scipy/scipy

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 """ ======================================================== Hierarchical clustering (:mod:scipy.cluster.hierarchy) ======================================================== .. currentmodule:: scipy.cluster.hierarchy These functions cut hierarchical clusterings into flat clusterings or find the roots of the forest formed by a cut by providing the flat cluster ids of each observation. .. autosummary:: :toctree: generated/ fcluster fclusterdata leaders These are routines for agglomerative clustering. .. autosummary:: :toctree: generated/ linkage single complete average weighted centroid median ward These routines compute statistics on hierarchies. .. autosummary:: :toctree: generated/ cophenet from_mlab_linkage inconsistent maxinconsts maxdists maxRstat to_mlab_linkage Routines for visualizing flat clusters. .. autosummary:: :toctree: generated/ dendrogram These are data structures and routines for representing hierarchies as tree objects. .. autosummary:: :toctree: generated/ ClusterNode leaves_list to_tree cut_tree 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 References ---------- .. [1] "Statistics toolbox." API Reference Documentation. The MathWorks. http://www.mathworks.com/access/helpdesk/help/toolbox/stats/. Accessed October 1, 2007. .. [2] "Hierarchical clustering." API Reference Documentation. The Wolfram Research, Inc. http://reference.wolfram.com/mathematica/HierarchicalClustering/tutorial/ HierarchicalClustering.html. Accessed October 1, 2007. .. [3] Gower, JC and Ross, GJS. "Minimum Spanning Trees and Single Linkage Cluster Analysis." Applied Statistics. 18(1): pp. 54--64. 1969. .. [4] Ward Jr, JH. "Hierarchical grouping to optimize an objective function." Journal of the American Statistical Association. 58(301): pp. 236--44. 1963. .. [5] Johnson, SC. "Hierarchical clustering schemes." Psychometrika. 32(2): pp. 241--54. 1966. .. [6] Sneath, PH and Sokal, RR. "Numerical taxonomy." Nature. 193: pp. 855--60. 1962. .. [7] Batagelj, V. "Comparing resemblance measures." Journal of Classification. 12: pp. 73--90. 1995. .. [8] Sokal, RR and Michener, CD. "A statistical method for evaluating systematic relationships." Scientific Bulletins. 38(22): pp. 1409--38. 1958. .. [9] Edelbrock, C. "Mixture model tests of hierarchical clustering algorithms: the problem of classifying everybody." Multivariate Behavioral Research. 14: pp. 367--84. 1979. .. [10] Jain, A., and Dubes, R., "Algorithms for Clustering Data." Prentice-Hall. Englewood Cliffs, NJ. 1988. .. [11] Fisher, RA "The use of multiple measurements in taxonomic problems." Annals of Eugenics, 7(2): 179-188. 1936 * MATLAB and MathWorks are registered trademarks of The MathWorks, Inc. * Mathematica is a registered trademark of The Wolfram Research, Inc. """ from __future__ import division, print_function, absolute_import # Copyright (C) Damian Eads, 2007-2008. New BSD License. # hierarchy.py (derived from cluster.py, http://scipy-cluster.googlecode.com) # # Author: Damian Eads # Date: September 22, 2007 # # Copyright (c) 2007, 2008, Damian Eads # # All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions # are met: # - Redistributions of source code must retain the above # copyright notice, this list of conditions and the # following disclaimer. # - Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer # in the documentation and/or other materials provided with the # distribution. # - Neither the name of the author nor the names of its # contributors may be used to endorse or promote products derived # from this software without specific prior written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS # "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT # LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR # A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT # OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, # SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT # LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, # DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY # THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT # (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE # OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. import warnings import bisect from collections import deque import numpy as np from . import _hierarchy import scipy.spatial.distance as distance from scipy._lib.six import string_types from scipy._lib.six import xrange _LINKAGE_METHODS = {'single': 0, 'complete': 1, 'average': 2, 'centroid': 3, 'median': 4, 'ward': 5, 'weighted': 6} _EUCLIDEAN_METHODS = ('centroid', 'median', 'ward') __all__ = ['ClusterNode', 'average', 'centroid', 'complete', 'cophenet', 'correspond', 'cut_tree', 'dendrogram', 'fcluster', 'fclusterdata', 'from_mlab_linkage', 'inconsistent', 'is_isomorphic', 'is_monotonic', 'is_valid_im', 'is_valid_linkage', 'leaders', 'leaves_list', 'linkage', 'maxRstat', 'maxdists', 'maxinconsts', 'median', 'num_obs_linkage', 'set_link_color_palette', 'single', 'to_mlab_linkage', 'to_tree', 'ward', 'weighted', 'distance'] class ClusterWarning(UserWarning): pass def _warning(s): warnings.warn('scipy.cluster: %s' % s, ClusterWarning, stacklevel=3) def _copy_array_if_base_present(a): """ Copies the array if its base points to a parent array. """ if a.base is not None: return a.copy() elif np.issubsctype(a, np.float32): return np.array(a, dtype=np.double) else: return a def _copy_arrays_if_base_present(T): """ Accepts a tuple of arrays T. Copies the array T[i] if its base array points to an actual array. Otherwise, the reference is just copied. This is useful if the arrays are being passed to a C function that does not do proper striding. """ l = [_copy_array_if_base_present(a) for a in T] return l def _randdm(pnts): """ Generates a random distance matrix stored in condensed form. A pnts * (pnts - 1) / 2 sized vector is returned. """ if pnts >= 2: D = np.random.rand(pnts * (pnts - 1) / 2) else: raise ValueError("The number of points in the distance matrix " "must be at least 2.") return D def single(y): """ Performs 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. """ return linkage(y, method='single', metric='euclidean') def complete(y): """ Performs 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 """ return linkage(y, method='complete', metric='euclidean') def average(y): """ Performs 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 the linkage function documentation for more information on its structure. See Also -------- linkage: for advanced creation of hierarchical clusterings. """ return linkage(y, method='average', metric='euclidean') def weighted(y): """ Performs 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 the linkage function documentation for more information on its structure. See Also -------- linkage : for advanced creation of hierarchical clusterings. """ return linkage(y, method='weighted', metric='euclidean') def centroid(y): """ Performs 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. See linkage for more information on the return structure and algorithm. 