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 """ Hierarchical clustering (:mod:scipy.cluster.hierarchy) ======================================================== .. currentmodule:: scipy.cluster.hierarchy These functions cut hierarchical clusterings into flat clusterings or find the roots of the forest formed by a cut by providing the flat cluster ids of each observation. .. autosummary:: :toctree: generated/ fcluster fclusterdata leaders These are routines for agglomerative clustering. .. autosummary:: :toctree: generated/ linkage single complete average weighted centroid median ward These routines compute statistics on hierarchies. .. autosummary:: :toctree: generated/ cophenet from_mlab_linkage inconsistent maxinconsts maxdists maxRstat to_mlab_linkage Routines for visualizing flat clusters. .. autosummary:: :toctree: generated/ dendrogram These are data structures and routines for representing hierarchies as tree objects. .. autosummary:: :toctree: generated/ ClusterNode leaves_list to_tree cut_tree optimal_leaf_ordering These are predicates for checking the validity of linkage and inconsistency matrices as well as for checking isomorphism of two flat cluster assignments. .. autosummary:: :toctree: generated/ is_valid_im is_valid_linkage is_isomorphic is_monotonic correspond num_obs_linkage Utility routines for plotting: .. autosummary:: :toctree: generated/ set_link_color_palette """ from __future__ import division, print_function, absolute_import # Copyright (C) Damian Eads, 2007-2008. New BSD License. # hierarchy.py (derived from cluster.py, http://scipy-cluster.googlecode.com) # # Author: Damian Eads # Date: September 22, 2007 # # Copyright (c) 2007, 2008, Damian Eads # # All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions # are met: # - Redistributions of source code must retain the above # copyright notice, this list of conditions and the # following disclaimer. # - Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer # in the documentation and/or other materials provided with the # distribution. # - Neither the name of the author nor the names of its # contributors may be used to endorse or promote products derived # from this software without specific prior written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS # "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT # LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR # A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT # OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, # SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT # LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, # DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY # THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT # (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE # OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. import warnings import bisect from collections import deque import numpy as np from . import _hierarchy, _optimal_leaf_ordering import scipy.spatial.distance as distance from scipy._lib.six import string_types from scipy._lib.six import xrange _LINKAGE_METHODS = {'single': 0, 'complete': 1, 'average': 2, 'centroid': 3, 'median': 4, 'ward': 5, 'weighted': 6} _EUCLIDEAN_METHODS = ('centroid', 'median', 'ward') __all__ = ['ClusterNode', 'average', 'centroid', 'complete', 'cophenet', 'correspond', 'cut_tree', 'dendrogram', 'fcluster', 'fclusterdata', 'from_mlab_linkage', 'inconsistent', 'is_isomorphic', 'is_monotonic', 'is_valid_im', 'is_valid_linkage', 'leaders', 'leaves_list', 'linkage', 'maxRstat', 'maxdists', 'maxinconsts', 'median', 'num_obs_linkage', 'optimal_leaf_ordering', 'set_link_color_palette', 'single', 'to_mlab_linkage', 'to_tree', 'ward', 'weighted', 'distance'] class ClusterWarning(UserWarning): pass def _warning(s): warnings.warn('scipy.cluster: %s' % s, ClusterWarning, stacklevel=3) def _copy_array_if_base_present(a): """ Copy the array if its base points to a parent array. """ if a.base is not None: return a.copy() elif np.issubsctype(a, np.float32): return np.array(a, dtype=np.double) else: return a def _copy_arrays_if_base_present(T): """ Accept a tuple of arrays T. Copies the array T[i] if its base array points to an actual array. Otherwise, the reference is just copied. This is useful if the arrays are being passed to a C function that does not do proper striding. """ l = [_copy_array_if_base_present(a) for a in T] return l def _randdm(pnts): """ Generate a random distance matrix stored in condensed form. Parameters ---------- pnts : int The number of points in the distance matrix. Has to be at least 2. Returns ------- D : ndarray A pnts * (pnts - 1) / 2 sized vector is returned. """ if pnts >= 2: D = np.random.rand(pnts * (pnts - 1) / 2) else: raise ValueError("The number of points in the distance matrix " "must be at least 2.") return D def single(y): """ Perform single/min/nearest linkage on the condensed distance matrix y. Parameters ---------- y : ndarray The upper triangular of the distance matrix. The result of pdist is returned in this form. Returns ------- Z : ndarray The linkage matrix. See Also -------- linkage: for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import single, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = single(y) >>> Z array([[ 0., 1., 1., 2.], [ 2., 12., 1., 3.], [ 3., 4., 1., 2.], [ 5., 14., 1., 3.], [ 6., 7., 1., 2.], [ 8., 16., 1., 3.], [ 9., 10., 1., 2.], [11., 18., 1., 3.], [13., 15., 2., 6.], [17., 20., 2., 9.], [19., 21., 2., 12.]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3], dtype=int32) >>> fcluster(Z, 1, criterion='distance') array([3, 3, 3, 4, 4, 4, 2, 2, 2, 1, 1, 1], dtype=int32) >>> fcluster(Z, 2, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='single', metric='euclidean') def complete(y): """ Perform complete/max/farthest point linkage on a condensed distance matrix. Parameters ---------- y : ndarray The upper triangular of the distance matrix. The result of pdist is returned in this form. Returns ------- Z : ndarray A linkage matrix containing the hierarchical clustering. See the linkage function documentation for more information on its structure. See Also -------- linkage: for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import complete, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = complete(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.41421356, 3. ], [ 5. , 13. , 1.41421356, 3. ], [ 8. , 14. , 1.41421356, 3. ], [11. , 15. , 1.41421356, 3. ], [16. , 17. , 4.12310563, 6. ], [18. , 19. , 4.12310563, 6. ], [20. , 21. , 5.65685425, 12. ]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], dtype=int32) >>> fcluster(Z, 1.5, criterion='distance') array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) >>> fcluster(Z, 4.5, criterion='distance') array([1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2], dtype=int32) >>> fcluster(Z, 6, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='complete', metric='euclidean') def average(y): """ Perform average/UPGMA linkage on a condensed distance matrix. Parameters ---------- y : ndarray The upper triangular of the distance matrix. The result of pdist is returned in this form. Returns ------- Z : ndarray A linkage matrix containing the hierarchical clustering. See linkage for more information on its structure. See Also -------- linkage: for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import average, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = average(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.20710678, 3. ], [ 5. , 13. , 1.20710678, 3. ], [ 8. , 14. , 1.20710678, 3. ], [11. , 15. , 1.20710678, 3. ], [16. , 17. , 3.39675184, 6. ], [18. , 19. , 3.39675184, 6. ], [20. , 21. , 4.09206523, 12. ]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], dtype=int32) >>> fcluster(Z, 1.5, criterion='distance') array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) >>> fcluster(Z, 4, criterion='distance') array([1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2], dtype=int32) >>> fcluster(Z, 6, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='average', metric='euclidean') def weighted(y): """ Perform weighted/WPGMA linkage on the condensed distance matrix. See linkage for more information on the return structure and algorithm. Parameters ---------- y : ndarray The upper triangular of the distance matrix. The result of pdist is returned in this form. Returns ------- Z : ndarray A linkage matrix containing the hierarchical clustering. See linkage for more information on its structure. See Also -------- linkage : for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import weighted, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = weighted(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 9. , 11. , 1. , 2. ], [ 2. , 12. , 1.20710678, 3. ], [ 8. , 13. , 1.20710678, 3. ], [ 5. , 14. , 1.20710678, 3. ], [10. , 15. , 1.20710678, 3. ], [18. , 19. , 3.05595762, 6. ], [16. , 17. , 3.32379407, 6. ], [20. , 21. , 4.06357713, 12. ]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 7, 8, 9, 1, 2, 3, 10, 11, 12, 4, 6, 5], dtype=int32) >>> fcluster(Z, 1.5, criterion='distance') array([3, 3, 3, 1, 1, 1, 4, 4, 4, 2, 2, 2], dtype=int32) >>> fcluster(Z, 4, criterion='distance') array([2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1], dtype=int32) >>> fcluster(Z, 6, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='weighted', metric='euclidean') def centroid(y): """ Perform centroid/UPGMC linkage. See linkage for more information on the input matrix, return structure, and algorithm. The following are common calling conventions: 1. Z = centroid(y) Performs centroid/UPGMC linkage on the condensed distance matrix y. 2. Z = centroid(X) Performs centroid/UPGMC linkage on the observation matrix X using Euclidean distance as the distance metric. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. Returns ------- Z : ndarray A linkage matrix containing the hierarchical clustering. See the linkage function documentation for more information on its structure. See Also -------- linkage: for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import centroid, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = centroid(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 2. , 12. , 1.11803399, 3. ], [ 5. , 13. , 1.11803399, 3. ], [ 8. , 15. , 1.11803399, 3. ], [11. , 14. , 1.11803399, 3. ], [18. , 19. , 3.33333333, 6. ], [16. , 17. , 3.33333333, 6. ], [20. , 21. , 3.33333333, 12. ]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6], dtype=int32) >>> fcluster(Z, 1.1, criterion='distance') array([5, 5, 6, 7, 7, 8, 1, 1, 2, 3, 3, 4], dtype=int32) >>> fcluster(Z, 2, criterion='distance') array([3, 3, 3, 4, 4, 4, 1, 1, 1, 2, 2, 2], dtype=int32) >>> fcluster(Z, 4, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='centroid', metric='euclidean') def median(y): """ Perform median/WPGMC linkage. See linkage for more information on the return structure and algorithm. The following are common calling conventions: 1. Z = median(y) Performs median/WPGMC linkage on the condensed distance matrix y. See linkage for more information on the return structure and algorithm. 2. Z = median(X) Performs median/WPGMC linkage on the observation matrix X using Euclidean distance as the distance metric. See linkage for more information on the return structure and algorithm. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. Returns ------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. See Also -------- linkage: for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import median, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = median(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 2. , 12. , 1.11803399, 3. ], [ 5. , 13. , 1.11803399, 3. ], [ 8. , 15. , 1.11803399, 3. ], [11. , 14. , 1.11803399, 3. ], [18. , 19. , 3. , 6. ], [16. , 17. , 3.5 , 6. ], [20. , 21. , 3.25 , 12. ]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6], dtype=int32) >>> fcluster(Z, 1.1, criterion='distance') array([5, 5, 6, 7, 7, 8, 1, 1, 2, 3, 3, 4], dtype=int32) >>> fcluster(Z, 2, criterion='distance') array([3, 3, 3, 4, 4, 4, 1, 1, 1, 2, 2, 2], dtype=int32) >>> fcluster(Z, 4, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='median', metric='euclidean') def ward(y): """ Perform Ward's linkage on a condensed distance matrix. See linkage for more information on the return structure and algorithm. The following are common calling conventions: 1. Z = ward(y) Performs Ward's linkage on the condensed distance matrix y. 2. Z = ward(X) Performs Ward's linkage on the observation matrix X using Euclidean distance as the distance metric. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. Returns ------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. See linkage for more information on the return structure and algorithm. See Also -------- linkage: for advanced creation of hierarchical clusterings. scipy.spatial.distance.pdist : pairwise distance metrics Examples -------- >>> from scipy.cluster.hierarchy import ward, fcluster >>> from scipy.spatial.distance import pdist First we need a toy dataset to play with:: x x x x x x x x x x x x >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] Then we get a condensed distance matrix from this dataset: >>> y = pdist(X) Finally, we can perform the clustering: >>> Z = ward(y) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) The linkage matrix Z represents a dendrogram - see scipy.cluster.hierarchy.linkage for a detailed explanation of its contents. We can use scipy.cluster.hierarchy.fcluster to see to which cluster each initial point would belong given a distance threshold: >>> fcluster(Z, 0.9, criterion='distance') array([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], dtype=int32) >>> fcluster(Z, 1.1, criterion='distance') array([1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8], dtype=int32) >>> fcluster(Z, 3, criterion='distance') array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) >>> fcluster(Z, 9, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) Also scipy.cluster.hierarchy.dendrogram can be used to generate a plot of the dendrogram. """ return linkage(y, method='ward', metric='euclidean') def linkage(y, method='single', metric='euclidean', optimal_ordering=False): """ Perform hierarchical/agglomerative clustering. The input y may be either a 1d condensed distance matrix or a 2d array of observation vectors. If y is a 1d condensed distance matrix, then y must be a :math:\\binom{n}{2} sized vector where n is the number of original observations paired in the distance matrix. The behavior of this function is very similar to the MATLAB linkage function. A :math:(n-1) by 4 matrix Z is returned. At the :math:i-th iteration, clusters with indices Z[i, 0] and Z[i, 1] are combined to form cluster :math:n + i. A cluster with an index less than :math:n corresponds to one of the :math:n original observations. The distance between clusters Z[i, 0] and Z[i, 1] is given by Z[i, 2]. The fourth value Z[i, 3] represents the number of original observations in the newly formed cluster. The following linkage methods are used to compute the distance :math:d(s, t) between two clusters :math:s and :math:t. The algorithm begins with a forest of clusters that have yet to be used in the hierarchy being formed. When two clusters :math:s and :math:t from this forest are combined into a single cluster :math:u, :math:s and :math:t are removed from the forest, and :math:u is added to the forest. When only one cluster remains in the forest, the algorithm stops, and this cluster becomes the root. A distance matrix is maintained at each iteration. The d[i,j] entry corresponds to the distance between cluster :math:i and :math:j in the original forest. At each iteration, the algorithm must update the distance matrix to reflect the distance of the newly formed cluster u with the remaining clusters in the forest. Suppose there are :math:|u| original observations :math:u[0], \\ldots, u[|u|-1] in cluster :math:u and :math:|v| original objects :math:v[0], \\ldots, v[|v|-1] in cluster :math:v. Recall :math:s and :math:t are combined to form cluster :math:u. Let :math:v be any remaining cluster in the forest that is not :math:u. The following are methods for calculating the distance between the newly formed cluster :math:u and each :math:v. * method='single' assigns .. math:: d(u,v) = \\min(dist(u[i],v[j])) for all points :math:i in cluster :math:u and :math:j in cluster :math:v. This is also known as the Nearest Point Algorithm. * method='complete' assigns .. math:: d(u, v) = \\max(dist(u[i],v[j])) for all points :math:i in cluster u and :math:j in cluster :math:v. This is also known by the Farthest Point Algorithm or Voor Hees Algorithm. * method='average' assigns .. math:: d(u,v) = \\sum_{ij} \\frac{d(u[i], v[j])} {(|u|*|v|)} for all points :math:i and :math:j where :math:|u| and :math:|v| are the cardinalities of clusters :math:u and :math:v, respectively. This is also called the UPGMA algorithm. * method='weighted' assigns .. math:: d(u,v) = (dist(s,v) + dist(t,v))/2 where cluster u was formed with cluster s and t and v is a remaining cluster in the forest. (also called WPGMA) * method='centroid' assigns .. math:: dist(s,t) = ||c_s-c_t||_2 where :math:c_s and :math:c_t are the centroids of clusters :math:s and :math:t, respectively. When two clusters :math:s and :math:t are combined into a new cluster :math:u, the new centroid is computed over all the original objects in clusters :math:s and :math:t. The distance then becomes the Euclidean distance between the centroid of :math:u and the centroid of a remaining cluster :math:v in the forest. This is also known as the UPGMC algorithm. * method='median' assigns :math:d(s,t) like the centroid method. When two clusters :math:s and :math:t are combined into a new cluster :math:u, the average of centroids s and t give the new centroid :math:u. This is also known as the WPGMC algorithm. * method='ward' uses the Ward variance minimization algorithm. The new entry :math:d(u,v) is computed as follows, .. math:: d(u,v) = \\sqrt{\\frac{|v|+|s|} {T}d(v,s)^2 + \\frac{|v|+|t|} {T}d(v,t)^2 - \\frac{|v|} {T}d(s,t)^2} where :math:u is the newly joined cluster consisting of clusters :math:s and :math:t, :math:v is an unused cluster in the forest, :math:T=|v|+|s|+|t|, and :math:|*| is the cardinality of its argument. This is also known as the incremental algorithm. Warning: When the minimum distance pair in the forest is chosen, there may be two or more pairs with the same minimum distance. This implementation may choose a different minimum than the MATLAB version. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of :math:m observation vectors in :math:n dimensions may be passed as an :math:m by :math:n array. All elements of the condensed distance matrix must be finite, i.e. no NaNs or infs. method : str, optional The linkage algorithm to use. See the Linkage Methods section below for full descriptions. metric : str or function, optional The distance metric to use in the case that y is a collection of observation vectors; ignored otherwise. See the pdist function for a list of valid distance metrics. A custom distance function can also be used. optimal_ordering : bool, optional If True, the linkage matrix will be reordered so that the distance between successive leaves is minimal. This results in a more intuitive tree structure when the data are visualized. defaults to False, because this algorithm can be slow, particularly on large datasets [2]_. See also the optimal_leaf_ordering function. .. versionadded:: 1.0.0 Returns ------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. Notes ----- 1. For method 'single' an optimized algorithm based on minimum spanning tree is implemented. It has time complexity :math:O(n^2). For methods 'complete', 'average', 'weighted' and 'ward' an algorithm called nearest-neighbors chain is implemented. It also has time complexity :math:O(n^2). For other methods a naive algorithm is implemented with :math:O(n^3) time complexity. All algorithms use :math:O(n^2) memory. Refer to [1]_ for details about the algorithms. 