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# Source code for astropy.modeling.polynomial

    # Licensed under a 3-clause BSD style license - see LICENSE.rst

    """
    This module contains predefined polynomial models.
    """

    from __future__ import division

    import collections

    from textwrap import dedent

    import numpy as np

    from .core import ParametricModel, Model, SerialCompositeModel, format_input
    from .functional_models import Shift
    from .parameters import Parameter
    from .utils import poly_map_domain, comb
    from ..logger import log
    from ..utils import lazyproperty, indent

    __all__ = [
        'Chebyshev1D', 'Chebyshev2D', 'InverseSIP',
        'Legendre1D', 'Legendre2D', 'Polynomial1D',
        'Polynomial2D', 'SIP', 'OrthoPolynomialBase',
        'PolynomialModel'
    ]

    class PolynomialBase(ParametricModel):
        """
        Base class for all polynomial-like models with an arbitrary number of
        parameters in the form of coeffecients.

        In this case Parameter instances are returned through the class's
        ``__getattr__`` rather than through class descriptors.
        """

        # Default _param_names list; this will be filled in by the implementation's
        # __init__
        _param_names = []

        linear = True
        col_deriv = False

        @lazyproperty
        def param_names(self):
            """Coefficient names generated based on the model's polynomial degree
            and number of dimensions.

            Subclasses should implement this to return parameter names in the
            desired format.

            On most `Model` classes this is a class attribute, but for polynomial
            models it is an instance attribute since each polynomial model instance
            can have different parameters depending on the degree of the polynomial
            and the number of dimensions, for example.
            """

            return self._param_names

        def __getattr__(self, attr):
            if self._param_names and attr in self._param_names:
                return Parameter(attr, default=0.0, model=self)
            else:
                return super(PolynomialBase, self).__getattr__(attr)

        def __setattr__(self, attr, value):
            # TODO: Support a means of specifying default values for coefficients
            # Check for self._ndim first--if it hasn't been defined then the
            # instance hasn't been initialized yet and self.param_names probably
            # won't work.
            # This has to vaguely duplicate the functionality of
            # Parameter.__set__.
            # TODO: I wonder if there might be a way around that though...
            if attr[0] != '_' and self._param_names and attr in self._param_names:
                param = Parameter(attr, default=0.0, model=self)
                # This is a little hackish, but we can actually reuse the
                # Parameter.__set__ method here
                param.__set__(self, value)
            else:
                super(PolynomialBase, self).__setattr__(attr, value)

        def _deriv_with_constraints(self, params=None, x=None, y=None):
            if y is None:
                d = np.array(self.deriv(x=x))
            else:
                d = np.array(self.deriv(x=x, y=y))

            if self.col_deriv:
                return d[self._fit_param_indices]
            else:
                return d[:, self._fit_param_indices]

    class PolynomialModel(PolynomialBase):
        """
        Base class for polynomial models.

        Its main purpose is to determine how many coefficients are needed
        based on the polynomial order and dimension and to provide their
        default values, names and ordering.
        """

        def __init__(self, degree, n_inputs=1, n_outputs=1, param_dim=1,
                     **params):
            self._degree = degree
            self._order = self.get_num_coeff(n_inputs)
            self._param_names = self._generate_coeff_names(n_inputs)
            self._n_inputs = n_inputs
            self._n_outputs = n_outputs

            if params:
                p = params.get('c0', params.get('c0_0'))
                if isinstance(p, collections.Sequence):
                    n_params = len(p)
                else:
                    n_params = 1
                if param_dim != n_params:
                    if param_dim == 1:
                        log.info("Inferred {0} dimensions when creating a {1} model. "
                                 "Resetting param_dim to {2}".format(
                                     n_params, self.__class__.__name__, n_params))
                        param_dim = n_params
                    else:
                        raise ValueError("Number of coefficient sets ({0}) does "
                                         "not match number of parameter sets "
                                         "({1}).".format(n_params, param_dim))

                self._validate_params(**params)

            super(PolynomialModel, self).__init__(param_dim=param_dim)

            for name, value in params.items():
                setattr(self, name, value)

        @property
        def degree(self):
            """TODO: Docstring for me"""

            return self._degree

        @property
        def n_inputs(self):
            """The number of input variables to model evaluation."""

            return self._n_inputs

        @property
        def n_outputs(self):
            """The number of outputs returned when model is evaluated."""