2. Z = centroid(X) Performs centroid/UPGMC linkage on the observation matrix X using Euclidean distance as the distance metric. See linkage for more information on the return structure and algorithm. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. Returns ------- Z : ndarray 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. """ return linkage(y, method='centroid', metric='euclidean') def median(y): """ Performs median/WPGMC linkage. See linkage for more information on the return structure and algorithm. The following are common calling conventions: 1. Z = median(y) Performs median/WPGMC linkage on the condensed distance matrix y. See linkage for more information on the return structure and algorithm. 2. Z = median(X) Performs median/WPGMC linkage on the observation matrix X using Euclidean distance as the distance metric. See linkage for more information on the return structure and algorithm. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. Returns ------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. See Also -------- linkage: for advanced creation of hierarchical clusterings. """ return linkage(y, method='median', metric='euclidean') def ward(y): """ Performs 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 Z. See linkage for more information on the return structure and algorithm. 2. Z = ward(X) Performs Ward's linkage on the observation matrix X using Euclidean distance as the distance metric. See linkage for more information on the return structure and algorithm. Parameters ---------- y : ndarray A condensed distance matrix. A condensed distance matrix is a flat array containing the upper triangular of the distance matrix. This is the form that pdist returns. Alternatively, a collection of m observation vectors in n dimensions may be passed as a m by n array. Returns ------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. See Also -------- linkage: for advanced creation of hierarchical clusterings. """ return linkage(y, method='ward', metric='euclidean') def linkage(y, method='single', metric='euclidean'): """ Performs hierarchical/agglomerative clustering. The input y may be either a 1d compressed distance matrix or a 2d array of observation vectors. If y is a 1d compressed distance matrix, then y must be a :math:{n \\choose 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 chose 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 n dimensions may be passed as an :math:m by :math:n array. 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 distance.pdist function for a list of valid distance metrics. A custom distance function can also be used. See the distance.pdist function for details. Returns ------- Z : ndarray The hierarchical clustering encoded as a linkage matrix. Notes ----- 1. For method 'single' an optimized algorithm called SLINK is implemented, which has :math:O(n^2) time complexity. For methods 'complete', 'average', 'weighted' and 'ward' an algorithm called nearest-neighbors chain is implemented, which too has time complexity :math:O(n^2). For other methods a naive algorithm is implemented with :math:O(n^3) time complexity. All algorithms use :math:O(n^2) memory. Refer to [1]_ for details about the algorithms. 2. Methods 'centroid', 'median' and 'ward' are correctly defined only if Euclidean pairwise metric is used. If y is passed as precomputed pairwise distances, then it is a user responsibility to assure that these distances are in fact Euclidean, otherwise the produced result will be incorrect. References ---------- .. [1] Daniel Mullner, "Modern hierarchical, agglomerative clustering algorithms", arXiv:1109.2378v1 _ , 2011. """ if method not in _LINKAGE_METHODS: raise ValueError("Invalid method: {0}".format(method)) y = _convert_to_double(np.asarray(y, order='c')) if y.ndim == 1: distance.is_valid_y(y, throw=True, name='y') [y] = _copy_arrays_if_base_present([y]) elif y.ndim == 2: if method in _EUCLIDEAN_METHODS and metric != 'euclidean': raise ValueError("Method '{0}' requires the distance metric " "to be Euclidean".format(method)) if y.shape[0] == y.shape[1] and np.allclose(np.diag(y), 0): if np.all(y >= 0) and np.allclose(y, y.T): _warning('The symmetric non-negative hollow observation ' 'matrix looks suspiciously like an uncondensed ' 'distance matrix') y = distance.pdist(y, metric) else: raise ValueError("y must be 1 or 2 dimensional.") n = int(distance.num_obs_y(y)) method_code = _LINKAGE_METHODS[method] if method == 'single': return _hierarchy.slink(y, n) elif method in ['complete', 'average', 'weighted', 'ward']: return _hierarchy.nn_chain(y, n, method_code) else: return _hierarchy.linkage(y, n, method_code) 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. See Also -------- to_tree : for converting a linkage matrix Z into a tree object. """ def __init__(self, id, left=None, right=None, dist=0, count=1): if id < 0: raise ValueError('The id must be non-negative.') if dist < 0: raise ValueError('The distance must be non-negative.') if (left is None and right is not None) or \ (left is not None and right is None): raise ValueError('Only full or proper binary trees are permitted.' ' This node has one child.') if count < 1: raise ValueError('A cluster must contain at least one original ' 'observation.') self.id = id self.left = left self.right = right self.dist = dist if self.left is None: self.count = count else: self.count = left.count + right.count def __lt__(self, node): if not isinstance(node, ClusterNode): raise ValueError("Can't compare ClusterNode " "to type {}".format(type(node))) return self.dist < node.dist def __gt__(self, node): if not isinstance(node, ClusterNode): raise ValueError("Can't compare ClusterNode " "to type {}".format(type(node))) return self.dist > node.dist def __eq__(self, node): if not isinstance(node, ClusterNode): raise ValueError("Can't compare