text: """Compatibility fixes for older version of python, numpy and scipy
  
  If you add content to this file, please give the version of the package
  at which the fix is no longer needed.
  
  # XXX : originally copied from scikit-learn
  
  """
  # Authors: Emmanuelle Gouillart <emmanuelle.gouillart@normalesup.org>
  #          Gael Varoquaux <gael.varoquaux@normalesup.org>
  #          Fabian Pedregosa <fpedregosa@acm.org>
  #          Lars Buitinck <L.J.Buitinck@uva.nl>
  # License: BSD
  
  from __future__ import division
  
  import inspect
  from distutils.version import LooseVersion
  import warnings
  
  import numpy as np
  from scipy import linalg, __version__ as sp_version
  
  from .externals.six import string_types, iteritems
  
  
  ###############################################################################
  # Misc
  
  # helpers to get function arguments
  if hasattr(inspect, 'signature'):  # py35
      def _get_args(function, varargs=False):
          params = inspect.signature(function).parameters
          args = [key for key, param in params.items()
                  if param.kind not in (param.VAR_POSITIONAL, param.VAR_KEYWORD)]
          if varargs:
              varargs = [param.name for param in params.values()
                         if param.kind == param.VAR_POSITIONAL]
              if len(varargs) == 0:
                  varargs = None
              return args, varargs
          else:
              return args
  else:
      def _get_args(function, varargs=False):
          out = inspect.getargspec(function)  # args, varargs, keywords, defaults
          if varargs:
              return out[:2]
          else:
              return out[0]
  
  
  def _safe_svd(A, **kwargs):
      """Wrapper to get around the SVD did not converge error of death"""
      # Intel has a bug with their GESVD driver:
      #     https://software.intel.com/en-us/forums/intel-distribution-for-python/topic/628049  # noqa: E501
      # For SciPy 0.18 and up, we can work around it by using
      # lapack_driver='gesvd' instead.
      if kwargs.get('overwrite_a', False):
          raise ValueError('Cannot set overwrite_a=True with this function')
      try:
          return linalg.svd(A, **kwargs)
      except np.linalg.LinAlgError as exp:
          from .utils import warn
          if 'lapack_driver' in _get_args(linalg.svd):
              warn('SVD error (%s), attempting to use GESVD instead of GESDD'
                   % (exp,))
              return linalg.svd(A, lapack_driver='gesvd', **kwargs)
          else:
              raise
  
  
  ###############################################################################
  # Backporting nibabel's read_geometry
  
  def _get_read_geometry():
      """Get the geometry reading function."""
      try:
          import nibabel as nib
          has_nibabel = True
      except ImportError:
          has_nibabel = False
      if has_nibabel and LooseVersion(nib.__version__) > LooseVersion('2.1.0'):
          from nibabel.freesurfer import read_geometry
      else:
          read_geometry = _read_geometry
      return read_geometry
  
  
  def _read_geometry(filepath, read_metadata=False, read_stamp=False):
      """Backport from nibabel."""
      from .surface import _fread3, _fread3_many
      volume_info = dict()
  
      TRIANGLE_MAGIC = 16777214
      QUAD_MAGIC = 16777215
      NEW_QUAD_MAGIC = 16777213
      with open(filepath, "rb") as fobj:
          magic = _fread3(fobj)
          if magic in (QUAD_MAGIC, NEW_QUAD_MAGIC):  # Quad file
              nvert = _fread3(fobj)
              nquad = _fread3(fobj)
              (fmt, div) = (">i2", 100.) if magic == QUAD_MAGIC else (">f4", 1.)
              coords = np.fromfile(fobj, fmt, nvert * 3).astype(np.float) / div
              coords = coords.reshape(-1, 3)
              quads = _fread3_many(fobj, nquad * 4)
              quads = quads.reshape(nquad, 4)
              #
              #   Face splitting follows
              #
              faces = np.zeros((2 * nquad, 3), dtype=np.int)
              nface = 0
              for quad in quads:
                  if (quad[0] % 2) == 0:
                      faces[nface] = quad[0], quad[1], quad[3]
                      nface += 1
                      faces[nface] = quad[2], quad[3], quad[1]
                      nface += 1
                  else:
                      faces[nface] = quad[0], quad[1], quad[2]
                      nface += 1
                      faces[nface] = quad[0], quad[2], quad[3]
                      nface += 1
  
          elif magic == TRIANGLE_MAGIC:  # Triangle file
              create_stamp = fobj.readline().rstrip(b'\n').decode('utf-8')
              fobj.readline()
              vnum = np.fromfile(fobj, ">i4", 1)[0]
              fnum = np.fromfile(fobj, ">i4", 1)[0]
              coords = np.fromfile(fobj, ">f4", vnum * 3).reshape(vnum, 3)
              faces = np.fromfile(fobj, ">i4", fnum * 3).reshape(fnum, 3)
  
              if read_metadata:
                  volume_info = _read_volume_info(fobj)
          else:
              raise ValueError("File does not appear to be a Freesurfer surface")
  
      coords = coords.astype(np.float)  # XXX: due to mayavi bug on mac 32bits
  
      ret = (coords, faces)
      if read_metadata:
          if len(volume_info) == 0:
              warnings.warn('No volume information contained in the file')
          ret += (volume_info,)
      if read_stamp:
          ret += (create_stamp,)
  
      return ret
  
  
  ###############################################################################
  # Backporting logsumexp from scipy which is imported from scipy.special (0.1.0.0)
  # instead of scipy.misc
  
  
  def _get_logsumexp():
      try:
          from scipy.special import logsumexp
      except ImportError:  # old SciPy
          from scipy.misc import logsumexp
      return logsumexp
  
  
  ###############################################################################
  # Backporting scipy.signal.sosfilt (0.17) and sosfiltfilt (0.18)
  
  
  def _sosfiltfilt(sos, x, axis=-1, padtype='odd', padlen=None):
      """copy of SciPy sosfiltfilt"""
      sos, n_sections = _validate_sos(sos)
  
