content: from __future__ import division, print_function, absolute_import
  
  import numpy as np
  from scipy._lib.six import callable, xrange
  from scipy._lib._numpy_compat import suppress_warnings
  from collections import namedtuple
  
  __all__ = ['binned_statistic',
             'binned_statistic_2d',
             'binned_statistic_dd']
  
  
  BinnedStatisticResult = namedtuple('BinnedStatisticResult',
                                     ('statistic', 'bin_edges', 'binnumber'))
  
  
  def binned_statistic(x, values, statistic='mean',
                       bins=10, range=None):
      """
      Compute a binned statistic for one or more sets of data.
  
      This is a generalization of a histogram function.  A histogram divides
      the space into bins, and returns the count of the number of points in
      each bin.  This function allows the computation of the sum, mean, median,
      or other statistic of the values (or set of values) within each bin.
  
      Parameters
      ----------
      x : (N,) array_like
          A sequence of values to be binned.
      values : (N,) array_like or list of (N,) array_like
          The data on which the statistic will be computed.  This must be
          the same shape as `x`, or a set of sequences - each the same shape as
          `x`.  If `values` is a set of sequences, the statistic will be computed
          on each independently.
      statistic : string or callable, optional
          The statistic to compute (default is 'mean').
          The following statistics are available:
  
            * 'mean' : compute the mean of values for points within each bin.
              Empty bins will be represented by NaN.
            * 'median' : compute the median of values for points within each
              bin. Empty bins will be represented by NaN.
            * 'count' : compute the count of points within each bin.  This is
              identical to an unweighted histogram.  `values` array is not
              referenced.
            * 'sum' : compute the sum of values for points within each bin.
              This is identical to a weighted histogram.
            * 'min' : compute the minimum of values for points within each bin.
              Empty bins will be represented by NaN.
            * 'max' : compute the maximum of values for point within each bin.
              Empty bins will be represented by NaN.
            * function : a user-defined function which takes a 1D array of
              values, and outputs a single numerical statistic. This function
              will be called on the values in each bin.  Empty bins will be
              represented by function([]), or NaN if this returns an error.
  
      bins : int or sequence of scalars, optional
          If `bins` is an int, it defines the number of equal-width bins in the
          given range (10 by default).  If `bins` is a sequence, it defines the
          bin edges, including the rightmost edge, allowing for non-uniform bin
          widths.  Values in `x` that are smaller than lowest bin edge are
          assigned to bin number 0, values beyond the highest bin are assigned to
          ``bins[-1]``.  If the bin edges are specified, the number of bins will
          be, (nx = len(bins)-1).
      range : (float, float) or [(float, float)], optional
          The lower and upper range of the bins.  If not provided, range
          is simply ``(x.min(), x.max())``.  Values outside the range are
          ignored.
  
      Returns
      -------
      statistic : array
          The values of the selected statistic in each bin.
      bin_edges : array of dtype float
          Return the bin edges ``(length(statistic)+1)``.
      binnumber: 1-D ndarray of ints
          Indices of the bins (corresponding to `bin_edges`) in which each value
          of `x` belongs.  Same length as `values`.  A binnumber of `i` means the
          corresponding value is between (bin_edges[i-1], bin_edges[i]).
  
      See Also
      --------
      numpy.digitize, numpy.histogram, binned_statistic_2d, binned_statistic_dd
  
      Notes
      -----
      All but the last (righthand-most) bin is half-open.  In other words, if
      `bins` is ``[1, 2, 3, 4]``, then the first bin is ``[1, 2)`` (including 1,
      but excluding 2) and the second ``[2, 3)``.  The last bin, however, is
      ``[3, 4]``, which *includes* 4.
  
