns
right_incid: list
the right incidence matrix of self as a list of lists: i.e. the list[[e.0.0, .., e.0.i(0)], .., [e.V.0, E.V.i(V)]] where e.i.j is the set of edge indexes so that e.i.j[1] = i
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set_edges(edges)¶ Sets the graph’s edges
Preconditions:
edges has a correct size
edges take values in [1..V]
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set_euclidian(X)¶ Compute the weights of the graph as the distances between the corresponding rows of X, which represents an embdedding of self
- Parameters
X array of shape (self.V, edim),
the coordinate matrix of the embedding
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set_gaussian(X, sigma=0)¶ Compute the weights of the graph as a gaussian function of the distance between the corresponding rows of X, which represents an embdedding of self
- Parameters
X array of shape (self.V, dim)
the coordinate matrix of the embedding
sigma=0, float: the parameter of the gaussian function
Notes
When sigma == 0, the following value is used:
sigma = sqrt(mean(||X[self.edges[:, 0], :]-X[self.edges[:, 1], :]||^2))
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set_weights(weights)¶ Set edge weights
- Parameters
weights: array
array shape(self.V): edges weights
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show(X=None, ax=None)¶ Plots the current graph in 2D
- Parameters
X : None or array of shape (self.V, 2)
a set of coordinates that can be used to embed the vertices in 2D. If X.shape[1]>2, a svd reduces X for display. By default, the graph is presented on a circle
ax: None or int, optional
ax handle
- Returns
ax: axis handle
Notes
This should be used only for small graphs.
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subforest(valid)¶ Creates a subforest with the vertices for which valid > 0
- Parameters
valid: array of shape (self.V): idicator of the selected nodes
- Returns
subforest: a new forest instance, with a reduced set of nodes
Notes
The children of deleted vertices become their own parent
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subgraph(valid)¶ Creates a subgraph with the vertices for which valid>0 and with the correponding set of edges
- Parameters
valid, array of shape (self.V): nonzero for vertices to be retained
- Returns
G, WeightedGraph instance, the desired subgraph of self
Notes
The vertices are renumbered as [1..p] where p = sum(valid>0) when sum(valid==0) then None is returned
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symmeterize()¶ Symmeterize self, modify edges and weights so that self.adjacency becomes the symmetric part of the current self.adjacency.
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to_coo_matrix()¶ Return adjacency matrix as coo sparse
- Returns
sp: scipy.sparse matrix instance
that encodes the adjacency matrix of self
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tree_depth()¶ Returns the number of hierarchical levels in the tree
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voronoi_diagram(seeds, samples)¶ Defines the graph as the Voronoi diagram (VD) that links the seeds. The VD is defined using the sample points.
- Parameters
seeds: array of shape (self.V, dim)
samples: array of shape (nsamples, dim)
Notes
By default, the weights are a Gaussian function of the distance The implementation is not optimal
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voronoi_labelling(seed)¶ Performs a voronoi labelling of the graph
- Parameters
seed: array of shape (nseeds), type (np.int),
vertices from which the cells are built
- Returns
labels: array of shape (self.V) the labelling of the vertices
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Functions¶
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nipy.algorithms.clustering.hierarchical_clustering.average_link_graph(G)[source]¶ Agglomerative function based on a (hopefully sparse) similarity graph
- Parameters
G the input graph
- Returns
t a weightForest structure that represents the dendrogram of the data
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nipy.algorithms.clustering.hierarchical_clustering.average_link_graph_segment(G, stop=0, qmax=1, verbose=False)[source]¶ Agglomerative function based on a (hopefully sparse) similarity graph
- Parameters
G the input graph
stop: float
the stopping criterion
qmax: int, optional
the number of desired clusters (in the limit of the stopping criterion)
verbose : bool, optional
If True, print diagnostic information
- Returns
u: array of shape (G.V)
a labelling of the graph vertices according to the criterion
cost: array of shape (G.V (?))
the cost of each merge step during the clustering procedure
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nipy.algorithms.clustering.hierarchical_clustering.fusion(K, pop, i, j, k)[source]¶ Modifies the graph K to merge nodes i and j into nodes k
The similarity values are weighted averaged, where pop[i] and pop[j] yield the relative weights. this is used in average_link_slow (deprecated)
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nipy.algorithms.clustering.hierarchical_clustering.ward(G, feature, verbose=False)[source]¶ Agglomerative function based on a topology-defining graph and a feature matrix.
