diff --git "a/venv/lib/python3.10/site-packages/scipy/cluster/hierarchy.py" "b/venv/lib/python3.10/site-packages/scipy/cluster/hierarchy.py" new file mode 100644--- /dev/null +++ "b/venv/lib/python3.10/site-packages/scipy/cluster/hierarchy.py" @@ -0,0 +1,4173 @@ +""" +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 + +Utility classes: + +.. autosummary:: + :toctree: generated/ + + DisjointSet -- data structure for incremental connectivity queries + +""" +# 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._array_api import array_namespace, _asarray, copy +from scipy._lib._disjoint_set import DisjointSet + + +_LINKAGE_METHODS = {'single': 0, 'complete': 1, 'average': 2, 'centroid': 3, + 'median': 4, 'ward': 5, 'weighted': 6} +_EUCLIDEAN_METHODS = ('centroid', 'median', 'ward') + +__all__ = ['ClusterNode', 'DisjointSet', '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'] + + +class ClusterWarning(UserWarning): + pass + + +def _warning(s): + warnings.warn('scipy.cluster: %s' % s, ClusterWarning, stacklevel=3) + + +def int_floor(arr, xp): + # array_api_strict is strict about not allowing `int()` on a float array. + # That's typically not needed, here it is - so explicitly convert + return int(xp.astype(xp.asarray(arr), xp.int64)) + + +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 + an 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. ]]) # may vary + + 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) # may vary + >>> fcluster(Z, 1.1, criterion='distance') + array([5, 5, 6, 7, 7, 8, 1, 1, 2, 3, 3, 4], dtype=int32) # may vary + >>> fcluster(Z, 2, criterion='distance') + array([3, 3, 3, 4, 4, 4, 1, 1, 1, 2, 2, 2], dtype=int32) # may vary + >>> 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 + an 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 + an 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 1-D condensed distance matrix + or a 2-D array of observation vectors. + + If y is a 1-D 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 the user's 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() + """ + xp = array_namespace(y) + y = _asarray(y, order='C', dtype=xp.float64, xp=xp) + + if method not in _LINKAGE_METHODS: + raise ValueError(f"Invalid method: {method}") + + if method in _EUCLIDEAN_METHODS and metric != 'euclidean' and y.ndim == 2: + msg = f"`method={method}` requires the distance metric to be Euclidean" + raise ValueError(msg) + + if y.ndim == 1: + distance.is_valid_y(y, throw=True, name='y') + elif y.ndim == 2: + if (y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0) and + xp.all(y >= 0) and np.allclose(y, y.T)): + warnings.warn('The symmetric non-negative hollow observation ' + 'matrix looks suspiciously like an uncondensed ' + 'distance matrix', + ClusterWarning, stacklevel=2) + y = distance.pdist(y, metric) + y = xp.asarray(y) + else: + raise ValueError("`y` must be 1 or 2 dimensional.") + + if not xp.all(xp.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] + + y = np.asarray(y) + 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) + result = xp.asarray(result) + + if optimal_ordering: + y = xp.asarray(y) + return optimal_leaf_ordering(result, y) + else: + return result + + +class ClusterNode: + """ + 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 " + f"to type {type(node)}") + return self.dist < node.dist + + def __gt__(self, node): + if not isinstance(node, ClusterNode): + raise ValueError("Can't compare ClusterNode " + f"to type {type(node)}") + return self.dist > node.dist + + def __eq__(self, node): + if not isinstance(node, ClusterNode): + raise ValueError("Can't compare ClusterNode " + f"to type {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` are given, the columns correspond to the columns of + `n_clusters` or `height`. + + Examples + -------- + >>> from scipy import cluster + >>> import numpy as np + >>> from numpy.random import default_rng + >>> rng = default_rng() + >>> X = rng.random((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]]) # random + + """ + xp = array_namespace(Z) + 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 = xp.arange(nobs) + elif height is not None: + height = xp.asarray(height) + heights = xp.asarray([x.dist for x in nodes]) + cols_idx = xp.searchsorted(heights, height) + else: + n_clusters = xp.asarray(n_clusters) + cols_idx = nobs - xp.searchsorted(xp.arange(nobs), n_clusters) + + try: + n_cols = len(cols_idx) + except TypeError: # scalar + n_cols = 1 + cols_idx = xp.asarray([cols_idx]) + + groups = xp.zeros((n_cols, nobs), dtype=xp.int64) + last_group = xp.arange(nobs) + if 0 in cols_idx: + groups[0] = last_group + + for i, node in enumerate(nodes): + idx = node.pre_order() + this_group = copy(last_group, xp=xp) + # TODO ARRAY_API complex indexing not supported + this_group[idx] = xp.min(last_group[idx]) + this_group[this_group > xp.max(last_group[idx])] -= 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 + -------- + >>> import numpy as np + >>> from scipy.cluster import hierarchy + >>> rng = np.random.default_rng() + >>> x = rng.random((5, 2)) + >>> Z = hierarchy.linkage(x) + >>> hierarchy.to_tree(Z) + >> rootnode, nodelist = hierarchy.to_tree(Z, rd=True) + >>> rootnode + >> len(nodelist) + 9 + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='c', xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + # Number of original objects is equal to the number of rows plus 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 range(0, n): + d[i] = ClusterNode(i) + + nd = None + + for i in range(Z.shape[0]): + row = Z[i, :] + + fi = int_floor(row[0], xp) + fj = int_floor(row[1], xp) + 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], row[2]) + # ^ id ^ left ^ right ^ dist + if row[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 an 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 + -------- + >>> import numpy as np + >>> from scipy.cluster import hierarchy + >>> rng = np.random.default_rng() + >>> X = rng.standard_normal((10, 10)) + >>> Z = hierarchy.ward(X) + >>> hierarchy.leaves_list(Z) + array([0, 3, 1, 9, 2, 5, 7, 4, 6, 8], dtype=int32) + >>> hierarchy.leaves_list(hierarchy.optimal_leaf_ordering(Z, X)) + array([3, 0, 2, 5, 7, 4, 8, 6, 9, 1], dtype=int32) + + """ + xp = array_namespace(Z, y) + Z = _asarray(Z, order='C', xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + y = _asarray(y, order='C', dtype=xp.float64, xp=xp) + + if y.ndim == 1: + distance.is_valid_y(y, throw=True, name='y') + elif y.ndim == 2: + if (y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0) and + np.all(y >= 0) and np.allclose(y, y.T)): + warnings.warn('The symmetric non-negative hollow observation ' + 'matrix looks suspiciously like an uncondensed ' + 'distance matrix', + ClusterWarning, stacklevel=2) + y = distance.pdist(y, metric) + y = xp.asarray(y) + else: + raise ValueError("`y` must be 1 or 2 dimensional.") + + if not xp.all(xp.isfinite(y)): + raise ValueError("The condensed distance matrix must contain only " + "finite values.") + + Z = np.asarray(Z) + y = np.asarray(y) + return xp.asarray(_optimal_leaf_ordering.optimal_leaf_ordering(Z, y)) + + +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. + + """ + xp = array_namespace(Z, Y) + # Ensure float64 C-contiguous array. Cython code doesn't deal with striding. + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + n = Z.shape[0] + 1 + zz = np.zeros((n * (n-1)) // 2, dtype=np.float64) + + Z = np.asarray(Z) + _hierarchy.cophenetic_distances(Z, zz, int(n)) + zz = xp.asarray(zz) + if Y is None: + return zz + + Y = _asarray(Y, order='C', xp=xp) + distance.is_valid_y(Y, throw=True, name='Y') + + z = xp.mean(zz) + y = xp.mean(Y) + Yy = Y - y + Zz = zz - z + numerator = (Yy * Zz) + denomA = Yy**2 + denomB = Zz**2 + c = xp.sum(numerator) / xp.sqrt(xp.sum(denomA) * xp.sum(denomB)) + 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]]) + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + 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.') + + n = Z.shape[0] + 1 + R = np.zeros((n - 1, 4), dtype=np.float64) + + Z = np.asarray(Z) + _hierarchy.inconsistent(Z, R, int(n), int(d)) + R = xp.asarray(R) + 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 indices are reduced by 1 + (MATLAB format uses 1-indexing, whereas SciPy uses 0-indexing). + + """ + xp = array_namespace(Z) + Z = _asarray(Z, dtype=xp.float64, order='C', xp=xp) + Zs = Z.shape + + # If it's empty, return it. + if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0): + return copy(Z, xp=xp) + + if len(Zs) != 2: + raise ValueError("The linkage array must be rectangular.") + + # If it contains no rows, return it. + if Zs[0] == 0: + return copy(Z, xp=xp) + + Zpart = copy(Z, xp=xp) + if xp.min(Zpart[:, 0:2]) != 1.0 and xp.max(Zpart[:, 0:2]) != 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.float64) + Zpart = np.asarray(Zpart) + _hierarchy.calculate_cluster_sizes(Zpart, CS, int(Zs[0]) + 1) + res = np.hstack([Zpart, CS.reshape(Zs[0], 1)]) + return xp.asarray(res) + + +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. + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + Zs = Z.shape + if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0): + return copy(Z, xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + ZP = copy(Z[:, 0:3], xp=xp) + 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 it 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. + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='c', xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + # We expect the i'th value to be greater than its successor. + return xp.all(Z[1:, 2] >= Z[:-1, 2]) + + +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 + + """ + xp = array_namespace(R) + R = _asarray(R, order='c', xp=xp) + valid = True + name_str = "%r " % name if name else '' + try: + if R.dtype != xp.float64: + 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 xp.any(R[:, 0] < 0): + raise ValueError('Inconsistency matrix %scontains negative link ' + 'height means.' % name_str) + if xp.any(R[:, 1] < 0): + raise ValueError('Inconsistency matrix %scontains negative link ' + 'height standard deviations.' % name_str) + if xp.any(R[:, 