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# Source code for astropy.modeling.core

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

    """
    This module defines base classes for all models.  The base class of all
    models is `~astropy.modeling.Model`. `~astropy.modeling.FittableModel` is
    the base class for all fittable models. Fittable models can be linear or
    nonlinear in a regression analysis sense.

    All models provide a `__call__` method which performs the transformation in
    a purely mathematical way, i.e. the models are unitless.  Model instances can
    represent either a single model, or a "model set" representing multiple copies
    of the same type of model, but with potentially different values of the
    parameters in each model making up the set.
    """

    from __future__ import (absolute_import, unicode_literals, division,
                            print_function)

    import abc
    import copy
    import inspect
    import functools
    import operator
    import sys
    import types
    import warnings

    from collections import defaultdict
    from itertools import chain, islice

    import numpy as np

    from ..utils import indent, isinstancemethod, metadata
    from ..extern import six
    from ..extern.six.moves import copyreg
    from ..table import Table
    from ..utils import (deprecated, sharedmethod, find_current_module,
                         InheritDocstrings, OrderedDescriptorContainer)
    from ..utils.codegen import make_function_with_signature
    from ..utils.compat import ignored
    from ..utils.compat.funcsigs import signature
    from ..utils.compat.odict import OrderedDict
    from ..utils.exceptions import AstropyDeprecationWarning
    from .utils import (array_repr_oneline, check_broadcast, combine_labels,
                        make_binary_operator_eval, ExpressionTree,
                        IncompatibleShapeError, AliasDict, get_inputs_and_params,
                        _BoundingBox)
    from ..nddata.utils import add_array, extract_array

    from .parameters import Parameter, InputParameterError

    __all__ = ['Model', 'FittableModel', 'Fittable1DModel', 'Fittable2DModel',
               'custom_model', 'ModelDefinitionError']

    class ModelDefinitionError(TypeError):
        """Used for incorrect models definitions"""

    def _model_oper(oper, **kwargs):
        """
        Returns a function that evaluates a given Python arithmetic operator
        between two models.  The operator should be given as a string, like ``'+'``
        or ``'**'``.

        Any additional keyword arguments passed in are passed to
        `_CompoundModelMeta._from_operator`.
        """

        # Note: Originally this used functools.partial, but that won't work when
        # used in the class definition of _CompoundModelMeta since
        # _CompoundModelMeta has not been defined yet.

        # Perform an arithmetic operation on two models.
        return lambda left, right: _CompoundModelMeta._from_operator(oper,
                left, right, **kwargs)

    class _ModelMeta(OrderedDescriptorContainer, InheritDocstrings, abc.ABCMeta):
        """
        Metaclass for Model.

        Currently just handles auto-generating the param_names list based on
        Parameter descriptors declared at the class-level of Model subclasses.
        """

        registry = set()
        """
        A registry of all known concrete (non-abstract) Model subclasses.
        """

        _is_dynamic = False
        """
        This flag signifies whether this class was created in the "normal" way,
        with a class statement in the body of a module, as opposed to a call to
        `type` or some other metaclass constructor, such that the resulting class
        does not belong to a specific module.  This is important for pickling of
        dynamic classes.

        This flag is always forced to False for new classes, so code that creates
        dynamic classes should manually set it to True on those classes when
        creating them.
        """

        # Default empty dict for _parameters_, which will be empty on model
        # classes that don't have any Parameters
        _parameters_ = OrderedDict()

        def __new__(mcls, name, bases, members):
            # See the docstring for _is_dynamic above
            if '_is_dynamic' not in members:
                members['_is_dynamic'] = mcls._is_dynamic

            return super(_ModelMeta, mcls).__new__(mcls, name, bases, members)

        def __init__(cls, name, bases, members):
            # Make sure OrderedDescriptorContainer gets to run before doing
            # anything else
            super(_ModelMeta, cls).__init__(name, bases, members)

            if cls._parameters_:
                if hasattr(cls, '_param_names'):
                    # Slight kludge to support compound models, where
                    # cls.param_names is a property; could be improved with a
                    # little refactoring but fine for now
                    cls._param_names = tuple(cls._parameters_)
                else:
                    cls.param_names = tuple(cls._parameters_)

            cls._create_inverse_property(members)
            cls._create_bounding_box_property(members)
            cls._handle_backwards_compat(name, members)
            cls._handle_special_methods(members)

            if not inspect.isabstract(cls) and not name.startswith('_'):
                cls.registry.add(cls)

        def __repr__(cls):
            """
            Custom repr for Model subclasses.
            """

            return cls._format_cls_repr()

        def _repr_pretty_(cls, p, cycle):
            """
            Repr for IPython's pretty printer.

            By default IPython "pretty prints" classes, so we need to implement
            this so that IPython displays the custom repr for Models.
            """

            p.text(repr(cls))

        def __reduce__(cls):
            if not cls._is_dynamic:
                # Just return a string specifying where the class can be imported
                # from
                return cls.__name__
            else:
                members = dict(cls.__dict__)
                # Delete any ABC-related attributes--these will be restored when
                # the class is reconstructed:
                for key in list(members):
                    if key.startswith('_abc_'):
                        del members[key]

                # Delete custom __init__ and __call__ if they exist:
                for key in ('__init__', '__call__'):
                    if key in members:
                        del members[key]

                return (type(cls), (cls.__name__, cls.__bases__, members))

        @property
        def name(cls):
            """
            The name of this model class--equivalent to ``cls.__name__``.

            This attribute is provided for symmetry with the `Model.name` attribute
            of model instances.
            """

            return cls.__name__

        @property
        def n_inputs(cls):
            return len(cls.inputs)

        @property
        def n_outputs(cls):
            return len(cls.outputs)

        @property
        def _is_concrete(cls):
            """
            A class-level property that determines whether the class is a concrete
            implementation of a Model--i.e. it is not some abstract base class or
            internal implementation detail (i.e. begins with '_').
            """
            return not (cls.__name__.startswith('_') or inspect.isabstract(cls))

        def rename(cls, name):
            """
            Creates a copy of this model class with a new name.