            return self._n_outputs

        def get_num_coeff(self, ndim):
            """
            Return the number of coefficients in one parameter set
            """

            if self.degree < 1  or self.degree > 16:
                raise ValueError("Degree of polynomial must be 1< deg < 16")
            # deg+1 is used to account for the difference between iraf using
            # degree and numpy using exact degree
            if ndim != 1:
                nmixed = comb(self.degree, ndim)
            else:
                nmixed = 0
            numc = self.degree * ndim + nmixed + 1
            return numc

        def _invlex(self):
            c = []
            lencoeff = self.degree + 1
            for i in range(lencoeff):
                for j in range(lencoeff):
                    if i + j <= self.degree:
                        c.append((j, i))
            return c[::-1]

        def _validate_params(self, **params):
            numcoeff = self._order
            assert(len(params) == numcoeff)
            valid_params = set(self._param_names)
            provided_params = set(params)
            diff = valid_params.difference(provided_params)
            if diff:
                raise TypeError('Unrecognized input parameters: %s' % diff)

        def _generate_coeff_names(self, ndim):
            names = []
            if ndim == 1:
                for n in range(self._order):
                    names.append('c{0}'.format(n))
            else:
                for i in range(self.degree + 1):
                    names.append('c{0}_{1}'.format(i, 0))
                for i in range(1, self.degree + 1):
                    names.append('c{0}_{1}'.format(0, i))
                for i in range(1, self.degree):
                    for j in range(1, self.degree):
                        if i + j < self.degree + 1:
                            names.append('c{0}_{1}'.format(i, j))
            return names

    class OrthoPolynomialBase(PolynomialBase):
        """
        This is a base class for the 2D Chebyshev and Legendre models.

        The polynomials implemented here require a maximum degree in x and y.

        Parameters
        ----------

        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : list or None
            domain of the x independent variable
        y_domain : list or None
            domain of the y independent variable
        x_window : list or None
            range of the x independent variable
        y_window : list or None
            range of the y independent variable
        param_dim : int
            number of parameter sets
        **params : dict
            {keyword: value} pairs, representing {parameter_name: value}
        """

        n_inputs = 2
        n_outputs = 1

        def __init__(self, x_degree, y_degree, x_domain=None, x_window=None,
                     y_domain=None, y_window=None, param_dim=1, **params):
            # TODO: Perhaps some of these other parameters should be properties?
            # TODO: An awful lot of the functionality in this method is still
            # shared by PolynomialModel; perhaps some of it can be generalized in
            # PolynomialBase
            self.x_degree = x_degree
            self.y_degree = y_degree
            self._order = self.get_num_coeff()
            self.x_domain = x_domain
            self.y_domain = y_domain
            self.x_window = x_window
            self.y_window = y_window
            self._param_names = self._generate_coeff_names()

            if params:
                p = params.get('c0_0')
                if isinstance(p, collections.Sequence):
                    n_params = len(p)
                else:
                    n_params = 1
                if param_dim != n_params:
                    if param_dim == 1:
                        log.info("Inferred {0} dimensions when creating a {1} "
                                 "model. Resetting param_dim to {2}".format(
                                     n_params, self.__class__.__name__, n_params))
                        param_dim = n_params
                    else:
                        raise ValueError("Number of coefficient sets {0} does "
                                         "not match number of parameter sets "
                                         "({1})".format(n_params, param_dim))

                self._validate_params(**params)

            super(OrthoPolynomialBase, self).__init__(param_dim=param_dim)

            for name, value in params.items():
                setattr(self, name, value)

        def get_num_coeff(self):
            """
            Determine how many coefficients are needed

            Returns
            -------
            numc : int
                number of coefficients
            """

            return (self.x_degree + 1) * (self.y_degree + 1)

        def _validate_params(self, **params):
            numcoeff = self.get_num_coeff()
            assert(len(params) == numcoeff)

        def _invlex(self):
            # TODO: This is a very slow way to do this; fix it and related methods
            # like _alpha
            c = []
            xvar = np.arange(self.x_degree + 1)
            yvar = np.arange(self.y_degree + 1)
            for j in yvar:
                for i in xvar:
                    c.append((i, j))
            return np.array(c[::-1])

        def invlex_coeff(self):
            coeff = []
            xvar = np.arange(self.x_degree + 1)
            yvar = np.arange(self.y_degree + 1)
            for j in yvar:
                for i in xvar:
                    name = 'c{0}_{1}'.format(i, j)
                    coeff.append(getattr(self, name))
            return np.array(coeff[::-1])

        def _alpha(self):
            invlexdeg = self._invlex()
            invlexdeg[:, 1] = invlexdeg[:, 1] + self.x_degree + 1
            nx = self.x_degree + 1
            ny = self.y_degree + 1
            alpha = np.zeros((ny * nx + 3, ny + nx))
            for n in range(len(invlexdeg)):
                alpha[n][invlexdeg[n]] = [1, 1]
                alpha[-2, 0] = 1
                alpha[-3, nx] = 1
            return alpha

        def imhorner(self, x, y, coeff):
            _coeff = list(coeff)
            _coeff.extend([0, 0, 0])
            alpha = self._alpha()
            r0 = _coeff[0]
            nalpha = len(alpha)

            karr = np.diff(alpha, axis=0)
            kfunc = self._fcache(x, y)
            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            nterms = x_terms + y_terms
            for n in range(1, nterms + 1 + 3):
                setattr(self, 'r' + str(n), 0.)