      .. versionadded:: 0.11.0
  
      Examples
      --------
      >>> from scipy import stats
      >>> import matplotlib.pyplot as plt
  
      First some basic examples:
  
      Create two evenly spaced bins in the range of the given sample, and sum the
      corresponding values in each of those bins:
  
      >>> values = [1.0, 1.0, 2.0, 1.5, 3.0]
      >>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
      (array([ 4. ,  4.5]), array([ 1.,  4.,  7.]), array([1, 1, 1, 2, 2]))
  
      Multiple arrays of values can also be passed.  The statistic is calculated
      on each set independently:
  
      >>> values = [[1.0, 1.0, 2.0, 1.5, 3.0], [2.0, 2.0, 4.0, 3.0, 6.0]]
      >>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
      (array([[ 4. ,  4.5], [ 8. ,  9. ]]), array([ 1.,  4.,  7.]),
          array([1, 1, 1, 2, 2]))
  
      >>> stats.binned_statistic([1, 2, 1, 2, 4], np.arange(5), statistic='mean',
      ...                        bins=3)
      (array([ 1.,  2.,  4.]), array([ 1.,  2.,  3.,  4.]),
          array([1, 2, 1, 2, 3]))
  
      As a second example, we now generate some random data of sailing boat speed
      as a function of wind speed, and then determine how fast our boat is for
      certain wind speeds:
  
      >>> windspeed = 8 * np.random.rand(500)
      >>> boatspeed = .3 * windspeed**.5 + .2 * np.random.rand(500)
      >>> bin_means, bin_edges, binnumber = stats.binned_statistic(windspeed,
      ...                 boatspeed, statistic='median', bins=[1,2,3,4,5,6,7])
      >>> plt.figure()
      >>> plt.plot(windspeed, boatspeed, 'b.', label='raw data')
      >>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=5,
      ...            label='binned statistic of data')
      >>> plt.legend()
  
      Now we can use ``binnumber`` to select all datapoints with a windspeed
      below 1:
  
      >>> low_boatspeed = boatspeed[binnumber == 0]
  
      As a final example, we will use ``bin_edges`` and ``binnumber`` to make a
      plot of a distribution that shows the mean and distribution around that
      mean per bin, on top of a regular histogram and the probability
      distribution function:
  
      >>> x = np.linspace(0, 5, num=500)
      >>> x_pdf = stats.maxwell.pdf(x)
      >>> samples = stats.maxwell.rvs(size=10000)
  
      >>> bin_means, bin_edges, binnumber = stats.binned_statistic(x, x_pdf,
      ...         statistic='mean', bins=25)
      >>> bin_width = (bin_edges[1] - bin_edges[0])
      >>> bin_centers = bin_edges[1:] - bin_width/2
  
      >>> plt.figure()
      >>> plt.hist(samples, bins=50, normed=True, histtype='stepfilled',
      ...          alpha=0.2, label='histogram of data')
      >>> plt.plot(x, x_pdf, 'r-', label='analytical pdf')
      >>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=2,
      ...            label='binned statistic of data')
      >>> plt.plot((binnumber - 0.5) * bin_width, x_pdf, 'g.', alpha=0.5)
      >>> plt.legend(fontsize=10)
      >>> plt.show()
  
      """
      try:
          N = len(bins)
      except TypeError:
          N = 1
  
      if N != 1:
          bins = [np.asarray(bins, float)]
  
      if range is not None:
          if len(range) == 2:
              range = [range]
  
      medians, edges, binnumbers = binned_statistic_dd(
          [x], values, statistic, bins, range)
  
      return BinnedStatisticResult(medians, edges[0], binnumbers)
  
  
  BinnedStatistic2dResult = namedtuple('BinnedStatistic2dResult',
                                       ('statistic', 'x_edge', 'y_edge',
                                        'binnumber'))
  
  
  def binned_statistic_2d(x, y, values, statistic='mean',
                          bins=10, range=None, expand_binnumbers=False):
      """
      Compute a bidimensional binned statistic for one or more sets of data.
  
      This is a generalization of a histogram2d function.  A histogram divides
      the space into bins, and returns the count of the number of points in
      each bin.  This function allows the computation of the sum, mean, median,
      or other statistic of the values (or set of values) within each bin.
  
