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# Hierarchical clustering (`scipy.cluster.hierarchy`)

Warning

This documentation is work-in-progress and unorganized.

## Function Reference

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.

`fcluster`(Z, t[, criterion, depth, R, monocrit]) | Forms flat clusters from the hierarchical clustering defined by
`fclusterdata`(X, t[, criterion, metric, ...]) | Cluster observation data using a given metric.
`leaders`(Z, T) | (L, M) = leaders(Z, T):

These are routines for agglomerative clustering.

`linkage`(y[, method, metric]) | Performs hierarchical/agglomerative clustering on the condensed distance matrix y.
`single`(y) | Performs single/min/nearest linkage on the condensed distance matrix `y`.
`complete`(y) | Performs complete complete/max/farthest point linkage on the condensed distance matrix `y`.
`average`(y) | Performs average/UPGMA linkage on the condensed distance matrix
`weighted`(y) | Performs weighted/WPGMA linkage on the condensed distance matrix
`centroid`(y) | Performs centroid/UPGMC linkage. See `linkage` for more
`median`(y) | Performs median/WPGMC linkage. See `linkage` for more
`ward`(y) | Performs Ward’s linkage on a condensed or redundant distance

These routines compute statistics on hierarchies.

`cophenet`(Z[, Y]) | Calculates the cophenetic distances between each observation in
`from_mlab_linkage`(Z) | Converts a linkage matrix generated by MATLAB(TM) to a new
`inconsistent`(Z[, d]) | Calculates inconsistency statistics on a linkage.
`maxinconsts`(Z, R) | Returns the maximum inconsistency coefficient for each non-singleton cluster and its descendents.
`maxdists`(Z) | Returns the maximum distance between any cluster for each non-singleton cluster.
`maxRstat`(Z, R, i) | Returns the maximum statistic for each non-singleton cluster and its descendents.
`to_mlab_linkage`(Z) | Converts a linkage matrix `Z` generated by the linkage function

Routines for visualizing flat clusters.

`dendrogram`(Z[, p, truncate_mode, ...]) | Plots the hiearchical clustering defined by the linkage Z as a

These are data structures and routines for representing hierarchies as tree objects.

`ClusterNode`(id[, left, right, dist, count]) | A tree node class for representing a cluster.
`leaves_list`(Z) | Returns a list of leaf node ids (corresponding to observation vector index) as they appear in the tree from left to right.
`to_tree`(Z[, rd]) | Converts a hierarchical clustering encoded in the matrix `Z` (by

These are predicates for checking the validity of linkage and inconsistency matrices as well as for checking isomorphism of two flat cluster assignments.

`is_valid_im`(R[, warning, throw, name]) | Returns True if the inconsistency matrix passed is valid. It must
`is_valid_linkage`(Z[, warning, throw, name]) | Checks the validity of a linkage matrix.
`is_isomorphic`(T1, T2) | Determines if two different cluster assignments `T1` and
`is_monotonic`(Z) | Returns `True` if the linkage passed is monotonic. The linkage
`correspond`(Z, Y) | Checks if a linkage matrix `Z` and condensed distance matrix
`num_obs_linkage`(Z) | Returns the number of original observations of the linkage matrix passed.

- MATLAB and MathWorks are registered trademarks of The MathWorks, Inc.
- Mathematica is a registered trademark of The Wolfram Research, Inc.

## References

[Sta07] | “Statistics toolbox.” API Reference Documentation. The MathWorks. http://www.mathworks.com/access/helpdesk/help/toolbox/stats/. Accessed October 1, 2007.

[Mti07] | “Hierarchical clustering.” API Reference Documentation. The Wolfram Research, Inc. http://reference.wolfram.com/mathematica/HierarchicalClustering/tutorial/HierarchicalClustering.html. Accessed October 1, 2007.

[Gow69] | Gower, JC and Ross, GJS. “Minimum Spanning Trees and Single Linkage Cluster Analysis.” Applied Statistics. 18(1): pp. 54–64. 1969.