      # `method` is "pad"...
      ntaps = 2 * n_sections + 1
      ntaps -= min((sos[:, 2] == 0).sum(), (sos[:, 5] == 0).sum())
      edge, ext = _validate_pad(padtype, padlen, x, axis,
                                ntaps=ntaps)
  
      # These steps follow the same form as filtfilt with modifications
      zi = sosfilt_zi(sos)  # shape (n_sections, 2) --> (n_sections, ..., 2, ...)
      zi_shape = [1] * x.ndim
      zi_shape[axis] = 2
      zi.shape = [n_sections] + zi_shape
      x_0 = axis_slice(ext, stop=1, axis=axis)
      (y, zf) = sosfilt(sos, ext, axis=axis, zi=zi * x_0)
      y_0 = axis_slice(y, start=-1, axis=axis)
      (y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=zi * y_0)
      y = axis_reverse(y, axis=axis)
      if edge > 0:
          y = axis_slice(y, start=edge, stop=-edge, axis=axis)
      return y
  
  
  def axis_slice(a, start=None, stop=None, step=None, axis=-1):
      """Take a slice along axis 'axis' from 'a'"""
      a_slice = [slice(None)] * a.ndim
      a_slice[axis] = slice(start, stop, step)
      b = a[a_slice]
      return b
  
  
  def axis_reverse(a, axis=-1):
      """Reverse the 1-d slices of `a` along axis `axis`."""
      return axis_slice(a, step=-1, axis=axis)
  
  
  def _validate_pad(padtype, padlen, x, axis, ntaps):
      """Helper to validate padding for filtfilt"""
      if padtype not in ['even', 'odd', 'constant', None]:
          raise ValueError(("Unknown value '%s' given to padtype.  padtype "
                            "must be 'even', 'odd', 'constant', or None.") %
                           padtype)
  
      if padtype is None:
          padlen = 0
  
      if padlen is None:
          # Original padding; preserved for backwards compatibility.
          edge = ntaps * 3
      else:
          edge = padlen
  
      # x's 'axis' dimension must be bigger than edge.
      if x.shape[axis] <= edge:
          raise ValueError("The length of the input vector x must be at least "
                           "padlen, which is %d." % edge)
  
      if padtype is not None and edge > 0:
          # Make an extension of length `edge` at each
          # end of the input array.
          if padtype == 'even':
              ext = even_ext(x, edge, axis=axis)
          elif padtype == 'odd':
              ext = odd_ext(x, edge, axis=axis)
          else:
              ext = const_ext(x, edge, axis=axis)
      else:
          ext = x
      return edge, ext
  
  
  def _validate_sos(sos):
      """Helper to validate a SOS input"""
      sos = np.atleast_2d(sos)
      if sos.ndim != 2:
          raise ValueError('sos array must be 2D')
      n_sections, m = sos.shape
      if m != 6:
          raise ValueError('sos array must be shape (n_sections, 6)')
      if not (sos[:, 3] == 1).all():
          raise ValueError('sos[:, 3] should be all ones')
      return sos, n_sections
  
  
  def odd_ext(x, n, axis=-1):
      """Generate a new ndarray by making an odd extension of x along an axis."""
      if n < 1:
          return x
      if n > x.shape[axis] - 1:
          raise ValueError(("The extension length n (%d) is too big. " +
                           "It must not exceed x.shape[axis]-1, which is %d.")
                           % (n, x.shape[axis] - 1))
      left_end = axis_slice(x, start=0, stop=1, axis=axis)
      left_ext = axis_slice(x, start=n, stop=0, step=-1, axis=axis)
      right_end = axis_slice(x, start=-1, axis=axis)
      right_ext = axis_slice(x, start=-2, stop=-(n + 2), step=-1, axis=axis)
      ext = np.concatenate((2 * left_end - left_ext,
                            x,
                            2 * right_end - right_ext),
                           axis=axis)
      return ext
  
  
  def even_ext(x, n, axis=-1):
      """Create an ndarray that is an even extension of x along an axis."""
      if n < 1:
          return x
      if n > x.shape[axis] - 1:
          raise ValueError(("The extension length n (%d) is too big. " +
                           "It must not exceed x.shape[axis]-1, which is %d.")
                           % (n, x.shape[axis] - 1))
      left_ext = axis_slice(x, start=n, stop=0, step=-1, axis=axis)
      right_ext = axis_slice(x, start=-2, stop=-(n + 2), step=-1, axis=axis)
      ext = np.concatenate((left_ext,
                            x,
                            right_ext),
                           axis=axis)
      return ext
  
  
  def const_ext(x, n, axis=-1):
      """Create an ndarray that is a constant extension of x along an axis"""
      if n < 1:
          return x
      left_end = axis_slice(x, start=0, stop=1, axis=axis)
      ones_shape = [1] * x.ndim
      ones_shape[axis] = n
      ones = np.ones(ones_shape, dtype=x.dtype)
      left_ext = ones * left_end
      right_end = axis_slice(x, start=-1, axis=axis)
      right_ext = ones * right_end
      ext = np.concatenate((left_ext,
                            x,
                            right_ext),
                           axis=axis)
      return ext
  
  
  def sosfilt_zi(sos):
      """Compute an initial state `zi` for the sosfilt function"""
      from scipy.signal import lfilter_zi
      sos = np.asarray(sos)
      if sos.ndim != 2 or sos.shape[1] != 6:
          raise ValueError('sos must be shape (n_sections, 6)')
  
      n_sections = sos.shape[0]
      zi = np.empty((n_sections, 2))
      scale = 1.0
      for section in range(n_sections):
          b = sos[section, :3]
          a = sos[section, 3:]
          zi[section] = scale * lfilter_zi(b, a)
          # If H(z) = B(z)/A(z) is this section's transfer function, then
          # b.sum()/a.sum() is H(1), the gain at omega=0.  That's the steady
          # state value of this section's step response.
          scale *= b.sum() / a.sum()
  
      return zi
  
  
  def sosfilt(sos, x, axis=-1, zi=None):
      """Filter data along one dimension using cascaded second-order sections"""
      from scipy.signal import lfilter
      x = np.asarray(x)
  
      sos = np.atleast_2d(sos)
      if sos.ndim != 2:
          raise ValueError('sos array must be 2D')
  
