content: """
  =========================
  Multi-dimensional scaling
  =========================
  
  An illustration of the metric and non-metric MDS on generated noisy data.
  
  The reconstructed points using the metric MDS and non metric MDS are slightly
  shifted to avoid overlapping.
  """
  
  # Author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
  # Licence: BSD
  
  print(__doc__)
  import numpy as np
  
  from matplotlib import pyplot as plt
  from matplotlib.collections import LineCollection
  
  from sklearn import manifold
  from sklearn.metrics import euclidean_distances
  from sklearn.decomposition import PCA
  
  n_samples = 20
  seed = np.random.RandomState(seed=3)
  X_true = seed.randint(0, 20, 2 * n_samples).astype(np.float)
  X_true = X_true.reshape((n_samples, 2))
  # Center the data
  X_true -= X_true.mean()
  
  similarities = euclidean_distances(X_true)
  
  # Add noise to the similarities
  noise = np.random.rand(n_samples, n_samples)
  noise = noise + noise.T
  noise[np.arange(noise.shape[0]), np.arange(noise.shape[0])] = 0
  similarities += noise
  
  mds = manifold.MDS(n_components=2, max_iter=3000, eps=1e-9, random_state=seed,
                     dissimilarity="precomputed", n_jobs=1)
  pos = mds.fit(similarities).embedding_
  
  nmds = manifold.MDS(n_components=2, metric=False, max_iter=3000, eps=1e-12,
                      dissimilarity="precomputed", random_state=seed, n_jobs=1,
                      n_init=1)
  npos = nmds.fit_transform(similarities, init=pos)
  
  # Rescale the data
  pos *= np.sqrt((X_true ** 2).sum()) / np.sqrt((pos ** 2).sum())
  npos *= np.sqrt((X_true ** 2).sum()) / np.sqrt((npos ** 2).sum())
  
  # Rotate the data
  clf = PCA(n_components=2)
  X_true = clf.fit_transform(X_true)
  
  pos = clf.fit_transform(pos)
  
  npos = clf.fit_transform(npos)
  
  fig = plt.figure(1)
  ax = plt.axes([0., 0., 1., 1.])
  
  plt.scatter(X_true[:, 0], X_true[:, 1], c='r', s=20)
  plt.scatter(pos[:, 0], pos[:, 1], s=20, c='g')
  plt.scatter(npos[:, 0], npos[:, 1], s=20, c='b')
  plt.legend(('True position', 'MDS', 'NMDS'), loc='best')
  
  similarities = similarities.max() / similarities * 100
  similarities[np.isinf(similarities)] = 0
  
  # Plot the edges
  start_idx, end_idx = np.where(pos)
  #a sequence of (*line0*, *line1*, *line2*), where::
  #            linen = (x0, y0), (x1, y1), ... (xm, ym)
  segments = [[X_true[i, :], X_true[j, :]]
              for i in range(len(pos)) for j in range(len(pos))]
  values = np.abs(similarities)
  lc = LineCollection(segments,
                      zorder=0, cmap=plt.cm.hot_r,
                      norm=plt.Normalize(0, values.max()))
  lc.set_array(similarities.flatten())
  lc.set_linewidths(0.5 * np.ones(len(segments)))
  ax.add_collection(lc)
  
  plt.show()
  