      Parameters
      ----------
      x : (N,) array_like
          A sequence of values to be binned along the first dimension.
      y : (N,) array_like
          A sequence of values to be binned along the second dimension.
      values : (N,) array_like or list of (N,) array_like
          The data on which the statistic will be computed.  This must be
          the same shape as `x`, or a list of sequences - each with the same
          shape as `x`.  If `values` is such a list, the statistic will be
          computed on each independently.
      statistic : string or callable, optional
          The statistic to compute (default is 'mean').
          The following statistics are available:
  
            * 'mean' : compute the mean of values for points within each bin.
              Empty bins will be represented by NaN.
            * 'median' : compute the median of values for points within each
              bin. Empty bins will be represented by NaN.
            * 'count' : compute the count of points within each bin.  This is
              identical to an unweighted histogram.  `values` array is not
              referenced.
            * 'sum' : compute the sum of values for points within each bin.
              This is identical to a weighted histogram.
            * 'min' : compute the minimum of values for points within each bin.
              Empty bins will be represented by NaN.
            * 'max' : compute the maximum of values for point within each bin.
              Empty bins will be represented by NaN.
            * function : a user-defined function which takes a 1D array of
              values, and outputs a single numerical statistic. This function
              will be called on the values in each bin.  Empty bins will be
              represented by function([]), or NaN if this returns an error.
  
      bins : int or [int, int] or array_like or [array, array], optional
          The bin specification:
  
            * the number of bins for the two dimensions (nx = ny = bins),
            * the number of bins in each dimension (nx, ny = bins),
            * the bin edges for the two dimensions (x_edge = y_edge = bins),
            * the bin edges in each dimension (x_edge, y_edge = bins).
  
          If the bin edges are specified, the number of bins will be,
          (nx = len(x_edge)-1, ny = len(y_edge)-1).
  
      range : (2,2) array_like, optional
          The leftmost and rightmost edges of the bins along each dimension
          (if not specified explicitly in the `bins` parameters):
          [[xmin, xmax], [ymin, ymax]]. All values outside of this range will be
          considered outliers and not tallied in the histogram.
      expand_binnumbers : bool, optional
          'False' (default): the returned `binnumber` is a shape (N,) array of
          linearized bin indices.
          'True': the returned `binnumber` is 'unraveled' into a shape (2,N)
          ndarray, where each row gives the bin numbers in the corresponding
          dimension.
          See the `binnumber` returned value, and the `Examples` section.
  
          .. versionadded:: 0.17.0
  
      Returns
      -------
      statistic : (nx, ny) ndarray
          The values of the selected statistic in each two-dimensional bin.
      x_edge : (nx + 1) ndarray
          The bin edges along the first dimension.
      y_edge : (ny + 1) ndarray
          The bin edges along the second dimension.
      binnumber : (N,) array of ints or (2,N) ndarray of ints
          This assigns to each element of `sample` an integer that represents the
          bin in which this observation falls.  The representation depends on the
          `expand_binnumbers` argument.  See `Notes` for details.
  
  
      See Also
      --------
      numpy.digitize, numpy.histogram2d, binned_statistic, binned_statistic_dd
  
      Notes
      -----
      Binedges:
      All but the last (righthand-most) bin is half-open.  In other words, if
      `bins` is ``[1, 2, 3, 4]``, then the first bin is ``[1, 2)`` (including 1,
      but excluding 2) and the second ``[2, 3)``.  The last bin, however, is
      ``[3, 4]``, which *includes* 4.
  
      `binnumber`:
      This returned argument assigns to each element of `sample` an integer that
      represents the bin in which it belongs.  The representation depends on the
      `expand_binnumbers` argument. If 'False' (default): The returned
      `binnumber` is a shape (N,) array of linearized indices mapping each
      element of `sample` to its corresponding bin (using row-major ordering).
      If 'True': The returned `binnumber` is a shape (2,N) ndarray where
      each row indicates bin placements for each dimension respectively.  In each
      dimension, a binnumber of `i` means the corresponding value is between
      (D_edge[i-1], D_edge[i]), where 'D' is either 'x' or 'y'.
  
      .. versionadded:: 0.11.0
  
      Examples
      --------
      >>> from scipy import stats
  
      Calculate the counts with explicit bin-edges:
  
      >>> x = [0.1, 0.1, 0.1, 0.6]
      >>> y = [2.1, 2.6, 2.1, 2.1]
      >>> binx = [0.0, 0.5, 1.0]
      >>> biny = [2.0, 2.5, 3.0]
      >>> ret = stats.binned_statistic_2d(x, y, None, 'count', bins=[binx,biny])
      >>> ret.statistic
      array([[ 2.,  1.],
             [ 1.,  0.]])
  