2] < 0): + 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 2-D 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, if 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 + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='c', xp=xp) + valid = True + name_str = "%r " % name if name else '' + try: + if Z.dtype != xp.float64: + 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 (xp.any(Z[:, 0] < 0) or xp.any(Z[:, 1] < 0)): + raise ValueError('Linkage %scontains negative indices.' % + name_str) + if xp.any(Z[:, 2] < 0): + raise ValueError('Linkage %scontains negative distances.' % + name_str) + if xp.any(Z[:, 3] < 0): + 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 range(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 range(0, n - 1): + used_more_than_once = ( + (float(Z[i, 0]) in chosen) + or (float(Z[i, 1]) in chosen) + or Z[i, 0] == Z[i, 1] + ) + if used_more_than_once: + return True + chosen.add(float(Z[i, 0])) + chosen.add(float(Z[i, 1])) + return False + + +def _check_hierarchy_not_all_clusters_used(Z): + n = Z.shape[0] + 1 + chosen = set() + for i in range(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 + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='c', xp=xp) + 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) + xp = array_namespace(Z, Y) + Z = _asarray(Z, order='c', xp=xp) + Y = _asarray(Y, order='c', xp=xp) + 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) + fcluster(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) + fcluster(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. + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + n = Z.shape[0] + 1 + T = np.zeros((n,), dtype='i') + + if monocrit is not None: + monocrit = np.asarray(monocrit, order='C', dtype=np.float64) + + Z = np.asarray(Z) + monocrit = np.asarray(monocrit) + if criterion == 'inconsistent': + if R is None: + R = inconsistent(Z, depth) + else: + R = _asarray(R, order='C', dtype=xp.float64, xp=xp) + is_valid_im(R, throw=True, name='R') + # Since the C code does not support striding using strides. + # The dimensions are used instead. + R = np.asarray(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), t) + elif criterion == 'monocrit': + _hierarchy.cluster_monocrit(Z, monocrit, T, float(t), int(n)) + elif criterion == 'maxclust_monocrit': + _hierarchy.cluster_maxclust_monocrit(Z, monocrit, T, int(n), int(t)) + else: + raise ValueError('Invalid cluster formation criterion: %s' + % str(criterion)) + return xp.asarray(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 1-D 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 or function, 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. + + """ + xp = array_namespace(X) + X = _asarray(X, order='C', dtype=xp.float64, xp=xp) + + if X.ndim != 2: + raise TypeError('The observation matrix X must be an n by m ' + 'array.') + + Y = distance.pdist(X, metric=metric) + Y = xp.asarray(Y) + Z = linkage(Y, method=method) + if R is None: + R = inconsistent(Z, d=depth) + else: + R = _asarray(R, order='c', xp=xp) + 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 + indices 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() + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='C', xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + n = Z.shape[0] + 1 + ML = np.zeros((n,), dtype='i') + Z = np.asarray(Z) + _hierarchy.prelist(Z, ML, n) + return xp.asarray(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='C0'): + # 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 as e: + raise ImportError("You must install the matplotlib library to plot " + "the dendrogram. Use no_plot=True to calculate the " + "dendrogram without plotting.") from e + + 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() + + +# C0 is used for above threshold color +_link_line_colors_default = ('C1', 'C2', 'C3', 'C4', 'C5', 'C6', 'C7', 'C8', 'C9') +_link_line_colors = list(_link_line_colors_default) + + +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 are matplotlib + default colors C1 to C9). + + Returns + ------- + None + + See Also + -------- + dendrogram + + Notes + ----- + Ability to reset the palette with ``None`` added in SciPy 0.17.0. + + Examples + -------- + >>> import numpy as np + >>> 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'] + ['C1', 'C0', 'C0', 'C0', 'C0'] + >>> hierarchy.set_link_color_palette(['c', 'm', 'y', 'k']) + >>> dn = hierarchy.dendrogram(Z, no_plot=True, above_threshold_color='b') + >>> 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 = _link_line_colors_default + elif not isinstance(palette, (list, tuple)): + raise TypeError("palette must be a list or tuple") + _ptypes = [isinstance(p, str) for p in palette] + + if False in _ptypes: + raise TypeError("all palette list elements must be color strings") + + global _link_line_colors + _link_line_colors = 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='C0'): + """ + 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 final 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 + with de default matplotlib color ``'C0'``. If :math:`t` is less + than or equal to zero, all nodes are colored ``'C0'``. + 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 + sequence, with ``n == Z.shape[0] + 1``. 