            for n in range(1, nalpha):
                k = karr[n - 1].nonzero()[0].max() + 1
                rsum = 0
                for i in range(1, k + 1):
                    rsum = rsum + getattr(self, 'r' + str(i))
                val = kfunc[k - 1] * (r0 + rsum)
                setattr(self, 'r' + str(k), val)
                r0 = _coeff[n]
                for i in range(1, k):
                    setattr(self, 'r' + str(i), 0.)
            result = r0
            for i in range(1, nterms + 1 + 3):
                result = result + getattr(self, 'r' + str(i))
            return result

        def _generate_coeff_names(self):
            names = []
            for j in range(self.y_degree + 1):
                for i in range(self.x_degree + 1):
                    names.append('c{0}_{1}'.format(i, j))
            return names

        def _fcache(self, x, y):
            # TODO: Write a docstring explaining the actual purpose of this method
            """To be implemented by subclasses"""

            raise NotImplementedError("Subclasses should implement this")

        @format_input
        def __call__(self, x, y):
            """
            Transforms data using this model.

            Parameters
            --------------
            x : scalar, list or array
            y : scalar, lis or array
            """

            assert x.shape == y.shape, \
                "Expected input arrays to have the same shape"
            if self.x_domain is not None:
                x = poly_map_domain(x, self.x_domain, self.x_window)
            if self.y_domain is not None:
                y = poly_map_domain(y, self.y_domain, self.y_window)
            invcoeff = self.invlex_coeff()

            return self.imhorner(x, y, invcoeff)

    class Chebyshev1D(PolynomialModel):
        """
        1D Chebyshev polynomial of the 1st kind.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : list or None
        window : list or None
            If None, it is set to [-1,1]
            Fitters will remap the domain to this window
        param_dim : int
            number of parameter sets
        **params : dict
            keyword : value pairs, representing parameter_name: value
        """

        def __init__(self, degree, domain=None, window=[-1, 1], param_dim=1,
                     **params):
            self.domain = domain
            self.window = window
            super(Chebyshev1D, self).__init__(degree, n_inputs=1, n_outputs=1,
                                              param_dim=param_dim,
                                              **params)

        def clenshaw(self, x, coeff):
            """
            Evaluates the polynomial using Clenshaw's algorithm.
            """

            if isinstance(x, tuple) or isinstance(x, list):
                x = np.asarray(x)
            if len(coeff) == 1:
                c0 = coeff[0]
                c1 = 0
            elif len(coeff) == 2:
                c0 = coeff[0]
                c1 = coeff[1]
            else:
                x2 = 2 * x
                c0 = coeff[-2]
                c1 = coeff[-1]
                for i in range(3, len(coeff) + 1):
                    tmp = c0
                    c0 = coeff[-i] - c1
                    c1 = tmp + c1 * x2
            return c0 + c1 * x

        def deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            params : throw away parameter
                parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """

            x = np.array(x, dtype=np.float, copy=False, ndmin=1)
            v = np.empty((self.degree + 1,) + x.shape, dtype=x.dtype)
            v[0] = x * 0 + 1
            x2 = 2 * x
            v[1] = x
            for i in range(2, self.degree + 1):
                v[i] = v[i - 1] * x2 - v[i - 2]
            return np.rollaxis(v, 0, v.ndim)

        @format_input
        def __call__(self, x):
            """
            Transforms data using this model.

            Parameters
            --------------
            x : scalar, list or array
                input
            """

            if self.domain is not None:
                x = poly_map_domain(x, self.domain, self.window)
            return self.clenshaw(x, self.param_sets)

    class Legendre1D(PolynomialModel):
        """
        1D Legendre polynomial.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : list or None
        window : list or None
            If None, it is set to [-1,1]
            Fitters will remap the domain to this window
        param_dim : int
            number of parameter sets
        **params : dict
            keyword: value pairs, representing parameter_name: value
        """

        def __init__(self, degree, domain=None, window=[-1, 1], param_dim=1,
                     **params):
            self.domain = domain
            self.window = window
            super(Legendre1D, self).__init__(degree, n_inputs=1, n_outputs=1,
                                             param_dim=param_dim,
                                             **params)