            The new class is technically a subclass of the original class, so that
            instance and type checks will still work.  For example::

                >>> from astropy.modeling.models import Rotation2D
                >>> SkyRotation = Rotation2D.rename('SkyRotation')
                >>> SkyRotation
                <class '__main__.SkyRotation'>
                Name: SkyRotation (Rotation2D)
                Inputs: ('x', 'y')
                Outputs: ('x', 'y')
                Fittable parameters: ('angle',)
                >>> issubclass(SkyRotation, Rotation2D)
                True
                >>> r = SkyRotation(90)
                >>> isinstance(r, Rotation2D)
                True
            """

            if six.PY2 and isinstance(name, six.text_type):
                # Unicode names are not allowed in Python 2, so just convert to
                # ASCII.  As such, for cross-compatibility all model names should
                # just be ASCII for now.
                name = name.encode('ascii')

            mod = find_current_module(2)
            if mod:
                modname = mod.__name__
            else:
                modname = '__main__'

            new_cls = type(name, (cls,), {})
            # On Python 2 __module__ must be a str, not unicode
            new_cls.__module__ = str(modname)

            if hasattr(cls, '__qualname__'):
                if new_cls.__module__ == '__main__':
                    # __main__ is not added to a class's qualified name
                    new_cls.__qualname__ = name
                else:
                    new_cls.__qualname__ = '{0}.{1}'.format(modname, name)

            return new_cls

        def _create_inverse_property(cls, members):
            inverse = members.get('inverse')
            if inverse is None or cls.__bases__[0] is object:
                # The latter clause is the prevent the below code from running on
                # the Model base class, which implements the default getter and
                # setter for .inverse
                return

            if isinstance(inverse, property):
                # We allow the @property decorator to be omitted entirely from
                # the class definition, though its use should be encouraged for
                # clarity
                inverse = inverse.fget

            # Store the inverse getter internally, then delete the given .inverse
            # attribute so that cls.inverse resolves to Model.inverse instead
            cls._inverse = inverse
            del cls.inverse

        def _create_bounding_box_property(cls, members):
            """
            Takes any bounding_box defined on a concrete Model subclass (either
            as a fixed tuple or a property or method) and wraps it in the generic
            getter/setter interface for the bounding_box attribute.
            """

            # TODO: Much of this is verbatim from _create_inverse_property--I feel
            # like there could be a way to generify properties that work this way,
            # but for the time being that would probably only confuse things more.
            bounding_box = members.get('bounding_box')
            if bounding_box is None or cls.__bases__[0] is object:
                return

            if isinstance(bounding_box, property):
                bounding_box = bounding_box.fget

            if not callable(bounding_box):
                # See if it's a hard-coded bounding_box (as a sequence) and
                # normalize it
                try:
                    bounding_box = _BoundingBox.validate(cls, bounding_box)
                except AssertionError as exc:
                    raise ModelDefinitionError(exc.args[0])
            else:
                sig = signature(bounding_box)
                # May be a method that only takes 'self' as an argument (like a
                # property, but the @property decorator was forgotten)
                # TODO: Maybe warn in the above case?
                #
                # However, if the method takes additional arguments then this is a
                # parameterized bounding box and should be callable
                if len(sig.parameters) > 1:
                    bounding_box = \
                            cls._create_bounding_box_subclass(bounding_box, sig)

            if six.PY2 and isinstance(bounding_box, types.MethodType):
                bounding_box = bounding_box.__func__

            # See the Model.bounding_box getter definition for how this attribute
            # is used
            cls._bounding_box = bounding_box
            del cls.bounding_box

        def _create_bounding_box_subclass(cls, func, sig):
            """
            For Models that take optional arguments for defining their bounding
            box, we create a subclass of _BoundingBox with a ``__call__`` method
            that supports those additional arguments.

            Takes the function's Signature as an argument since that is already
            computed in _create_bounding_box_property, so no need to duplicate that
            effort.
            """

            # TODO: Might be convenient if calling the bounding box also
            # automatically sets the _user_bounding_box.  So that
            #
            #    >>> model.bounding_box(arg=1)
            #
            # in addition to returning the computed bbox, also sets it, so that
            # it's a shortcut for
            #
            #    >>> model.bounding_box = model.bounding_box(arg=1)
            #
            # Not sure if that would be non-obvious / confusing though...

            def __call__(self, **kwargs):
                return func(self._model, **kwargs)

            kwargs = []
            for idx, param in enumerate(sig.parameters.values()):
                if idx == 0:
                    # Presumed to be a 'self' argument
                    continue

                if param.default is param.empty:
                    raise ModelDefinitionError(
                        'The bounding_box method for {0} is not correctly '
                        'defined: If defined as a method all arguments to that '
                        'method (besides self) must be keyword arguments with '
                        'default values that can be used to compute a default '
                        'bounding box.'.format(cls.name))

                kwargs.append((param.name, param.default))

            __call__ = make_function_with_signature(__call__, ('self',), kwargs)

            return type(str('_{0}BoundingBox'.format(cls.name)), (_BoundingBox,),
                        {'__call__': __call__})

        def _handle_backwards_compat(cls, name, members):
            # Backwards compatibility check for 'eval' -> 'evaluate'
            # TODO: Remove sometime after Astropy 1.0 release.
            if 'eval' in members and 'evaluate' not in members:
                warnings.warn(
                    "Use of an 'eval' method when defining subclasses of "
                    "FittableModel is deprecated; please rename this method to "
                    "'evaluate'.  Otherwise its semantics remain the same.",
                    AstropyDeprecationWarning)
                cls.evaluate = members['eval']
            elif ('evaluate' in members and callable(members['evaluate']) and
                    not getattr(members['evaluate'], '__isabstractmethod__',
                                False)):
                # Don't bother making a deprecated eval() except for concrete
                # implementations of evaluate, so that we don't end up with an eval
                # abstractmethod as well
                alt = '.'.join((name, 'evaluate'))
                deprecate = deprecated('1.0', alternative=alt, name='eval')
                cls.eval = deprecate(members['evaluate'])