      n_sections, m = sos.shape
      if m != 6:
          raise ValueError('sos array must be shape (n_sections, 6)')
  
      use_zi = zi is not None
      if use_zi:
          zi = np.asarray(zi)
          x_zi_shape = list(x.shape)
          x_zi_shape[axis] = 2
          x_zi_shape = tuple([n_sections] + x_zi_shape)
          if zi.shape != x_zi_shape:
              raise ValueError('Invalid zi shape.  With axis=%r, an input with '
                               'shape %r, and an sos array with %d sections, zi '
                               'must have shape %r.' %
                               (axis, x.shape, n_sections, x_zi_shape))
          zf = np.zeros_like(zi)
  
      for section in range(n_sections):
          if use_zi:
              x, zf[section] = lfilter(sos[section, :3], sos[section, 3:],
                                       x, axis, zi=zi[section])
          else:
              x = lfilter(sos[section, :3], sos[section, 3:], x, axis)
      out = (x, zf) if use_zi else x
      return out
  
  
  def get_sosfiltfilt():
      """Helper to get sosfiltfilt from scipy"""
      try:
          from scipy.signal import sosfiltfilt
      except ImportError:
          sosfiltfilt = _sosfiltfilt
      return sosfiltfilt
  
  
  def minimum_phase(h):
      """Convert a linear-phase FIR filter to minimum phase.
  
      Parameters
      ----------
      h : array
          Linear-phase FIR filter coefficients.
  
      Returns
      -------
      h_minimum : array
          The minimum-phase version of the filter, with length
          ``(length(h) + 1) // 2``.
      """
      try:
          from scipy.signal import minimum_phase
      except Exception:
          pass
      else:
          return minimum_phase(h)
      from scipy.fftpack import fft, ifft
      h = np.asarray(h)
      if np.iscomplexobj(h):
          raise ValueError('Complex filters not supported')
      if h.ndim != 1 or h.size <= 2:
          raise ValueError('h must be 1D and at least 2 samples long')
      n_half = len(h) // 2
      if not np.allclose(h[-n_half:][::-1], h[:n_half]):
          warnings.warn('h does not appear to by symmetric, conversion may '
                        'fail', RuntimeWarning)
      n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01)))
      # zero-pad; calculate the DFT
      h_temp = np.abs(fft(h, n_fft))
      # take 0.25*log(|H|**2) = 0.5*log(|H|)
      h_temp += 1e-7 * h_temp[h_temp > 0].min()  # don't let log blow up
      np.log(h_temp, out=h_temp)
      h_temp *= 0.5
      # IDFT
      h_temp = ifft(h_temp).real
      # multiply pointwise by the homomorphic filter
      # lmin[n] = 2u[n] - d[n]
      win = np.zeros(n_fft)
      win[0] = 1
      stop = (len(h) + 1) // 2
      win[1:stop] = 2
      if len(h) % 2:
          win[stop] = 1
      h_temp *= win
      h_temp = ifft(np.exp(fft(h_temp)))
      h_minimum = h_temp.real
      n_out = n_half + len(h) % 2
      return h_minimum[:n_out]
  
  
  ###############################################################################
  # scipy.special.sph_harm ()
  
  def _sph_harm(order, degree, az, pol):
      """Evaluate point in specified multipolar moment.
  
      When using, pay close attention to inputs. Spherical harmonic notation for
      order/degree, and theta/phi are both reversed in original SSS work compared
      to many other sources. See mathworld.wolfram.com/SphericalHarmonic.html for
      more discussion.
  
      Note that scipy has ``scipy.special.sph_harm``, but that function is
      too slow on old versions (< 0.15) for heavy use.
  
      Parameters
      ----------
      order : int
          Order of spherical harmonic. (Usually) corresponds to 'm'.
      degree : int
          Degree of spherical harmonic. (Usually) corresponds to 'l'.
      az : float
          Azimuthal (longitudinal) spherical coordinate [0, 2*pi]. 0 is aligned
          with x-axis.
      pol : float
          Polar (or colatitudinal) spherical coordinate [0, pi]. 0 is aligned
          with z-axis.
      norm : bool
          If True, include normalization factor.
  
      Returns
      -------
      base : complex float
          The spherical harmonic value.
      """
      from scipy.special import lpmv
      from .preprocessing.maxwell import _sph_harm_norm
  
      # Error checks
      if np.abs(order) > degree:
          raise ValueError('Absolute value of order must be <= degree')
      # Ensure that polar and azimuth angles are arrays
      az = np.asarray(az)
      pol = np.asarray(pol)
      if (np.abs(az) > 2 * np.pi).any():
          raise ValueError('Azimuth coords must lie in [-2*pi, 2*pi]')
      if(pol < 0).any() or (pol > np.pi).any():
          raise ValueError('Polar coords must lie in [0, pi]')
      # This is the "seismology" convention on Wikipedia, w/o Condon-Shortley
      sph = lpmv(order, degree, np.cos(pol)) * np.exp(1j * order * az)
      sph *= _sph_harm_norm(order, degree)
      return sph
  
  
  def _get_sph_harm():
      """Helper to get a usable spherical harmonic function."""
      if LooseVersion(sp_version) < LooseVersion('0.17.1'):
          sph_harm = _sph_harm
      else:
          from scipy.special import sph_harm
      return sph_harm
  
  
  ###############################################################################
  # Scipy spectrogram (for mne.time_frequency.psd_welch) needed for scipy < 0.16
  
  def _spectrogram(x, fs=1.0, window=('tukey',.25), nperseg=256, noverlap=None,
                  nfft=None, detrend='constant', return_onesided=True,
                  scaling='density', axis=-1, mode='psd'):
      """
      Compute a spectrogram with consecutive Fourier transforms.
      Spectrograms can be used as a way of visualizing the change of a
      nonstationary signal's frequency content over time.
  