      The bin in which each sample is placed is given by the `binnumber`
      returned parameter.  By default, these are the linearized bin indices:
  
      >>> ret.binnumber
      array([5, 6, 5, 9])
  
      The bin indices can also be expanded into separate entries for each
      dimension using the `expand_binnumbers` parameter:
  
      >>> ret = stats.binned_statistic_2d(x, y, None, 'count', bins=[binx,biny],
      ...                                 expand_binnumbers=True)
      >>> ret.binnumber
      array([[1, 1, 1, 2],
             [1, 2, 1, 1]])
  
      Which shows that the first three elements belong in the xbin 1, and the
      fourth into xbin 2; and so on for y.
  
      """
  
      # This code is based on np.histogram2d
      try:
          N = len(bins)
      except TypeError:
          N = 1
  
      if N != 1 and N != 2:
          xedges = yedges = np.asarray(bins, float)
          bins = [xedges, yedges]
  
      medians, edges, binnumbers = binned_statistic_dd(
          [x, y], values, statistic, bins, range,
          expand_binnumbers=expand_binnumbers)
  
      return BinnedStatistic2dResult(medians, edges[0], edges[1], binnumbers)
  
  
  BinnedStatisticddResult = namedtuple('BinnedStatisticddResult',
                                       ('statistic', 'bin_edges',
                                        'binnumber'))
  
  
  def binned_statistic_dd(sample, values, statistic='mean',
                          bins=10, range=None, expand_binnumbers=False):
      """
      Compute a multidimensional binned statistic for a set of data.
  
      This is a generalization of a histogramdd function.  A histogram divides
      the space into bins, and returns the count of the number of points in
      each bin.  This function allows the computation of the sum, mean, median,
      or other statistic of the values within each bin.
  
      Parameters
      ----------
      sample : array_like
          Data to histogram passed as a sequence of D arrays of length N, or
          as an (N,D) array.
      values : (N,) array_like or list of (N,) array_like
          The data on which the statistic will be computed.  This must be
          the same shape as `x`, or a list of sequences - each with the same
          shape as `x`.  If `values` is such a list, the statistic will be
          computed on each independently.
      statistic : string or callable, optional
          The statistic to compute (default is 'mean').
          The following statistics are available:
  
            * 'mean' : compute the mean of values for points within each bin.
              Empty bins will be represented by NaN.
            * 'median' : compute the median of values for points within each
              bin. Empty bins will be represented by NaN.
            * 'count' : compute the count of points within each bin.  This is
              identical to an unweighted histogram.  `values` array is not
              referenced.
            * 'sum' : compute the sum of values for points within each bin.
              This is identical to a weighted histogram.
            * 'min' : compute the minimum of values for points within each bin.
              Empty bins will be represented by NaN.
            * 'max' : compute the maximum of values for point within each bin.
              Empty bins will be represented by NaN.
            * function : a user-defined function which takes a 1D array of
              values, and outputs a single numerical statistic. This function
              will be called on the values in each bin.  Empty bins will be
              represented by function([]), or NaN if this returns an error.
  
      bins : sequence or int, optional
          The bin specification must be in one of the following forms:
  
            * A sequence of arrays describing the bin edges along each dimension.
            * The number of bins for each dimension (nx, ny, ... = bins).
            * The number of bins for all dimensions (nx = ny = ... = bins).
  
      range : sequence, optional
          A sequence of lower and upper bin edges to be used if the edges are
          not given explicitely in `bins`. Defaults to the minimum and maximum
          values along each dimension.
      expand_binnumbers : bool, optional
          'False' (default): the returned `binnumber` is a shape (N,) array of
          linearized bin indices.
          'True': the returned `binnumber` is 'unraveled' into a shape (D,N)
          ndarray, where each row gives the bin numbers in the corresponding
          dimension.
          See the `binnumber` returned value, and the `Examples` section of
          `binned_statistic_2d`.
  
