class_docstr: Dilation mapping transformer.
  
  A transformer for applying an index grid dilation mapping to time series data,
  in the terminology of [1]_.
  
  This transformation is motivated by kernel dilation, it
  reorders the timesteps of a time series to simulate the effect of dilation.
  For instance, in a pipeline, it enables a dilation-like effect for downstream
  models that do not inherently support such a feature.
  
  Mathematically, the mapping operates on sequences :math:`x_1, \dots, x_k`.
  The dilation with factor :math:`d` is defined as the sequence
  :math:`x_1, x_{1+d}, x_{1+2d}, \dots, x_2, x_{2+d}, x_{2+2d}, \dots, x_d, x_{2d}, \dots`,
  where the subsequences with grid spacing :math:`d` are maximal.
  
  The resulting sequence is of equal length to the input sequence.
  
  This transformer reorders the values, and resets the sequence index
  to a ``RangeIndex``, if the mtype is ``pandas`` based.
  
  Parameters
  ----------
  dilation : int, default=2
      The dilation factor. Determines the spacing between original data points in the
      transformed series. Must be an integer greater than 0. A dilation of 1 means no
      change, while higher values increase the spacing.
  
  References
  ----------
  .. [1] Patrick Schäfer and Ulf Leser,
     "WEASEL 2.0--A Random Dilated Dictionary Transform for Fast,
     Accurate and Memory Constrained Time Series Classification", 2023,
     arXiv preprint arXiv:2301.10194.
  
  Example
  ----------
  >>> from sktime.transformations.series.dilation_mapping import \
  ...     DilationMappingTransformer
  >>> from sktime.datasets import load_airline
  >>> y = load_airline()
  >>> y_transform = DilationMappingTransformer(dilation=2).fit_transform(y)
code: def _dilate_series(self, x, d):
      x_dilated = pd.Series(name=x.name)
      for i in range(0, d):
          x_dilated = pd.concat((x_dilated, x[i::d]), axis=0)
      return x_dilated.reset_index(drop=True)
  
imports: import pandas
  from sktime.transformations.base import BaseTransformer
label_desc: Use image node_id 3 for calling the DilationMappingTransformer obj's underlying member method code with example usage: obj._dilate_series(x, d) and returns: x_dilated