        def _handle_special_methods(cls, members):
            # Handle init creation from inputs
            def update_wrapper(wrapper, cls):
                # Set up the new __call__'s metadata attributes as though it were
                # manually defined in the class definition
                # A bit like functools.update_wrapper but uses the class instead of
                # the wrapped function
                wrapper.__module__ = cls.__module__
                wrapper.__doc__ = getattr(cls, wrapper.__name__).__doc__
                if hasattr(cls, '__qualname__'):
                    wrapper.__qualname__ = '{0}.{1}'.format(
                            cls.__qualname__, wrapper.__name__)

            if ('__call__' not in members and 'inputs' in members and
                    isinstance(members['inputs'], tuple)):
                inputs = members['inputs']
                # Done create a custom __call__ for classes that already have one
                # explicitly defined (this includes the Model base class, and any
                # other classes that manually override __call__
                def __call__(self, *inputs, **kwargs):
                    """Evaluate this model on the supplied inputs."""

                    return super(cls, self).__call__(*inputs, **kwargs)

                args = ('self',) + inputs
                new_call = make_function_with_signature(
                        __call__, args, [('model_set_axis', None)])
                update_wrapper(new_call, cls)
                cls.__call__ = new_call

            if ('__init__' not in members and not inspect.isabstract(cls) and
                    cls._parameters_):
                # If *all* the parameters have default values we can make them
                # keyword arguments; otherwise they must all be positional
                # arguments
                if all(p.default is not None
                       for p in six.itervalues(cls._parameters_)):
                    args = ('self',)
                    kwargs = [(name, cls._parameters_[name].default)
                              for name in cls.param_names]
                else:
                    args = ('self',) + cls.param_names
                    kwargs = {}

                def __init__(self, *params, **kwargs):
                    return super(cls, self).__init__(*params, **kwargs)

                new_init = make_function_with_signature(
                        __init__, args, kwargs, varkwargs='kwargs')
                update_wrapper(new_init, cls)
                cls.__init__ = new_init

        # *** Arithmetic operators for creating compound models ***
        __add__ =     _model_oper('+')
        __sub__ =     _model_oper('-')
        __mul__ =     _model_oper('*')
        __truediv__ = _model_oper('/')
        __pow__ =     _model_oper('**')
        __or__ =      _model_oper('|')
        __and__ =     _model_oper('&')

        if not six.PY3:
            # The classic __div__ operator need only be implemented for Python 2
            # without from __future__ import division
            __div__ = _model_oper('/')

        # *** Other utilities ***

        def _format_cls_repr(cls, keywords=[]):
            """
            Internal implementation of ``__repr__``.

            This is separated out for ease of use by subclasses that wish to
            override the default ``__repr__`` while keeping the same basic
            formatting.
            """

            # For the sake of familiarity start the output with the standard class
            # __repr__
            parts = [super(_ModelMeta, cls).__repr__()]

            if not cls._is_concrete:
                return parts[0]

            def format_inheritance(cls):
                bases = []
                for base in cls.mro()[1:]:
                    if not issubclass(base, Model):
                        continue
                    elif (inspect.isabstract(base) or
                            base.__name__.startswith('_')):
                        break
                    bases.append(base.name)
                if bases:
                    return '{0} ({1})'.format(cls.name, ' -> '.join(bases))
                else:
                    return cls.name

            try:
                default_keywords = [
                    ('Name', format_inheritance(cls)),
                    ('Inputs', cls.inputs),
                    ('Outputs', cls.outputs),
                ]

                if cls.param_names:
                    default_keywords.append(('Fittable parameters',
                                             cls.param_names))

                for keyword, value in default_keywords + keywords:
                    if value is not None:
                        parts.append('{0}: {1}'.format(keyword, value))

                return '\n'.join(parts)
            except:
                # If any of the above formatting fails fall back on the basic repr
                # (this is particularly useful in debugging)
                return parts[0]

    @six.add_metaclass(_ModelMeta)
    class Model(object):
        """
        Base class for all models.

        This is an abstract class and should not be instantiated directly.

        This class sets the constraints and other properties for all individual
        parameters and performs parameter validation.

        The following initialization arguments apply to the majority of Model
        subclasses by default (exceptions include specialized utility models
        like `~astropy.modeling.mappings.Mapping`).  Parametric models take all
        their parameters as arguments, followed by any of the following optional
        keyword arguments:

        Parameters
        ----------
        name : str, optional
            A human-friendly name associated with this model instance
            (particularly useful for identifying the individual components of a
            compound model).

        meta : dict, optional
            An optional dict of user-defined metadata to attach to this model.
            How this is used and interpreted is up to the user or individual use
            case.

        n_models : int, optional
            If given an integer greater than 1, a *model set* is instantiated
            instead of a single model.  This affects how the parameter arguments
            are interpreted.  In this case each parameter must be given as a list
            or array--elements of this array are taken along the first axis (or
            ``model_set_axis`` if specified), such that the Nth element is the
            value of that parameter for the Nth model in the set.

            See the section on model sets in the documentation for more details.

        model_set_axis : int, optional
            This argument only applies when creating a model set (i.e. ``n_models >
            1``).  It changes how parameter values are interpreted.  Normally the
            first axis of each input parameter array (properly the 0th axis) is
            taken as the axis corresponding to the model sets.  However, any axis
            of an input array may be taken as this "model set axis".  This accepts
            negative integers as well--for example use ``model_set_axis=-1`` if the
            last (most rapidly changing) axis should be associated with the model
            sets.

        fixed : dict, optional
            Dictionary ``{parameter_name: bool}`` setting the fixed constraint
            for one or more parameters.  `True` means the parameter is held fixed
            during fitting and is prevented from updates once an instance of the
            model has been created.