ClusterNode " "to type {}".format(type(node))) return self.dist == node.dist def get_id(self): """ The identifier of the target node. For 0 <= i < n, i corresponds to original observation i. For n <= i < 2n-1, i corresponds to non-singleton cluster formed at iteration i-n. Returns ------- id : int The identifier of the target node. """ return self.id def get_count(self): """ The number of leaf nodes (original observations) belonging to the cluster node nd. If the target node is a leaf, 1 is returned. Returns ------- get_count : int The number of leaf nodes below the target node. """ return self.count def get_left(self): """ Return a reference to the left child tree object. Returns ------- left : ClusterNode The left child of the target node. If the node is a leaf, None is returned. """ return self.left def get_right(self): """ Returns 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): """ Returns 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)): """ Performs 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): """ Returns clustering nodes in bottom-up order by distance. Parameters ---------- Z : scipy.cluster.linkage array The linkage matrix. Returns ------- nodes : list A list of ClusterNode objects. """ q = deque() tree = to_tree(Z) q.append(tree) nodes = [] while q: node = q.popleft() if not node.is_leaf(): bisect.insort_left(nodes, node) q.append(node.get_right()) q.append(node.get_left()) return nodes def cut_tree(Z, n_clusters=None, height=None): """ Given a linkage matrix Z, return the cut tree. Parameters ---------- Z : scipy.cluster.linkage array The linkage matrix. n_clusters : array_like, optional Number of clusters in the tree at the cut point. height : array_like, optional The height at which to cut the tree. Only possible for ultrametric trees. Returns ------- cutree : array An array indicating group membership at each agglomeration step. I.e., for a full cut tree, in the first column each data point is in its own cluster. At the next step, two nodes are merged. Finally all singleton and non-singleton clusters are in one group. If n_clusters or height is given, the columns correspond to the columns of n_clusters or height. Examples -------- >>> from scipy import cluster >>> np.random.seed(23) >>> X = np.random.randn(50, 4) >>> Z = cluster.hierarchy.ward(X) >>> cutree = cluster.hierarchy.cut_tree(Z, n_clusters=[5, 10]) >>> cutree[:10] array([[0, 0], [1, 1], [2, 2], [3, 3], [3, 4], [2, 2], [0, 0], [1, 5], [3, 6], [4, 7]]) """ nobs = num_obs_linkage(Z) nodes = _order_cluster_tree(Z) if height is not None and n_clusters is not None: raise ValueError("At least one of either height or n_clusters " "must be None") elif height is None and n_clusters is None: # return the full cut tree cols_idx = np.arange(nobs) elif height is not None: heights = np.array([x.dist for x in nodes]) cols_idx = np.searchsorted(heights, height) else: cols_idx = nobs - np.searchsorted(np.arange(nobs), n_clusters) try: n_cols = len(cols_idx) except TypeError: # scalar n_cols = 1 cols_idx = np.array([cols_idx]) groups = np.zeros((n_cols, nobs), dtype=int) last_group = np.arange(nobs) if 0 in cols_idx: groups[0] = last_group for i, node in enumerate(nodes): idx = node.pre_order() this_group = last_group.copy() this_group[idx] = last_group[idx].min() this_group[this_group > last_group[idx].max()] -= 1 if i + 1 in cols_idx: groups[np.where(i + 1 == cols_idx)[0]] = this_group last_group = this_group return groups.T def to_tree(Z, rd=False): """ Converts a hierarchical clustering encoded in the matrix Z (by linkage) into an easy-to-use tree object. The reference r to the root ClusterNode object is returned. 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, 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 dictionary mapping cluster ids to ClusterNode references. If a cluster id is less than n, then it corresponds to a singleton cluster (leaf node). See linkage for more information on the assignment of cluster ids to clusters. Returns ------- L : list The pre-order traversal. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') # Number of original objects is equal to the number of rows minus 1. n = Z.shape[0] + 1 # Create a list full of None's to store the node objects d = [None] * (n * 2 - 1) # Create the nodes corresponding to the n original objects. for i in xrange(0, n): d[i] = ClusterNode(i) nd = None for i in xrange(0, n - 1): fi = int(Z[i, 0]) fj = int(Z[i, 1]) if fi > i + n: raise ValueError(('Corrupt matrix Z. Index to derivative cluster ' 'is used before it is formed. See row %d, ' 'column 0') % fi) if fj > i + n: raise ValueError(('Corrupt matrix Z. Index to derivative cluster ' 'is used before it is formed. See row %d, ' 'column 1') % fj) nd = ClusterNode(i + n, d[fi], d[fj], Z[i, 2]) # ^ id ^ left ^ right ^ dist if Z[i, 3] != nd.count: raise ValueError(('Corrupt matrix Z. The count Z[%d,3] is ' 'incorrect.') % i) d[n + i] = nd if rd: return (nd, d) else: return nd def _convert_to_bool(X): if X.dtype != bool: X = X.astype(bool) if not X.flags.contiguous: X = X.copy() return X def _convert_to_double(X): if X.dtype != np.double: X = X.astype(np.double) if not X.flags.contiguous: X = X.copy() return X def cophenet(Z, Y=None): """ Calculates 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. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') Zs = Z.shape n = Zs[0] + 1 zz = np.zeros((n * (n-1)) // 2, dtype=np.double) # Since the C code does not support striding using strides. # The dimensions are used instead. Z = _convert_to_double(Z) _hierarchy.cophenetic_distances(Z, zz, int(n)) if Y is None: return zz Y = np.asarray(Y, order='c') distance.is_valid_y(Y, throw=True, name='Y') z = zz.mean() y = Y.mean() Yy = Y - y Zz = zz - z numerator = (Yy * Zz) denomA = Yy**2 denomB = Zz**2 c = numerator.sum() / np.sqrt((denomA.sum() * denomB.sum())) return (c, zz) def inconsistent(Z, d=2): r""" Calculates inconsistency statistics on a linkage. Note: This function behaves similarly to the MATLAB(TM) inconsistent function. 