2. Methods 'centroid', 'median' and 'ward' are correctly defined only if Euclidean pairwise metric is used. If y is passed as precomputed pairwise distances, then it is a user responsibility to assure that these distances are in fact Euclidean, otherwise the produced result will be incorrect. See Also -------- scipy.spatial.distance.pdist : pairwise distance metrics References ---------- .. [1] Daniel Mullner, "Modern hierarchical, agglomerative clustering algorithms", :arXiv:1109.2378v1. .. [2] Ziv Bar-Joseph, David K. Gifford, Tommi S. Jaakkola, "Fast optimal leaf ordering for hierarchical clustering", 2001. Bioinformatics :doi:10.1093/bioinformatics/17.suppl_1.S22 Examples -------- >>> from scipy.cluster.hierarchy import dendrogram, linkage >>> from matplotlib import pyplot as plt >>> X = [[i] for i in [2, 8, 0, 4, 1, 9, 9, 0]] >>> Z = linkage(X, 'ward') >>> fig = plt.figure(figsize=(25, 10)) >>> dn = dendrogram(Z) >>> Z = linkage(X, 'single') >>> fig = plt.figure(figsize=(25, 10)) >>> dn = dendrogram(Z) >>> plt.show() """ if method not in _LINKAGE_METHODS: raise ValueError("Invalid method: {0}".format(method)) y = _convert_to_double(np.asarray(y, order='c')) if y.ndim == 1: distance.is_valid_y(y, throw=True, name='y') [y] = _copy_arrays_if_base_present([y]) elif y.ndim == 2: if method in _EUCLIDEAN_METHODS and metric != 'euclidean': raise ValueError("Method '{0}' requires the distance metric " "to be Euclidean".format(method)) if y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0): if np.all(y >= 0) and np.allclose(y, y.T): _warning('The symmetric non-negative hollow observation ' 'matrix looks suspiciously like an uncondensed ' 'distance matrix') y = distance.pdist(y, metric) else: raise ValueError("y must be 1 or 2 dimensional.") if not np.all(np.isfinite(y)): raise ValueError("The condensed distance matrix must contain only " "finite values.") n = int(distance.num_obs_y(y)) method_code = _LINKAGE_METHODS[method] if method == 'single': result = _hierarchy.mst_single_linkage(y, n) elif method in ['complete', 'average', 'weighted', 'ward']: result = _hierarchy.nn_chain(y, n, method_code) else: result = _hierarchy.fast_linkage(y, n, method_code) if optimal_ordering: return optimal_leaf_ordering(result, y) else: return result class ClusterNode(object): """ A tree node class for representing a cluster. Leaf nodes correspond to original observations, while non-leaf nodes correspond to non-singleton clusters. The to_tree function converts a matrix returned by the linkage function into an easy-to-use tree representation. All parameter names are also attributes. Parameters ---------- id : int The node id. left : ClusterNode instance, optional The left child tree node. right : ClusterNode instance, optional The right child tree node. dist : float, optional Distance for this cluster in the linkage matrix. count : int, optional The number of samples in this cluster. See Also -------- to_tree : for converting a linkage matrix Z into a tree object. """ def __init__(self, id, left=None, right=None, dist=0, count=1): if id < 0: raise ValueError('The id must be non-negative.') if dist < 0: raise ValueError('The distance must be non-negative.') if (left is None and right is not None) or \ (left is not None and right is None): raise ValueError('Only full or proper binary trees are permitted.' ' This node has one child.') if count < 1: raise ValueError('A cluster must contain at least one original ' 'observation.') self.id = id self.left = left self.right = right self.dist = dist if self.left is None: self.count = count else: self.count = left.count + right.count def __lt__(self, node): if not isinstance(node, ClusterNode): raise ValueError("Can't compare ClusterNode " "to type {}".format(type(node))) return self.dist < node.dist def __gt__(self, node): if not isinstance(node, ClusterNode): raise ValueError("Can't compare ClusterNode " "to type {}".format(type(node))) return self.dist > node.dist def __eq__(self, node): if not isinstance(node, ClusterNode): raise ValueError("Can't compare ClusterNode " "to type {}".format(type(node))) return self.dist == node.dist def get_id(self): """ The identifier of the target node. For 0 <= i < n, i corresponds to original observation i. For n <= i < 2n-1, i corresponds to non-singleton cluster formed at iteration i-n. Returns ------- id : int The identifier of the target node. """ return self.id def get_count(self): """ The number of leaf nodes (original observations) belonging to the cluster node nd. If the target node is a leaf, 1 is returned. Returns ------- get_count : int The number of leaf nodes below the target node. """ return self.count def get_left(self): """ Return a reference to the left child tree object. Returns ------- left : ClusterNode The left child of the target node. If the node is a leaf, None is returned. """ return self.left def get_right(self): """ Return a reference to the right child tree object. Returns ------- right : ClusterNode The left child of the target node. If the node is a leaf, None is returned. """ return self.right def is_leaf(self): """ Return True if the target node is a leaf. Returns ------- leafness : bool True if the target node is a leaf node. """ return self.left is None def pre_order(self, func=(lambda x: x.id)): """ Perform pre-order traversal without recursive function calls. When a leaf node is first encountered, func is called with the leaf node as its argument, and its result is appended to the list. For example, the statement:: ids = root.pre_order(lambda x: x.id) returns a list of the node ids corresponding to the leaf nodes of the tree as they appear from left to right. Parameters ---------- func : function Applied to each leaf ClusterNode object in the pre-order traversal. Given the i-th leaf node in the pre-order traversal n[i], the result of func(n[i]) is stored in L[i]. If not provided, the index of the original observation to which the node corresponds is used. Returns ------- L : list The pre-order traversal. """ # Do a preorder traversal, caching the result. To avoid having to do # recursion, we'll store the previous index we've visited in a vector. n = self.count curNode = [None] * (2 * n) lvisited = set() rvisited = set() curNode[0] = self k = 0 preorder = [] while k >= 0: nd = curNode[k] ndid = nd.id if nd.is_leaf(): preorder.append(func(nd)) k = k - 1 else: if ndid not in lvisited: curNode[k + 1] = nd.left lvisited.add(ndid) k = k + 1 elif ndid not in rvisited: curNode[k + 1] = nd.right rvisited.add(ndid) k = k + 1 # If we've visited the left and right of this non-leaf # node already, go up in the tree. else: k = k - 1 return preorder _cnode_bare = ClusterNode(0) _cnode_type = type(ClusterNode) def _order_cluster_tree(Z): """ Return clustering nodes in bottom-up order by distance. Parameters ---------- Z : scipy.cluster.linkage array The linkage matrix. Returns ------- nodes : list A list of ClusterNode objects. """ q = deque() tree = to_tree(Z) q.append(tree) nodes = [] while q: node = q.popleft() if not node.is_leaf(): bisect.insort_left(nodes, node) q.append(node.get_right()) q.append(node.get_left()) return nodes def cut_tree(Z, n_clusters=None, height=None): """ Given a linkage matrix Z, return the cut tree. Parameters ---------- Z : scipy.cluster.linkage array The linkage matrix. n_clusters : array_like, optional Number of clusters in the tree at the cut point. height : array_like, optional The height at which to cut the tree. Only possible for ultrametric trees. Returns ------- cutree : array An array indicating group membership at each agglomeration step. I.e., for a full cut tree, in the first column each data point is in its own cluster. At the next step, two nodes are merged. Finally all singleton and non-singleton clusters are in one group. If n_clusters or height is given, the columns correspond to the columns of n_clusters or height. Examples -------- >>> from scipy import cluster >>> np.random.seed(23) >>> X = np.random.randn(50, 4) >>> Z = cluster.hierarchy.ward(X) >>> cutree = cluster.hierarchy.cut_tree(Z, n_clusters=[5, 10]) >>> cutree[:10] array([[0, 0], [1, 1], [2, 2], [3, 3], [3, 4], [2, 2], [0, 0], [1, 5], [3, 6], [4, 7]]) """ nobs = num_obs_linkage(Z) nodes = _order_cluster_tree(Z) if height is not None and n_clusters is not None: raise ValueError("At least one of either height or n_clusters " "must be None") elif height is None and n_clusters is None: # return the full cut tree cols_idx = np.arange(nobs) elif height is not None: heights = np.array([x.dist for x in nodes]) cols_idx = np.searchsorted(heights, height) else: cols_idx = nobs - np.searchsorted(np.arange(nobs), n_clusters) try: n_cols = len(cols_idx) except TypeError: # scalar n_cols = 1 cols_idx = np.array([cols_idx]) groups = np.zeros((n_cols, nobs), dtype=int) last_group = np.arange(nobs) if 0 in cols_idx: groups[0] = last_group for i, node in enumerate(nodes): idx = node.pre_order() this_group = last_group.copy() this_group[idx] = last_group[idx].min() this_group[this_group > last_group[idx].max()] -= 1 if i + 1 in cols_idx: groups[np.nonzero(i + 1 == cols_idx)[0]] = this_group last_group = this_group return groups.T def to_tree(Z, rd=False): """ Convert a linkage matrix into an easy-to-use tree object. The reference to the root ClusterNode object is returned (by default). Each ClusterNode object has a left, right, dist, id, and count attribute. The left and right attributes point to ClusterNode objects that were combined to generate the cluster. If both are None then the ClusterNode object is a leaf node, its count must be 1, and its distance is meaningless but set to 0. *Note: This function is provided for the convenience of the library user. ClusterNodes are not used as input to any of the functions in this library.