            Alternatively the `~astropy.modeling.Parameter.fixed` property of a
            parameter may be used to lock or unlock individual parameters.

        tied : dict, optional
            Dictionary ``{parameter_name: callable}`` of parameters which are
            linked to some other parameter. The dictionary values are callables
            providing the linking relationship.

            Alternatively the `~astropy.modeling.Parameter.tied` property of a
            parameter may be used to set the ``tied`` constraint on individual
            parameters.

        bounds : dict, optional
            Dictionary ``{parameter_name: value}`` of lower and upper bounds of
            parameters. Keys are parameter names. Values are a list of length 2
            giving the desired range for the parameter.

            Alternatively the `~astropy.modeling.Parameter.min` and
            `~astropy.modeling.Parameter.max` or
            ~astropy.modeling.Parameter.bounds` properties of a parameter may be
            used to set bounds on individual parameters.

        eqcons : list, optional
            List of functions of length n such that ``eqcons[j](x0, *args) == 0.0``
            in a successfully optimized problem.

        ineqcons : list, optional
            List of functions of length n such that ``ieqcons[j](x0, *args) >=
            0.0`` is a successfully optimized problem.

        Examples
        --------
        >>> from astropy.modeling import models
        >>> def tie_center(model):
        ...         mean = 50 * model.stddev
        ...         return mean
        >>> tied_parameters = {'mean': tie_center}

        Specify that ``'mean'`` is a tied parameter in one of two ways:

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3,
        ...                        tied=tied_parameters)

        or

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3)
        >>> g1.mean.tied
        False
        >>> g1.mean.tied = tie_center
        >>> g1.mean.tied
        <function tie_center at 0x...>

        Fixed parameters:

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3,
        ...                        fixed={'stddev': True})
        >>> g1.stddev.fixed
        True

        or

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3)
        >>> g1.stddev.fixed
        False
        >>> g1.stddev.fixed = True
        >>> g1.stddev.fixed
        True
        """

        parameter_constraints = Parameter.constraints
        """
        Primarily for informational purposes, these are the types of constraints
        that can be set on a model's parameters.
        """
        model_constraints = ('eqcons', 'ineqcons')
        """
        Primarily for informational purposes, these are the types of constraints
        that constrain model evaluation.
        """

        param_names = ()
        """
        Names of the parameters that describe models of this type.

        The parameters in this tuple are in the same order they should be passed in
        when initializing a model of a specific type.  Some types of models, such
        as polynomial models, have a different number of parameters depending on
        some other property of the model, such as the degree.

        When defining a custom model class the value of this attribute is
        automatically set by the `~astropy.modeling.Parameter` attributes defined
        in the class body.
        """

        inputs = ()
        """The name(s) of the input variable(s) on which a model is evaluated."""
        outputs = ()
        """The name(s) of the output(s) of the model."""

        standard_broadcasting = True
        fittable = False
        linear = True

        meta = metadata.MetaData()
        """A dict-like object to store optional information."""

        # By default models either use their own inverse property or have no
        # inverse at all, but users may also assign a custom inverse to a model,
        # optionally; in that case it is of course up to the user to determine
        # whether their inverse is *actually* an inverse to the model they assign
        # it to.
        _inverse = None
        _user_inverse = None

        _bounding_box = None
        _user_bounding_box = None

        # Default n_models attribute, so that __len__ is still defined even when a
        # model hasn't completed initialization yet
        _n_models = 1

        def __init__(self, *args, **kwargs):
            super(Model, self).__init__()
            meta = kwargs.pop('meta', None)
            if meta is not None:
                self.meta = meta

            self._name = kwargs.pop('name', None)

            self._initialize_constraints(kwargs)
            # Remaining keyword args are either parameter values or invalid
            # Parameter values must be passed in as keyword arguments in order to
            # distinguish them
            self._initialize_parameters(args, kwargs)

        def __repr__(self):
            return self._format_repr()

        def __str__(self):
            return self._format_str()

        def __len__(self):
            return self._n_models

        def __call__(self, *inputs, **kwargs):
            """
            Evaluate this model using the given input(s) and the parameter values
            that were specified when the model was instantiated.
            """

            inputs, format_info = self.prepare_inputs(*inputs, **kwargs)
            parameters = self._param_sets(raw=True)

            outputs = self.evaluate(*chain(inputs, parameters))

            if self.n_outputs == 1:
                outputs = (outputs,)

            return self.prepare_outputs(format_info, *outputs, **kwargs)

        # *** Arithmetic operators for creating compound models ***
        __add__ =     _model_oper('+')
        __sub__ =     _model_oper('-')
        __mul__ =     _model_oper('*')
        __truediv__ = _model_oper('/')
        __pow__ =     _model_oper('**')
        __or__ =      _model_oper('|')
        __and__ =     _model_oper('&')

        if not six.PY3:
            __div__ = _model_oper('/')

        # *** Properties ***
        @property
        def name(self):
            """User-provided name for this model instance."""

            return self._name

        @property
        @deprecated('0.4', alternative='len(model)')
        def param_dim(self):
            return self._n_models

        @property
        def n_inputs(self):
            """
            The number of inputs to this model.

            Equivalent to ``len(model.inputs)``.
            """

            return len(self.inputs)

        @property
        def n_outputs(self):
            """
            The number of outputs from this model.

            Equivalent to ``len(model.outputs)``.
            """
            return len(self.outputs)

        @property
        def model_set_axis(self):
            """
            The index of the model set axis--that is the axis of a parameter array
            that pertains to which model a parameter value pertains to--as
            specified when the model was initialized.

            See the documentation on `Model Sets
            <http://docs.astropy.org/en/stable/modeling/models.html#model-sets>`_
            for more details.
            """

            return self._model_set_axis

        @property
        def param_sets(self):
            """
            Return parameters as a pset.

            This is a list with one item per parameter set, which is an array of
            that parameter's values across all parameter sets, with the last axis
            associated with the parameter set.
            """

            return self._param_sets()

        @property
        def parameters(self):
            """
            A flattened array of all parameter values in all parameter sets.