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 5 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]} """ Z = np.asarray(Z, order='c') Zs = Z.shape is_valid_linkage(Z, throw=True, name='Z') if (not d == np.floor(d)) or d < 0: raise ValueError('The second argument d must be a nonnegative ' 'integer value.') # Since the C code does not support striding using strides. # The dimensions are used instead. [Z] = _copy_arrays_if_base_present([Z]) n = Zs[0] + 1 R = np.zeros((n - 1, 4), dtype=np.double) _hierarchy.inconsistent(Z, R, int(n), int(d)) return R def from_mlab_linkage(Z): """ Converts 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] is 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 this library. """ Z = np.asarray(Z, dtype=np.double, order='c') Zs = Z.shape # If it's empty, return it. if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0): return Z.copy() if len(Zs) != 2: raise ValueError("The linkage array must be rectangular.") # If it contains no rows, return it. if Zs[0] == 0: return Z.copy() Zpart = Z.copy() if Zpart[:, 0:2].min() != 1.0 and Zpart[:, 0:2].max() != 2 * Zs[0]: raise ValueError('The format of the indices is not 1..N') Zpart[:, 0:2] -= 1.0 CS = np.zeros((Zs[0],), dtype=np.double) _hierarchy.calculate_cluster_sizes(Zpart, CS, int(Zs[0]) + 1) return np.hstack([Zpart, CS.reshape(Zs[0], 1)]) def to_mlab_linkage(Z): """ Converts 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 this library. 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. """ Z = np.asarray(Z, order='c', dtype=np.double) Zs = Z.shape if len(Zs) == 0 or (len(Zs) == 1 and Zs[0] == 0): return Z.copy() is_valid_linkage(Z, throw=True, name='Z') ZP = Z[:, 0:3].copy() ZP[:, 0:2] += 1.0 return ZP def is_monotonic(Z): """ Returns 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. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') # We expect the i'th value to be greater than its successor. return (Z[1:, 2] >= Z[:-1, 2]).all() def is_valid_im(R, warning=False, throw=False, name=None): """Returns True if the inconsistency matrix passed is valid. It must be a :math:n by 4 numpy 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. """ R = np.asarray(R, order='c') valid = True name_str = "%r " % name if name else '' try: if type(R) != np.ndarray: raise TypeError('Variable %spassed as inconsistency matrix is not ' 'a numpy array.' % name_str) if R.dtype != np.double: raise TypeError('Inconsistency matrix %smust contain doubles ' '(double).' % name_str) if len(R.shape) != 2: raise ValueError('Inconsistency matrix %smust have shape=2 (i.e. ' 'be two-dimensional).' % name_str) if R.shape[1] != 4: raise ValueError('Inconsistency matrix %smust have 4 columns.' % name_str) if R.shape[0] < 1: raise ValueError('Inconsistency matrix %smust have at least one ' 'row.' % name_str) if (R[:, 0] < 0).any(): raise ValueError('Inconsistency matrix %scontains negative link ' 'height means.' % name_str) if (R[:, 1] < 0).any(): raise ValueError('Inconsistency matrix %scontains negative link ' 'height standard deviations.' % name_str) if (R[:, 2] < 0).any(): raise ValueError('Inconsistency matrix %scontains negative link ' 'counts.' % name_str) except Exception as e: if throw: raise if warning: _warning(str(e)) valid = False return valid def is_valid_linkage(Z, warning=False, throw=False, name=None): """ Checks the validity of a linkage matrix. A linkage matrix is valid if it is a two dimensional array (type double) with :math:n rows and 4 columns. The first two columns must contain indices between 0 and :math:2n-1. For a given row i, the following two expressions have to hold: .. math:: 0 \\leq \\mathtt{Z[i,0]} \\leq i+n-1 0 \\leq Z[i,1] \\leq i+n-1 I.e. a cluster cannot join another cluster unless the cluster being joined has been generated. Parameters ---------- Z : array_like Linkage matrix. warning : bool, optional When True, issues a Python warning if the linkage matrix passed is invalid. throw : bool, optional When True, throws a Python exception if the linkage matrix passed is invalid. name : str, optional This string refers to the variable name of the invalid linkage matrix. Returns ------- b : bool True if the inconsistency matrix is valid. """ Z = np.asarray(Z, order='c') valid = True name_str = "%r " % name if name else '' try: if type(Z) != np.ndarray: raise TypeError('Passed linkage argument %sis not a valid array.' % name_str) if Z.dtype != np.double: raise TypeError('Linkage matrix %smust contain doubles.' % name_str) if len(Z.shape) != 2: raise ValueError('Linkage matrix %smust have shape=2 (i.e. be ' 'two-dimensional).' % name_str) if Z.shape[1] != 4: raise ValueError('Linkage matrix %smust have 4 columns.' % name_str) if Z.shape[0] == 0: raise ValueError('Linkage must be computed on at least two ' 'observations.') n = Z.shape[0] if n > 1: if ((Z[:, 0] < 0).any() or (Z[:, 1] < 0).any()): raise ValueError('Linkage %scontains negative indices.' % name_str) if (Z[:, 2] < 0).any(): raise ValueError('Linkage %scontains negative distances.' % name_str) if (Z[:, 3] < 0).any(): raise ValueError('Linkage %scontains negative counts.' % name_str) if _check_hierarchy_uses_cluster_before_formed(Z): raise ValueError('Linkage %suses non-singleton cluster before ' 'it is formed.' % name_str) if _check_hierarchy_uses_cluster_more_than_once(Z): raise ValueError('Linkage %suses the same cluster more than once.' % name_str) except Exception as e: if throw: raise if warning: _warning(str(e)) valid = False return valid def _check_hierarchy_uses_cluster_before_formed(Z): n = Z.shape[0] + 1 for i in xrange(0, n - 1): if Z[i, 0] >= n + i or Z[i, 1] >= n + i: return True return False def _check_hierarchy_uses_cluster_more_than_once(Z): n = Z.shape[0] + 1 chosen = set([]) for i in xrange(0, n - 1): if (Z[i, 0] in chosen) or (Z[i, 1] in chosen) or Z[i, 0] == Z[i, 1]: return True chosen.add(Z[i, 0]) chosen.add(Z[i, 1]) return False def _check_hierarchy_not_all_clusters_used(Z): n = Z.shape[0] + 1 chosen = set([]) for i in xrange(0, n - 1): chosen.add(int(Z[i, 0])) chosen.add(int(Z[i, 1])) must_chosen = set(range(0, 2 * n - 2)) return len(must_chosen.difference(chosen)) > 0 def num_obs_linkage(Z): """ Returns 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. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') return (Z.shape[0] + 1) def correspond(Z, Y): """ Checks 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. """ is_valid_linkage(Z, throw=True) distance.is_valid_y(Y, throw=True) Z = np.asarray(Z, order='c') Y = np.asarray(Y, order='c') return distance.num_obs_y(Y) == num_obs_linkage(Z) def fcluster(Z, t, criterion='inconsistent', depth=2, R=None, monocrit=None): """ Forms flat clusters from the hierarchical clustering defined by the linkage matrix Z. Parameters ---------- Z : ndarray The hierarchical clustering encoded with the matrix returned by the linkage function. t : float The threshold to apply when forming flat clusters. criterion : str, optional The criterion to use in forming flat clusters. This can be any of the following values: inconsistent : If a cluster node and all its descendants have an inconsistent value less than or equal to t then all its leaf descendants belong to the same flat cluster. When no non-singleton cluster meets this criterion, every node is assigned to its own cluster. (Default) distance : Forms flat clusters so that the original observations in each flat cluster have no greater a cophenetic distance than t. maxclust : Finds a minimum threshold r so that the cophenetic distance between any two original observations in the same flat cluster is no more than r and no more than t flat clusters are formed. monocrit : Forms a flat cluster from a cluster node c with index i when monocrit[j] <= t. For example, to threshold on the maximum mean distance as computed in the inconsistency matrix R with a threshold of 0.8 do:: MR = maxRstat(Z, R, 3) cluster(Z, t=0.8, criterion='monocrit', monocrit=MR) maxclust_monocrit : Forms a flat cluster from a non-singleton cluster node c when monocrit[i] <= r for all cluster indices i below and including c. r is minimized such that no more than t flat clusters are formed. monocrit must be monotonic. For example, to minimize the threshold t on maximum inconsistency values so that no more than 3 flat clusters are formed, do:: MI = maxinconsts(Z, R) cluster(Z, t=3, criterion='maxclust_monocrit', monocrit=MI) depth : int, optional The maximum depth to perform the inconsistency calculation. It has no meaning for the other criteria. Default is 2. R : ndarray, optional The inconsistency matrix to use for the 'inconsistent' criterion. This matrix is computed if not provided. monocrit : ndarray, optional An array of length n-1. monocrit[i] is the statistics upon which non-singleton i is thresholded. The monocrit vector must be monotonic, i.e. given a node c with index i, for all node indices j corresponding to nodes below c, monocrit[i] >= monocrit[j]. Returns ------- fcluster : ndarray An array of length n. T[i] is the flat cluster number to which original observation i belongs. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') n = Z.shape[0] + 1 T = np.zeros((n,), dtype='i') # Since the C code does not support striding using strides. # The dimensions are used instead. [Z] = _copy_arrays_if_base_present([Z]) if criterion == 'inconsistent': if R is None: R = inconsistent(Z, depth) else: R = np.asarray(R, order='c') is_valid_im(R, throw=True, name='R') # Since the C code does not support striding using strides. # The dimensions are used instead. [R] = _copy_arrays_if_base_present([R]) _hierarchy.cluster_in(Z, R, T, float(t), int(n)) elif criterion == 'distance': _hierarchy.cluster_dist(Z, T, float(t), int(n)) elif criterion == 'maxclust': _hierarchy.cluster_maxclust_dist(Z, T, int(n), int(t)) elif criterion == 'monocrit': [monocrit] = _copy_arrays_if_base_present([monocrit]) _hierarchy.cluster_monocrit(Z, monocrit, T, float(t), int(n)) elif criterion == 'maxclust_monocrit': [monocrit] = _copy_arrays_if_base_present([monocrit]) _hierarchy.cluster_maxclust_monocrit(Z, monocrit, T, int(n), int(t)) else: raise ValueError('Invalid cluster formation criterion: %s' % str(criterion)) return T def fclusterdata(X, t, criterion='inconsistent', metric='euclidean', depth=2, method='single', R=None): """ Cluster observation data using a given metric. Clusters the original observations in the n-by-m data matrix X (n observations in m dimensions), using the euclidean distance metric to calculate distances between original observations, performs hierarchical clustering using the single linkage algorithm, and forms flat clusters using the inconsistency method with t as the cut-off threshold. A one-dimensional array T of length n is returned. T[i] is the index of the flat cluster to which the original observation i belongs. Parameters ---------- X : (N, M) ndarray N by M data matrix with N observations in M dimensions. t : float The threshold to apply when forming flat clusters. criterion : str, optional Specifies the criterion for forming flat clusters. Valid values are 'inconsistent' (default), 'distance', or 'maxclust' cluster formation algorithms. See fcluster for descriptions. metric : str, optional The distance metric for calculating pairwise distances. See distance.pdist for descriptions and linkage to verify compatibility with the linkage method. depth : int, optional The maximum depth for the inconsistency calculation. See inconsistent for more information. method : str, optional The linkage method to use (single, complete, average, weighted, median centroid, ward). See linkage for more information. Default is "single". R : ndarray, optional The inconsistency matrix. It will be computed if necessary if it is not passed. Returns ------- fclusterdata : ndarray A vector of length n. T[i] is the flat cluster number to which original observation i belongs. Notes ----- This function is similar to the MATLAB function clusterdata. """ X = np.asarray(X, order='c', dtype=np.double) if type(X) != np.ndarray or len(X.shape) != 2: raise TypeError('The observation matrix X must be an n by m numpy ' 'array.') Y = distance.pdist(X, metric=metric) Z = linkage(Y, method=method) if R is None: R = inconsistent(Z, d=depth) else: R = np.asarray(R, order='c') T = fcluster(Z, criterion=criterion, depth=depth, R=R, t=t) return T def leaves_list(Z): """ Returns 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. """ Z = np.asarray(Z, order='c') is_valid_linkage(Z, throw=True, name='Z') n = Z.shape[0] + 1 ML = np.zeros((n,), dtype='i') [Z] = _copy_arrays_if_base_present([Z]) _hierarchy.prelist(Z, ML, int(n)) return ML # Maps number of leaves to text size. # # p <= 20, size="12" # 20 < p <= 30, size="10" # 30 < p <= 50, size="8" # 50 < p <= np.inf, size="6" _dtextsizes = {20: 12, 30: 10, 50: 8, 85: 6, np.inf: 5} _drotation = {20: 0, 40: 45, np.inf: 90} _dtextsortedkeys = list(_dtextsizes.keys()) _dtextsortedkeys.sort() _drotationsortedkeys = list(_drotation.keys()) _drotationsortedkeys.sort() def _remove_dups(L): """ Removes duplicates AND preserves the original order of the elements. The set class is not guaranteed to do this. """ seen_before = set([]) L2 = [] for i in L: if i not in seen_before: seen_before.add(i) L2.append(i) return L2 def _get_tick_text_size(p): for k in _dtextsortedkeys: if p <= k: return _dtextsizes[k] def _get_tick_rotation(p): for k in _drotationsortedkeys: if p <= k: return _drotation[k] def _plot_dendrogram(icoords, dcoords, ivl, p, n, mh, orientation, no_labels, color_list, leaf_font_size=None, leaf_rotation=None, contraction_marks=None, ax=None, above_threshold_color='b'): # Import matplotlib here so that it's not imported unless dendrograms # are plotted. Raise an informative error if importing fails. try: # if an axis is provided, don't use pylab at all if ax is None: import matplotlib.pylab import matplotlib.patches import matplotlib.collections except ImportError: raise ImportError("You must install the matplotlib library to plot " "the dendrogram. Use no_plot=True to calculate the " "dendrogram without plotting.") if ax is None: ax = matplotlib.pylab.gca() # if we're using pylab, we want to trigger a draw at the end trigger_redraw = True else: trigger_redraw = False # Independent variable plot width ivw = len(ivl) * 10 # Dependent variable plot height dvw = mh + mh * 0.05 iv_ticks = np.arange(5, len(ivl) * 10 + 5, 10) if orientation in ('top', 'bottom'): if orientation == 'top': ax.set_ylim([0, dvw]) ax.set_xlim([0, ivw]) else: ax.set_ylim([dvw, 0]) ax.set_xlim([0, ivw]) xlines = icoords ylines = dcoords if no_labels: ax.set_xticks([]) ax.set_xticklabels([]) else: ax.set_xticks(iv_ticks) if orientation == 'top': ax.xaxis.set_ticks_position('bottom') else: ax.xaxis.set_ticks_position('top') # Make the tick marks invisible because they cover up the links for line in ax.get_xticklines(): line.set_visible(False) leaf_rot = float(_get_tick_rotation(len(ivl))) if ( leaf_rotation is None) else leaf_rotation leaf_font = float(_get_tick_text_size(len(ivl))) if ( leaf_font_size is None) else leaf_font_size ax.set_xticklabels(ivl, rotation=leaf_rot, size=leaf_font) elif orientation in ('left', 'right'): if orientation == 'left': ax.set_xlim([dvw, 0]) ax.set_ylim([0, ivw]) else: ax.set_xlim([0, dvw]) ax.set_ylim([0, ivw]) xlines = dcoords ylines = icoords if no_labels: ax.set_yticks([]) ax.set_yticklabels([]) else: ax.set_yticks(iv_ticks) if orientation == 'left': ax.yaxis.set_ticks_position('right') else: ax.yaxis.set_ticks_position('left') # Make the tick marks invisible because they cover up the links for line in ax.get_yticklines(): line.set_visible(False) leaf_font = float(_get_tick_text_size(len(ivl))) if ( leaf_font_size is None) else leaf_font_size if leaf_rotation is not None: ax.set_yticklabels(ivl, rotation=leaf_rotation, size=leaf_font) else: ax.set_yticklabels(ivl, size=leaf_font) # Let's use collections instead. This way there is a separate legend item # for each tree grouping, rather than stupidly one for each line segment. colors_used = _remove_dups(color_list) color_to_lines = {} for color in colors_used: color_to_lines[color] = [] for (xline, yline, color) in zip(xlines, ylines, color_list): color_to_lines[color].append(list(zip(xline, yline))) colors_to_collections = {} # Construct the collections. for color in colors_used: coll = matplotlib.collections.LineCollection(color_to_lines[color], colors=(color,)) colors_to_collections[color] = coll # Add all the groupings below the color threshold. for color in colors_used: if color != above_threshold_color: ax.add_collection(colors_to_collections[color]) # If there's a grouping of links above the color threshold, it goes last. if above_threshold_color in colors_to_collections: ax.add_collection(colors_to_collections[above_threshold_color]) if contraction_marks is not None: Ellipse = matplotlib.patches.Ellipse for (x, y) in contraction_marks: if orientation in ('left', 'right'): e = Ellipse((y, x), width=dvw / 100, height=1.0) else: e = Ellipse((x, y), width=1.0, height=dvw / 100) ax.add_artist(e) e.set_clip_box(ax.bbox) e.set_alpha(0.5) e.set_facecolor('k') if trigger_redraw: matplotlib.pylab.draw_if_interactive() _link_line_colors = ['g', 'r', 'c', 'm', 'y', 'k'] def set_link_color_palette(palette): """ Set list of matplotlib color codes for use by dendrogram. Note that this palette is global (i.e. setting it once changes the colors for all subsequent calls to dendrogram) and that it affects only the the colors below color_threshold. Note that dendrogram also accepts a custom coloring function through its link_color_func keyword, which is more flexible and non-global. Parameters ---------- palette : list of str or None A list of matplotlib color codes. The order of the color codes is the order in which the colors are cycled through when color thresholding in the dendrogram. If None, resets the palette to its default (which is ['g', 'r', 'c', 'm', 'y', 'k']). Returns ------- None See Also -------- dendrogram Notes ----- Ability to reset the palette with None added in Scipy 0.17.0. Examples -------- >>> from scipy.cluster import hierarchy >>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268., 400., ... 754., 564., 138., 219., 869., 669.]) >>> Z = hierarchy.linkage(ytdist, 'single') >>> dn = hierarchy.dendrogram(Z, no_plot=True) >>> dn['color_list'] ['g', 'b', 'b', 'b', 'b'] >>> hierarchy.set_link_color_palette(['c', 'm', 'y', 'k']) >>> dn = hierarchy.dendrogram(Z, no_plot=True) >>> dn['color_list'] ['c', 'b', 'b', 'b', 'b'] >>> dn = hierarchy.dendrogram(Z, no_plot=True, color_threshold=267, ... above_threshold_color='k') >>> dn['color_list'] ['c', 'm', 'm', 'k', 'k'] Now reset the color palette to its default: >>> hierarchy.set_link_color_palette(None) """ if palette is None: # reset to its default palette = ['g', 'r', 'c', 'm', 'y', 'k'] elif type(palette) not in (list, tuple): raise TypeError("palette must be a list or tuple") _ptypes = [isinstance(p, string_types) for p in palette] if False in _ptypes: raise TypeError("all palette list elements must be color strings") for i in list(_link_line_colors): _link_line_colors.remove(i) _link_line_colors.extend(list(palette)) def dendrogram(Z, p=30, truncate_mode=None, color_threshold=None, get_leaves=True, orientation='top', labels=None, count_sort=False, distance_sort=False, show_leaf_counts=True, no_plot=False, no_labels=False, leaf_font_size=None, leaf_rotation=None, leaf_label_func=None, show_contracted=False, link_color_func=None, ax=None, above_threshold_color='b'): """ Plots 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 height of the top of the U-link is the distance between its children clusters. It is also the cophenetic distance between original observations in the two children clusters. It is expected that the distances in Z[:,2] be monotonic, otherwise crossings appear in the dendrogram. 