* Parameters ---------- Z : ndarray The linkage matrix in proper form (see the linkage function documentation). rd : bool, optional When False (default), a reference to the root ClusterNode object is returned. Otherwise, a tuple (r, d) is returned. r is a reference to the root node while d is a list of ClusterNode objects - one per original entry in the linkage matrix plus entries for all clustering steps. If a cluster id is less than the number of samples n in the data that the linkage matrix describes, then it corresponds to a singleton cluster (leaf node). See linkage for more information on the assignment of cluster ids to clusters. Returns ------- tree : ClusterNode or tuple (ClusterNode, list of ClusterNode) If rd is False, a ClusterNode. If rd is True, a list of length 2*n - 1, with n the number of samples. See the description of rd above for more details. See Also -------- linkage, is_valid_linkage, ClusterNode Examples -------- >>> from scipy.cluster import hierarchy >>> x = np.random.rand(10).reshape(5, 2) >>> Z = hierarchy.linkage(x) >>> hierarchy.to_tree(Z) >> rootnode, nodelist = hierarchy.to_tree(Z, rd=True) >>> rootnode >> len(nodelist) 9 """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') # Number of original objects is equal to the number of rows minus 1. n = Z.shape[0] + 1 # Create a list full of None's to store the node objects d = [None] * (n * 2 - 1) # Create the nodes corresponding to the n original objects. for i in xrange(0, n): d[i] = ClusterNode(i) nd = None for i in xrange(0, n - 1): fi = int(Z[i, 0]) fj = int(Z[i, 1]) if fi > i + n: raise ValueError(('Corrupt matrix Z. Index to derivative cluster ' 'is used before it is formed. See row %d, ' 'column 0') % fi) if fj > i + n: raise ValueError(('Corrupt matrix Z. Index to derivative cluster ' 'is used before it is formed. See row %d, ' 'column 1') % fj) nd = ClusterNode(i + n, d[fi], d[fj], Z[i, 2]) # ^ id ^ left ^ right ^ dist if Z[i, 3] != nd.count: raise ValueError(('Corrupt matrix Z. The count Z[%d,3] is ' 'incorrect.') % i) d[n + i] = nd if rd: return (nd, d) else: return nd def optimal_leaf_ordering(Z, y, metric='euclidean'): """ Given a linkage matrix Z and distance, reorder the cut tree. Parameters ---------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. See linkage for more information on the return structure and algorithm. y : ndarray The condensed distance matrix from which Z was generated. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. metric : str or function, optional The distance metric to use in the case that y is a collection of observation vectors; ignored otherwise. See the pdist function for a list of valid distance metrics. A custom distance function can also be used. Returns ------- Z_ordered : ndarray A copy of the linkage matrix Z, reordered to minimize the distance between adjacent leaves. Examples -------- >>> from scipy.cluster import hierarchy >>> np.random.seed(23) >>> X = np.random.randn(10,10) >>> Z = hierarchy.ward(X) >>> hierarchy.leaves_list(Z) array([0, 5, 3, 9, 6, 8, 1, 4, 2, 7], dtype=int32) >>> hierarchy.leaves_list(hierarchy.optimal_leaf_ordering(Z, X)) array([3, 9, 0, 5, 8, 2, 7, 4, 1, 6], dtype=int32) """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') y = _convert_to_double(np.asarray(y, order='c')) if y.ndim == 1: distance.is_valid_y(y, throw=True, name='y') [y] = _copy_arrays_if_base_present([y]) elif y.ndim == 2: if y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0): if np.all(y >= 0) and np.allclose(y, y.T): _warning('The symmetric non-negative hollow observation ' 'matrix looks suspiciously like an uncondensed ' 'distance matrix') y = distance.pdist(y, metric) else: raise ValueError("y must be 1 or 2 dimensional.") if not np.all(np.isfinite(y)): raise ValueError("The condensed distance matrix must contain only " "finite values.") return _optimal_leaf_ordering.optimal_leaf_ordering(Z, y) def _convert_to_bool(X): if X.dtype != bool: X = X.astype(bool) if not X.flags.contiguous: X = X.copy() return X def _convert_to_double(X): if X.dtype != np.double: X = X.astype(np.double) if not X.flags.contiguous: X = X.copy() return X def cophenet(Z, Y=None): """ Calculate the cophenetic distances between each observation in the hierarchical clustering defined by the linkage Z. Suppose p and q are original observations in disjoint clusters s and t, respectively and s and t are joined by a direct parent cluster u. The cophenetic distance between observations i and j is simply the distance between clusters s and t. Parameters ---------- Z : ndarray The hierarchical clustering encoded as an array (see linkage function). Y : ndarray (optional) Calculates the cophenetic correlation coefficient c of a hierarchical clustering defined by the linkage matrix Z of a set of :math:n observations in :math:m dimensions. Y is the condensed distance matrix from which Z was generated. Returns ------- c : ndarray The cophentic correlation distance (if Y is passed). d : ndarray The cophenetic distance matrix in condensed form. The :math:ij th entry is the cophenetic distance between original observations :math:i and :math:j. See Also -------- linkage: for a description of what a linkage matrix is. scipy.spatial.distance.squareform: transforming condensed matrices into square ones. Examples -------- >>> from scipy.cluster.hierarchy import single, cophenet >>> from scipy.spatial.distance import pdist, squareform Given a dataset X and a linkage matrix Z, the cophenetic distance between two points of X is the distance between the largest two distinct clusters that each of the points: >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] X corresponds to this dataset :: x x x x x x x x x x x x >>> Z = single(pdist(X)) >>> Z array([[ 0., 1., 1., 2.], [ 2., 12., 1., 3.], [ 3., 4., 1., 2.], [ 5., 14., 1., 3.], [ 6., 7., 1., 2.], [ 8., 16., 1., 3.], [ 9., 10., 1., 2.], [11., 18., 1., 3.], [13., 15., 2., 6.], [17., 20., 2., 9.], [19., 21., 2., 12.]]) >>> cophenet(Z) array([1., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 1., 2., 2., 2., 2., 2., 2., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 1., 2., 2., 2., 1., 2., 2., 2., 2., 2., 2., 1., 1., 1.]) The output of the scipy.cluster.hierarchy.cophenet method is represented in condensed form. We can use scipy.spatial.distance.squareform to see the output as a regular matrix (where each element ij denotes the cophenetic distance between each i, j pair of points in X): >>> squareform(cophenet(Z)) array([[0., 1., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2.], [1., 0., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2.], [1., 1., 0., 2., 2., 2., 2., 2., 2., 2., 2., 2.], [2., 2., 2., 0., 1., 1., 2., 2., 2., 2., 2., 2.], [2., 2., 2., 1., 0., 1., 2., 2., 2., 2., 2., 2.], [2., 2., 2., 1., 1., 0., 2., 2., 2., 2., 2., 2.], [2., 2., 2., 2., 2., 2., 0., 1., 1., 2., 2., 2.], [2., 2., 2., 2., 2., 2., 1., 0., 1., 2., 2., 2.], [2., 2., 2., 2., 2., 2., 1., 1., 0., 2., 2., 2.], [2., 2., 2., 2., 2., 2., 2., 2., 2., 0., 1., 1.], [2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 0., 1.], [2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 1., 0.]]) In this example, the cophenetic distance between points on X that are very close (i.e. in the same corner) is 1. For other pairs of points is 2, because the points will be located in clusters at different corners - thus the distance between these clusters will be larger. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') Zs = Z.shape n = Zs[0] + 1 zz = np.zeros((n * (n-1)) // 2, dtype=np.double) # Since the C code does not support striding using strides. # The dimensions are used instead. Z = _convert_to_double(Z) _hierarchy.cophenetic_distances(Z, zz, int(n)) if Y is None: return zz Y = np.asarray(Y, order='c') distance.is_valid_y(Y, throw=True, name='Y') z = zz.mean() y = Y.mean() Yy = Y - y Zz = zz - z numerator = (Yy * Zz) denomA = Yy**2 denomB = Zz**2 c = numerator.sum() / np.sqrt((denomA.sum() * denomB.sum())) return (c, zz) def inconsistent(Z, d=2): r""" Calculate inconsistency statistics on a linkage matrix. Parameters ---------- Z : ndarray The :math:(n-1) by 4 matrix encoding the linkage (hierarchical clustering). See linkage documentation for more information on its form. d : int, optional The number of links up to d levels below each non-singleton cluster. Returns ------- R : ndarray A :math:(n-1) by 4 matrix where the i'th row contains the link statistics for the non-singleton cluster i. The link statistics are computed over the link heights for links :math:d levels below the cluster i. R[i,0] and R[i,1] are the mean and standard deviation of the link heights, respectively; R[i,2] is the number of links included in the calculation; and R[i,3] is the inconsistency coefficient, .. math:: \frac{\mathtt{Z[i,2]} - \mathtt{R[i,0]}} {R[i,1]} Notes ----- This function behaves similarly to the MATLAB(TM) inconsistent function. Examples -------- >>> from scipy.cluster.hierarchy import inconsistent, linkage >>> from matplotlib import pyplot as plt >>> X = [[i] for i in [2, 8, 0, 4, 1, 9, 9, 0]] >>> Z = linkage(X, 'ward') >>> print(Z) [[ 5. 6. 0. 2. ] [ 2. 7. 0. 2. ] [ 0. 4. 1. 2. ] [ 1. 8. 1.15470054 3. ] [ 9. 10. 2.12132034 4. ] [ 3. 12. 4.11096096 5. ] [11. 13. 14.07183949 8. ]] >>> inconsistent(Z) array([[ 0. , 0. , 1. , 0. ], [ 0. , 0. , 1. , 0. ], [ 1. , 0. , 1. , 0. ], [ 0.57735027, 0.81649658, 2. , 0.70710678], [ 1.04044011, 1.06123822, 3. , 1.01850858], [ 3.11614065, 1.40688837, 2. , 0.70710678], [ 6.44583366, 6.76770586, 3. , 1.12682288]]) """ Z = np.asarray(Z, order='c') Zs = Z.shape is_valid_linkage(Z, throw=True, name='Z') if (not d == np.floor(d)) or d < 0: raise ValueError('The second argument d must be a nonnegative ' 'integer value.') # Since the C code does not support striding using strides. # The dimensions are used instead. [Z] = _copy_arrays_if_base_present([Z]) n = Zs[0] + 1 R = np.zeros((n - 1, 4), dtype=np.double) _hierarchy.inconsistent(Z, R, int(n), int(d)) return R def from_mlab_linkage(Z): """ Convert a linkage matrix generated by MATLAB(TM) to a new linkage matrix compatible with this module. The conversion does two things: * the indices are converted from 1..N to 0..