content: """ Collection of Model instances for use with the odrpack fitting package.
  """
  import numpy as np
  from scipy.odr.odrpack import Model
  
  __all__ = ['Model', 'exponential', 'multilinear', 'unilinear', 'quadratic',
             'polynomial']
  
  
  def _lin_fcn(B, x):
      a, b = B[0], B[1:]
      b.shape = (b.shape[0], 1)
  
      return a + (x*b).sum(axis=0)
  
  
  def _lin_fjb(B, x):
      a = np.ones(x.shape[-1], float)
      res = np.concatenate((a, x.ravel()))
      res.shape = (B.shape[-1], x.shape[-1])
      return res
  
  
  def _lin_fjd(B, x):
      b = B[1:]
      b = np.repeat(b, (x.shape[-1],)*b.shape[-1], axis=0)
      b.shape = x.shape
      return b
  
  
  def _lin_est(data):
      # Eh. The answer is analytical, so just return all ones.
      # Don't return zeros since that will interfere with
      # ODRPACK's auto-scaling procedures.
  
      if len(data.x.shape) == 2:
          m = data.x.shape[0]
      else:
          m = 1
  
      return np.ones((m + 1,), float)
  
  
  def _poly_fcn(B, x, powers):
      a, b = B[0], B[1:]
      b.shape = (b.shape[0], 1)
  
      return a + np.sum(b * np.power(x, powers), axis=0)
  
  
  def _poly_fjacb(B, x, powers):
      res = np.concatenate((np.ones(x.shape[-1], float),
                            np.power(x, powers).flat))
      res.shape = (B.shape[-1], x.shape[-1])
      return res
  
  
  def _poly_fjacd(B, x, powers):
      b = B[1:]
      b.shape = (b.shape[0], 1)
  
      b = b * powers
  
      return np.sum(b * np.power(x, powers-1), axis=0)
  
  
  def _exp_fcn(B, x):
      return B[0] + np.exp(B[1] * x)
  
  
  def _exp_fjd(B, x):
      return B[1] * np.exp(B[1] * x)
  
  
  def _exp_fjb(B, x):
      res = np.concatenate((np.ones(x.shape[-1], float), x * np.exp(B[1] * x)))
      res.shape = (2, x.shape[-1])
      return res
  
  
  def _exp_est(data):
      # Eh.
      return np.array([1., 1.])
  
  
  class _MultilinearModel(Model):
      r"""
      Arbitrary-dimensional linear model
  
      This model is defined by :math:`y=\beta_0 + \sum_{i=1}^m \beta_i x_i`
  
      Examples
      --------
      We can calculate orthogonal distance regression with an arbitrary
      dimensional linear model:
  
      >>> from scipy import odr
      >>> x = np.linspace(0.0, 5.0)
      >>> y = 10.0 + 5.0 * x
      >>> data = odr.Data(x, y)
      >>> odr_obj = odr.ODR(data, odr.multilinear)
      >>> output = odr_obj.run()
      >>> print(output.beta)
      [10.  5.]
  
      """
      def __init__(self):
          super().__init__(
              _lin_fcn, fjacb=_lin_fjb, fjacd=_lin_fjd, estimate=_lin_est,
              meta={'name': 'Arbitrary-dimensional Linear',
                    'equ': 'y = B_0 + Sum[i=1..m, B_i * x_i]',
                    'TeXequ': r'$y=\beta_0 + \sum_{i=1}^m \beta_i x_i$'})
  
  
  multilinear = _MultilinearModel()
  
  
  def polynomial(order):
      """
      Factory function for a general polynomial model.
  
      Parameters
      ----------
      order : int or sequence
          If an integer, it becomes the order of the polynomial to fit. If
          a sequence of numbers, then these are the explicit powers in the
          polynomial.
          A constant term (power 0) is always included, so don't include 0.
          Thus, polynomial(n) is equivalent to polynomial(range(1, n+1)).
  
      Returns
      -------
      polynomial : Model instance
          Model instance.
  
      Examples
      --------
      We can fit an input data using orthogonal distance regression (ODR) with
      a polynomial model:
  
      >>> import matplotlib.pyplot as plt
      >>> from scipy import odr
      >>> x = np.linspace(0.0, 5.0)
      >>> y = np.sin(x)
      >>> poly_model = odr.polynomial(3)  # using third order polynomial model
      >>> data = odr.Data(x, y)
      >>> odr_obj = odr.ODR(data, poly_model)
      >>> output = odr_obj.run()  # running ODR fitting
      >>> poly = np.poly1d(output.beta[::-1])
      >>> poly_y = poly(x)
      >>> plt.plot(x, y, label="input data")
      >>> plt.plot(x, poly_y, label="polynomial ODR")
      >>> plt.legend()
      >>> plt.show()
  