- Parameters
G : graph
the input graph (a topological graph essentially)
feature : array of shape (G.V,dim_feature)
vectorial information related to the graph vertices
verbose : bool, optional
If True, print diagnostic information
- Returns
t :
WeightedForestinstancestructure that represents the dendrogram
Notes
When G has more than 1 connected component, t is no longer a tree. This case is handled cleanly now
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nipy.algorithms.clustering.hierarchical_clustering.ward_field_segment(F, stop=-1, qmax=-1, verbose=False)[source]¶ Agglomerative function based on a field structure
- Parameters
F the input field (graph+feature)
stop: float, optional
the stopping crterion. if stop==-1, then no stopping criterion is used
qmax: int, optional
the maximum number of desired clusters (in the limit of the stopping criterion)
verbose : bool, optional
If True, print diagnostic information
- Returns
u: array of shape (F.V)
labelling of the graph vertices according to the criterion
cost array of shape (F.V - 1)
the cost of each merge step during the clustering procedure
Notes
See ward_quick_segment for more information
Caveat : only approximate
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nipy.algorithms.clustering.hierarchical_clustering.ward_quick(G, feature, verbose=False)[source]¶ Agglomerative function based on a topology-defining graph and a feature matrix.
- Parameters
G : graph instance
topology-defining graph
feature: array of shape (G.V,dim_feature)
some vectorial information related to the graph vertices
verbose : bool, optional
If True, print diagnostic information
- Returns
t: weightForest instance,
that represents the dendrogram of the data
Notes
Hopefully a quicker version
A euclidean distance is used in the feature space
Caveat : only approximate
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nipy.algorithms.clustering.hierarchical_clustering.ward_quick_segment(G, feature, stop=-1, qmax=1, verbose=False)[source]¶ Agglomerative function based on a topology-defining graph and a feature matrix.
- Parameters
G: labs.graph.WeightedGraph instance
the input graph (a topological graph essentially)
feature array of shape (G.V,dim_feature)
vectorial information related to the graph vertices
stop1 : int or float, optional
the stopping crterion if stop==-1, then no stopping criterion is used
qmax : int, optional
the maximum number of desired clusters (in the limit of the stopping criterion)
verbose : bool, optional
If True, print diagnostic information
- Returns
u: array of shape (G.V)
labelling of the graph vertices according to the criterion
cost: array of shape (G.V - 1)
the cost of each merge step during the clustering procedure
Notes
Hopefully a quicker version
A euclidean distance is used in the feature space
Caveat : only approximate
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nipy.algorithms.clustering.hierarchical_clustering.ward_segment(G, feature, stop=-1, qmax=1, verbose=False)[source]¶ Agglomerative function based on a topology-defining graph and a feature matrix.
- Parameters
G : graph object
the input graph (a topological graph essentially)
feature : array of shape (G.V,dim_feature)
some vectorial information related to the graph vertices
stop : int or float, optional
the stopping crterion. if stop==-1, then no stopping criterion is used
qmax : int, optional
the maximum number of desired clusters (in the limit of the stopping criterion)
verbose : bool, optional
If True, print diagnostic information
- Returns
u: array of shape (G.V):
a labelling of the graph vertices according to the criterion
cost: array of shape (G.V - 1)
the cost of each merge step during the clustering procedure
Notes
A euclidean distance is used in the feature space
Caveat : when the number of cc in G (nbcc) is greter than qmax, u contains nbcc values, not qmax !