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scipy.cluster.hierarchy.

# dendrogram

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, leaf_font_size=None, leaf_rotation=None, leaf_label_func=None, show_contracted=False, link_color_func=None, ax=None, above_threshold_color='C0')

  Plot the hierarchical clustering as a dendrogram.

  The dendrogram illustrates how each cluster is composed by drawing a U-shaped link between a non-singleton cluster and its children. The top of the U-link indicates a cluster merge. The two legs of the U-link indicate which clusters were merged. The length of the two legs of the U-link represents the distance between the child clusters. It is also the cophenetic distance between original observations in the two children clusters.

  Parameters:
    Z ndarray

      The linkage matrix encoding the hierarchical clustering to render as a dendrogram. See the `linkage` function for more information on the format of `Z`.

    p int, optional

      The `p` parameter for `truncate_mode`.

    truncate_mode str, optional

      The dendrogram can be hard to read when the original observation matrix from which the linkage is derived is large. Truncation is used to condense the dendrogram. There are several modes:

      `None`

        No truncation is performed (default). Note: `'none'` is an alias for `None` that’s kept for backward compatibility.

      `'lastp'`

        The last `p` non-singleton clusters formed in the linkage are the only non-leaf nodes in the linkage; they correspond to rows `Z[n-p-2:end]` in `Z`. All other non-singleton clusters are contracted into leaf nodes.

      `'level'`

        No more than `p` levels of the dendrogram tree are displayed. A “level” includes all nodes with `p` merges from the final merge.

        Note: `'mtica'` is an alias for `'level'` that’s kept for backward compatibility.

    color_threshold double, optional

      For brevity, let \(t\) be the `color_threshold`. Colors all the descendent links below a cluster node \(k\) the same color if \(k\) is the first node below the cut threshold \(t\). All links connecting nodes with distances greater than or equal to the threshold are colored with de default matplotlib color `'C0'`. If \(t\) is less than or equal to zero, all nodes are colored `'C0'`. If `color_threshold` is None or ‘default’, corresponding with MATLAB(TM) behavior, the threshold is set to `0.7*max(Z[:,2])`.

    get_leaves bool, optional

      Includes a list `R['leaves']=H` in the result dictionary. For each \(i\), `H[i] == j`, cluster node `j` appears in position `i` in the left-to-right traversal of the leaves, where \(j < 2n-1\) and \(i < n\).

    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 \(n\)-sized sequence, with `n == Z.shape[0] + 1`. The `labels[i]` value is the text to put under the \(i\) th leaf node only if it corresponds to an original observation and not a non-singleton cluster.

    count_sort str or bool, optional

      For each node n, the order (visually, from left-to-right) n’s two descendent links are plotted is determined by this parameter, which can be any of the following values:

      `False`

        Nothing is done.

      `'ascending'` or `True`

        The child with the minimum number of original objects in its cluster is plotted first.

      `'descending'`

        The child with the maximum number of original objects in its cluster is plotted first.

      Note, `distance_sort` and `count_sort` cannot both be True.

    distance_sort str or bool, optional

      For each node n, the order (visually, from left-to-right) n’s two descendent links are plotted is determined by this parameter, which can be any of the following values:

      `False`

        Nothing is done.

      `'ascending'` or `True`

        The child with the minimum distance between its direct descendents is plotted first.

      `'descending'`

        The child with the maximum distance between its direct descendents is plotted first.

      Note `distance_sort` and `count_sort` cannot both be True.

    show_leaf_counts bool, optional

      When True, leaf nodes representing \(k>1\) 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 \(k < 2n-1\). The function is expected to return a string with the label for the leaf.

      Indices \(k < n\) correspond to original observations while indices \(k \geq n\) correspond to non-singleton clusters.