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/'none' No truncation is performed (Default). 'lastp' The last p non-singleton 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. 'mlab' This corresponds to MATLAB(TM) behavior. (not implemented yet) 'level'/'mtica' No more than p levels of the dendrogram tree are displayed. This corresponds to Mathematica(TM) behavior. color_threshold : double, optional For brevity, let :math:t be the color_threshold. Colors all the descendent links below a cluster node :math:k the same color if :math:k is the first node below the cut threshold :math:t. All links connecting nodes with distances greater than or equal to the threshold are colored blue. If :math:t is less than or equal to zero, all nodes are colored blue. If color_threshold is None or 'default', corresponding with MATLAB(TM) behavior, the threshold is set to 0.7*max(Z[:,2]). get_leaves : bool, optional Includes a list R['leaves']=H in the result dictionary. For each :math:i, H[i] == j, cluster node j appears in position i in the left-to-right traversal of the leaves, where :math:j < 2n-1 and :math:i < n. orientation : str, optional The direction to plot the dendrogram, which can be any of the following strings: 'top' Plots the root at the top, and plot descendent links going downwards. (default). 'bottom' Plots the root at the bottom, and plot descendent links going upwards. 'left' Plots the root at the left, and plot descendent links going right. 'right' Plots the root at the right, and plot descendent links going left. labels : ndarray, optional By default labels is None so the index of the original observation is used to label the leaf nodes. Otherwise, this is an :math:n -sized list (or tuple). The labels[i] value is the text to put under the :math:i th leaf node only if it corresponds to an original observation and not a non-singleton cluster. count_sort : str or bool, optional For each node n, the order (visually, from left-to-right) n's two descendent links are plotted is determined by this parameter, which can be any of the following values: False Nothing is done. 'ascending' or True The child with the minimum number of original objects in its cluster is plotted first. 'descendent' The child with the maximum number of original objects in its cluster is plotted first. Note distance_sort and count_sort cannot both be True. distance_sort : str or bool, optional For each node n, the order (visually, from left-to-right) n's two descendent links are plotted is determined by this parameter, which can be any of the following values: False Nothing is done. 'ascending' or True The child with the minimum distance between its direct descendents is plotted first. 'descending' The child with the maximum distance between its direct descendents is plotted first. Note distance_sort and count_sort cannot both be True. show_leaf_counts : bool, optional When True, leaf nodes representing :math:k>1 original observation are labeled with the number of observations they contain in parentheses. no_plot : bool, optional When True, the final rendering is not performed. This is useful if only the data structures computed for the rendering are needed or if matplotlib is not available. no_labels : bool, optional When True, no labels appear next to the leaf nodes in the rendering of the dendrogram. leaf_rotation : double, optional Specifies the angle (in degrees) to rotate the leaf labels. When unspecified, the rotation is based on the number of nodes in the dendrogram (default is 0). leaf_font_size : int, optional Specifies the font size (in points) of the leaf labels. When unspecified, the size based on the number of nodes in the dendrogram. leaf_label_func : lambda or function, optional When leaf_label_func is a callable function, for each leaf with cluster index :math:k < 2n-1. The function is expected to return a string with the label for the leaf. Indices :math:k < n correspond to original observations while indices :math:k \\geq n correspond to non-singleton clusters. For example, to label singletons with their node id and non-singletons with their id, count, and inconsistency coefficient, simply do:: # First define the leaf label function. def llf(id): if id < n: return str(id) else: return '[%d %d %1.2f]' % (id, count, R[n-id,3]) # The text for the leaf nodes is going to be big so force # a rotation of 90 degrees. dendrogram(Z, leaf_label_func=llf, leaf_rotation=90) show_contracted : bool, optional When True the heights of non-singleton nodes contracted into a leaf node are plotted as crosses along the link connecting that leaf node. This really is only useful when truncation is used (see truncate_mode parameter). link_color_func : callable, optional If given, link_color_function is called with each non-singleton id corresponding to each U-shaped link it will paint. The function is expected to return the color to paint the link, encoded as a matplotlib color string code. For example:: dendrogram(Z, link_color_func=lambda k: colors[k]) colors the direct links below each untruncated non-singleton node k using colors[k]. ax : matplotlib Axes instance, optional If None and no_plot is not True, the dendrogram will be plotted on the current axes. Otherwise if no_plot is not True the dendrogram will be plotted on the given Axes instance. This can be useful if the dendrogram is part of a more complex figure. above_threshold_color : str, optional This matplotlib color string sets the color of the links above the color_threshold. The default is 'b'. Returns ------- R : dict A dictionary of data structures computed to render the dendrogram. Its has the following keys: 'color_list' A list of color names. The k'th element represents the color of the k'th link. 'icoord' and 'dcoord' Each of them is a list of lists. Let icoord = [I1, I2, ..., Ip] where Ik = [xk1, xk2, xk3, xk4] and dcoord = [D1, D2, ..., Dp] where Dk = [yk1, yk2, yk3, yk4], then the k'th link painted is (xk1, yk1) - (xk2, yk2) - (xk3, yk3) - (xk4, yk4). 'ivl' A list of labels corresponding to the leaf nodes. 'leaves' For each i, H[i] == j, cluster node j appears in position i in the left-to-right traversal of the leaves, where :math:j < 2n-1 and :math:i < n. If j is less than n, the i-th leaf node corresponds to an original observation. Otherwise, it corresponds to a non-singleton cluster. See Also -------- linkage, set_link_color_palette Examples -------- >>> from scipy.cluster import hierarchy >>> import matplotlib.pyplot as plt A very basic example: >>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268., ... 