        def clenshaw(self, x, coeff):
            if isinstance(x, tuple) or isinstance(x, list):
                x = np.asarray(x)
            if len(coeff) == 1:
                c0 = coeff[0]
                c1 = 0
            elif len(coeff) == 2:
                c0 = coeff[0]
                c1 = coeff[1]
            else:
                nd = len(coeff)
                c0 = coeff[-2]
                c1 = coeff[-1]
                for i in range(3, len(coeff) + 1):
                    tmp = c0
                    nd = nd - 1
                    c0 = coeff[-i] - (c1 * (nd - 1)) / nd
                    c1 = tmp + (c1 * x * (2 * nd - 1)) / nd
            return c0 + c1 * x

        def deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            params : throw away parameter
                parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """

            x = np.array(x, dtype=np.float, copy=False, ndmin=1)
            v = np.empty((self.degree + 1,) + x.shape, dtype=x.dtype)
            v[0] = x * 0 + 1
            v[1] = x
            for i in range(2, self.degree + 1):
                v[i] = (v[i - 1] * x * (2 * i - 1) - v[i - 2] * (i - 1)) / i
            return np.rollaxis(v, 0, v.ndim)

        @format_input
        def __call__(self, x):
            """
            Transforms data using this model.

            Parameters
            --------------
            x : scalar, list or array
                input
            """

            if self.domain is not None:
                x = poly_map_domain(x, self.domain, self.window)
            return self.clenshaw(x, self.param_sets)

    class Polynomial1D(PolynomialModel):
        """
        1D Polynomial model.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : list or None
        window : list or None
            If None, it is set to [-1,1]
            Fitters will remap the domain to this window
        param_dim : int
            number of parameter sets
        **params : dict
            keyword: value pairs, representing parameter_name: value
        """

        def __init__(self, degree, domain=[-1, 1], window=[-1, 1], param_dim=1,
                     **params):
            self.domain = domain
            self.window = window
            super(Polynomial1D, self).__init__(degree, n_inputs=1,
                                               n_outputs=1,
                                               param_dim=param_dim, **params)

        def deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            params : throw away parameter
                parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """

            v = np.empty((self.degree + 1,) + x.shape, dtype=np.float)
            v[0] = x * 0 + 1
            v[1] = x
            for i in range(2, self.degree + 1):
                v[i] = v[i - 1] * x
            return np.rollaxis(v, 0, v.ndim)

        def horner(self, x, coef):
            c0 = coef[-1] + x * 0
            for i in range(2, len(coef) + 1):
                c0 = coef[-i] + c0 * x
            return c0

        @format_input
        def __call__(self, x):
            """
            Transforms data using this model.

            Parameters
            --------------
            x : scalar, list or array
                input
            """

            return self.horner(x, self.param_sets)

    class Polynomial2D(PolynomialModel):
        """
        2D Polynomial  model.

        Represents a general polynomial of degree n:

        .. math::
            P(x,y) = c_{0_0} + c_{1_0}*x + ...+ c_{n_0}*x^n + c_{0_1}*y + ...+ c_{0_n}*y^n
            + c_{1_1}*x*y + c_{1_2}*x*y^2 + ... + c_{1_(n-1)}*x*y^{n-1}+ ... + c_{(n-1)_1}*x^{n-1}*y

        Parameters
        ----------
        degree : int
            highest power of the polynomial,
            the number of terms is degree+1
        x_domain : list or None
            domain of the x independent variable
        y_domain : list or None
            domain of the y independent variable
        x_window : list or None
            range of the x independent variable
        y_window : list or None
            range of the y independent variable
        param_dim : int
            number of parameter sets
        **params : dict
            keyword: value pairs, representing parameter_name: value
        """

        def __init__(self, degree, x_domain=[-1, 1], y_domain=[-1, 1],
                     x_window=[-1, 1], y_window=[-1, 1],
                     param_dim=1, **params):
            super(Polynomial2D, self).__init__(degree, n_inputs=2,
                                               n_outputs=1,
                                               param_dim=param_dim, **params)
            self.x_domain = x_domain
            self.y_domain = y_domain
            self.x_window = x_window
            self.y_window = y_window

        def mhorner(self, x, y, coeff):
            """
            Multivariate Horner's scheme