            Fittable parameters maintain this list and fitters modify it.
            """

            # Currently the sequence of a model's parameters must be contiguous
            # within the _parameters array (which may be a view of a larger array,
            # for example when taking a sub-expression of a compound model), so
            # the assumption here is reliable:
            if not self.param_names:
                # Trivial, but not unheard of
                return self._parameters

            start = self._param_metrics[self.param_names[0]]['slice'].start
            stop = self._param_metrics[self.param_names[-1]]['slice'].stop

            return self._parameters[start:stop]

        @parameters.setter
        def parameters(self, value):
            """
            Assigning to this attribute updates the parameters array rather than
            replacing it.
            """

            if not self.param_names:
                return

            start = self._param_metrics[self.param_names[0]]['slice'].start
            stop = self._param_metrics[self.param_names[-1]]['slice'].stop

            try:
                value = np.array(value).flatten()
                self._parameters[start:stop] = value
            except ValueError as e:
                raise InputParameterError(
                    "Input parameter values not compatible with the model "
                    "parameters array: {0}".format(e))

        @property
        def fixed(self):
            """
            A `dict` mapping parameter names to their fixed constraint.
            """

            return self._constraints['fixed']

        @property
        def tied(self):
            """
            A `dict` mapping parameter names to their tied constraint.
            """

            return self._constraints['tied']

        @property
        def bounds(self):
            """
            A `dict` mapping parameter names to their upper and lower bounds as
            ``(min, max)`` tuples.
            """

            return self._constraints['bounds']

        @property
        def eqcons(self):
            """List of parameter equality constraints."""

            return self._constraints['eqcons']

        @property
        def ineqcons(self):
            """List of parameter inequality constraints."""

            return self._constraints['ineqcons']

        @property
        def inverse(self):
            """
            Returns a new `~astropy.modeling.Model` instance which performs the
            inverse transform, if an analytic inverse is defined for this model.

            Even on models that don't have an inverse defined, this property can be
            set with a manually-defined inverse, such a pre-computed or
            experimentally determined inverse (often given as a
            `~astropy.modeling.polynomial.PolynomialModel`, but not by
            requirement).

            A custom inverse can be deleted with ``del model.inverse``.  In this
            case the model's inverse is reset to its default, if a default exists
            (otherwise the default is to raise `NotImplementedError`).

            Note to authors of `~astropy.modeling.Model` subclasses:  To define an
            inverse for a model simply override this property to return the
            appropriate model representing the inverse.  The machinery that will
            make the inverse manually-overridable is added automatically by the
            base class.
            """

            if self._user_inverse is not None:
                return self._user_inverse
            elif self._inverse is not None:
                return self._inverse()

            raise NotImplementedError("An analytical inverse transform has not "
                                      "been implemented for this model.")

        @inverse.setter
        def inverse(self, value):
            if not isinstance(value, (Model, type(None))):
                raise ValueError(
                    "The ``inverse`` attribute may be assigned a `Model` "
                    "instance or `None` (where `None` restores the default "
                    "inverse for this model if one is defined.")

            if value is None:
                warnings.warn(
                    "Currently setting `model.inverse = None` resets the inverse "
                    "to the default inverse (if one exists).  However, starting "
                    "in Astropy 1.2, setting `model.inverse = None` explicitly "
                    "forces a model to have no inverse (such that accessing "
                    "`model.inverse` raises a NotImplementedError) even if that "
                    "model's class has a default inverse.\n\n"
                    "Instead, call `del model.inverse` to reset the inverse to "
                    "its default (if a default exists for that model's class--"
                    "otherwise the model is reset to having no inverse.",
                    AstropyDeprecationWarning)

            self._user_inverse = value

        @inverse.deleter
        def inverse(self):
            """
            Resets the model's inverse to its default (if one exists, otherwise
            the model will have no inverse).
            """

            del self._user_inverse

        @property
        def has_user_inverse(self):
            """
            A flag indicating whether or not a custom inverse model has been
            assigned to this model by a user, via assignment to ``model.inverse``.
            """

            return self._user_inverse is not None

        @property
        def _custom_inverse(self):
            # Deprecated alias for _user_inverse--included solely for temporary
            # backwards compatibility for pyasdf--remove after Astropy v1.1
            # release.
            return self._user_inverse

        @property
        def bounding_box(self):
            """
            A `tuple` of length `n_inputs` defining the bounding box limits, or
            `None` for no bounding box.

            The default limits are given by a ``bounding_box`` property or method
            defined in the class body of a specific model.  If not defined then
            this property just raises `NotImplementedError` by default (but may be
            assigned a custom value by a user).  ``bounding_box`` can be set
            manually to an array-like object of shape ``(model.n_inputs, 2)``. For
            further usage, see :ref:`bounding-boxes`

            The limits are ordered according to the `numpy` indexing
            convention, and are the reverse of the model input order,
            e.g. for inputs ``('x', 'y', 'z')``, ``bounding_box`` is defined:

            * for 1D: ``(x_low, x_high)``
            * for 2D: ``((y_low, y_high), (x_low, x_high))``
            * for 3D: ``((z_low, z_high), (y_low, y_high), (x_low, x_high))``

            Examples
            --------

            Setting the ``bounding_box`` limits for a 1D and 2D model:

            >>> from astropy.modeling.models import Gaussian1D, Gaussian2D
            >>> model_1d = Gaussian1D()
            >>> model_2d = Gaussian2D(x_stddev=1, y_stddev=1)
            >>> model_1d.bounding_box = (-5, 5)
            >>> model_2d.bounding_box = ((-6, 6), (-5, 5))

            Setting the bounding_box limits for a user-defined 3D `custom_model`:

            >>> from astropy.modeling.models import custom_model
            >>> def const3d(x, y, z, amp=1):
            ...    return amp
            ...
            >>> Const3D = custom_model(const3d)
            >>> model_3d = Const3D()
            >>> model_3d.bounding_box = ((-6, 6), (-5, 5), (-4, 4))