      Parameters
      ----------
      x : array_like
          Time series of measurement values
      fs : float, optional
          Sampling frequency of the `x` time series. Defaults to 1.0.
      window : str or tuple or array_like, optional
          Desired window to use. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length will be used for nperseg.
          Defaults to a Tukey window with shape parameter of 0.25.
      nperseg : int, optional
          Length of each segment.  Defaults to 256.
      noverlap : int, optional
          Number of points to overlap between segments. If None,
          ``noverlap = nperseg // 8``.  Defaults to None.
      nfft : int, optional
          Length of the FFT used, if a zero padded FFT is desired.  If None,
          the FFT length is `nperseg`. Defaults to None.
      detrend : str or function or False, optional
          Specifies how to detrend each segment. If `detrend` is a string,
          it is passed as the ``type`` argument to `detrend`.  If it is a
          function, it takes a segment and returns a detrended segment.
          If `detrend` is False, no detrending is done.  Defaults to 'constant'.
      return_onesided : bool, optional
          If True, return a one-sided spectrum for real data. If False return
          a two-sided spectrum. Note that for complex data, a two-sided
          spectrum is always returned.
      scaling : { 'density', 'spectrum' }, optional
          Selects between computing the power spectral density ('density')
          where `Pxx` has units of V**2/Hz and computing the power spectrum
          ('spectrum') where `Pxx` has units of V**2, if `x` is measured in V
          and fs is measured in Hz.  Defaults to 'density'
      axis : int, optional
          Axis along which the spectrogram is computed; the default is over
          the last axis (i.e. ``axis=-1``).
      mode : str, optional
          Defines what kind of return values are expected. Options are ['psd',
          'complex', 'magnitude', 'angle', 'phase'].
  
      Returns
      -------
      f : ndarray
          Array of sample frequencies.
      t : ndarray
          Array of segment times.
      Sxx : ndarray
          Spectrogram of x. By default, the last axis of Sxx corresponds to the
          segment times.
  
      See Also
      --------
      periodogram: Simple, optionally modified periodogram
      lombscargle: Lomb-Scargle periodogram for unevenly sampled data
      welch: Power spectral density by Welch's method.
      csd: Cross spectral density by Welch's method.
  
      Notes
      -----
      An appropriate amount of overlap will depend on the choice of window
      and on your requirements. In contrast to welch's method, where the entire
      data stream is averaged over, one may wish to use a smaller overlap (or
      perhaps none at all) when computing a spectrogram, to maintain some
      statistical independence between individual segments.
      .. versionadded:: 0.16.0
  
      References
      ----------
      .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck "Discrete-Time
             Signal Processing", Prentice Hall, 1999.
      """
      # Less overlap than welch, so samples are more statisically independent
      if noverlap is None:
          noverlap = nperseg // 8
  
      freqs, time, Pxy = _spectral_helper(x, x, fs, window, nperseg, noverlap,
                                          nfft, detrend, return_onesided, scaling,
                                          axis, mode=mode)
  
      return freqs, time, Pxy
  
  
  def _spectral_helper(x, y, fs=1.0, window='hann', nperseg=256,
                      noverlap=None, nfft=None, detrend='constant',
                      return_onesided=True, scaling='spectrum', axis=-1,
                      mode='psd'):
      """
      Calculate various forms of windowed FFTs for PSD, CSD, etc.
      This is a helper function that implements the commonality between the
      psd, csd, and spectrogram functions. It is not designed to be called
      externally. The windows are not averaged over; the result from each window
      is returned.
  
      Parameters
      ---------
      x : array_like
          Array or sequence containing the data to be analyzed.
      y : array_like
          Array or sequence containing the data to be analyzed. If this is
          the same object in memoery as x (i.e. _spectral_helper(x, x, ...)),
          the extra computations are spared.
      fs : float, optional
          Sampling frequency of the time series. Defaults to 1.0.
      window : str or tuple or array_like, optional
          Desired window to use. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length will be used for nperseg.
          Defaults to 'hann'.
      nperseg : int, optional
          Length of each segment.  Defaults to 256.
      noverlap : int, optional
          Number of points to overlap between segments. If None,
          ``noverlap = nperseg // 2``.  Defaults to None.
      nfft : int, optional
          Length of the FFT used, if a zero padded FFT is desired.  If None,
          the FFT length is `nperseg`. Defaults to None.
      detrend : str or function or False, optional
          Specifies how to detrend each segment. If `detrend` is a string,
          it is passed as the ``type`` argument to `detrend`.  If it is a
          function, it takes a segment and returns a detrended segment.
          If `detrend` is False, no detrending is done.  Defaults to 'constant'.
      return_onesided : bool, optional
          If True, return a one-sided spectrum for real data. If False return
          a two-sided spectrum. Note that for complex data, a two-sided
          spectrum is always returned.
      scaling : { 'density', 'spectrum' }, optional
          Selects between computing the cross spectral density ('density')
          where `Pxy` has units of V**2/Hz and computing the cross spectrum
          ('spectrum') where `Pxy` has units of V**2, if `x` and `y` are
          measured in V and fs is measured in Hz.  Defaults to 'density'
      axis : int, optional
          Axis along which the periodogram is computed; the default is over
          the last axis (i.e. ``axis=-1``).
      mode : str, optional
          Defines what kind of return values are expected. Options are ['psd',
          'complex', 'magnitude', 'angle', 'phase'].
  
      Returns
      -------
      freqs : ndarray
          Array of sample frequencies.
      t : ndarray
          Array of times corresponding to each data segment
      result : ndarray
          Array of output data, contents dependent on *mode* kwarg.
  