(N-1) form, and * a fourth column Z[:,3] is added where Z[i,3] represents the number of original observations (leaves) in the non-singleton cluster i. This function is useful when loading in linkages from legacy data files generated by MATLAB. Parameters ---------- Z : ndarray A linkage matrix generated by MATLAB(TM). Returns ------- ZS : ndarray A linkage matrix compatible with scipy.cluster.hierarchy. See Also -------- linkage: for a description of what a linkage matrix is. to_mlab_linkage: transform from SciPy to MATLAB format. Examples -------- >>> import numpy as np >>> from scipy.cluster.hierarchy import ward, from_mlab_linkage Given a linkage matrix in MATLAB format mZ, we can use scipy.cluster.hierarchy.from_mlab_linkage to import it into SciPy format: >>> mZ = np.array([[1, 2, 1], [4, 5, 1], [7, 8, 1], ... [10, 11, 1], [3, 13, 1.29099445], ... [6, 14, 1.29099445], ... [9, 15, 1.29099445], ... [12, 16, 1.29099445], ... [17, 18, 5.77350269], ... [19, 20, 5.77350269], ... [21, 22, 8.16496581]]) >>> Z = from_mlab_linkage(mZ) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [ 11. , 15. , 1.29099445, 3. ], [ 16. , 17. , 5.77350269, 6. ], [ 18. , 19. , 5.77350269, 6. ], [ 20. , 21. , 8.16496581, 12. ]]) As expected, the linkage matrix Z returned includes an additional column counting the number of original samples in each cluster. Also, all cluster indexes are reduced by 1 (MATLAB format uses 1-indexing, whereas SciPy uses 0-indexing). """ Z = np.asarray(Z, dtype=np.double, order='c') Zs = Z.shape # If it's empty, return it. if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0): return Z.copy() if len(Zs) != 2: raise ValueError("The linkage array must be rectangular.") # If it contains no rows, return it. if Zs[0] == 0: return Z.copy() Zpart = Z.copy() if Zpart[:, 0:2].min() != 1.0 and Zpart[:, 0:2].max() != 2 * Zs[0]: raise ValueError('The format of the indices is not 1..N') Zpart[:, 0:2] -= 1.0 CS = np.zeros((Zs[0],), dtype=np.double) _hierarchy.calculate_cluster_sizes(Zpart, CS, int(Zs[0]) + 1) return np.hstack([Zpart, CS.reshape(Zs[0], 1)]) def to_mlab_linkage(Z): """ Convert a linkage matrix to a MATLAB(TM) compatible one. Converts a linkage matrix Z generated by the linkage function of this module to a MATLAB(TM) compatible one. The return linkage matrix has the last column removed and the cluster indices are converted to 1..N indexing. Parameters ---------- Z : ndarray A linkage matrix generated by scipy.cluster.hierarchy. Returns ------- to_mlab_linkage : ndarray A linkage matrix compatible with MATLAB(TM)'s hierarchical clustering functions. The return linkage matrix has the last column removed and the cluster indices are converted to 1..N indexing. See Also -------- linkage: for a description of what a linkage matrix is. from_mlab_linkage: transform from Matlab to SciPy format. Examples -------- >>> from scipy.cluster.hierarchy import ward, to_mlab_linkage >>> from scipy.spatial.distance import pdist >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) After a linkage matrix Z has been created, we can use scipy.cluster.hierarchy.to_mlab_linkage to convert it into MATLAB format: >>> mZ = to_mlab_linkage(Z) >>> mZ array([[ 1. , 2. , 1. ], [ 4. , 5. , 1. ], [ 7. , 8. , 1. ], [ 10. , 11. , 1. ], [ 3. , 13. , 1.29099445], [ 6. , 14. , 1.29099445], [ 9. , 15. , 1.29099445], [ 12. , 16. , 1.29099445], [ 17. , 18. , 5.77350269], [ 19. , 20. , 5.77350269], [ 21. , 22. , 8.16496581]]) The new linkage matrix mZ uses 1-indexing for all the clusters (instead of 0-indexing). Also, the last column of the original linkage matrix has been dropped. """ Z = np.asarray(Z, order='c', dtype=np.double) Zs = Z.shape if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0): return Z.copy() is_valid_linkage(Z, throw=True, name='Z') ZP = Z[:, 0:3].copy() ZP[:, 0:2] += 1.0 return ZP def is_monotonic(Z): """ Return True if the linkage passed is monotonic. The linkage is monotonic if for every cluster :math:s and :math:t joined, the distance between them is no less than the distance between any previously joined clusters. Parameters ---------- Z : ndarray The linkage matrix to check for monotonicity. Returns ------- b : bool A boolean indicating whether the linkage is monotonic. See Also -------- linkage: for a description of what a linkage matrix is. Examples -------- >>> from scipy.cluster.hierarchy import median, ward, is_monotonic >>> from scipy.spatial.distance import pdist By definition, some hierarchical clustering algorithms - such as scipy.cluster.hierarchy.ward - produce monotonic assignments of samples to clusters; however, this is not always true for other hierarchical methods - e.g. scipy.cluster.hierarchy.median. Given a linkage matrix Z (as the result of a hierarchical clustering method) we can test programmatically whether if is has the monotonicity property or not, using scipy.cluster.hierarchy.is_monotonic: >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) >>> is_monotonic(Z) True >>> Z = median(pdist(X)) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 2. , 12. , 1.11803399, 3. ], [ 5. , 13. , 1.11803399, 3. ], [ 8. , 15. , 1.11803399, 3. ], [11. , 14. , 1.11803399, 3. ], [18. , 19. , 3. , 6. ], [16. , 17. , 3.5 , 6. ], [20. , 21. , 3.25 , 12. ]]) >>> is_monotonic(Z) False Note that this method is equivalent to just verifying that the distances in the third column of the linkage matrix appear in a monotonically increasing order. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') # We expect the i'th value to be greater than its successor. return (Z[1:, 2] >= Z[:-1, 2]).all() def is_valid_im(R, warning=False, throw=False, name=None): """Return True if the inconsistency matrix passed is valid. It must be a :math:n by 4 array of doubles. The standard deviations R[:,1] must be nonnegative. The link counts R[:,2] must be positive and no greater than :math:n-1. Parameters ---------- R : ndarray The inconsistency matrix to check for validity. warning : bool, optional When True, issues a Python warning if the linkage matrix passed is invalid. throw : bool, optional When True, throws a Python exception if the linkage matrix passed is invalid. name : str, optional This string refers to the variable name of the invalid linkage matrix. Returns ------- b : bool True if the inconsistency matrix is valid. See Also -------- linkage: for a description of what a linkage matrix is. inconsistent: for the creation of a inconsistency matrix. Examples -------- >>> from scipy.cluster.hierarchy import ward, inconsistent, is_valid_im >>> from scipy.spatial.distance import pdist Given a data set X, we can apply a clustering method to obtain a linkage matrix Z. scipy.cluster.hierarchy.inconsistent can be also used to obtain the inconsistency matrix R associated to this clustering process: >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) >>> R = inconsistent(Z) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) >>> R array([[1. , 0. , 1. , 0. ], [1. , 0. , 1. , 0. ], [1. , 0. , 1. , 0. ], [1. , 0. , 1. , 0. ], [1.14549722, 0.20576415, 2. , 0.70710678], [1.14549722, 0.20576415, 2. , 0.70710678], [1.14549722, 0.20576415, 2. , 0.70710678], [1.14549722, 0.20576415, 2. , 0.70710678], [2.78516386, 2.58797734, 3. , 1.15470054], [2.78516386, 2.58797734, 3. , 1.15470054], [6.57065706, 1.38071187, 3. , 1.15470054]]) Now we can use scipy.cluster.hierarchy.is_valid_im to verify that R is correct: >>> is_valid_im(R) True However, if R is wrongly constructed (e.g one of the standard deviations is set to a negative value) then the check will fail: >>> R[-1,1] = R[-1,1] * -1 >>> is_valid_im(R) False """ R = np.asarray(R, order='c') valid = True name_str = "%r " % name if name else '' try: if type(R) != np.ndarray: raise TypeError('Variable %spassed as inconsistency matrix is not ' 'a numpy array.' % name_str) if R.dtype != np.double: raise TypeError('Inconsistency matrix %smust contain doubles ' '(double).' % name_str) if len(R.shape) != 2: raise ValueError('Inconsistency matrix %smust have shape=2 (i.e. ' 'be two-dimensional).' % name_str) if R.shape[1] != 4: raise ValueError('Inconsistency matrix %smust have 4 columns.' % name_str) if R.shape[0] < 1: raise ValueError('Inconsistency matrix %smust have at least one ' 'row.' % name_str) if (R[:, 0] < 0).any(): raise ValueError('Inconsistency matrix %scontains negative link ' 'height means.' % name_str) if (R[:, 1] < 0).any(): raise ValueError('Inconsistency matrix %scontains negative link ' 'height standard deviations.' % name_str) if (R[:, 2] < 0).any(): raise ValueError('Inconsistency matrix %scontains negative link ' 'counts.' % name_str) except Exception as e: if throw: raise if warning: _warning(str(e)) valid = False return valid def is_valid_linkage(Z, warning=False, throw=False, name=None): """ Check the validity of a linkage matrix. A linkage matrix is valid if it is a two dimensional array (type double) with :math:n rows and 4 columns. The first two columns must contain indices between 0 and :math:2n-1. For a given row i, the following two expressions have to hold: .. math:: 0 \\leq \\mathtt{Z[i,0]} \\leq i+n-1 0 \\leq Z[i,1] \\leq i+n-1 I.e. a cluster cannot join another cluster unless the cluster being joined has been generated. Parameters ---------- Z : array_like Linkage matrix. warning : bool, optional When True, issues a Python warning if the linkage matrix passed is invalid. throw : bool, optional When True, throws a Python exception if the linkage matrix passed is invalid. name : str, optional This string refers to the variable name of the invalid linkage matrix. Returns ------- b : bool True if the inconsistency matrix is valid. See Also -------- linkage: for a description of what a linkage matrix is. Examples -------- >>> from scipy.cluster.hierarchy import ward, is_valid_linkage >>> from scipy.spatial.distance import pdist All linkage matrices generated by the clustering methods in this module will be valid (i.e. they will have the appropriate dimensions and the two required expressions will hold for all the rows). We can check this using scipy.cluster.hierarchy.is_valid_linkage: >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) >>> is_valid_linkage(Z) True However, is we create a linkage matrix in a wrong way - or if we modify a valid one in a way that any of the required expressions don't hold anymore, then the check will fail: >>> Z[3][1] = 20 # the cluster number 20 is not defined at this point >>> is_valid_linkage(Z) False """ Z = np.asarray(Z, order='c') valid = True name_str = "%r " % name if name else '' try: if type(Z) != np.ndarray: raise TypeError('Passed linkage argument %sis not a valid array.' % name_str) if Z.dtype != np.double: raise TypeError('Linkage matrix %smust contain doubles.' % name_str) if len(Z.shape) != 2: raise ValueError('Linkage matrix %smust have shape=2 (i.e. be ' 'two-dimensional).' % name_str) if Z.shape[1] != 4: raise ValueError('Linkage matrix %smust have 4 columns.' % name_str) if Z.shape[0] == 0: raise ValueError('Linkage must be computed on at