400., 754., 564., 138., 219., 869., 669.]) >>> Z = hierarchy.linkage(ytdist, 'single') >>> plt.figure() >>> dn = hierarchy.dendrogram(Z) Now plot in given axes, improve the color scheme and use both vertical and horizontal orientations: >>> hierarchy.set_link_color_palette(['m', 'c', 'y', 'k']) >>> fig, axes = plt.subplots(1, 2, figsize=(8, 3)) >>> dn1 = hierarchy.dendrogram(Z, ax=axes[0], above_threshold_color='y', ... orientation='top') >>> dn2 = hierarchy.dendrogram(Z, ax=axes[1], above_threshold_color='#bcbddc', ... orientation='right') >>> hierarchy.set_link_color_palette(None) # reset to default after use >>> plt.show() """ # This feature was thought about but never implemented (still useful?): # # ... = dendrogram(..., leaves_order=None) # # Plots the leaves in the order specified by a vector of # original observation indices. If the vector contains duplicates # or results in a crossing, an exception will be thrown. Passing # None orders leaf nodes based on the order they appear in the # pre-order traversal. Z = np.asarray(Z, order='c') if orientation not in ["top", "left", "bottom", "right"]: raise ValueError("orientation must be one of 'top', 'left', " "'bottom', or 'right'") is_valid_linkage(Z, throw=True, name='Z') Zs = Z.shape n = Zs[0] + 1 if type(p) in (int, float): p = int(p) else: raise TypeError('The second argument must be a number') if truncate_mode not in ('lastp', 'mlab', 'mtica', 'level', 'none', None): raise ValueError('Invalid truncation mode.') if truncate_mode == 'lastp' or truncate_mode == 'mlab': if p > n or p == 0: p = n if truncate_mode == 'mtica' or truncate_mode == 'level': if p <= 0: p = np.inf if get_leaves: lvs = [] else: lvs = None icoord_list = [] dcoord_list = [] color_list = [] current_color = [0] currently_below_threshold = [False] ivl = [] # list of leaves if color_threshold is None or (isinstance(color_threshold, string_types) and color_threshold == 'default'): color_threshold = max(Z[:, 2]) * 0.7 R = {'icoord': icoord_list, 'dcoord': dcoord_list, 'ivl': ivl, 'leaves': lvs, 'color_list': 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) return R 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(int(Z[i - n, 3])) + ")") else: ivl.append("") def _append_contraction_marks(Z, iv, i, n, contraction_marks): _append_contraction_marks_sub(Z, iv, int(Z[i - n, 0]), n, contraction_marks) _append_contraction_marks_sub(Z, iv, int(Z[i - n, 1]), n, contraction_marks) def _append_contraction_marks_sub(Z, iv, i, n, contraction_marks): if i >= n: contraction_marks.append((iv, Z[i - n, 2])) _append_contraction_marks_sub(Z, iv, int(Z[i - n, 0]), n, contraction_marks) _append_contraction_marks_sub(Z, iv, int(Z[i - n, 1]), n, contraction_marks) 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='b'): """ Calculates 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. """ 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-single 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) 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 in ('mtica', '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) 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 in ('mlab',): pass # Otherwise, only truncate if we have a leaf node. # # If the truncate_mode is mlab, the linkage has been modified # with the truncated tree. # # 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(Z[i - n, 0]) ab = int(Z[i - n, 1]) 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 == 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 == 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, string_types): 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): """ Determines 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. """ T1 = np.asarray(T1, order='c') T2 = np.asarray(T2, order='c') if type(T1) != np.ndarray: raise TypeError('T1 must be a numpy array.') if type(T2) != np.ndarray: raise TypeError('T2 must be a numpy array.') 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] d = {} for i in xrange(0, n): if T1[i] in d: if d[T1[i]] != T2[i]: return False else: d[T1[i]] = T2[i] return True def maxdists(Z): """ Returns 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. """ Z = np.asarray(Z, order='c', dtype=np.double) is_valid_linkage(Z, throw=True, name='Z') n = Z.shape[0] + 1 MD = np.zeros((n - 1,)) [Z] = _copy_arrays_if_base_present([Z]) _hierarchy.get_max_dist_for_each_cluster(Z, MD, int(n)) return MD def maxinconsts(Z, R): """ Returns the maximum inconsistency coefficient for each non-singleton cluster and its descendents. 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. """ Z = np.asarray(Z, order='c') R = np.asarray(R, order='c') 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, R] = _copy_arrays_if_base_present([Z, R]) _hierarchy.get_max_Rfield_for_each_cluster(Z, R, MI, int(n), 3) return MI def maxRstat(Z, R, i): """ Returns the maximum statistic for each non-singleton cluster and its descendents. 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. """ Z = np.asarray(Z, order='c') R = np.asarray(R, order='c') is_valid_linkage(Z, throw=True, name='Z') is_valid_im(R, throw=True, name='R') if type(i) is not 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, R] = _copy_arrays_if_base_present([Z, R]) _hierarchy.get_max_Rfield_for_each_cluster(Z, R, MR, int(n), i) return MR def leaders(Z, T): """ Returns 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 descendents 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 leaf nodes descendent with cluster node :math:i) * there does not exist a leaf that is not descendent 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. For example: if L[3]=2 and M[3]=8, the flat cluster with id 8's leader is linkage node 2. 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. """ Z = np.asarray(Z, order='c') T = np.asarray(T, order='c') if type(T) != np.ndarray or T.dtype != 'i': raise TypeError('T must be a one-dimensional numpy array of integers.') is_valid_linkage(Z, throw=True, name='Z') if len(T) != Z.shape[0] + 1: raise ValueError('Mismatch: len(T)!=Z.shape[0] + 1.') Cl = np.unique(T) kk = len(Cl) L = np.zeros((kk,), dtype='i') M = np.zeros((kk,), dtype='i') n = Z.shape[0] + 1 [Z, T] = _copy_arrays_if_base_present([Z, T]) s = _hierarchy.leaders(Z, T, L, M, int(kk), int(n)) if s >= 0: raise ValueError(('T is not a valid assignment vector. Error found ' 'when examining linkage node %d (< 2n-1).') % s) return (L, M)