      References
      ----------
      .. [1] Stack Overflow, "Rolling window for 1D arrays in Numpy?",
          http://stackoverflow.com/a/6811241
      .. [2] Stack Overflow, "Using strides for an efficient moving average
          filter", http://stackoverflow.com/a/4947453
  
      Notes
      -----
      Adapted from matplotlib.mlab
      .. versionadded:: 0.16.0
      """
      from scipy import fftpack
      from scipy.signal import signaltools
      from scipy.signal.windows import get_window
  
      if mode not in ['psd', 'complex', 'magnitude', 'angle', 'phase']:
          raise ValueError("Unknown value for mode %s, must be one of: "
                           "'default', 'psd', 'complex', "
                           "'magnitude', 'angle', 'phase'" % mode)
  
      # If x and y are the same object we can save ourselves some computation.
      same_data = y is x
  
      if not same_data and mode != 'psd':
          raise ValueError("x and y must be equal if mode is not 'psd'")
  
      axis = int(axis)
  
      # Ensure we have np.arrays, get outdtype
      x = np.asarray(x)
      if not same_data:
          y = np.asarray(y)
          outdtype = np.result_type(x,y,np.complex64)
      else:
          outdtype = np.result_type(x,np.complex64)
  
      if not same_data:
          # Check if we can broadcast the outer axes together
          xouter = list(x.shape)
          youter = list(y.shape)
          xouter.pop(axis)
          youter.pop(axis)
          try:
              outershape = np.broadcast(np.empty(xouter), np.empty(youter)).shape
          except ValueError:
              raise ValueError('x and y cannot be broadcast together.')
  
      if same_data:
          if x.size == 0:
              return np.empty(x.shape), np.empty(x.shape), np.empty(x.shape)
      else:
          if x.size == 0 or y.size == 0:
              outshape = outershape + (min([x.shape[axis], y.shape[axis]]),)
              emptyout = np.rollaxis(np.empty(outshape), -1, axis)
              return emptyout, emptyout, emptyout
  
      if x.ndim > 1:
          if axis != -1:
              x = np.rollaxis(x, axis, len(x.shape))
              if not same_data and y.ndim > 1:
                  y = np.rollaxis(y, axis, len(y.shape))
  
      # Check if x and y are the same length, zero-pad if necessary
      if not same_data:
          if x.shape[-1] != y.shape[-1]:
              if x.shape[-1] < y.shape[-1]:
                  pad_shape = list(x.shape)
                  pad_shape[-1] = y.shape[-1] - x.shape[-1]
                  x = np.concatenate((x, np.zeros(pad_shape)), -1)
              else:
                  pad_shape = list(y.shape)
                  pad_shape[-1] = x.shape[-1] - y.shape[-1]
                  y = np.concatenate((y, np.zeros(pad_shape)), -1)
  
      # X and Y are same length now, can test nperseg with either
      if x.shape[-1] < nperseg:
          warnings.warn('nperseg = {0:d}, is greater than input length = {1:d}, '
                        'using nperseg = {1:d}'.format(nperseg, x.shape[-1]))
          nperseg = x.shape[-1]
  
      nperseg = int(nperseg)
      if nperseg < 1:
          raise ValueError('nperseg must be a positive integer')
  
      if nfft is None:
          nfft = nperseg
      elif nfft < nperseg:
          raise ValueError('nfft must be greater than or equal to nperseg.')
      else:
          nfft = int(nfft)
  
      if noverlap is None:
          noverlap = nperseg//2
      elif noverlap >= nperseg:
          raise ValueError('noverlap must be less than nperseg.')
      else:
          noverlap = int(noverlap)
  
      # Handle detrending and window functions
      if not detrend:
          def detrend_func(d):
              return d
      elif not hasattr(detrend, '__call__'):
          def detrend_func(d):
              return signaltools.detrend(d, type=detrend, axis=-1)
      elif axis != -1:
          # Wrap this function so that it receives a shape that it could
          # reasonably expect to receive.
          def detrend_func(d):
              d = np.rollaxis(d, -1, axis)
              d = detrend(d)
              return np.rollaxis(d, axis, len(d.shape))
      else:
          detrend_func = detrend
  
      if isinstance(window, string_types) or type(window) is tuple:
          win = get_window(window, nperseg)
      else:
          win = np.asarray(window)
          if len(win.shape) != 1:
              raise ValueError('window must be 1-D')
          if win.shape[0] != nperseg:
              raise ValueError('window must have length of nperseg')
  
      if np.result_type(win,np.complex64) != outdtype:
          win = win.astype(outdtype)
  
      if mode == 'psd':
          if scaling == 'density':
              scale = 1.0 / (fs * (win*win).sum())
          elif scaling == 'spectrum':
              scale = 1.0 / win.sum()**2
          else:
              raise ValueError('Unknown scaling: %r' % scaling)
      else:
          scale = 1
  
      if return_onesided is True:
          if np.iscomplexobj(x):
              sides = 'twosided'
          else:
              sides = 'onesided'
              if not same_data:
                  if np.iscomplexobj(y):
                      sides = 'twosided'
      else:
          sides = 'twosided'
  
      if sides == 'twosided':
          num_freqs = nfft
      elif sides == 'onesided':
          if nfft % 2:
              num_freqs = (nfft + 1)//2
          else:
              num_freqs = nfft//2 + 1
  
      # Perform the windowed FFTs
      result = _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft)
      result = result[..., :num_freqs]
      freqs = fftpack.fftfreq(nfft, 1/fs)[:num_freqs]
  
      if not same_data:
          # All the same operations on the y data
          result_y = _fft_helper(y, win, detrend_func, nperseg, noverlap, nfft)
          result_y = result_y[..., :num_freqs]
          result = np.conjugate(result) * result_y
      elif mode == 'psd':
          result = np.conjugate(result) * result
      elif mode == 'magnitude':
          result = np.absolute(result)
      elif mode == 'angle' or mode == 'phase':
          result = np.angle(result)
      elif mode == 'complex':
          pass
  
      result *= scale
      if sides == 'onesided':
          if nfft % 2:
              result[...,1:] *= 2
          else:
              # Last point is unpaired Nyquist freq point, don't double
              result[...,1:-1] *= 2
  
      t = np.arange(nperseg/2, x.shape[-1] - nperseg/2 + 1, nperseg - noverlap)/float(fs)
  
      if sides != 'twosided' and not nfft % 2:
          # get the last value correctly, it is negative otherwise
          freqs[-1] *= -1
  
      # we unwrap the phase here to handle the onesided vs. twosided case
      if mode == 'phase':
          result = np.unwrap(result, axis=-1)
  
      result = result.astype(outdtype)
  
      # All imaginary parts are zero anyways
      if same_data and mode != 'complex':
          result = result.real
  
      # Output is going to have new last axis for window index
      if axis != -1:
          # Specify as positive axis index
          if axis < 0:
              axis = len(result.shape)-1-axis
  
          # Roll frequency axis back to axis where the data came from
          result = np.rollaxis(result, -1, axis)
      else:
          # Make sure window/time index is last axis
          result = np.rollaxis(result, -1, -2)
  
      return freqs, t, result
  
  
  def _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft):
      """
      Calculate windowed FFT, for internal use by scipy.signal._spectral_helper
      This is a helper function that does the main FFT calculation for
      _spectral helper. All input valdiation is performed there, and the data
      axis is assumed to be the last axis of x. It is not designed to be called
      externally. The windows are not averaged over; the result from each window
      is returned.
  