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) + + # leaf_label_func can also be used together with ``truncate_mode``, + # in which case you will get your leaves labeled after truncation: + dendrogram(Z, leaf_label_func=llf, leaf_rotation=90, + truncate_mode='level', p=2) + + 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 ``'C0'``. + + 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. + + ``'leaves_color_list'`` + A list of color names. The k'th element represents the color of the + k'th leaf. + + 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 + -------- + >>> import numpy as np + >>> 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. + xp = array_namespace(Z) + Z = _asarray(Z, order='c', xp=xp) + + if orientation not in ["top", "left", "bottom", "right"]: + raise ValueError("orientation must be one of 'top', 'left', " + "'bottom', or 'right'") + + if labels is not None: + try: + len_labels = len(labels) + except (TypeError, AttributeError): + len_labels = labels.shape[0] + if Z.shape[0] + 1 != len_labels: + raise ValueError("Dimensions of Z and labels must be consistent.") + + is_valid_linkage(Z, throw=True, name='Z') + Zs = Z.shape + n = Zs[0] + 1 + if isinstance(p, (int, float)): + p = int(p) + else: + raise TypeError('The second argument must be a number') + + if truncate_mode not in ('lastp', 'mtica', 'level', 'none', None): + # 'mtica' is kept working for backwards compat. + raise ValueError('Invalid truncation mode.') + + if truncate_mode == 'lastp': + 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, str) and + color_threshold == 'default'): + color_threshold = max(Z[:, 2]) * 0.7 + + R = {'icoord': icoord_list, 'dcoord': dcoord_list, 'ivl': ivl, + 'leaves': lvs, 'color_list': color_list} + + # Empty list will be filled in _dendrogram_calculate_info + contraction_marks = [] if show_contracted else None + + _dendrogram_calculate_info( + Z=Z, p=p, + truncate_mode=truncate_mode, + color_threshold=color_threshold, + get_leaves=get_leaves, + orientation=orientation, + labels=labels, + count_sort=count_sort, + distance_sort=distance_sort, + show_leaf_counts=show_leaf_counts, + i=2*n - 2, + iv=0.0, + ivl=ivl, + n=n, + icoord_list=icoord_list, + dcoord_list=dcoord_list, + lvs=lvs, + current_color=current_color, + color_list=color_list, + currently_below_threshold=currently_below_threshold, + leaf_label_func=leaf_label_func, + contraction_marks=contraction_marks, + link_color_func=link_color_func, + above_threshold_color=above_threshold_color) + + if not no_plot: + mh = max(Z[:, 2]) + _plot_dendrogram(icoord_list, dcoord_list, ivl, p, n, mh, orientation, + no_labels, color_list, + leaf_font_size=leaf_font_size, + leaf_rotation=leaf_rotation, + contraction_marks=contraction_marks, + ax=ax, + above_threshold_color=above_threshold_color) + + R["leaves_color_list"] = _get_leaves_color_list(R) + + return R + + +def _get_leaves_color_list(R): + leaves_color_list = [None] * len(R['leaves']) + for link_x, link_y, link_color in zip(R['icoord'], + R['dcoord'], + R['color_list']): + for (xi, yi) in zip(link_x, link_y): + if yi == 0.0 and (xi % 5 == 0 and xi % 2 == 1): + # if yi is 0.0 and xi is divisible by 5 and odd, + # the point is a leaf + # xi of leaves are 5, 15, 25, 35, ... (see `iv_ticks`) + # index of leaves are 0, 1, 2, 3, ... as below + leaf_index = (int(xi) - 5) // 10 + # each leaf has a same color of its link. + leaves_color_list[leaf_index] = link_color + return leaves_color_list + + +def _append_singleton_leaf_node(Z, p, n, level, lvs, ivl, leaf_label_func, + i, labels): + # If the leaf id structure is not None and is a list then the caller + # to dendrogram has indicated that cluster id's corresponding to the + # leaf nodes should be recorded. + + if lvs is not None: + lvs.append(int(i)) + + # If leaf node labels are to be displayed... + if ivl is not None: + # If a leaf_label_func has been provided, the label comes from the + # string returned from the leaf_label_func, which is a function + # passed to dendrogram. + if leaf_label_func: + ivl.append(leaf_label_func(int(i))) + else: + # Otherwise, if the dendrogram caller has passed a labels list + # for the leaf nodes, use it. + if labels is not None: + ivl.append(labels[int(i - n)]) + else: + # Otherwise, use the id as the label for the leaf.x + ivl.append(str(int(i))) + + +def _append_nonsingleton_leaf_node(Z, p, n, level, lvs, ivl, leaf_label_func, + i, labels, show_leaf_counts): + # If the leaf id structure is not None and is a list then the caller + # to dendrogram has indicated that cluster id's corresponding to the + # leaf nodes should be recorded. + + if lvs is not None: + lvs.append(int(i)) + if ivl is not None: + if leaf_label_func: + ivl.append(leaf_label_func(int(i))) + else: + if show_leaf_counts: + ivl.append("(" + str(np.asarray(Z[i - n, 3], dtype=np.int64)) + ")") + else: + ivl.append("") + + +def _append_contraction_marks(Z, iv, i, n, contraction_marks, xp): + _append_contraction_marks_sub(Z, iv, int_floor(Z[i - n, 0], xp), + n, contraction_marks, xp) + _append_contraction_marks_sub(Z, iv, int_floor(Z[i - n, 1], xp), + n, contraction_marks, xp) + + +def _append_contraction_marks_sub(Z, iv, i, n, contraction_marks, xp): + if i >= n: + contraction_marks.append((iv, Z[i - n, 2])) + _append_contraction_marks_sub(Z, iv, int_floor(Z[i - n, 0], xp), + n, contraction_marks, xp) + _append_contraction_marks_sub(Z, iv, int_floor(Z[i - n, 1], xp), + n, contraction_marks, xp) + + +def _dendrogram_calculate_info(Z, p, truncate_mode, + color_threshold=np.inf, get_leaves=True, + orientation='top', labels=None, + count_sort=False, distance_sort=False, + show_leaf_counts=False, i=-1, iv=0.0, + ivl=[], n=0, icoord_list=[], dcoord_list=[], + lvs=None, mhr=False, + current_color=[], color_list=[], + currently_below_threshold=[], + leaf_label_func=None, level=0, + contraction_marks=None, + link_color_func=None, + above_threshold_color='C0'): + """ + Calculate the endpoints of the links as well as the labels for the + the dendrogram rooted at the node with index i. iv is the independent + variable value to plot the left-most leaf node below the root node i + (if orientation='top', this would be the left-most x value where the + plotting of this root node i and its descendents should begin). + + ivl is a list to store the labels of the leaf nodes. The leaf_label_func + is called whenever ivl != None, labels == None, and + leaf_label_func != None. When ivl != None and labels != None, the + labels list is used only for labeling the leaf nodes. When + ivl == None, no labels are generated for leaf nodes. + + When get_leaves==True, a list of leaves is built as they are visited + in the dendrogram. + + Returns a tuple with l being the independent variable coordinate that + corresponds to the midpoint of cluster to the left of cluster i if + i is non-singleton, otherwise the independent coordinate of the leaf + node if i is a leaf node. + + Returns + ------- + A tuple (left, w, h, md), where: + * left is the independent variable coordinate of the center of the + the U of the subtree + + * w is the amount of space used for the subtree (in independent + variable units) + + * h is the height of the subtree in dependent variable units + + * md is the ``max(Z[*,2]``) for all nodes ``*`` below and including + the target node. + + """ + xp = array_namespace(Z) + if n == 0: + raise ValueError("Invalid singleton cluster count n.") + + if i == -1: + raise ValueError("Invalid root cluster index i.") + + if truncate_mode == 'lastp': + # If the node is a leaf node but corresponds to a non-singleton + # cluster, its label is either the empty string or the number of + # original observations belonging to cluster i. + if 2*n - p > i >= n: + d = Z[i - n, 2] + _append_nonsingleton_leaf_node(Z, p, n, level, lvs, ivl, + leaf_label_func, i, labels, + show_leaf_counts) + if contraction_marks is not None: + _append_contraction_marks(Z, iv + 5.0, i, n, contraction_marks, xp) + return (iv + 5.0, 10.0, 0.0, d) + elif i < n: + _append_singleton_leaf_node(Z, p, n, level, lvs, ivl, + leaf_label_func, i, labels) + return (iv + 5.0, 10.0, 0.0, 0.0) + elif truncate_mode == 'level': + if i > n and level > p: + d = Z[i - n, 2] + _append_nonsingleton_leaf_node(Z, p, n, level, lvs, ivl, + leaf_label_func, i, labels, + show_leaf_counts) + if contraction_marks is not None: + _append_contraction_marks(Z, iv + 5.0, i, n, contraction_marks, xp) + return (iv + 5.0, 10.0, 0.0, d) + elif i < n: + _append_singleton_leaf_node(Z, p, n, level, lvs, ivl, + leaf_label_func, i, labels) + return (iv + 5.0, 10.0, 0.0, 0.0) + + # Otherwise, only truncate if we have a leaf node. + # + # Only place leaves if they correspond to original observations. + if i < n: + _append_singleton_leaf_node(Z, p, n, level, lvs, ivl, + leaf_label_func, i, labels) + return (iv + 5.0, 10.0, 0.0, 0.0) + + # !!! Otherwise, we don't have a leaf node, so work on plotting a + # non-leaf node. + # Actual indices of a and b + aa = int_floor(Z[i - n, 0], xp) + ab = int_floor(Z[i - n, 1], xp) + if aa >= n: + # The number of singletons below cluster a + na = Z[aa - n, 3] + # The distance between a's two direct children. + da = Z[aa - n, 2] + else: + na = 1 + da = 0.0 + if ab >= n: + nb = Z[ab - n, 3] + db = Z[ab - n, 2] + else: + nb = 1 + db = 0.0 + + if count_sort == 'ascending' or count_sort is True: + # If a has a count greater than b, it and its descendents should + # be drawn to the right. Otherwise, to the left. + if na > nb: + # The cluster index to draw to the left (ua) will be ab + # and the one to draw to the right (ub) will be aa + ua = ab + ub = aa + else: + ua = aa + ub = ab + elif count_sort == 'descending': + # If a has a count less than or equal to b, it and its + # descendents should be drawn to the left. Otherwise, to + # the right. + if na > nb: + ua = aa + ub = ab + else: + ua = ab + ub = aa + elif distance_sort == 'ascending' or distance_sort is True: + # If a has a distance greater than b, it and its