            To reset ``bounding_box`` to its default limits just delete the
            user-defined value--this will reset it back to the default defined
            on the class:

            >>> del model_1d.bounding_box

            To disable the bounding box entirely (including the default),
            set ``bounding_box`` to `None`:

            >>> model_1d.bounding_box = None
            >>> model_1d.bounding_box  # doctest: +IGNORE_EXCEPTION_DETAIL
            Traceback (most recent call last):
              File "<stdin>", line 1, in <module>
              File "astropy\modeling\core.py", line 980, in bounding_box
                "No bounding box is defined for this model (note: the "
            NotImplementedError: No bounding box is defined for this model (note:
            the bounding box was explicitly disabled for this model; use `del
            model.bounding_box` to restore the default bounding box, if one is
            defined for this model).
            """

            if self._user_bounding_box is not None:
                if self._user_bounding_box is NotImplemented:
                    raise NotImplementedError(
                        "No bounding box is defined for this model (note: the "
                        "bounding box was explicitly disabled for this model; "
                        "use `del model.bounding_box` to restore the default "
                        "bounding box, if one is defined for this model).")
                return self._user_bounding_box
            elif self._bounding_box is None:
                raise NotImplementedError(
                        "No bounding box is defined for this model.")
            elif isinstance(self._bounding_box, _BoundingBox):
                # This typically implies a hard-coded bounding box.  This will
                # probably be rare, but it is an option
                return self._bounding_box
            elif isinstance(self._bounding_box, types.MethodType):
                return self._bounding_box()
            else:
                # The only other allowed possibility is that it's a _BoundingBox
                # subclass, so we call it with its default arguments and return an
                # instance of it (that can be called to recompute the bounding box
                # with any optional parameters)
                # (In other words, in this case self._bounding_box is a *class*)
                bounding_box = self._bounding_box((), _model=self)()
                return self._bounding_box(bounding_box, _model=self)

        @bounding_box.setter
        def bounding_box(self, bounding_box):
            """
            Assigns the bounding box limits.
            """

            if bounding_box is None:
                cls = None
                # We use this to explicitly set an unimplemented bounding box (as
                # opposed to no user bounding box defined)
                bounding_box = NotImplemented
            elif (isinstance(self._bounding_box, type) and
                    issubclass(self._bounding_box, _BoundingBox)):
                cls = self._bounding_box
            else:
                cls = _BoundingBox

            if cls is not None:
                try:
                    bounding_box = cls.validate(self, bounding_box)
                except AssertionError as exc:
                    raise ValueError(exc.args[0])

            self._user_bounding_box = bounding_box

        @bounding_box.deleter
        def bounding_box(self):
            self._user_bounding_box = None

        @property
        def has_user_bounding_box(self):
            """
            A flag indicating whether or not a custom bounding_box has been
            assigned to this model by a user, via assignment to
            ``model.bounding_box``.
            """

            return self._user_bounding_box is not None

        # *** Public methods ***

        @abc.abstractmethod
        def evaluate(self, *args, **kwargs):
            """Evaluate the model on some input variables."""

        def render(self, out=None, coords=None):
            """
            Evaluates a model on an input array. Evaluation is limited to
            a bounding box if the `Model.bounding_box` attribute is set.

            Parameters
            ----------
            out : `numpy.ndarray`, optional
                The array on which the model is to be evaluated.
            coords : array-like, optional
                Coordinate arrays mapping to ``arr``, such that
                ``arr[coords] == arr``.

            Returns
            -------
            out : `numpy.ndarray`
                The model evaluated on the input array if given, or else a new array from
                ``coords``.
                If ``out`` and ``coords`` are both `None`, the returned array is
                limited to the `Model.bounding_box` limits. If
                `Model.bounding_box` is `None`, ``arr`` or ``coords`` must be passed.

            Examples
            --------
            :ref:`bounding-boxes`
            """

            try:
                bbox = self.bounding_box
            except NotImplementedError:
                bbox = None

            ndim = self.n_inputs

            if (coords is None) and (out is None) and (bbox is None):
                raise ValueError('If no bounding_box is set, '
                                 'coords or out must be input.')

            # for consistent indexing
            if ndim == 1:
                if coords is not None:
                    coords = [coords]
                if bbox is not None:
                    bbox = [bbox]

            if coords is not None:
                # Check dimensions match out and model
                assert len(coords) == ndim
                if out is not None:
                    assert coords[0].shape == out.shape
                else:
                    out = np.zeros(coords[0].shape)

            if out is not None:
                try:
                    assert out.ndim == ndim
                except AssertionError:
                    raise AssertionError(
                        'The array and model must have the same number '
                        'of dimensions.')

            if bbox is not None:

                # assures position is at center pixel, important when using add_array
                pd = np.array([(np.mean(bb), np.ceil((bb[1] - bb[0]) / 2))
                               for bb in bbox]).astype(int).T
                pos, delta = pd

                if coords is not None:
                    sub_shape = tuple(delta * 2 + 1)
                    sub_coords = np.array([extract_array(c, sub_shape, pos)
                                           for c in coords])
                else:
                    limits = [slice(p - d, p + d + 1, 1) for p, d in pd.T]
                    sub_coords = np.mgrid[limits]

                sub_coords = sub_coords[::-1]

                if out is None:
                    out = self(*sub_coords)
                else:
                    try:
                        out = add_array(out, self(*sub_coords), pos)
                    except ValueError:
                        raise ValueError(
                            'The `bounding_box` is larger than the input out in '
                            'one or more dimensions. Set '
                            '`model.bounding_box = None`.')
            else:
                if coords is None:
                    im_shape = out.shape
                    limits = [slice(i) for i in im_shape]
                    coords = np.mgrid[limits]

                coords = coords[::-1]

                out += self(*coords)

            return out

        def prepare_inputs(self, *inputs, **kwargs):
            """
            This method is used in `~astropy.modeling.Model.__call__` to ensure
            that all the inputs to the model can be broadcast into compatible
            shapes (if one or both of them are input as arrays), particularly if
            there are more than one parameter sets.
            """

            model_set_axis = kwargs.pop('model_set_axis', None)

            if model_set_axis is None:
                # By default the model_set_axis for the input is assumed to be the
                # same as that for the parameters the model was defined with
                # TODO: Ensure that negative model_set_axis arguments are respected
                model_set_axis = self.model_set_axis

            n_models = len(self)

            params = [getattr(self, name) for name in self.param_names]
            inputs = [np.asanyarray(_input, dtype=float) for _input in inputs]