      Returns
      -------
      result : ndarray
          Array of FFT data
  
      References
      ----------
      .. [1] Stack Overflow, "Repeat NumPy array without replicating data?",
          http://stackoverflow.com/a/5568169
  
      Notes
      -----
      Adapted from matplotlib.mlab
      .. versionadded:: 0.16.0
      """
      from scipy import fftpack
  
      # Created strided array of data segments
      if nperseg == 1 and noverlap == 0:
          result = x[..., np.newaxis]
      else:
          step = nperseg - noverlap
          shape = x.shape[:-1]+((x.shape[-1]-noverlap)//step, nperseg)
          strides = x.strides[:-1]+(step*x.strides[-1], x.strides[-1])
          result = np.lib.stride_tricks.as_strided(x, shape=shape,
                                                   strides=strides)
  
      # Detrend each data segment individually
      result = detrend_func(result)
  
      # Apply window by multiplication
      result = win * result
  
      # Perform the fft. Acts on last axis by default. Zero-pads automatically
      result = fftpack.fft(result, n=nfft)
  
      return result
  
  
  def get_spectrogram():
      '''helper function to get relevant spectrogram'''
      from .utils import check_version
      if check_version('scipy', '0.16.0'):
          from scipy.signal import spectrogram
      else:
          spectrogram = _spectrogram
      return spectrogram
  
  
  ###############################################################################
  # Misc utilities
  
  def assert_true(expr, msg='False is not True'):
      """Fake assert_true without message"""
      if not expr:
          raise AssertionError(msg)
  
  
  def assert_is(expr1, expr2, msg=None):
      """Fake assert_is without message"""
      assert_true(expr2 is expr2, msg)
  
  
  def assert_is_not(expr1, expr2, msg=None):
      """Fake assert_is_not without message"""
      assert_true(expr1 is not expr2, msg)
  
  
  def _read_volume_info(fobj):
      """An implementation of nibabel.freesurfer.io._read_volume_info, since old
      versions of nibabel (<=2.1.0) don't have it.
      """
      volume_info = dict()
      head = np.fromfile(fobj, '>i4', 1)
      if not np.array_equal(head, [20]):  # Read two bytes more
          head = np.concatenate([head, np.fromfile(fobj, '>i4', 2)])
          if not np.array_equal(head, [2, 0, 20]):
              warnings.warn("Unknown extension code.")
              return volume_info
  
      volume_info['head'] = head
      for key in ['valid', 'filename', 'volume', 'voxelsize', 'xras', 'yras',
                  'zras', 'cras']:
          pair = fobj.readline().decode('utf-8').split('=')
          if pair[0].strip() != key or len(pair) != 2:
              raise IOError('Error parsing volume info.')
          if key in ('valid', 'filename'):
              volume_info[key] = pair[1].strip()
          elif key == 'volume':
              volume_info[key] = np.array(pair[1].split()).astype(int)
          else:
              volume_info[key] = np.array(pair[1].split()).astype(float)
      # Ignore the rest
      return volume_info
  
  
  def _serialize_volume_info(volume_info):
      """An implementation of nibabel.freesurfer.io._serialize_volume_info, since
      old versions of nibabel (<=2.1.0) don't have it."""
      keys = ['head', 'valid', 'filename', 'volume', 'voxelsize', 'xras', 'yras',
              'zras', 'cras']
      diff = set(volume_info.keys()).difference(keys)
      if len(diff) > 0:
          raise ValueError('Invalid volume info: %s.' % diff.pop())
  
      strings = list()
      for key in keys:
          if key == 'head':
              if not (np.array_equal(volume_info[key], [20]) or np.array_equal(
                      volume_info[key], [2, 0, 20])):
                  warnings.warn("Unknown extension code.")
              strings.append(np.array(volume_info[key], dtype='>i4').tostring())
          elif key in ('valid', 'filename'):
              val = volume_info[key]
              strings.append('{0} = {1}\n'.format(key, val).encode('utf-8'))
          elif key == 'volume':
              val = volume_info[key]
              strings.append('{0} = {1} {2} {3}\n'.format(
                  key, val[0], val[1], val[2]).encode('utf-8'))
          else:
              val = volume_info[key]
              strings.append('{0} = {1:0.10g} {2:0.10g} {3:0.10g}\n'.format(
                  key.ljust(6), val[0], val[1], val[2]).encode('utf-8'))
      return b''.join(strings)
  
  
  ##############################################################################
  # adapted from scikit-learn
  
  
  def is_classifier(estimator):
      """Returns True if the given estimator is (probably) a classifier.
  
      Parameters
      ----------
      estimator : object
          Estimator object to test.
  
      Returns
      -------
      out : bool
          True if estimator is a classifier and False otherwise.
      """
      return getattr(estimator, "_estimator_type", None) == "classifier"
  
  
  def is_regressor(estimator):
      """Returns True if the given estimator is (probably) a regressor.
  
      Parameters
      ----------
      estimator : object
          Estimator object to test.
  