[War63] | Ward Jr, JH. “Hierarchical grouping to optimize an objective function.” Journal of the American Statistical Association. 58(301): pp. 236–44. 1963.

[Joh66] | Johnson, SC. “Hierarchical clustering schemes.” Psychometrika. 32(2): pp. 241–54. 1966.

[Sne62] | Sneath, PH and Sokal, RR. “Numerical taxonomy.” Nature. 193: pp. 855–60. 1962.

[Bat95] | Batagelj, V. “Comparing resemblance measures.” Journal of Classification. 12: pp. 73–90. 1995.

[Sok58] | Sokal, RR and Michener, CD. “A statistical method for evaluating systematic relationships.” Scientific Bulletins. 38(22): pp. 1409–38. 1958.

[Ede79] | Edelbrock, C. “Mixture model tests of hierarchical clustering algorithms: the problem of classifying everybody.” Multivariate Behavioral Research. 14: pp. 367–84. 1979.

[Jai88] | Jain, A., and Dubes, R., “Algorithms for Clustering Data.” Prentice-Hall. Englewood Cliffs, NJ. 1988.

[Fis36] | Fisher, RA “The use of multiple measurements in taxonomic problems.” Annals of Eugenics, 7(2): 179-188. 1936

## Copyright Notice

Copyright (C) Damian Eads, 2007-2008. New BSD License.

class `scipy.cluster.hierarchy.``ClusterNode`(id, left=None, right=None, dist=0, count=1)

  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.

  Seealso : | to_tree: for converting a linkage matrix `Z` into a tree object.

  Methods

  `get_count` |
  `get_id` |
  `get_left` |
  `get_right` |
  `is_leaf` |
  `pre_order` |

  `get_count`()

    The number of leaf nodes (original observations) belonging to the cluster node nd. If the target node is a leaf, 1 is returned.

    Returns : | c : int The number of leaf nodes below the target node.

  `get_id`()

    The identifier of the target node. For , corresponds to original observation . For < , corresponds to non-singleton cluster formed at iteration .

    Returns : | id : int The identifier of the target node.

  `get_left`()

    Returns a reference to the left child tree object. If the node is a leaf, None is returned.

    Returns : | left : ClusterNode The left child of the target node.

  `get_right`()

    Returns a reference to the right child tree object. If the node is a leaf, None is returned.

    Returns : | right : ClusterNode The left child of the target node.

  `is_leaf`()

    Returns True iff the target node is a leaf.

    Returns : | leafness : bool True if the target node is a leaf node.

  `pre_order`(func=<function <lambda> at 0x49a0b18>)

    Performs preorder 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.

`scipy.cluster.hierarchy.``average`(y)

  Performs average/UPGMA linkage on the condensed distance matrix `y`. 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.
  Seealso : | linkage: for advanced creation of hierarchical clusterings.

`scipy.cluster.hierarchy.``centroid`(y)

  Performs centroid/UPGMC linkage. See `linkage` for more information on the 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 : | Q : ndarray A condensed or redundant 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.
  Seealso : | linkage: for advanced creation of hierarchical clusterings.

`scipy.cluster.hierarchy.``complete`(y)

  Performs complete complete/max/farthest point linkage on the condensed distance matrix `y`. 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.

`scipy.cluster.hierarchy.``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 observations in dimensions. `Y` is the condensed distance matrix from which `Z` was generated.
  Returns : | (c, {d}) - c : ndarray The cophentic correlation distance (if `y` is passed). d : ndarray The cophenetic distance matrix in condensed form. The th entry is the cophenetic distance between original observations and .

`scipy.cluster.hierarchy.``correspond`(Z, Y)

  Checks if a linkage matrix `Z` and condensed distance matrix `Y` could possibly correspond to one another.

  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.

  Arguments : | Z : ndarray The linkage matrix to check for correspondance. Y : ndarray The condensed distance matrix to check for correspondance.
  Returns : | b : bool A boolean indicating whether the linkage matrix and distance matrix could possibly correspond to one another.