least two ' 'observations.') n = Z.shape[0] if n > 1: if ((Z[:, 0] < 0).any() or (Z[:, 1] < 0).any()): raise ValueError('Linkage %scontains negative indices.' % name_str) if (Z[:, 2] < 0).any(): raise ValueError('Linkage %scontains negative distances.' % name_str) if (Z[:, 3] < 0).any(): raise ValueError('Linkage %scontains negative counts.' % name_str) if _check_hierarchy_uses_cluster_before_formed(Z): raise ValueError('Linkage %suses non-singleton cluster before ' 'it is formed.' % name_str) if _check_hierarchy_uses_cluster_more_than_once(Z): raise ValueError('Linkage %suses the same cluster more than once.' % name_str) except Exception as e: if throw: raise if warning: _warning(str(e)) valid = False return valid def _check_hierarchy_uses_cluster_before_formed(Z): n = Z.shape[0] + 1 for i in xrange(0, n - 1): if Z[i, 0] >= n + i or Z[i, 1] >= n + i: return True return False def _check_hierarchy_uses_cluster_more_than_once(Z): n = Z.shape[0] + 1 chosen = set([]) for i in xrange(0, n - 1): if (Z[i, 0] in chosen) or (Z[i, 1] in chosen) or Z[i, 0] == Z[i, 1]: return True chosen.add(Z[i, 0]) chosen.add(Z[i, 1]) return False def _check_hierarchy_not_all_clusters_used(Z): n = Z.shape[0] + 1 chosen = set([]) for i in xrange(0, n - 1): chosen.add(int(Z[i, 0])) chosen.add(int(Z[i, 1])) must_chosen = set(range(0, 2 * n - 2)) return len(must_chosen.difference(chosen)) > 0 def num_obs_linkage(Z): """ Return the number of original observations of the linkage matrix passed. Parameters ---------- Z : ndarray The linkage matrix on which to perform the operation. Returns ------- n : int The number of original observations in the linkage. Examples -------- >>> from scipy.cluster.hierarchy import ward, num_obs_linkage >>> from scipy.spatial.distance import pdist >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) Z is a linkage matrix obtained after using the Ward clustering method with X, a dataset with 12 data points. >>> num_obs_linkage(Z) 12 """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') return (Z.shape[0] + 1) def correspond(Z, Y): """ Check for correspondence between linkage and condensed distance matrices. They must have the same number of original observations for the check to succeed. This function is useful as a sanity check in algorithms that make extensive use of linkage and distance matrices that must correspond to the same set of original observations. Parameters ---------- Z : array_like The linkage matrix to check for correspondence. Y : array_like The condensed distance matrix to check for correspondence. Returns ------- b : bool A boolean indicating whether the linkage matrix and distance matrix could possibly correspond to one another. See Also -------- linkage: for a description of what a linkage matrix is. Examples -------- >>> from scipy.cluster.hierarchy import ward, correspond >>> from scipy.spatial.distance import pdist This method can be used to check if a given linkage matrix Z has been obtained from the application of a cluster method over a dataset X: >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> X_condensed = pdist(X) >>> Z = ward(X_condensed) Here we can compare Z and X (in condensed form): >>> correspond(Z, X_condensed) True """ is_valid_linkage(Z, throw=True) distance.is_valid_y(Y, throw=True) Z = np.asarray(Z, order='c') Y = np.asarray(Y, order='c') return distance.num_obs_y(Y) == num_obs_linkage(Z) def fcluster(Z, t, criterion='inconsistent', depth=2, R=None, monocrit=None): """ Form flat clusters from the hierarchical clustering defined by the given linkage matrix. Parameters ---------- Z : ndarray The hierarchical clustering encoded with the matrix returned by the linkage function. t : scalar For criteria 'inconsistent', 'distance' or 'monocrit', this is the threshold to apply when forming flat clusters. For 'maxclust' or 'maxclust_monocrit' criteria, this would be max number of clusters requested. criterion : str, optional The criterion to use in forming flat clusters. This can be any of the following values: inconsistent : If a cluster node and all its descendants have an inconsistent value less than or equal to t then all its leaf descendants belong to the same flat cluster. When no non-singleton cluster meets this criterion, every node is assigned to its own cluster. (Default) distance : Forms flat clusters so that the original observations in each flat cluster have no greater a cophenetic distance than t. maxclust : Finds a minimum threshold r so that the cophenetic distance between any two original observations in the same flat cluster is no more than r and no more than t flat clusters are formed. monocrit : Forms a flat cluster from a cluster node c with index i when monocrit[j] <= t. For example, to threshold on the maximum mean distance as computed in the inconsistency matrix R with a threshold of 0.8 do:: MR = maxRstat(Z, R, 3) cluster(Z, t=0.8, criterion='monocrit', monocrit=MR) maxclust_monocrit : Forms a flat cluster from a non-singleton cluster node c when monocrit[i] <= r for all cluster indices i below and including c. r is minimized such that no more than t flat clusters are formed. monocrit must be monotonic. For example, to minimize the threshold t on maximum inconsistency values so that no more than 3 flat clusters are formed, do:: MI = maxinconsts(Z, R) cluster(Z, t=3, criterion='maxclust_monocrit', monocrit=MI) depth : int, optional The maximum depth to perform the inconsistency calculation. It has no meaning for the other criteria. Default is 2. R : ndarray, optional The inconsistency matrix to use for the 'inconsistent' criterion. This matrix is computed if not provided. monocrit : ndarray, optional An array of length n-1. monocrit[i] is the statistics upon which non-singleton i is thresholded. The monocrit vector must be monotonic, i.e. given a node c with index i, for all node indices j corresponding to nodes below c, monocrit[i] >= monocrit[j]. Returns ------- fcluster : ndarray An array of length n. T[i] is the flat cluster number to which original observation i belongs. See Also -------- linkage : for information about hierarchical clustering methods work. Examples -------- >>> from scipy.cluster.hierarchy import ward, fcluster >>> from scipy.spatial.distance import pdist All cluster linkage methods - e.g. scipy.cluster.hierarchy.ward generate a linkage matrix Z as their output: >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) >>> Z array([[ 0. , 1. , 1. , 2. ], [ 3. , 4. , 1. , 2. ], [ 6. , 7. , 1. , 2. ], [ 9. , 10. , 1. , 2. ], [ 2. , 12. , 1.29099445, 3. ], [ 5. , 13. , 1.29099445, 3. ], [ 8. , 14. , 1.29099445, 3. ], [11. , 15. , 1.29099445, 3. ], [16. , 17. , 5.77350269, 6. ], [18. , 19. , 5.77350269, 6. ], [20. , 21. , 8.16496581, 12. ]]) This matrix represents a dendrogram, where the first and second elements are the two clusters merged at each step, the third element is the distance between these clusters, and the fourth element is the size of the new cluster - the number of original data points included. scipy.cluster.hierarchy.fcluster can be used to flatten the dendrogram, obtaining as a result an assignation of the original data points to single clusters. This assignation mostly depends on a distance threshold t - the maximum inter-cluster distance allowed: >>> fcluster(Z, t=0.9, criterion='distance') array([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], dtype=int32) >>> fcluster(Z, t=1.1, criterion='distance') array([1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8], dtype=int32) >>> fcluster(Z, t=3, criterion='distance') array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) >>> fcluster(Z, t=9, criterion='distance') array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32) In the first case, the threshold t is too small to allow any two samples in the data to form a cluster, so 12 different clusters are returned. In the second case, the threshold is large enough to allow the first 4 points to be merged with their nearest neighbors. So here only 8 clusters are returned. The third case, with a much higher threshold, allows for up to 8 data points to be connected - so 4 clusters are returned here. Lastly, the threshold of the fourth case is large enough to allow for all data points to be merged together - so a single cluster is returned. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') n = Z.shape[0] + 1 T = np.zeros((n,), dtype='i') # Since the C code does not support striding using strides. # The dimensions are used instead. [Z] = _copy_arrays_if_base_present([Z]) if criterion == 'inconsistent': if R is None: R = inconsistent(Z, depth) else: R = np.asarray(R, order='c') is_valid_im(R, throw=True, name='R') # Since the C code does not support striding using strides. # The dimensions are used instead. [R] = _copy_arrays_if_base_present([R]) _hierarchy.cluster_in(Z, R, T, float(t), int(n)) elif criterion == 'distance': _hierarchy.cluster_dist(Z, T, float(t), int(n)) elif criterion == 'maxclust': _hierarchy.cluster_maxclust_dist(Z, T, int(n), int(t)) elif criterion == 'monocrit': [monocrit] = _copy_arrays_if_base_present([monocrit]) _hierarchy.cluster_monocrit(Z, monocrit, T, float(t), int(n)) elif criterion == 'maxclust_monocrit': [monocrit] = _copy_arrays_if_base_present([monocrit]) _hierarchy.cluster_maxclust_monocrit(Z, monocrit, T, int(n), int(t)) else: raise ValueError('Invalid cluster formation criterion: %s' % str(criterion)) return T def fclusterdata(X, t, criterion='inconsistent', metric='euclidean', depth=2, method='single', R=None): """ Cluster observation data using a given metric. Clusters the original observations in the n-by-m data matrix X (n observations in m dimensions), using the euclidean distance metric to calculate distances between original observations, performs hierarchical clustering using the single linkage algorithm, and forms flat clusters using the inconsistency method with t as the cut-off threshold. A one-dimensional array T of length n is returned. T[i] is the index of the flat cluster to which the original observation i belongs. Parameters ---------- X : (N, M) ndarray N by M data matrix with N observations in M dimensions. t : scalar For criteria 'inconsistent', 'distance' or 'monocrit', this is the