            _validate_input_shapes(inputs, self.inputs, n_models,
                                   model_set_axis, self.standard_broadcasting)

            # The input formatting required for single models versus a multiple
            # model set are different enough that they've been split into separate
            # subroutines
            if n_models == 1:
                return _prepare_inputs_single_model(self, params, inputs,
                                                    **kwargs)
            else:
                return _prepare_inputs_model_set(self, params, inputs, n_models,
                                                 model_set_axis, **kwargs)

        def prepare_outputs(self, format_info, *outputs, **kwargs):
            if len(self) == 1:
                return _prepare_outputs_single_model(self, outputs, format_info)
            else:
                return _prepare_outputs_model_set(self, outputs, format_info)

        @deprecated('1.0',
                    alternative='Use Model operators (TODO: link to compound '
                                'model docs')
        def add_model(self, model, mode):
            """
            Create a CompositeModel by chaining the current model with the new one
            using the specified mode.

            Parameters
            ----------
            model : an instance of a subclass of Model
            mode :  string
                   'parallel', 'serial', 'p' or 's'
                   a flag indicating whether to combine the models
                   in series or in parallel

            Returns
            -------
            model : CompositeModel
                an instance of CompositeModel
            """

            from ._compound_deprecated import (SummedCompositeModel,
                                               SerialCompositeModel)

            if mode in ['parallel', 'p']:
                return SummedCompositeModel([self, model])
            elif mode in ['serial', 's']:
                return SerialCompositeModel([self, model])
            else:
                raise InputParameterError("Unrecognized mode {0}".format(mode))

        def copy(self):
            """
            Return a copy of this model.

            Uses a deep copy so that all model attributes, including parameter
            values, are copied as well.
            """

            return copy.deepcopy(self)

        @sharedmethod
        def rename(self, name):
            """
            Return a copy of this model with a new name.
            """

            new_model = self.copy()
            new_model._name = name
            return new_model

        # *** Internal methods ***
        @sharedmethod
        def _from_existing(self, existing, param_names):
            """
            Creates a new instance of ``cls`` that shares its underlying parameter
            values with an existing model instance given by ``existing``.

            This is used primarily by compound models to return a view of an
            individual component of a compound model.  ``param_names`` should be
            the names of the parameters in the *existing* model to use as the
            parameters in this new model.  Its length should equal the number of
            parameters this model takes, so that it can map parameters on the
            existing model to parameters on this model one-to-one.
            """

            # Basically this is an alternative __init__
            if isinstance(self, type):
                # self is a class, not an instance
                needs_initialization = True
                dummy_args = (0,) * len(param_names)
                self = self.__new__(self, *dummy_args)
            else:
                needs_initialization = False
                self = self.copy()

            aliases = dict(zip(self.param_names, param_names))
            # This is basically an alternative _initialize_constraints
            constraints = {}
            for cons_type in self.parameter_constraints:
                orig = existing._constraints[cons_type]
                constraints[cons_type] = AliasDict(orig, aliases)

            self._constraints = constraints

            self._n_models = existing._n_models
            self._model_set_axis = existing._model_set_axis
            self._parameters = existing._parameters

            self._param_metrics = defaultdict(dict)
            for param_a, param_b in six.iteritems(aliases):
                # Take the param metrics info for the giving parameters in the
                # existing model, and hand them to the appropriate parameters in
                # the new model
                self._param_metrics[param_a] = existing._param_metrics[param_b]

            if needs_initialization:
                self.__init__(*dummy_args)

            return self

        def _initialize_constraints(self, kwargs):
            """
            Pop parameter constraint values off the keyword arguments passed to
            `Model.__init__` and store them in private instance attributes.
            """

            if hasattr(self, '_constraints'):
                # Skip constraint initialization if it has already been handled via
                # an alternate initialization
                return

            self._constraints = {}
            # Pop any constraints off the keyword arguments
            for constraint in self.parameter_constraints:
                values = kwargs.pop(constraint, {})
                self._constraints[constraint] = values.copy()

                # Update with default parameter constraints
                for param_name in self.param_names:
                    param = getattr(self, param_name)

                    # Parameters don't have all constraint types
                    value = getattr(param, constraint)
                    if value is not None:
                        self._constraints[constraint][param_name] = value

            for constraint in self.model_constraints:
                values = kwargs.pop(constraint, [])
                self._constraints[constraint] = values

        def _initialize_parameters(self, args, kwargs):
            """
            Initialize the _parameters array that stores raw parameter values for
            all parameter sets for use with vectorized fitting algorithms; on
            FittableModels the _param_name attributes actually just reference
            slices of this array.
            """

            if hasattr(self, '_parameters'):
                # Skip parameter initialization if it has already been handled via
                # an alternate initialization
                return

            n_models = None
            # Pop off param_dim and handle backwards compatibility
            if 'param_dim' in kwargs:
                n_models = kwargs.pop('param_dim')
                warnings.warn(
                    'The param_dim argument to {0}.__init__ is deprecated; '
                    'use n_models instead.  See also the model_set_axis argument '
                    'and related discussion in the docstring for Model.'.format(
                        self.__class__.__name__), AstropyDeprecationWarning)
                if 'n_models' in kwargs:
                    raise TypeError(
                        "param_dim and n_models cannot both be specified; use "
                        "n_models, as param_dim is deprecated")
            else:
                n_models = kwargs.pop('n_models', None)

            if not (n_models is None or
                        (isinstance(n_models, int) and n_models >=1)):
                raise ValueError(
                    "n_models must be either None (in which case it is "
                    "determined from the model_set_axis of the parameter initial "
                    "values) or it must be a positive integer "
                    "(got {0!r})".format(n_models))

            model_set_axis = kwargs.pop('model_set_axis', None)
            if model_set_axis is None:
                if n_models is not None and n_models > 1:
                    # Default to zero
                    model_set_axis = 0
                else:
                    # Otherwise disable
                    model_set_axis = False
            else:
                if not (model_set_axis is False or
                        (isinstance(model_set_axis, int) and
                            not isinstance(model_set_axis, bool))):
                    raise ValueError(
                        "model_set_axis must be either False or an integer "
                        "specifying the parameter array axis to map to each "
                        "model in a set of models (got {0!r}).".format(
                            model_set_axis))