            Parameters
            --------------
            x, y : array
            coeff : array of coefficients in inverse lexical order
            """
            alpha = np.array(self._invlex())
            r0 = coeff[0]
            r1 = r0 * 0.0
            r2 = r0 * 0.0
            karr = np.diff(alpha, axis=0)
            for n in range(len(karr)):
                if karr[n, 1] != 0:
                    r2 = y * (r0 + r1 + r2)
                    r1 = coeff[0] * 0.
                else:
                    r1 = x * (r0 + r1)
                r0 = coeff[n + 1]
            return r0 + r1 + r2

        def deriv(self, x, y, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            params : throw away parameter
                parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """

            if x.ndim == 2:
                x = x.flatten()
            if y.ndim == 2:
                y = y.flatten()
            if x.size != y.size:
                raise ValueError('Expected x and y to be of equal size')

            designx = x[:, None] ** np.arange(self.degree + 1)
            designy = y[:, None] ** np.arange(1, self.degree + 1)

            designmixed = []
            for i in range(1, self.degree):
                for j in range(1, self.degree):
                    if i + j <= self.degree:
                        designmixed.append((x ** i) * (y ** j))
            designmixed = np.array(designmixed).T
            if designmixed.any():
                v = np.hstack([designx, designy, designmixed])
            else:
                v = np.hstack([designx, designy])
            return v

        def invlex_coeff(self):
            coeff = []
            lencoeff = range(self.degree + 1)
            for i in lencoeff:
                for j in lencoeff:
                    if i + j <= self.degree:
                        name = 'c{0}_{1}'.format(j, i)
                        coeff.append(getattr(self, name))
            return np.array(coeff[::-1])

        @format_input
        def __call__(self, x, y):
            """
            Transforms data using this model.

            Parameters
            --------------
            x : scalar, list or array
                input
            y : scalar, list or array
                input
            """

            invcoeff = self.invlex_coeff()
            assert x.shape == y.shape, \
                "Expected input arrays to have the same shape"

            return self.mhorner(x, y, invcoeff)

    class Chebyshev2D(OrthoPolynomialBase):
        """
        2D Chebyshev polynomial of the 1st kind.

        It is defined as

        .. math:: P_{n_m}(x,y) = \sum C_{n_m}  T_n(x) T_m(y)

        Parameters
        ----------

        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : list or None
            domain of the x independent variable
        y_domain : list or None
            domain of the y independent variable
        x_window : list or None
            range of the x independent variable
        y_window : list or None
            range of the y independent variable
        param_dim : int
            number of parameter sets
        **params : dict
            keyword: value pairs, representing parameter_name: value

        """

        def __init__(self, x_degree, y_degree, x_domain=None, x_window=[-1, 1],
                     y_domain=None, y_window=[-1,1], param_dim=1, **params):
            super(Chebyshev2D, self).__init__(x_degree, y_degree,
                                              x_domain=x_domain,
                                              y_domain=y_domain,
                                              x_window=x_window,
                                              y_window=y_window,
                                              param_dim=param_dim, **params)

        def _fcache(self, x, y):
            """
            Calculate the individual Chebyshev functions once and store them in a
            dictionary to be reused.
            """

            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            kfunc = {}
            kfunc[0] = np.ones(x.shape)
            kfunc[1] = x.copy()
            kfunc[x_terms] = np.ones(y.shape)
            kfunc[x_terms + 1] = y.copy()
            for n in range(2, x_terms):
                kfunc[n] = 2 * x * kfunc[n - 1] - kfunc[n - 2]
            for n in range(x_terms + 2, x_terms + y_terms):
                kfunc[n] = 2 * y * kfunc[n - 1] - kfunc[n - 2]
            return kfunc

        def deriv(self, x, y, *params):
            """
            Derivatives with respect to the coefficients.

            This is an array with Chebyshev polynomials:

            Tx0Ty0  Tx1Ty0...TxnTy0...TxnTym

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            params : throw away parameter
                parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """

            if x.shape != y.shape:
                raise ValueError("x and y must have the same shape")

            x = x.flatten()
            y = y.flatten()
            x_deriv = self._chebderiv1d(x, self.x_degree + 1).T
            y_deriv = self._chebderiv1d(y, self.y_degree + 1).T

            ij = []
            for i in range(self.y_degree + 1):
                for j in range(self.x_degree + 1):
                    ij.append(x_deriv[j] * y_deriv[i])

            v = np.array(ij)
            return v.T

        def _chebderiv1d(self, x, deg):
            """
            Derivative of 1D Chebyshev series
            """

            x = np.array(x, dtype=np.float, copy=False, ndmin=1)
            d = np.empty((deg + 1, len(x)), dtype=x.dtype)
            d[0] = x * 0 + 1
            if deg > 0:
                x2 = 2 * x
                d[1] = x
                for i in range(2, deg + 1):
                    d[i] = d[i - 1] * x2 - d[i - 2]
            return np.rollaxis(d, 0, d.ndim)

    class Legendre2D(OrthoPolynomialBase):
        """
        Legendre 2D polynomial.