      Returns
      -------
      out : bool
          True if estimator is a regressor and False otherwise.
      """
      return getattr(estimator, "_estimator_type", None) == "regressor"
  
  
  class BaseEstimator(object):
      """Base class for all estimators in scikit-learn
  
      Notes
      -----
      All estimators should specify all the parameters that can be set
      at the class level in their ``__init__`` as explicit keyword
      arguments (no ``*args`` or ``**kwargs``).
      """
  
      @classmethod
      def _get_param_names(cls):
          """Get parameter names for the estimator"""
          try:
              from inspect import signature
          except ImportError:
              from .externals.funcsigs import signature
          # fetch the constructor or the original constructor before
          # deprecation wrapping if any
          init = getattr(cls.__init__, 'deprecated_original', cls.__init__)
          if init is object.__init__:
              # No explicit constructor to introspect
              return []
  
          # introspect the constructor arguments to find the model parameters
          # to represent
          init_signature = signature(init)
          # Consider the constructor parameters excluding 'self'
          parameters = [p for p in init_signature.parameters.values()
                        if p.name != 'self' and p.kind != p.VAR_KEYWORD]
          for p in parameters:
              if p.kind == p.VAR_POSITIONAL:
                  raise RuntimeError("scikit-learn estimators should always "
                                     "specify their parameters in the signature"
                                     " of their __init__ (no varargs)."
                                     " %s with constructor %s doesn't "
                                     " follow this convention."
                                     % (cls, init_signature))
          # Extract and sort argument names excluding 'self'
          return sorted([p.name for p in parameters])
  
      def get_params(self, deep=True):
          """Get parameters for this estimator.
  
          Parameters
          ----------
          deep : boolean, optional
              If True, will return the parameters for this estimator and
              contained subobjects that are estimators.
  
          Returns
          -------
          params : mapping of string to any
              Parameter names mapped to their values.
          """
          out = dict()
          for key in self._get_param_names():
              # We need deprecation warnings to always be on in order to
              # catch deprecated param values.
              # This is set in utils/__init__.py but it gets overwritten
              # when running under python3 somehow.
              warnings.simplefilter("always", DeprecationWarning)
              try:
                  with warnings.catch_warnings(record=True) as w:
                      value = getattr(self, key, None)
                  if len(w) and w[0].category == DeprecationWarning:
                      # if the parameter is deprecated, don't show it
                      continue
              finally:
                  warnings.filters.pop(0)
  
              # XXX: should we rather test if instance of estimator?
              if deep and hasattr(value, 'get_params'):
                  deep_items = value.get_params().items()
                  out.update((key + '__' + k, val) for k, val in deep_items)
              out[key] = value
          return out
  
      def set_params(self, **params):
          """Set the parameters of this estimator.
          The method works on simple estimators as well as on nested objects
          (such as pipelines). The latter have parameters of the form
          ``<component>__<parameter>`` so that it's possible to update each
          component of a nested object.
          Returns
          -------
          self
          """
          if not params:
              # Simple optimisation to gain speed (inspect is slow)
              return self
          valid_params = self.get_params(deep=True)
          for key, value in iteritems(params):
              split = key.split('__', 1)
              if len(split) > 1:
                  # nested objects case
                  name, sub_name = split
                  if name not in valid_params:
                      raise ValueError('Invalid parameter %s for estimator %s. '
                                       'Check the list of available parameters '
                                       'with `estimator.get_params().keys()`.' %
                                       (name, self))
                  sub_object = valid_params[name]
                  sub_object.set_params(**{sub_name: value})
              else:
                  # simple objects case
                  if key not in valid_params:
                      raise ValueError('Invalid parameter %s for estimator %s. '
                                       'Check the list of available parameters '
                                       'with `estimator.get_params().keys()`.' %
                                       (key, self.__class__.__name__))
                  setattr(self, key, value)
          return self
  
      def __repr__(self):
          from sklearn.base import _pprint
          class_name = self.__class__.__name__
          return '%s(%s)' % (class_name, _pprint(self.get_params(deep=False),
                                                 offset=len(class_name),),)
  
      # __getstate__ and __setstate__ are omitted because they only contain
      # conditionals that are not satisfied by our objects (e.g.,
      # ``if type(self).__module__.startswith('sklearn.')``.
  
  
  ###############################################################################
  # Copied from sklearn to simplify code paths
  
  def empirical_covariance(X, assume_centered=False):
      """Computes the Maximum likelihood covariance estimator
  
  
      Parameters
      ----------
      X : ndarray, shape (n_samples, n_features)
          Data from which to compute the covariance estimate
  
      assume_centered : Boolean
          If True, data are not centered before computation.
          Useful when working with data whose mean is almost, but not exactly
          zero.
          If False, data are centered before computation.
  
      Returns
      -------
      covariance : 2D ndarray, shape (n_features, n_features)
          Empirical covariance (Maximum Likelihood Estimator).
  
      """
      X = np.asarray(X)
      if X.ndim == 1:
          X = np.reshape(X, (1, -1))
  
      if X.shape[0] == 1:
          warnings.warn("Only one sample available. "
                        "You may want to reshape your data array")
  
      if assume_centered:
          covariance = np.dot(X.T, X) / X.shape[0]
      else:
          covariance = np.cov(X.T, bias=1)
  
      if covariance.ndim == 0:
          covariance = np.array([[covariance]])
      return covariance
  
  
  class EmpiricalCovariance(BaseEstimator):
      """Maximum likelihood covariance estimator
  
      Read more in the :ref:`User Guide <covariance>`.
  
      Parameters
      ----------
      store_precision : bool
          Specifies if the estimated precision is stored.
  
      assume_centered : bool
          If True, data are not centered before computation.
          Useful when working with data whose mean is almost, but not exactly
          zero.
          If False (default), data are centered before computation.
  
      Attributes
      ----------
      covariance_ : 2D ndarray, shape (n_features, n_features)
          Estimated covariance matrix
  
      precision_ : 2D ndarray, shape (n_features, n_features)
          Estimated pseudo-inverse matrix.
          (stored only if store_precision is True)
  
      """
      def __init__(self, store_precision=True, assume_centered=False):
          self.store_precision = store_precision
          self.assume_centered = assume_centered
  
      def _set_covariance(self, covariance):
          """Saves the covariance and precision estimates
  
          Storage is done accordingly to `self.store_precision`.
          Precision stored only if invertible.
  
          Parameters
          ----------
          covariance : 2D ndarray, shape (n_features, n_features)
              Estimated covariance matrix to be stored, and from which precision
              is computed.
  