descendents should + # be drawn to the right. Otherwise, to the left. + if da > db: + ua = ab + ub = aa + else: + ua = aa + ub = ab + elif distance_sort == 'descending': + # If a has a distance less than or equal to b, it and its + # descendents should be drawn to the left. Otherwise, to + # the right. + if da > db: + ua = aa + ub = ab + else: + ua = ab + ub = aa + else: + ua = aa + ub = ab + + # Updated iv variable and the amount of space used. + (uiva, uwa, uah, uamd) = \ + _dendrogram_calculate_info( + Z=Z, p=p, + truncate_mode=truncate_mode, + color_threshold=color_threshold, + get_leaves=get_leaves, + orientation=orientation, + labels=labels, + count_sort=count_sort, + distance_sort=distance_sort, + show_leaf_counts=show_leaf_counts, + i=ua, iv=iv, ivl=ivl, n=n, + icoord_list=icoord_list, + dcoord_list=dcoord_list, lvs=lvs, + current_color=current_color, + color_list=color_list, + currently_below_threshold=currently_below_threshold, + leaf_label_func=leaf_label_func, + level=level + 1, contraction_marks=contraction_marks, + link_color_func=link_color_func, + above_threshold_color=above_threshold_color) + + h = Z[i - n, 2] + if h >= color_threshold or color_threshold <= 0: + c = above_threshold_color + + if currently_below_threshold[0]: + current_color[0] = (current_color[0] + 1) % len(_link_line_colors) + currently_below_threshold[0] = False + else: + currently_below_threshold[0] = True + c = _link_line_colors[current_color[0]] + + (uivb, uwb, ubh, ubmd) = \ + _dendrogram_calculate_info( + Z=Z, p=p, + truncate_mode=truncate_mode, + color_threshold=color_threshold, + get_leaves=get_leaves, + orientation=orientation, + labels=labels, + count_sort=count_sort, + distance_sort=distance_sort, + show_leaf_counts=show_leaf_counts, + i=ub, iv=iv + uwa, ivl=ivl, n=n, + icoord_list=icoord_list, + dcoord_list=dcoord_list, lvs=lvs, + current_color=current_color, + color_list=color_list, + currently_below_threshold=currently_below_threshold, + leaf_label_func=leaf_label_func, + level=level + 1, contraction_marks=contraction_marks, + link_color_func=link_color_func, + above_threshold_color=above_threshold_color) + + max_dist = max(uamd, ubmd, h) + + icoord_list.append([uiva, uiva, uivb, uivb]) + dcoord_list.append([uah, h, h, ubh]) + if link_color_func is not None: + v = link_color_func(int(i)) + if not isinstance(v, str): + raise TypeError("link_color_func must return a matplotlib " + "color string!") + color_list.append(v) + else: + color_list.append(c) + + return (((uiva + uivb) / 2), uwa + uwb, h, max_dist) + + +def is_isomorphic(T1, T2): + """ + Determine if two different cluster assignments are equivalent. + + Parameters + ---------- + T1 : array_like + An assignment of singleton cluster ids to flat cluster ids. + T2 : array_like + An assignment of singleton cluster ids to flat cluster ids. + + Returns + ------- + b : bool + Whether the flat cluster assignments `T1` and `T2` are + equivalent. + + See Also + -------- + linkage : for a description of what a linkage matrix is. + fcluster : for the creation of flat cluster assignments. + + Examples + -------- + >>> from scipy.cluster.hierarchy import fcluster, is_isomorphic + >>> from scipy.cluster.hierarchy import single, complete + >>> from scipy.spatial.distance import pdist + + Two flat cluster assignments can be isomorphic if they represent the same + cluster assignment, with different labels. + + For example, we can use the `scipy.cluster.hierarchy.single`: method + and flatten the output to four clusters: + + >>> 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 = single(pdist(X)) + >>> T = fcluster(Z, 1, criterion='distance') + >>> T + array([3, 3, 3, 4, 4, 4, 2, 2, 2, 1, 1, 1], dtype=int32) + + We can then do the same using the + `scipy.cluster.hierarchy.complete`: method: + + >>> Z = complete(pdist(X)) + >>> T_ = fcluster(Z, 1.5, criterion='distance') + >>> T_ + array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) + + As we can see, in both cases we obtain four clusters and all the data + points are distributed in the same way - the only thing that changes + are the flat cluster labels (3 => 1, 4 =>2, 2 =>3 and 4 =>1), so both + cluster assignments are isomorphic: + + >>> is_isomorphic(T, T_) + True + + """ + T1 = np.asarray(T1, order='c') + T2 = np.asarray(T2, order='c') + + T1S = T1.shape + T2S = T2.shape + + if len(T1S) != 1: + raise ValueError('T1 must be one-dimensional.') + if len(T2S) != 1: + raise ValueError('T2 must be one-dimensional.') + if T1S[0] != T2S[0]: + raise ValueError('T1 and T2 must have the same number of elements.') + n = T1S[0] + d1 = {} + d2 = {} + for i in range(0, n): + if T1[i] in d1: + if T2[i] not in d2: + return False + if d1[T1[i]] != T2[i] or d2[T2[i]] != T1[i]: + return False + elif T2[i] in d2: + return False + else: + d1[T1[i]] = T2[i] + d2[T2[i]] = T1[i] + return True + + +def maxdists(Z): + """ + Return the maximum distance between any non-singleton cluster. + + Parameters + ---------- + Z : ndarray + The hierarchical clustering encoded as a matrix. See + ``linkage`` for more information. + + Returns + ------- + maxdists : ndarray + A ``(n-1)`` sized numpy array of doubles; ``MD[i]`` represents + the maximum distance between any cluster (including + singletons) below and including the node with index i. More + specifically, ``MD[i] = Z[Q(i)-n, 2].max()`` where ``Q(i)`` is the + set of all node indices below and including node i. + + See Also + -------- + linkage : for a description of what a linkage matrix is. + is_monotonic : for testing for monotonicity of a linkage matrix. + + Examples + -------- + >>> from scipy.cluster.hierarchy import median, maxdists + >>> from scipy.spatial.distance import pdist + + Given a linkage matrix ``Z``, `scipy.cluster.hierarchy.maxdists` + computes for each new cluster generated (i.e., for each row of the linkage + matrix) what is the maximum distance between any two child clusters. + + Due to the nature of hierarchical clustering, in many cases this is going + to be just the distance between the two child clusters that were merged + to form the current one - that is, Z[:,2]. + + However, for non-monotonic cluster assignments such as + `scipy.cluster.hierarchy.median` clustering this is not always the + case: There may be cluster formations were the distance between the two + clusters merged is smaller than the distance between their children. + + We can see this in an example: + + >>> 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 = 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. ]]) + >>> maxdists(Z) + array([1. , 1. , 1. , 1. , 1.11803399, + 1.11803399, 1.11803399, 1.11803399, 3. , 3.5 , + 3.5 ]) + + Note that while the distance between the two clusters merged when creating the + last cluster is 3.25, there are two children (clusters 16 and 17) whose distance + is larger (3.5). Thus, `scipy.cluster.hierarchy.maxdists` returns 3.5 in + this case. + + """ + xp = array_namespace(Z) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + n = Z.shape[0] + 1 + MD = np.zeros((n - 1,)) + Z = np.asarray(Z) + _hierarchy.get_max_dist_for_each_cluster(Z, MD, int(n)) + MD = xp.asarray(MD) + return MD + + +def maxinconsts(Z, R): + """ + Return the maximum inconsistency coefficient for each + non-singleton cluster and its children. + + Parameters + ---------- + Z : ndarray + The hierarchical clustering encoded as a matrix. See + `linkage` for more information. + R : ndarray + The inconsistency matrix. + + Returns + ------- + MI : ndarray + A monotonic ``(n-1)``-sized numpy array of doubles. + + 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 median, inconsistent, maxinconsts + >>> 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 = median(pdist(X)) + >>> R = inconsistent(Z) + >>> 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. ]]) + >>> R + array([[1. , 0. , 1. , 0. ], + [1. , 0. , 1. , 0. ], + [1. , 0. , 1. , 0. ], + [1. , 0. , 1. , 0. ], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.74535599, 1.08655358, 3. , 1.15470054], + [1.91202266, 1.37522872, 3. , 1.15470054], + [3.25 , 0.25 , 3. , 0. ]]) + + Here, `scipy.cluster.hierarchy.maxinconsts` can be used to compute + the maximum value of the inconsistency statistic (the last column of + ``R``) for each non-singleton cluster and its children: + + >>> maxinconsts(Z, R) + array([0. , 0. , 0. , 0. , 0.70710678, + 0.70710678, 0.70710678, 0.70710678, 1.15470054, 1.15470054, + 1.15470054]) + + """ + xp = array_namespace(Z, R) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + R = _asarray(R, order='C', dtype=xp.float64, xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + is_valid_im(R, throw=True, name='R') + + n = Z.shape[0] + 1 + if Z.shape[0] != R.shape[0]: + raise ValueError("The inconsistency matrix and linkage matrix each " + "have a different number of rows.") + MI = np.zeros((n - 1,)) + Z = np.asarray(Z) + R = np.asarray(R) + _hierarchy.get_max_Rfield_for_each_cluster(Z, R, MI, int(n), 3) + MI = xp.asarray(MI) + return MI + + +def maxRstat(Z, R, i): + """ + Return the maximum statistic for each non-singleton cluster and its + children. + + Parameters + ---------- + Z : array_like + The hierarchical clustering encoded as a matrix. See `linkage` for more + information. + R : array_like + The inconsistency matrix. + i : int + The column of `R` to use as the statistic. + + Returns + ------- + MR : ndarray + Calculates the maximum statistic for the i'th column of the + inconsistency matrix `R` for each non-singleton cluster + node. ``MR[j]`` is the maximum over ``R[Q(j)-n, i]``, where + ``Q(j)`` the set of all node ids corresponding to nodes below + and including ``j``. + + 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 median, inconsistent, maxRstat + >>> 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 = median(pdist(X)) + >>> R = inconsistent(Z) + >>> R + array([[1. , 0. , 1. , 0. ], + [1. , 0. , 1. , 0. ], + [1. , 0. , 1. , 0. ], + [1. , 0. , 1. , 0. ], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.05901699, 0.08346263, 2. , 0.70710678], + [1.74535599, 1.08655358, 3. , 1.15470054], + [1.91202266, 1.37522872, 3. , 1.15470054], + [3.25 , 0.25 , 3. , 