        Defined as:

        .. math:: P_{nm}(x,y) = C_{n_m}  L_n(x ) L_m(y)

        Parameters
        ----------

        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : list or None
            domain of the x independent variable
        y_domain : list or None
            domain of the y independent variable
        x_window : list or None
            range of the x independent variable
        y_window : list or None
            range of the y independent variable
        param_dim : int
            number of parameter sets
        **params : dict
            keyword: value pairs, representing parameter_name: value
        """

        def __init__(self, x_degree, y_degree, x_domain=None, x_window=[-1, 1],
                     y_domain=None, y_window=[-1, 1], param_dim=1, **params):
            super(Legendre2D, self).__init__(x_degree, y_degree,
                                             x_domain=x_domain,
                                             y_domain=y_domain,
                                             x_window=x_window,
                                             y_window=y_window,
                                             param_dim=param_dim, **params)

        def _fcache(self, x, y):
            """
            Calculate the individual Legendre functions once and store them in a
            dictionary to be reused.
            """

            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            kfunc = {}
            kfunc[0] = np.ones(x.shape)
            kfunc[1] = x.copy()
            kfunc[x_terms] = np.ones(y.shape)
            kfunc[x_terms + 1] = y.copy()
            for n in range(2, x_terms):
                kfunc[n] = (((2 * (n - 1) + 1) * x * kfunc[n - 1] -
                            (n - 1) * kfunc[n - 2]) / n)
            for n in range(2, y_terms):
                kfunc[n + x_terms] = ((2 * (n - 1) + 1) * y * kfunc[n + x_terms - 1] -
                                      (n - 1) * kfunc[n + x_terms - 2]) / (n)
            return kfunc

        def deriv(self, x, y, *params):
            """
            Derivatives with repect to the coefficients.
            This is an array with Legendre polynomials:

            Lx0Ly0  Lx1Ly0...LxnLy0...LxnLym

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            params : throw away parameter
                parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            if x.shape != y.shape:
                raise ValueError("x and y must have the same shape")
            x = x.flatten()
            y = y.flatten()
            x_deriv = self._legendderiv1d(x, self.x_degree + 1).T
            y_deriv = self._legendderiv1d(y, self.y_degree + 1).T

            ij = []
            for i in range(self.y_degree + 1):
                for j in range(self.x_degree + 1):
                    ij.append(x_deriv[j] * y_deriv[i])

            v = np.array(ij)
            return v.T

        def _legendderiv1d(self, x, deg):
            """Derivative of 1D Legendre polynomial"""

            x = np.array(x, dtype=np.float, copy=False, ndmin=1)
            d = np.empty((deg + 1,) + x.shape, dtype=x.dtype)
            d[0] = x * 0 + 1
            if deg > 0:
                d[1] = x
                for i in range(2, deg + 1):
                    d[i] = (d[i - 1] * x * (2 * i - 1) - d[i - 2] * (i - 1)) / i
            return np.rollaxis(d, 0, d.ndim)

    class _SIP1D(Model):
        """
        This implements the Simple Imaging Polynomial Model (SIP) in 1D.

        It's unlikely it will be used in 1D so this class is private
        and SIP should be used instead.
        """

        n_inputs = 2

        def __init__(self, order, coeff_prefix, param_dim=1, **params):
            self.order = order
            self.coeff_prefix = coeff_prefix
            self._param_names = self._generate_coeff_names(coeff_prefix)
            coeffname = '{0}_0_2'.format(coeff_prefix)

            if params:
                p = params.get(coeffname, None)
                if isinstance(p, collections.Sequence):
                    n_params = len(p)
                else:
                    n_params = 1

                # TODO: This pattern is repeated in so many models and seems like a
                # check that could be moved into a method
                if param_dim != n_params:
                    if param_dim == 1:
                        log.info("Inferred {0} dimensions when creating a {1} "
                                 "model. Resetting param_dim to {2}".format(
                                     n_params, self.__class__.__name__, n_params))
                        param_dim = n_params
                    else:
                        raise ValueError(
                            "Number of coefficient sets ({0}) does not match the "
                            "number of parameter sets ({1}).".format(
                                n_params, param_dim))

                self._validate_params(ndim=2, **params)

            super(_SIP1D, self).__init__(param_dim=param_dim, **params)

        def __repr__(self):
            fmt = """
            Model: {0}
            order: {1}
            coeff_prefix: {2}
            param_dim: {3}
            params: {4}
            """.format(
                  self.__class__.__name__,
                  self.order,
                  self.coeff_prefix,
                  self.param_dim,
                  indent('\n'.join('{0}: {1}'.format(n, getattr(self, '_' + n))
                         for n in self.param_names), width=19))

            return dedent(fmt[1:])