      """
  
      powers = np.asarray(order)
      if powers.shape == ():
          # Scalar.
          powers = np.arange(1, powers + 1)
  
      powers.shape = (len(powers), 1)
      len_beta = len(powers) + 1
  
      def _poly_est(data, len_beta=len_beta):
          # Eh. Ignore data and return all ones.
          return np.ones((len_beta,), float)
  
      return Model(_poly_fcn, fjacd=_poly_fjacd, fjacb=_poly_fjacb,
                   estimate=_poly_est, extra_args=(powers,),
                   meta={'name': 'Sorta-general Polynomial',
                   'equ': 'y = B_0 + Sum[i=1..%s, B_i * (x**i)]' % (len_beta-1),
                   'TeXequ': r'$y=\beta_0 + \sum_{i=1}^{%s} \beta_i x^i$' %
                          (len_beta-1)})
  
  
  class _ExponentialModel(Model):
      r"""
      Exponential model
  
      This model is defined by :math:`y=\beta_0 + e^{\beta_1 x}`
  
      Examples
      --------
      We can calculate orthogonal distance regression with an exponential model:
  
      >>> from scipy import odr
      >>> x = np.linspace(0.0, 5.0)
      >>> y = -10.0 + np.exp(0.5*x)
      >>> data = odr.Data(x, y)
      >>> odr_obj = odr.ODR(data, odr.exponential)
      >>> output = odr_obj.run()
      >>> print(output.beta)
      [-10.    0.5]
  
      """
      def __init__(self):
          super().__init__(_exp_fcn, fjacd=_exp_fjd, fjacb=_exp_fjb,
                           estimate=_exp_est,
                           meta={'name': 'Exponential',
                                 'equ': 'y= B_0 + exp(B_1 * x)',
                                 'TeXequ': r'$y=\beta_0 + e^{\beta_1 x}$'})
  
  
  exponential = _ExponentialModel()
  
  
  def _unilin(B, x):
      return x*B[0] + B[1]
  
  
  def _unilin_fjd(B, x):
      return np.ones(x.shape, float) * B[0]
  
  
  def _unilin_fjb(B, x):
      _ret = np.concatenate((x, np.ones(x.shape, float)))
      _ret.shape = (2,) + x.shape
  
      return _ret
  
  
  def _unilin_est(data):
      return (1., 1.)
  
  
  def _quadratic(B, x):
      return x*(x*B[0] + B[1]) + B[2]
  
  
  def _quad_fjd(B, x):
      return 2*x*B[0] + B[1]
  
  
  def _quad_fjb(B, x):
      _ret = np.concatenate((x*x, x, np.ones(x.shape, float)))
      _ret.shape = (3,) + x.shape
  
      return _ret
  
  
  def _quad_est(data):
      return (1.,1.,1.)
  
  
  class _UnilinearModel(Model):
      r"""
      Univariate linear model
  
      This model is defined by :math:`y = \beta_0 x + \beta_1`
  
      Examples
      --------
      We can calculate orthogonal distance regression with an unilinear model:
  
      >>> from scipy import odr
      >>> x = np.linspace(0.0, 5.0)
      >>> y = 1.0 * x + 2.0
      >>> data = odr.Data(x, y)
      >>> odr_obj = odr.ODR(data, odr.unilinear)
      >>> output = odr_obj.run()
      >>> print(output.beta)
      [1. 2.]
  
      """
      def __init__(self):
          super().__init__(_unilin, fjacd=_unilin_fjd, fjacb=_unilin_fjb,
                           estimate=_unilin_est,
                           meta={'name': 'Univariate Linear',
                                 'equ': 'y = B_0 * x + B_1',
                                 'TeXequ': '$y = \\beta_0 x + \\beta_1$'})
  
  
  unilinear = _UnilinearModel()
  
  
  class _QuadraticModel(Model):
      r"""
      Quadratic model
  
      This model is defined by :math:`y = \beta_0 x^2 + \beta_1 x + \beta_2`
  
      Examples
      --------
      We can calculate orthogonal distance regression with a quadratic model:
  
      >>> from scipy import odr
      >>> x = np.linspace(0.0, 5.0)
      >>> y = 1.0 * x ** 2 + 2.0 * x + 3.0
      >>> data = odr.Data(x, y)
      >>> odr_obj = odr.ODR(data, odr.quadratic)
      >>> output = odr_obj.run()
      >>> print(output.beta)
      [1. 2. 3.]
  
      """
      def __init__(self):
          super().__init__(
              _quadratic, fjacd=_quad_fjd, fjacb=_quad_fjb, estimate=_quad_est,
              meta={'name': 'Quadratic',
                    'equ': 'y = B_0*x**2 + B_1*x + B_2',
                    'TeXequ': '$y = \\beta_0 x^2 + \\beta_1 x + \\beta_2'})
  
  
  quadratic = _QuadraticModel()
  