0. ]]) + + `scipy.cluster.hierarchy.maxRstat` can be used to compute + the maximum value of each column of ``R``, for each non-singleton + cluster and its children: + + >>> maxRstat(Z, R, 0) + array([1. , 1. , 1. , 1. , 1.05901699, + 1.05901699, 1.05901699, 1.05901699, 1.74535599, 1.91202266, + 3.25 ]) + >>> maxRstat(Z, R, 1) + array([0. , 0. , 0. , 0. , 0.08346263, + 0.08346263, 0.08346263, 0.08346263, 1.08655358, 1.37522872, + 1.37522872]) + >>> maxRstat(Z, R, 3) + array([0. , 0. , 0. , 0. , 0.70710678, + 0.70710678, 0.70710678, 0.70710678, 1.15470054, 1.15470054, + 1.15470054]) + + """ + xp = array_namespace(Z, R) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + R = _asarray(R, order='C', dtype=xp.float64, xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + is_valid_im(R, throw=True, name='R') + + if not isinstance(i, int): + raise TypeError('The third argument must be an integer.') + + if i < 0 or i > 3: + raise ValueError('i must be an integer between 0 and 3 inclusive.') + + if Z.shape[0] != R.shape[0]: + raise ValueError("The inconsistency matrix and linkage matrix each " + "have a different number of rows.") + + n = Z.shape[0] + 1 + MR = np.zeros((n - 1,)) + Z = np.asarray(Z) + R = np.asarray(R) + _hierarchy.get_max_Rfield_for_each_cluster(Z, R, MR, int(n), i) + MR = xp.asarray(MR) + return MR + + +def leaders(Z, T): + """ + Return the root nodes in a hierarchical clustering. + + Returns the root nodes in a hierarchical clustering corresponding + to a cut defined by a flat cluster assignment vector ``T``. See + the ``fcluster`` function for more information on the format of ``T``. + + For each flat cluster :math:`j` of the :math:`k` flat clusters + represented in the n-sized flat cluster assignment vector ``T``, + this function finds the lowest cluster node :math:`i` in the linkage + tree Z, such that: + + * leaf descendants belong only to flat cluster j + (i.e., ``T[p]==j`` for all :math:`p` in :math:`S(i)`, where + :math:`S(i)` is the set of leaf ids of descendant leaf nodes + with cluster node :math:`i`) + + * there does not exist a leaf that is not a descendant with + :math:`i` that also belongs to cluster :math:`j` + (i.e., ``T[q]!=j`` for all :math:`q` not in :math:`S(i)`). If + this condition is violated, ``T`` is not a valid cluster + assignment vector, and an exception will be thrown. + + Parameters + ---------- + Z : ndarray + The hierarchical clustering encoded as a matrix. See + `linkage` for more information. + T : ndarray + The flat cluster assignment vector. + + Returns + ------- + L : ndarray + The leader linkage node id's stored as a k-element 1-D array, + where ``k`` is the number of flat clusters found in ``T``. + + ``L[j]=i`` is the linkage cluster node id that is the + leader of flat cluster with id M[j]. If ``i < n``, ``i`` + corresponds to an original observation, otherwise it + corresponds to a non-singleton cluster. + M : ndarray + The leader linkage node id's stored as a k-element 1-D array, where + ``k`` is the number of flat clusters found in ``T``. This allows the + set of flat cluster ids to be any arbitrary set of ``k`` integers. + + For example: if ``L[3]=2`` and ``M[3]=8``, the flat cluster with + id 8's leader is linkage node 2. + + See Also + -------- + fcluster : for the creation of flat cluster assignments. + + Examples + -------- + >>> from scipy.cluster.hierarchy import ward, fcluster, leaders + >>> from scipy.spatial.distance import pdist + + Given a linkage matrix ``Z`` - obtained after apply a clustering method + to a dataset ``X`` - and a flat cluster assignment array ``T``: + + >>> 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. ]]) + + >>> T = fcluster(Z, 3, criterion='distance') + >>> T + array([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4], dtype=int32) + + `scipy.cluster.hierarchy.leaders` returns the indices of the nodes + in the dendrogram that are the leaders of each flat cluster: + + >>> L, M = leaders(Z, T) + >>> L + array([16, 17, 18, 19], dtype=int32) + + (remember that indices 0-11 point to the 12 data points in ``X``, + whereas indices 12-22 point to the 11 rows of ``Z``) + + `scipy.cluster.hierarchy.leaders` also returns the indices of + the flat clusters in ``T``: + + >>> M + array([1, 2, 3, 4], dtype=int32) + + """ + xp = array_namespace(Z, T) + Z = _asarray(Z, order='C', dtype=xp.float64, xp=xp) + T = _asarray(T, order='C', xp=xp) + is_valid_linkage(Z, throw=True, name='Z') + + if T.dtype != xp.int32: + raise TypeError('T must be a 1-D array of dtype int32.') + + if T.shape[0] != Z.shape[0] + 1: + raise ValueError('Mismatch: len(T)!=Z.shape[0] + 1.') + + n_clusters = int(xp.unique_values(T).shape[0]) + n_obs = int(Z.shape[0] + 1) + L = np.zeros(n_clusters, dtype=np.int32) + M = np.zeros(n_clusters, dtype=np.int32) + Z = np.asarray(Z) + T = np.asarray(T, dtype=np.int32) + s = _hierarchy.leaders(Z, T, L, M, n_clusters, n_obs) + if s >= 0: + raise ValueError(('T is not a valid assignment vector. Error found ' + 'when examining linkage node %d (< 2n-1).') % s) + L, M = xp.asarray(L), xp.asarray(M) + return (L, M)