threshold to apply when forming flat clusters. For 'maxclust' or 'maxclust_monocrit' criteria, this would be max number of clusters requested. criterion : str, optional Specifies the criterion for forming flat clusters. Valid values are 'inconsistent' (default), 'distance', or 'maxclust' cluster formation algorithms. See fcluster for descriptions. metric : str, optional The distance metric for calculating pairwise distances. See distance.pdist for descriptions and linkage to verify compatibility with the linkage method. depth : int, optional The maximum depth for the inconsistency calculation. See inconsistent for more information. method : str, optional The linkage method to use (single, complete, average, weighted, median centroid, ward). See linkage for more information. Default is "single". R : ndarray, optional The inconsistency matrix. It will be computed if necessary if it is not passed. Returns ------- fclusterdata : ndarray A vector of length n. T[i] is the flat cluster number to which original observation i belongs. See Also -------- scipy.spatial.distance.pdist : pairwise distance metrics Notes ----- This function is similar to the MATLAB function clusterdata. Examples -------- >>> from scipy.cluster.hierarchy import fclusterdata This is a convenience method that abstracts all the steps to perform in a typical SciPy's hierarchical clustering workflow. * Transform the input data into a condensed matrix with scipy.spatial.distance.pdist. * Apply a clustering method. * Obtain flat clusters at a user defined distance threshold t using scipy.cluster.hierarchy.fcluster. >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> fclusterdata(X, t=1) array([3, 3, 3, 4, 4, 4, 2, 2, 2, 1, 1, 1], dtype=int32) The output here (for the dataset X, distance threshold t, and the default settings) is four clusters with three data points each. """ X = np.asarray(X, order='c', dtype=np.double) if type(X) != np.ndarray or len(X.shape) != 2: raise TypeError('The observation matrix X must be an n by m numpy ' 'array.') Y = distance.pdist(X, metric=metric) Z = linkage(Y, method=method) if R is None: R = inconsistent(Z, d=depth) else: R = np.asarray(R, order='c') T = fcluster(Z, criterion=criterion, depth=depth, R=R, t=t) return T def leaves_list(Z): """ Return a list of leaf node ids. The return corresponds to the observation vector index as it appears in the tree from left to right. Z is a linkage matrix. Parameters ---------- Z : ndarray The hierarchical clustering encoded as a matrix. Z is a linkage matrix. See linkage for more information. Returns ------- leaves_list : ndarray The list of leaf node ids. See Also -------- dendrogram: for information about dendrogram structure. Examples -------- >>> from scipy.cluster.hierarchy import ward, dendrogram, leaves_list >>> from scipy.spatial.distance import pdist >>> from matplotlib import pyplot as plt >>> X = [[0, 0], [0, 1], [1, 0], ... [0, 4], [0, 3], [1, 4], ... [4, 0], [3, 0], [4, 1], ... [4, 4], [3, 4], [4, 3]] >>> Z = ward(pdist(X)) The linkage matrix Z represents a dendrogram, that is, a tree that encodes the structure of the clustering performed. scipy.cluster.hierarchy.leaves_list shows the mapping between indexes in the X dataset and leaves in the dendrogram: >>> leaves_list(Z) array([ 2, 0, 1, 5, 3, 4, 8, 6, 7, 11, 9, 10], dtype=int32) >>> fig = plt.figure(figsize=(25, 10)) >>> dn = dendrogram(Z) >>> plt.show() """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') n = Z.shape[0] + 1 ML = np.zeros((n,), dtype='i') [Z] = _copy_arrays_if_base_present([Z]) _hierarchy.prelist(Z, ML, int(n)) return ML # Maps number of leaves to text size. # # p <= 20, size="12" # 20 < p <= 30, size="10" # 30 < p <= 50, size="8" # 50 < p <= np.inf, size="6" _dtextsizes = {20: 12, 30: 10, 50: 8, 85: 6, np.inf: 5} _drotation = {20: 0, 40: 45, np.inf: 90} _dtextsortedkeys = list(_dtextsizes.keys()) _dtextsortedkeys.sort() _drotationsortedkeys = list(_drotation.keys()) _drotationsortedkeys.sort() def _remove_dups(L): """ Remove duplicates AND preserve the original order of the elements. The set class is not guaranteed to do this. """ seen_before = set([]) L2 = [] for i in L: if i not in seen_before: seen_before.add(i) L2.append(i) return L2 def _get_tick_text_size(p): for k in _dtextsortedkeys: if p <= k: return _dtextsizes[k] def _get_tick_rotation(p): for k in _drotationsortedkeys: if p <= k: return _drotation[k] def _plot_dendrogram(icoords, dcoords, ivl, p, n, mh, orientation, no_labels, color_list, leaf_font_size=None, leaf_rotation=None, contraction_marks=None, ax=None, above_threshold_color='b'): # Import matplotlib here so that it's not imported unless dendrograms # are plotted. Raise an informative error if importing fails. try: # if an axis is provided, don't use pylab at all if ax is None: import matplotlib.pylab import matplotlib.patches import matplotlib.collections except ImportError: raise ImportError("You must install the matplotlib library to plot " "the dendrogram. Use no_plot=True to calculate the " "dendrogram without plotting.") if ax is None: ax = matplotlib.pylab.gca() # if we're using pylab, we want to trigger a draw at the end trigger_redraw = True else: trigger_redraw = False # Independent variable plot width ivw = len(ivl) * 10 # Dependent variable plot height dvw = mh + mh * 0.05 iv_ticks = np.arange(5, len(ivl) * 10 + 5, 10) if orientation in ('top', 'bottom'): if orientation == 'top': ax.set_ylim([0, dvw]) ax.set_xlim([0, ivw]) else: ax.set_ylim([dvw, 0]) ax.set_xlim([0, ivw]) xlines = icoords ylines = dcoords if no_labels: ax.set_xticks([]) ax.set_xticklabels([]) else: ax.set_xticks(iv_ticks) if orientation == 'top': ax.xaxis.set_ticks_position('bottom') else: ax.xaxis.set_ticks_position('top') # Make the tick marks invisible because they cover up the links for line in ax.get_xticklines(): line.set_visible(False) leaf_rot = (float(_get_tick_rotation(len(ivl))) if (leaf_rotation is None) else leaf_rotation) leaf_font = (float(_get_tick_text_size(len(ivl))) if (leaf_font_size is None) else leaf_font_size) ax.set_xticklabels(ivl, rotation=leaf_rot, size=leaf_font) elif orientation in ('left', 'right'): if orientation == 'left': ax.set_xlim([dvw, 0]) ax.set_ylim([0, ivw]) else: ax.set_xlim([0, dvw]) ax.set_ylim([0, ivw]) xlines = dcoords ylines = icoords if no_labels: ax.set_yticks([]) ax.set_yticklabels([]) else: ax.set_yticks(iv_ticks) if orientation == 'left': ax.yaxis.set_ticks_position('right') else: ax.yaxis.set_ticks_position('left') # Make the tick marks invisible because they cover up the links for line in ax.get_yticklines(): line.set_visible(False) leaf_font = (float(_get_tick_text_size(len(ivl))) if (leaf_font_size is None) else leaf_font_size) if leaf_rotation is not None: ax.set_yticklabels(ivl, rotation=leaf_rotation, size=leaf_font) else: ax.set_yticklabels(ivl, size=leaf_font) # Let's use collections instead. This way there is a separate legend item # for each tree grouping, rather than stupidly one for each line segment. colors_used = _remove_dups(color_list) color_to_lines = {} for color in colors_used: color_to_lines[color] = [] for (xline, yline, color) in zip(xlines, ylines, color_list): color_to_lines[color].append(list(zip(xline, yline))) colors_to_collections = {} # Construct the collections. for color in colors_used: coll = matplotlib.collections.LineCollection(color_to_lines[color], colors=(color,)) colors_to_collections[color] = coll # Add all the groupings below the color threshold. for color in colors_used: if color != above_threshold_color: ax.add_collection(colors_to_collections[color]) # If there's a grouping of links above the color threshold, it goes last. if above_threshold_color in colors_to_collections: ax.add_collection(colors_to_collections[above_threshold_color]) if contraction_marks is not None: Ellipse = matplotlib.patches.Ellipse for (x, y) in contraction_marks: if orientation in ('left', 'right'): e = Ellipse((y, x), width=dvw / 100, height=1.0) else: e = Ellipse((x, y), width=1.0, height=dvw / 100) ax.add_artist(e) e.set_clip_box(ax.bbox) e.set_alpha(0.5) e.set_facecolor('k') if trigger_redraw: matplotlib.pylab.draw_if_interactive() _link_line_colors = ['g', 'r', 'c', 'm', 'y', 'k'] def set_link_color_palette(palette): """ Set list of matplotlib color codes for use by dendrogram. Note that this palette is global (i.e. setting it once changes the colors for all subsequent calls to dendrogram) and that it affects only the the colors below color_threshold. Note that dendrogram also accepts a custom coloring function through its link_color_func keyword, which is more flexible and non-global. Parameters ---------- palette : list of str or None A list of matplotlib color codes. The order of the color codes is the order in which the colors are cycled through when color thresholding in the dendrogram. If None, resets the palette to its default (which is ['g', 'r', 'c', 'm', 'y', 'k']). Returns ------- None See Also -------- dendrogram Notes ----- Ability to reset the palette with None added in SciPy 0.17.0. Examples -------- >>> from scipy.cluster import hierarchy >>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268., ... 