`scipy.cluster.hierarchy.``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, color_list=None, leaf_font_size=None, leaf_rotation=None, leaf_label_func=None, no_leaves=False, show_contracted=False, link_color_func=None)

  > Plots the hiearchical clustering defined by the linkage Z 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.

  > Arguments: | 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`. truncate_mode : string 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 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. p : int The `p` parameter for `truncate_mode`.

  `

    > - color_threshold : double For brevity, let be the `color_threshold`. Colors all the descendent links below a cluster node the same color if is the first node below the cut threshold . All links connecting nodes with distances greater than or equal to the threshold are colored blue. If 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 Includes a list `R['leaves']=H` in the result dictionary. For each , `H[i] == j`, cluster node appears in the th position in the left-to-right traversal of the leaves, where and .

    > - orientation : string 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 By default `labels` is `None` so the index of the original observation is used to label the leaf nodes.

    >   Otherwise, this is an -sized list (or tuple). The `labels[i]` value is the text to put under the th leaf node only if it corresponds to an original observation and not a non-singleton cluster.

    > - count_sort : string/bool 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’/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 : string/bool 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’/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

    >   When `True`, leaf nodes representing original observation are labeled with the number of observations they contain in parentheses.

    > - no_plot : bool 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 When `True`, no labels appear next to the leaf nodes in the rendering of the dendrogram.

    > - leaf_label_rotation : double

    >   Specifies the angle (in degrees) to rotate the leaf labels. When unspecified, the rotation based on the number of nodes in the dendrogram. (Default=0)

    > - leaf_font_size : int 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

    >   When leaf_label_func is a callable function, for each leaf with cluster index . The function is expected to return a string with the label for the leaf.

    >   Indices correspond to original observations while indices 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 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 : lambda/function When a callable function, 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]`.

    Returns: | R : dict A dictionary of data structures computed to render the dendrogram. Its has the following keys: ‘icoords’: a list of lists `[I1, I2, ..., Ip]` where `Ik` is a list of 4 independent variable coordinates corresponding to the line that represents the k’th link painted. ‘dcoords’: a list of lists `[I2, I2, ..., Ip]` where `Ik` is a list of 4 independent variable coordinates corresponding to the line that represents the k’th link painted. ‘ivl’: a list of labels corresponding to the leaf nodes. ‘leaves’: for each i, `H[i] == j`, cluster node appears in the th position in the left-to-right traversal of the leaves, where and . If is less than , the th leaf node corresponds to an original observation. Otherwise, it corresponds to a non-singleton cluster.

`scipy.cluster.hierarchy.``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.

`scipy.cluster.hierarchy.``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 : 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. method : str, optional The linkage method to use (single, complete, average, weighted, median centroid, ward). See linkage for more information. Default is “single”. metric : str, optional The distance metric for calculating pairwise distances. See distance.pdist for descriptions and linkage to verify compatibility with the linkage method. t : double, optional The cut-off threshold for the cluster function or the maximum number of clusters (criterion=’maxclust’). depth : int, optional The maximum depth for the inconsistency calculation. See inconsistent for more information. R : ndarray, optional The inconsistency matrix. It will be computed if necessary if it is not passed.
  Returns : | T : 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.

`scipy.cluster.hierarchy.``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.

  Arguments : | Z : ndarray A linkage matrix generated by MATLAB(TM)
  Returns : | ZS : ndarray A linkage matrix compatible with this library.

`scipy.cluster.hierarchy.``inconsistent`(Z, d=2)

  Calculates inconsistency statistics on a linkage.

  Note: This function behaves similarly to the MATLAB(TM) inconsistent function.

  Parameters : | d : int The number of links up to `d` levels below each non-singleton cluster Z : ndarray The by 4 matrix encoding the linkage (hierarchical clustering). See `linkage` documentation for more information on its form.
  Returns : | R : ndarray A 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 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,

`scipy.cluster.hierarchy.``is_isomorphic`(T1, T2)

  Determines if two different cluster assignments `T1` and `T2` are equivalent.