        def __str__(self):
            fmt = """
            Model: {0}
            order: {1}
            coeff_prefix: {2}
            Parameter sets: {3}
            Parameters:
                       {3}
            """.format(
                  self.__class__.__name__,
                  self.order,
                  self.coeff_prefix,
                  self.param_dim,
                  indent('\n'.join('{0}: {1}'.format(n, getattr(self, '_' + n))
                         for n in self.param_names), width=19))

            return dedent(fmt[1:])

        def get_num_coeff(self, ndim):
            """
            Return the number of coefficients in one param set
            """

            if self.order < 2 or self.order > 9:
                raise ValueError("Degree of polynomial must be 2< deg < 9")

            nmixed = comb(self.order - 1, ndim)
            numc = self.order * ndim + nmixed + 1
            return numc

        def _generate_coeff_names(self, coeff_prefix):
            names = []
            for i in range(2, self.order + 1):
                names.append('{0}_{1}_{2}'.format(coeff_prefix, i, 0))
            for i in range(2, self.order + 1):
                names.append('{0}_{1}_{2}'.format(coeff_prefix, 0, i))
            for i in range(1, self.order):
                for j in range(1, self.order):
                    if i + j < self.order + 1:
                        names.append('{0}_{1}_{2}'.format(coeff_prefix, i, j))
            return names

        def _validate_params(self, ndim, **params):
            numcoeff = self.get_num_coeff(ndim)
            assert(len(params) == numcoeff)

        def _coef_matrix(self, coeff_prefix):
            mat = np.zeros((self.order + 1, self.order + 1))
            for i in range(2, self.order + 1):
                attr = '_{0}_{1}_{2}'.format(coeff_prefix, i, 0)
                mat[i, 0] = getattr(self, attr).value
            for i in range(2, self.order + 1):
                attr = '_{0}_{1}_{2}'.format(coeff_prefix, 0, i)
                mat[0, i] = getattr(self, attr).value
            for i in range(1, self.order):
                for j in range(1, self.order):
                    if i + j < self.order + 1:
                        attr = '_{0}_{1}_{2}'.format(coeff_prefix, i, j)
                        mat[i, j] = getattr(self, attr).value
            return mat

        def _eval_sip(self, x, y, coef):
            x = np.asarray(x, dtype=np.float64)
            y = np.asarray(y, dtype=np.float64)
            if self.coeff_prefix == 'a':
                result = np.zeros(x.shape)
            else:
                result = np.zeros(y.shape)

            for i in range(coef.shape[0]):
                for j in range(coef.shape[1]):
                    if i + j > 1 and i + j < self.order + 1:
                        result = result + coef[i, j] * x ** i * y ** j
            return result

        def __call__(self, x, y):
            mcoef = self._coef_matrix(self.coeff_prefix)
            return self._eval_sip(x, y, mcoef)

    class _SIPModel(SerialCompositeModel):
        """
        Simple Imaging Polynomial (SIP) model.

        The SIP convention is used to represent distortions in FITS image headers.
        See [1]_ for a description of the SIP convention.

        Parameters
        ----------
        crpix : list or ndarray of length(2)
            CRPIX values
        a_order : int
            SIP polynomial order for first axis
        a_coeff : dict
            SIP coefficients for first axis
        coeff_prefix : str: 'a', 'b', 'A' or 'B'
            SIP coefficient preffix
        b_order : int
            SIP order for second axis
        b_coeff : dict
            SIP coefficients for the second axis
        a_inv_order : int
            order for the inverse transformation (AP coefficients)
        a_inv_coeff : dict
            coefficients for the inverse transform
        b_inv_order : int
            order for the inverse transformation (BP coefficients)
        b_inv_coeff : dict
            coefficients for the inverse transform
        param_dim : int
            number of parameter sets

        References
        ----------
        .. [1] `David Shupe, et al, ADASS, ASP Conference Series, Vol. 347, 2005 <http://adsabs.harvard.edu/abs/2005ASPC..347..491S>`_
        """

        n_inputs = 2
        n_outputs = 2

        def __init__(self, crpix, a_order, a_coeff, b_order, b_coeff,
                     ap_order=None, ap_coeff=None, bp_order=None, bp_coeff=None,
                     param_dim=1):
            self.crpix = crpix
            self.a_order = a_order
            self.b_order = b_order
            self.a_coeff = a_coeff
            self.b_coeff = b_coeff
            self.ap_order = ap_order
            self.bp_order = bp_order
            self.ap_coeff = ap_coeff
            self.bp_coeff = bp_coeff
            self.shift_a = Shift(-crpix[0])
            self.shift_b = Shift(-crpix[1])
            self.sip1d_a = _SIP1D(a_order, coeff_prefix='A',
                                  param_dim=param_dim, **a_coeff)
            self.sip1d_b = _SIP1D(b_order, coeff_prefix='B',
                                  param_dim=param_dim, **b_coeff)
            if (ap_order is not None and ap_coeff is not None and
                    bp_order is not None and bp_coeff is not None):
                self.sip1d_ap = _SIP1D(ap_order, coeff_prefix='AP', **ap_coeff)
                self.sip1d_bp = _SIP1D(bp_order, coeff_prefix='BP', **bp_coeff)
            else:
                self.sip1d_ap = None
                self.sip1d_bp = None