          """
          # covariance = check_array(covariance)
          # set covariance
          self.covariance_ = covariance
          # set precision
          if self.store_precision:
              self.precision_ = linalg.pinvh(covariance)
          else:
              self.precision_ = None
  
      def get_precision(self):
          """Getter for the precision matrix.
  
          Returns
          -------
          precision_ : array-like,
              The precision matrix associated to the current covariance object.
  
          """
          if self.store_precision:
              precision = self.precision_
          else:
              precision = linalg.pinvh(self.covariance_)
          return precision
  
      def fit(self, X, y=None):
          """Fits the Maximum Likelihood Estimator covariance model
          according to the given training data and parameters.
  
          Parameters
          ----------
          X : array-like, shape = [n_samples, n_features]
            Training data, where n_samples is the number of samples and
            n_features is the number of features.
  
          y : not used, present for API consistence purpose.
  
          Returns
          -------
          self : object
              Returns self.
  
          """
          # X = check_array(X)
          if self.assume_centered:
              self.location_ = np.zeros(X.shape[1])
          else:
              self.location_ = X.mean(0)
          covariance = empirical_covariance(
              X, assume_centered=self.assume_centered)
          self._set_covariance(covariance)
  
          return self
  
      def score(self, X_test, y=None):
          """Computes the log-likelihood of a Gaussian data set with
          `self.covariance_` as an estimator of its covariance matrix.
  
          Parameters
          ----------
          X_test : array-like, shape = [n_samples, n_features]
              Test data of which we compute the likelihood, where n_samples is
              the number of samples and n_features is the number of features.
              X_test is assumed to be drawn from the same distribution than
              the data used in fit (including centering).
  
          y : not used, present for API consistence purpose.
  
          Returns
          -------
          res : float
              The likelihood of the data set with `self.covariance_` as an
              estimator of its covariance matrix.
  
          """
          # compute empirical covariance of the test set
          test_cov = empirical_covariance(
              X_test - self.location_, assume_centered=True)
          # compute log likelihood
          res = log_likelihood(test_cov, self.get_precision())
  
          return res
  
      def error_norm(self, comp_cov, norm='frobenius', scaling=True,
                     squared=True):
          """Computes the Mean Squared Error between two covariance estimators.
          (In the sense of the Frobenius norm).
  
          Parameters
          ----------
          comp_cov : array-like, shape = [n_features, n_features]
              The covariance to compare with.
  
          norm : str
              The type of norm used to compute the error. Available error types:
              - 'frobenius' (default): sqrt(tr(A^t.A))
              - 'spectral': sqrt(max(eigenvalues(A^t.A))
              where A is the error ``(comp_cov - self.covariance_)``.
  
          scaling : bool
              If True (default), the squared error norm is divided by n_features.
              If False, the squared error norm is not rescaled.
  
          squared : bool
              Whether to compute the squared error norm or the error norm.
              If True (default), the squared error norm is returned.
              If False, the error norm is returned.
  
          Returns
          -------
          The Mean Squared Error (in the sense of the Frobenius norm) between
          `self` and `comp_cov` covariance estimators.
  
          """
          # compute the error
          error = comp_cov - self.covariance_
          # compute the error norm
          if norm == "frobenius":
              squared_norm = np.sum(error ** 2)
          elif norm == "spectral":
              squared_norm = np.amax(linalg.svdvals(np.dot(error.T, error)))
          else:
              raise NotImplementedError(
                  "Only spectral and frobenius norms are implemented")
          # optionally scale the error norm
          if scaling:
              squared_norm = squared_norm / error.shape[0]
          # finally get either the squared norm or the norm
          if squared:
              result = squared_norm
          else:
              result = np.sqrt(squared_norm)
  
          return result
  
      def mahalanobis(self, observations):
          """Computes the squared Mahalanobis distances of given observations.
  
          Parameters
          ----------
          observations : array-like, shape = [n_observations, n_features]
              The observations, the Mahalanobis distances of the which we
              compute. Observations are assumed to be drawn from the same
              distribution than the data used in fit.
  
          Returns
          -------
          mahalanobis_distance : array, shape = [n_observations,]
              Squared Mahalanobis distances of the observations.
  
          """
          precision = self.get_precision()
          # compute mahalanobis distances
          centered_obs = observations - self.location_
          mahalanobis_dist = np.sum(
              np.dot(centered_obs, precision) * centered_obs, 1)
  
          return mahalanobis_dist
  
  
  def log_likelihood(emp_cov, precision):
      """Computes the sample mean of the log_likelihood under a covariance model
  
      computes the empirical expected log-likelihood (accounting for the
      normalization terms and scaling), allowing for universal comparison (beyond
      this software package)
  
      Parameters
      ----------
      emp_cov : 2D ndarray (n_features, n_features)
          Maximum Likelihood Estimator of covariance
  
      precision : 2D ndarray (n_features, n_features)
          The precision matrix of the covariance model to be tested
  
      Returns
      -------
      sample mean of the log-likelihood
      """
      p = precision.shape[0]
      log_likelihood_ = - np.sum(emp_cov * precision) + _logdet(precision)
      log_likelihood_ -= p * np.log(2 * np.pi)
      log_likelihood_ /= 2.
      return log_likelihood_
  
  
  # sklearn uses np.linalg for this, but ours is more robust to zero eigenvalues
  
  def _logdet(A):
      """Compute the log det of a positive semidefinite matrix."""
      vals = linalg.eigh(A)[0]
      # avoid negative (numerical errors) or zero (semi-definite matrix) values
      tol = vals.max() * vals.size * np.finfo(np.float64).eps
      vals = np.where(vals > tol, vals, tol)
      return np.sum(np.log(vals))
  
  
  ###############################################################################
  # NumPy einsum backward compat (allow "optimize" arg and fix 1.14.0 bug)
  # XXX eventually we should hand-tune our `einsum` calls given our array sizes!
  
  _has_optimize = (LooseVersion(np.__version__) >= '1.12')
  
  
  def einsum(*args, **kwargs):
      if 'optimize' in kwargs:
          if not _has_optimize:
              kwargs.pop('optimize')
      elif _has_optimize:
          kwargs['optimize'] = False
      return np.einsum(*args, **kwargs)
  