  Arguments : | T1 : ndarray An assignment of singleton cluster ids to flat cluster ids. T2 : ndarray An assignment of singleton cluster ids to flat cluster ids.
  Returns: | b : boolean Whether the flat cluster assignments `T1` and `T2` are equivalent.

`scipy.cluster.hierarchy.``is_monotonic`(Z)

  Returns `True` if the linkage passed is monotonic. The linkage is monotonic if for every cluster and joined, the distance between them is no less than the distance between any previously joined clusters.

  Arguments : | Z : ndarray The linkage matrix to check for monotonicity.
  Returns : | b : bool A boolean indicating whether the linkage is monotonic.

`scipy.cluster.hierarchy.``is_valid_im`(R, warning=False, throw=False, name=None)

  Returns True if the inconsistency matrix passed is valid. It must be a 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 .

  Arguments : | R : ndarray The inconsistency matrix to check for validity. warning : bool When `True`, issues a Python warning if the linkage matrix passed is invalid. throw : bool When `True`, throws a Python exception if the linkage matrix passed is invalid. name : string This string refers to the variable name of the invalid linkage matrix.
  Returns : | b : bool True iff the inconsistency matrix is valid.

`scipy.cluster.hierarchy.``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 nd-array (type double) with rows and 4 columns. The first two columns must contain indices between 0 and . For a given row `i`, and (i.e. a cluster cannot join another cluster unless the cluster being joined has been generated.)

  Arguments : | warning : bool When `True`, issues a Python warning if the linkage matrix passed is invalid. throw : bool When `True`, throws a Python exception if the linkage matrix passed is invalid. name : string This string refers to the variable name of the invalid linkage matrix.
  Returns : | b : bool True iff the inconsistency matrix is valid.

`scipy.cluster.hierarchy.``leaders`(Z, T)

  (L, M) = leaders(Z, T):

  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 of the flat clusters represented in the n-sized flat cluster assignment vector `T`, this function finds the lowest cluster node in the linkage tree Z such that:

  > - leaf descendents belong only to flat cluster j (i.e. `T[p]==j` for all in where is the set of leaf ids of leaf nodes descendent with cluster node )
  > - there does not exist a leaf that is not descendent with that also belongs to cluster (i.e. `T[q]!=j` for all not in ). If this condition is violated, `T` is not a valid cluster assignment vector, and an exception will be thrown.

  Arguments : | Z : ndarray The hierarchical clustering encoded as a matrix. See `linkage` for more information. T : ndarray The flat cluster assignment vector.
  Returns : | (L, M) L : ndarray The leader linkage node id’s stored as a k-element 1D array where 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 1D array where is the number of flat clusters found in `T`. This allows the set of flat cluster ids to be any arbitrary set of integers.

`scipy.cluster.hierarchy.``leaves_list`(Z)

  Returns a list of leaf node ids (corresponding to observation vector index) as they appear in the tree from left to right. Z is a linkage matrix.

  Arguments : | Z : ndarray The hierarchical clustering encoded as a matrix. See `linkage` for more information.
  Returns : | L : ndarray The list of leaf node ids.

`scipy.cluster.hierarchy.``linkage`(y, method='single', metric='euclidean')
  Performs hierarchical/agglomerative clustering on the

    condensed distance matrix y. y must be a 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(TM) linkage function.

    A 4 by matrix `Z` is returned. At the -th iteration, clusters with indices `Z[i, 0]` and `Z[i, 1]` are combined to form cluster . A cluster with an index less than corresponds to one of the 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 between two clusters and . The algorithm begins with a forest of clusters that have yet to be used in the hierarchy being formed. When two clusters and from this forest are combined into a single cluster , and are removed from the forest, and 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 and 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 original observations in cluster and original objects in cluster . Recall and are combined to form cluster . Let be any remaining cluster in the forest that is not .