400., 754., 564., 138., 219., 869., 669.]) >>> Z = hierarchy.linkage(ytdist, 'single') >>> dn = hierarchy.dendrogram(Z, no_plot=True) >>> dn['color_list'] ['g', 'b', 'b', 'b', 'b'] >>> hierarchy.set_link_color_palette(['c', 'm', 'y', 'k']) >>> dn = hierarchy.dendrogram(Z, no_plot=True) >>> dn['color_list'] ['c', 'b', 'b', 'b', 'b'] >>> dn = hierarchy.dendrogram(Z, no_plot=True, color_threshold=267, ... above_threshold_color='k') >>> dn['color_list'] ['c', 'm', 'm', 'k', 'k'] Now reset the color palette to its default: >>> hierarchy.set_link_color_palette(None) """ if palette is None: # reset to its default palette = ['g', 'r', 'c', 'm', 'y', 'k'] elif type(palette) not in (list, tuple): raise TypeError("palette must be a list or tuple") _ptypes = [isinstance(p, string_types) for p in palette] if False in _ptypes: raise TypeError("all palette list elements must be color strings") for i in list(_link_line_colors): _link_line_colors.remove(i) _link_line_colors.extend(list(palette)) def dendrogram(Z, p=30, truncate_mode=None, color_threshold=None, get_leaves=True, orientation='top', labels=None, count_sort=False, distance_sort=False, show_leaf_counts=True, no_plot=False, no_labels=False, leaf_font_size=None, leaf_rotation=None, leaf_label_func=None, show_contracted=False, link_color_func=None, ax=None, above_threshold_color='b'): """ Plot the hierarchical clustering as a dendrogram. The dendrogram illustrates how each cluster is composed by drawing a U-shaped link between a non-singleton cluster and its children. The top of the U-link indicates a cluster merge. The two legs of the U-link indicate which clusters were merged. The length of the two legs of the U-link represents the distance between the child clusters. It is also the cophenetic distance between original observations in the two children clusters. Parameters ---------- Z : ndarray The linkage matrix encoding the hierarchical clustering to render as a dendrogram. See the linkage function for more information on the format of Z. p : int, optional The p parameter for truncate_mode. truncate_mode : str, optional The dendrogram can be hard to read when the original observation matrix from which the linkage is derived is large. Truncation is used to condense the dendrogram. There are several modes: None No truncation is performed (default). Note: 'none' is an alias for None that's kept for backward compatibility. 'lastp' The last p non-singleton clusters formed in the linkage are the only non-leaf nodes in the linkage; they correspond to rows Z[n-p-2:end] in Z. All other non-singleton clusters are contracted into leaf nodes. 'level' No more than p levels of the dendrogram tree are displayed. A "level" includes all nodes with p merges from the last merge. Note: 'mtica' is an alias for 'level' that's kept for backward compatibility. color_threshold : double, optional For brevity, let :math:t be the color_threshold. Colors all the descendent links below a cluster node :math:k the same color if :math:k is the first node below the cut threshold :math:t. All links connecting nodes with distances greater than or equal to the threshold are colored blue. If :math:t is less than or equal to zero, all nodes are colored blue. If color_threshold is None or 'default', corresponding with MATLAB(TM) behavior, the threshold is set to 0.7*max(Z[:,2]). get_leaves : bool, optional Includes a list R['leaves']=H in the result dictionary. For each :math:i, H[i] == j, cluster node j appears in position i in the left-to-right traversal of the leaves, where :math:j < 2n-1 and :math:i < n. orientation : str, optional The direction to plot the dendrogram, which can be any of the following strings: 'top' Plots the root at the top, and plot descendent links going downwards. (default). 'bottom' Plots the root at the bottom, and plot descendent links going upwards. 'left' Plots the root at the left, and plot descendent links going right. 'right' Plots the root at the right, and plot descendent links going left. labels : ndarray, optional By default labels is None so the index of the original observation is used to label the leaf nodes. Otherwise, this is an :math:n -sized list (or tuple). The labels[i] value is the text to put under the :math:i th leaf node only if it corresponds to an original observation and not a non-singleton cluster. count_sort : str or bool, optional For each node n, the order (visually, from left-to-right) n's two descendent links are plotted is determined by this parameter, which can be any of the following values: False Nothing is done. 'ascending' or True The child with the minimum number of original objects in its cluster is plotted first. 'descending' The child with the maximum number of original objects in its cluster is plotted first. Note distance_sort and count_sort cannot both be True. distance_sort : str or bool, optional For each node n, the order (visually, from left-to-right) n's two descendent links are plotted is determined by this parameter, which can be any of the following values: False Nothing is done. 'ascending' or True The child with the minimum distance between its direct descendents is plotted first. 'descending' The child with the maximum distance between its direct descendents is plotted first. Note distance_sort and count_sort cannot both be True. show_leaf_counts : bool, optional When True, leaf nodes representing :math:k>1 original observation are labeled with the number of observations they contain in parentheses. no_plot : bool, optional When True, the final rendering is not performed. This is useful if only the data structures computed for the rendering are needed or if matplotlib is not available. no_labels : bool, optional When True, no labels appear next to the leaf nodes in the rendering of the dendrogram. leaf_rotation : double, optional Specifies the angle (in degrees) to rotate the leaf labels. When unspecified, the rotation is based on the number of nodes in the dendrogram (default is 0). leaf_font_size : int, optional Specifies the font size (in points) of the leaf labels. When unspecified, the size based on the number of nodes in the dendrogram. leaf_label_func : lambda or function, optional When leaf_label_func is a callable function, for each leaf with cluster index :math:k < 2n-1. The function is expected to return a string with the label for the leaf. Indices :math:k < n correspond to original observations while indices :math:k \\geq n correspond to non-singleton clusters. For example, to label singletons with their node id and non-singletons with their id, count, and inconsistency coefficient, simply do:: # First define the leaf label function. def llf(id): if id < n: return str(id) else: return '[%d %d %1.2f]' % (id, count, R[n-id,3]) # The text for the leaf nodes is going to be big so force # a rotation of 90 degrees. dendrogram(Z, leaf_label_func=llf, leaf_rotation=90) show_contracted : bool, optional When True the heights of non-singleton nodes contracted into a leaf node are plotted as crosses along the link connecting that leaf node. This really is only useful when truncation is used (see truncate_mode parameter). link_color_func : callable, optional If given, link_color_function is called with each non-singleton id corresponding to each U-shaped link it will paint. The function is expected to return the color to paint the link, encoded as a matplotlib color string code. For example:: dendrogram(Z, link_color_func=lambda k: colors[k]) colors the direct links below each untruncated non-singleton node k using colors[k]. ax : matplotlib Axes instance, optional If None and no_plot is not True, the dendrogram will be plotted on the current axes. Otherwise if no_plot is not True the dendrogram will be plotted on the given Axes instance. This can be useful if the dendrogram is part of a more complex figure. above_threshold_color : str, optional This matplotlib color string sets the color of the links above the color_threshold. The default is 'b'. Returns ------- R : dict A dictionary of data structures computed to render the dendrogram. Its has the following keys: 'color_list' A list of color names. The k'th element represents the color of the k'th link. 'icoord' and 'dcoord' Each of them is a list of lists. Let icoord = [I1, I2, ..., Ip] where Ik = [xk1, xk2, xk3, xk4] and dcoord = [D1, D2, ..., Dp] where Dk = [yk1, yk2, yk3, yk4], then the k'th link painted is (xk1, yk1) - (xk2, yk2) - (xk3, yk3) - (xk4, yk4). 'ivl' A list of labels corresponding to the leaf nodes. 'leaves' For each i, H[i] == j, cluster node j appears in position i in the left-to-right traversal of the leaves, where :math:j < 2n-1 and :math:i < n. If j is less than n, the i-th leaf node corresponds to an original observation. Otherwise, it corresponds to a non-singleton cluster. See Also -------- linkage, set_link_color_palette Notes ----- It is expected that the distances in Z[:,2] be monotonic, otherwise crossings appear in the dendrogram. Examples -------- >>> from scipy.cluster import hierarchy >>> import matplotlib.pyplot as plt A very basic example: >>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268., ... 400., 754., 564., 138., 219., 869., 669.]) >>> Z = hierarchy.linkage(ytdist, 'single') >>> plt.figure() >>> dn = hierarchy.dendrogram(Z) Now plot in given axes, improve the color scheme and use both vertical and horizontal orientations: >>> hierarchy.set_link_color_palette(['m', 'c', 'y', 'k']) >>> fig, axes = plt.subplots(1, 2, figsize=(8, 3)) >>> dn1 = hierarchy.dendrogram(Z, ax=axes[0], above_threshold_color='y', ... orientation='top') >>> dn2 = hierarchy.dendrogram(Z, ax=axes[1], ... above_threshold_color='#bcbddc', ... orientation='right') >>> hierarchy.set_link_color_palette(None) # reset to default after use >>> plt.show() """ # This feature was thought about but never implemented (still useful?): # # ... = dendrogram(..., leaves_order=None) # # Plots the leaves in the order specified by a vector of # original observation indices. If the vector contains duplicates # or results in a crossing, an exception will be thrown. Passing # None orders leaf nodes based on the order they appear in the # pre-order traversal. Z = np.asarray(Z, order='c') if orientation not in ["top", "left", "bottom", "right"]: raise ValueError("orientation must be one of 'top', 'left', " "'bottom', or 'right'") is_valid_linkage(Z, throw=True, name='Z') Zs = Z.shape n = Zs[0] + 1 if type(p) in (int, float): p = int(p) else: raise TypeError('The second argument must be a number') if truncate_mode not in ('lastp', 'mlab', 'mtica', 'level', 'none', None): # 'mlab' and 'mtica' are kept working for backwards compat. raise ValueError('Invalid truncation mode.') if truncate_mode == 'lastp' or truncate_mode == 'mlab': if p > n or p == 0: p = n if truncate_mode == 'mtica': # 'mtica' is an alias truncate_mode = 'level' if truncate_mode == 'level': if p <= 0: p = np.inf if get_leaves: lvs = [] else: lvs = None icoord_list = [] dcoord_list = [] color_list = [] current_color = [0] currently_below_threshold = [False] ivl = [] # list of leaves if color_threshold is None or (isinstance(color_threshold, string_types) and color_threshold == 'default'): color_threshold = max(Z[:, 2]) * 0.7 R = {'icoord': icoord_list, 'dcoord': dcoord_list, 'ivl': ivl, 'leaves': lvs, 'color_list