          .. versionadded:: 0.17.0
  
      Returns
      -------
      statistic : ndarray, shape(nx1, nx2, nx3,...)
          The values of the selected statistic in each two-dimensional bin.
      bin_edges : list of ndarrays
          A list of D arrays describing the (nxi + 1) bin edges for each
          dimension.
      binnumber : (N,) array of ints or (D,N) ndarray of ints
          This assigns to each element of `sample` an integer that represents the
          bin in which this observation falls.  The representation depends on the
          `expand_binnumbers` argument.  See `Notes` for details.
  
  
      See Also
      --------
      numpy.digitize, numpy.histogramdd, binned_statistic, binned_statistic_2d
  
      Notes
      -----
      Binedges:
      All but the last (righthand-most) bin is half-open in each dimension.  In
      other words, if `bins` is ``[1, 2, 3, 4]``, then the first bin is
      ``[1, 2)`` (including 1, but excluding 2) and the second ``[2, 3)``.  The
      last bin, however, is ``[3, 4]``, which *includes* 4.
  
      `binnumber`:
      This returned argument assigns to each element of `sample` an integer that
      represents the bin in which it belongs.  The representation depends on the
      `expand_binnumbers` argument. If 'False' (default): The returned
      `binnumber` is a shape (N,) array of linearized indices mapping each
      element of `sample` to its corresponding bin (using row-major ordering).
      If 'True': The returned `binnumber` is a shape (D,N) ndarray where
      each row indicates bin placements for each dimension respectively.  In each
      dimension, a binnumber of `i` means the corresponding value is between
      (bin_edges[D][i-1], bin_edges[D][i]), for each dimension 'D'.
  
      .. versionadded:: 0.11.0
  
      """
      known_stats = ['mean', 'median', 'count', 'sum', 'std','min','max']
      if not callable(statistic) and statistic not in known_stats:
          raise ValueError('invalid statistic %r' % (statistic,))
  
      # `Ndim` is the number of dimensions (e.g. `2` for `binned_statistic_2d`)
      # `Dlen` is the length of elements along each dimension.
      # This code is based on np.histogramdd
      try:
          # `sample` is an ND-array.
          Dlen, Ndim = sample.shape
      except (AttributeError, ValueError):
          # `sample` is a sequence of 1D arrays.
          sample = np.atleast_2d(sample).T
          Dlen, Ndim = sample.shape
  
      # Store initial shape of `values` to preserve it in the output
      values = np.asarray(values)
      input_shape = list(values.shape)
      # Make sure that `values` is 2D to iterate over rows
      values = np.atleast_2d(values)
      Vdim, Vlen = values.shape
  
      # Make sure `values` match `sample`
      if(statistic != 'count' and Vlen != Dlen):
          raise AttributeError('The number of `values` elements must match the '
                               'length of each `sample` dimension.')
  
      nbin = np.empty(Ndim, int)    # Number of bins in each dimension
      edges = Ndim * [None]         # Bin edges for each dim (will be 2D array)
      dedges = Ndim * [None]        # Spacing between edges (will be 2D array)
  
      try:
          M = len(bins)
          if M != Ndim:
              raise AttributeError('The dimension of bins must be equal '
                                   'to the dimension of the sample x.')
      except TypeError:
          bins = Ndim * [bins]
  
      # Select range for each dimension
      # Used only if number of bins is given.
      if range is None:
          smin = np.atleast_1d(np.array(sample.min(axis=0), float))
          smax = np.atleast_1d(np.array(sample.max(axis=0), float))
      else:
          smin = np.zeros(Ndim)
          smax = np.zeros(Ndim)
          for i in xrange(Ndim):
              smin[i], smax[i] = range[i]
  
      # Make sure the bins have a finite width.
      for i in xrange(len(smin)):
          if smin[i] == smax[i]:
              smin[i] = smin[i] - .5
              smax[i] = smax[i] + .5
  
      # Create edge arrays
      for i in xrange(Ndim):
          if np.isscalar(bins[i]):
              nbin[i] = bins[i] + 2  # +2 for outlier bins
              edges[i] = np.linspace(smin[i], smax[i], nbin[i] - 1)
          else:
              edges[i] = np.asarray(bins[i], float)
              nbin[i] = len(edges[i]) + 1  # +1 for outlier bins
          dedges[i] = np.diff(edges[i])
  
      nbin = np.asarray(nbin)
  