      For example, to label singletons with their node id and non-singletons with their id, count, and inconsistency coefficient, simply do:

          # First define the leaf label function.
          def llf(id):
              if id < n:
                  return str(id)
              else:
                  return '[%d %d %1.2f]' % (id, count, R[n-id,3])

          # The text for the leaf nodes is going to be big so force
          # a rotation of 90 degrees.
          dendrogram(Z, leaf_label_func=llf, leaf_rotation=90)

          # leaf_label_func can also be used together with ``truncate_mode``,
          # in which case you will get your leaves labeled after truncation:
          dendrogram(Z, leaf_label_func=llf, leaf_rotation=90,
                     truncate_mode='level', p=2)

    show_contracted bool, optional

      When True the heights of non-singleton nodes contracted into a leaf node are plotted as crosses along the link connecting that leaf node. This really is only useful when truncation is used (see `truncate_mode` parameter).

    link_color_func callable, optional

      If given, link_color_function is called with each non-singleton id corresponding to each U-shaped link it will paint. The function is expected to return the color to paint the link, encoded as a matplotlib color string code. For example:

          dendrogram(Z, link_color_func=lambda k: colors[k])

      colors the direct links below each untruncated non-singleton node `k` using `colors[k]`.

    ax matplotlib Axes instance, optional

      If None and no_plot is not True, the dendrogram will be plotted on the current axes. Otherwise if no_plot is not True the dendrogram will be plotted on the given `Axes` instance. This can be useful if the dendrogram is part of a more complex figure.

    above_threshold_color str, optional

      This matplotlib color string sets the color of the links above the color_threshold. The default is `'C0'`.

  Returns:
    R dict

      A dictionary of data structures computed to render the dendrogram. Its has the following keys:

      `'color_list'`

        A list of color names. The k’th element represents the color of the k’th link.

      `'icoord'` and `'dcoord'`

        Each of them is a list of lists. Let `icoord = [I1, I2, ..., Ip]` where `Ik = [xk1, xk2, xk3, xk4]` and `dcoord = [D1, D2, ..., Dp]` where `Dk = [yk1, yk2, yk3, yk4]`, then the k’th link painted is `(xk1, yk1)` - `(xk2, yk2)` - `(xk3, yk3)` - `(xk4, yk4)`.

      `'ivl'`

        A list of labels corresponding to the leaf nodes.

      `'leaves'`

        For each i, `H[i] == j`, cluster node `j` appears in position `i` in the left-to-right traversal of the leaves, where \(j < 2n-1\) and \(i < n\). If `j` is less than `n`, the `i`-th leaf node corresponds to an original observation. Otherwise, it corresponds to a non-singleton cluster.

      `'leaves_color_list'`

        A list of color names. The k’th element represents the color of the k’th leaf.

  See also

  `linkage`, `set_link_color_palette`

  Notes

  It is expected that the distances in `Z[:,2]` be monotonic, otherwise crossings appear in the dendrogram.

  Array API Standard Support

  `dendrogram` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ⛔
  PyTorch | ✅ | ⛔
  JAX | ⚠️ no JIT | ⛔
  Dask | ⚠️ computes graph | n/a

  > See Support for the array API standard for more information.

  Examples

      >>> import numpy as np
      >>> from scipy.cluster import hierarchy
      >>> import matplotlib.pyplot as plt

  A very basic example:

      >>> ytdist = np.array([662., 877., 255., 412., 996., 295., 468., 268.,
      ...                    400., 754., 564., 138., 219., 869., 669.])
      >>> Z = hierarchy.linkage(ytdist, 'single')
      >>> plt.figure()
      >>> dn = hierarchy.dendrogram(Z)

  Now, plot in given axes, improve the color scheme and use both vertical and horizontal orientations:

      >>> hierarchy.set_link_color_palette(['m', 'c', 'y', 'k'])
      >>> fig, axes = plt.subplots(1, 2, figsize=(8, 3))
      >>> dn1 = hierarchy.dendrogram(Z, ax=axes[0], above_threshold_color='y',
      ...                            orientation='top')
      >>> dn2 = hierarchy.dendrogram(Z, ax=axes[1],
      ...                            above_threshold_color='#bcbddc',
      ...                            orientation='right')
      >>> hierarchy.set_link_color_palette(None)  # reset to default after use
      >>> plt.show()