            super(_SIPModel, self).__init__([self.shift_a, self.shift_b,
                                             self.sip1d_a, self.sip1d_b])

    class SIP(_SIPModel):
        def __init__(self, crpix, a_order, a_coeff, b_order, b_coeff,
                     ap_order=None, ap_coeff=None, bp_order=None, bp_coeff=None,
                     param_dim=1):

            super(SIP, self).__init__(crpix, a_order, a_coeff,
                                      b_order, b_coeff, ap_order, ap_coeff,
                                      bp_order, bp_coeff, param_dim=1)

        def inverse(self):
            if (self.ap_order is not None and self.ap_coeff is not None and
                    self.bp_order is not None and self.bp_coeff is not None):
                return InverseSIP(self.crpix, self.a_order, self.a_coeff,
                                  self.b_order, self.b_coeff,
                                  self.ap_order, self.ap_coeff,
                                  self.bp_order, self.bp_coeff)
            else:
                raise NotImplementedError("An analytical inverse transform has "
                                          "not been implemented for this model.")

        def __repr__(self):
            models = [self.shift_a, self.shift_b, self.sip1d_a, self.sip1d_b]
            fmt = """
                Model:  {0}
                """.format(self.__class__.__name__)
            fmt1 = " %s  " * len(models) % tuple([repr(model) for model in models])
            fmt = fmt + fmt1
            return fmt

        def __str__(self):
            models = [self.shift_a, self.shift_b, self.sip1d_a, self.sip1d_b]
            fmt = """
                Model:  {0}
                """.format(self.__class__.__name__)
            fmt1 = " %s  " * len(models) % tuple([str(model) for model in models])
            fmt = fmt + fmt1
            return fmt

        def __call__(self, x, y):
            """
            Transforms data using this model.

            Parameters
            ----------
            x : scalar, list, array
                input
            y : scalar, list or array
                input
            """
            x = self.shift_a(x)
            y = self.shift_b(y)
            x1 = self.sip1d_a(x, y)
            y1 = self.sip1d_b(x, y)
            return x1, y1

    class InverseSIP(_SIPModel):
        def __init__(self, crpix, a_order, a_coeff, b_order, b_coeff,
                     ap_order, ap_coeff, bp_order, bp_coeff,
                     param_dim=1):

            super(InverseSIP, self).__init__(crpix, a_order, a_coeff,
                                             b_order, b_coeff,
                                             ap_order, ap_coeff,
                                             bp_order, bp_coeff, param_dim=1)

        def inverse(self):
            if (self.a_order is not None and self.a_coeff is not None and
                    self.b_order is not None and self.bp_coeff is not None):
                return SIP(self.crpix, self.a_order, self.a_coeff,
                           self.b_order, self.b_coeff,
                           self.ap_order, self.ap_coeff,
                           self.bp_order, self.bp_coeff)
            else:
                raise NotImplementedError("An analytical inverse transform has "
                                          "not been implemented for this model.")

        def __call__(self, x, y):
            if self.sip1d_ap is not None and self.sip1d_bp is not None:
                x1 = self.sip1d_ap(x, y)
                y1 = self.sip1d_bp(x, y)
                x = self.shift_a.inverse()(x1)
                y = self.shift_b.inverse()(y1)
                return x, y
            else:
                raise NotImplementedError("An analytical inverse transform has "
                                          "not been implemented for this model.")

        def __repr__(self):
            models = [self.sip1d_ap, self.sip1d_bp, self.shift_a.inverse(),
                      self.shift_b.inverse()]
            fmt = """
                Model:  {0}
                """.format(self.__class__.__name__)
            fmt1 = " %s  " * len(models) % tuple([repr(model) for model in models])
            fmt = fmt + fmt1
            return fmt

        def __str__(self):
            models = [self.sip1d_ap, self.sip1d_bp, self.shift_a.inverse(),
                      self.shift_b.inverse()]
            fmt = """
                Model:  {0}
                """.format(self.__class__.__name__)
            fmt1 = " %s  " * len(models) % tuple([str(model) for model in models])
            fmt = fmt + fmt1
            return fmt