      # Compute the bin number each sample falls into, in each dimension
      sampBin = [
          np.digitize(sample[:, i], edges[i])
          for i in xrange(Ndim)
      ]
  
      # Using `digitize`, values that fall on an edge are put in the right bin.
      # For the rightmost bin, we want values equal to the right
      # edge to be counted in the last bin, and not as an outlier.
      for i in xrange(Ndim):
          # Find the rounding precision
          decimal = int(-np.log10(dedges[i].min())) + 6
          # Find which points are on the rightmost edge.
          on_edge = np.where(np.around(sample[:, i], decimal) ==
                             np.around(edges[i][-1], decimal))[0]
          # Shift these points one bin to the left.
          sampBin[i][on_edge] -= 1
  
      # Compute the sample indices in the flattened statistic matrix.
      binnumbers = np.ravel_multi_index(sampBin, nbin)
  
      result = np.empty([Vdim, nbin.prod()], float)
  
      if statistic == 'mean':
          result.fill(np.nan)
          flatcount = np.bincount(binnumbers, None)
          a = flatcount.nonzero()
          for vv in xrange(Vdim):
              flatsum = np.bincount(binnumbers, values[vv])
              result[vv, a] = flatsum[a] / flatcount[a]
      elif statistic == 'std':
          result.fill(0)
          flatcount = np.bincount(binnumbers, None)
          a = flatcount.nonzero()
          for vv in xrange(Vdim):
              flatsum = np.bincount(binnumbers, values[vv])
              flatsum2 = np.bincount(binnumbers, values[vv] ** 2)
              result[vv, a] = np.sqrt(flatsum2[a] / flatcount[a] -
                                      (flatsum[a] / flatcount[a]) ** 2)
      elif statistic == 'count':
          result.fill(0)
          flatcount = np.bincount(binnumbers, None)
          a = np.arange(len(flatcount))
          result[:, a] = flatcount[np.newaxis, :]
      elif statistic == 'sum':
          result.fill(0)
          for vv in xrange(Vdim):
              flatsum = np.bincount(binnumbers, values[vv])
              a = np.arange(len(flatsum))
              result[vv, a] = flatsum
      elif statistic == 'median':
          result.fill(np.nan)
          for i in np.unique(binnumbers):
              for vv in xrange(Vdim):
                  result[vv, i] = np.median(values[vv, binnumbers == i])
      elif statistic == 'min':
          result.fill(np.nan)
          for i in np.unique(binnumbers):
              for vv in xrange(Vdim):
                  result[vv, i] = np.min(values[vv, binnumbers == i])
      elif statistic == 'max':
          result.fill(np.nan)
          for i in np.unique(binnumbers):
              for vv in xrange(Vdim):
                  result[vv, i] = np.max(values[vv, binnumbers == i])
      elif callable(statistic):
          with np.errstate(invalid='ignore'), suppress_warnings() as sup:
              sup.filter(RuntimeWarning)
              try:
                  null = statistic([])
              except:
                  null = np.nan
          result.fill(null)
          for i in np.unique(binnumbers):
              for vv in xrange(Vdim):
                  result[vv, i] = statistic(values[vv, binnumbers == i])
  
      # Shape into a proper matrix
      result = result.reshape(np.append(Vdim, nbin))
  
      # Remove outliers (indices 0 and -1 for each bin-dimension).
      core = [slice(None)] + Ndim * [slice(1, -1)]
      result = result[core]
  
      # Unravel binnumbers into an ndarray, each row the bins for each dimension
      if(expand_binnumbers and Ndim > 1):
          binnumbers = np.asarray(np.unravel_index(binnumbers, nbin))
  
      if np.any(result.shape[1:] != nbin - 2):
          raise RuntimeError('Internal Shape Error')
  
      # Reshape to have output (`reulst`) match input (`values`) shape
      result = result.reshape(input_shape[:-1] + list(nbin-2))
  
      return BinnedStatisticddResult(result, edges, binnumbers)
  