    The following are methods for calculating the distance between the newly formed cluster and each .

    > - method=’single’ assigns

    >   for all points in cluster and in cluster . This is also known as the Nearest Point Algorithm.

    > - method=’complete’ assigns

    >   for all points in cluster u and in cluster . This is also known by the Farthest Point Algorithm or Voor Hees Algorithm.

    > - method=’average’ assigns

    >   for all points and where and are the cardinalities of clusters and , respectively. This is also called the UPGMA algorithm. This is called UPGMA.

    > - method=’weighted’ assigns

    >   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

    >   where and are the centroids of clusters and , respectively. When two clusters and are combined into a new cluster , the new centroid is computed over all the original objects in clusters and . The distance then becomes the Euclidean distance between the centroid of and the centroid of a remaining cluster 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 and are combined into a new cluster , the average of centroids s and t give the new centroid . This is also known as the WPGMC algorithm.

    > - method=’ward’ uses the Ward variance minimization algorithm. The new entry is computed as follows,

    >   where is the newly joined cluster consisting of clusters and , is an unused cluster in the forest, , and 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(TM) version.

    Parameters: | y : ndarray A condensed or redundant 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 observation vectors in n dimensions may be passed as an by array. method : string The linkage algorithm to use. See the `Linkage Methods` section below for full descriptions. metric : string The distance metric to use. See the `distance.pdist` function for a list of valid distance metrics.

  Returns : | Z : ndarray The hierarchical clustering encoded as a linkage matrix.

`scipy.cluster.hierarchy.``maxRstat`(Z, R, i)

  Returns the maximum statistic for each non-singleton cluster and its descendents.

  Arguments : | Z : ndarray The hierarchical clustering encoded as a matrix. See `linkage` for more information. R : ndarray 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`.

`scipy.cluster.hierarchy.``maxdists`(Z)

  > Returns the maximum distance between any cluster for each non-singleton cluster.

  Arguments : | Z : ndarray The hierarchical clustering encoded as a matrix. See `linkage` for more information.
  Returns : | MD : 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.

`scipy.cluster.hierarchy.``maxinconsts`(Z, R)

  Returns the maximum inconsistency coefficient for each non-singleton cluster and its descendents.

  Arguments : | 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.

`scipy.cluster.hierarchy.``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 : | Q : ndarray A condensed or redundant 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.
  Seealso : | linkage: for advanced creation of hierarchical clusterings.

`scipy.cluster.hierarchy.``num_obs_linkage`(Z)

  Returns the number of original observations of the linkage matrix passed.

  Arguments : | Z : ndarray The linkage matrix on which to perform the operation.
  Returns : | n : int The number of original observations in the linkage.

`scipy.cluster.hierarchy.``set_link_color_palette`(palette)

  Changes the list of matplotlib color codes to use when coloring links with the dendrogram color_threshold feature.

  Arguments : | palette : 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.

`scipy.cluster.hierarchy.``single`(y)

  Performs single/min/nearest linkage on the condensed distance matrix `y`. 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 The linkage matrix.
  Seealso : | linkage: for advanced creation of hierarchical clusterings.

`scipy.cluster.hierarchy.``to_mlab_linkage`(Z)

  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.

  Arguments : | Z : ndarray A linkage matrix generated by this library.
  Returns : | ZM : ndarray A linkage matrix compatible with MATLAB(TM)’s hierarchical clustering functions.

`scipy.cluster.hierarchy.``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). r : bool 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.

`scipy.cluster.hierarchy.``ward`(y)

  Performs Ward’s linkage on a condensed or redundant 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 : | Q : ndarray A condensed or redundant 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.
  Seealso : | linkage: for advanced creation of hierarchical clusterings.

`scipy.cluster.hierarchy.``weighted`(y)

  Performs weighted/WPGMA linkage on the condensed distance matrix `y`. 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.
  Seealso : | linkage: for advanced creation of hierarchical clusterings.

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