            # Process positional arguments by matching them up with the
            # corresponding parameters in self.param_names--if any also appear as
            # keyword arguments this presents a conflict
            params = {}
            if len(args) > len(self.param_names):
                raise TypeError(
                    "{0}.__init__() takes at most {1} positional arguments ({2} "
                    "given)".format(self.__class__.__name__, len(self.param_names),
                                    len(args)))

            for idx, arg in enumerate(args):
                if arg is None:
                    # A value of None implies using the default value, if exists
                    continue
                params[self.param_names[idx]] = np.asanyarray(arg, dtype=np.float)

            # At this point the only remaining keyword arguments should be
            # parameter names; any others are in error.
            for param_name in self.param_names:
                if param_name in kwargs:
                    if param_name in params:
                        raise TypeError(
                            "{0}.__init__() got multiple values for parameter "
                            "{1!r}".format(self.__class__.__name__, param_name))
                    value = kwargs.pop(param_name)
                    if value is None:
                        continue
                    params[param_name] = np.asanyarray(value, dtype=np.float)

            if kwargs:
                # If any keyword arguments were left over at this point they are
                # invalid--the base class should only be passed the parameter
                # values, constraints, and param_dim
                for kwarg in kwargs:
                    # Just raise an error on the first unrecognized argument
                    raise TypeError(
                        '{0}.__init__() got an unrecognized parameter '
                        '{1!r}'.format(self.__class__.__name__, kwarg))

            self._model_set_axis = model_set_axis
            self._param_metrics = defaultdict(dict)

            # Determine the number of model sets: If the model_set_axis is
            # None then there is just one parameter set; otherwise it is determined
            # by the size of that axis on the first parameter--if the other
            # parameters don't have the right number of axes or the sizes of their
            # model_set_axis don't match an error is raised
            if model_set_axis is not False and n_models != 1 and params:
                max_ndim = 0
                if model_set_axis < 0:
                    min_ndim = abs(model_set_axis)
                else:
                    min_ndim = model_set_axis + 1

                for name, value in six.iteritems(params):
                    param_ndim = np.ndim(value)
                    if param_ndim < min_ndim:
                        raise InputParameterError(
                            "All parameter values must be arrays of dimension "
                            "at least {0} for model_set_axis={1} (the value "
                            "given for {2!r} is only {3}-dimensional)".format(
                                min_ndim, model_set_axis, name, param_ndim))

                    max_ndim = max(max_ndim, param_ndim)

                    if n_models is None:
                        # Use the dimensions of the first parameter to determine
                        # the number of model sets
                        n_models = value.shape[model_set_axis]
                    elif value.shape[model_set_axis] != n_models:
                        raise InputParameterError(
                            "Inconsistent dimensions for parameter {0!r} for "
                            "{1} model sets.  The length of axis {2} must be the "
                            "same for all input parameter values".format(
                            name, n_models, model_set_axis))

                self._check_param_broadcast(params, max_ndim)
            else:
                if n_models is None:
                    n_models = 1

                self._check_param_broadcast(params, None)

            self._n_models = n_models
            self._initialize_parameter_values(params)

        def _initialize_parameter_values(self, params):
            # self._param_metrics should have been initialized in
            # self._initialize_parameters
            param_metrics = self._param_metrics
            total_size = 0

            for name in self.param_names:
                if params.get(name) is None:
                    default = getattr(self, name).default

                    if default is None:
                        # No value was supplied for the parameter, and the
                        # parameter does not have a default--therefor the model is
                        # underspecified
                        raise TypeError(
                            "{0}.__init__() requires a value for parameter "
                            "{1!r}".format(self.__class__.__name__, name))

                    value = params[name] = default
                else:
                    value = params[name]

                param_size = np.size(value)
                param_shape = np.shape(value)

                param_slice = slice(total_size, total_size + param_size)

                param_metrics[name]['slice'] = param_slice
                param_metrics[name]['shape'] = param_shape
                total_size += param_size

            self._param_metrics = param_metrics
            self._parameters = np.empty(total_size, dtype=np.float64)

            # Now set the parameter values (this will also fill
            # self._parameters)
            # TODO: This is a bit ugly, but easier to deal with than how this was
            # done previously.  There's still lots of opportunity for refactoring
            # though, in particular once we move the _get/set_model_value methods
            # out of Parameter and into Model (renaming them
            # _get/set_parameter_value)
            for name, value in params.items():
                param_descr = getattr(self, name)
                value = np.array(value)
                if param_descr._setter is not None:
                    value = param_descr._setter(value)
                self._parameters[param_metrics[name]['slice']] = value.ravel()

            # Finally validate all the parameters; we do this last so that
            # validators that depend on one of the other parameters' values will
            # work
            for name in params:
                param_descr = getattr(self, name)
                param_descr.validator(param_descr.value)

        def _check_param_broadcast(self, params, max_ndim):
            """
            This subroutine checks that all parameter arrays can be broadcast
            against each other, and determines the shapes parameters must have in
            order to broadcast correctly.

            If model_set_axis is None this merely checks that the parameters
            broadcast and returns an empty dict if so.  This mode is only used for
            single model sets.
            """

            all_shapes = []
            param_names = []
            model_set_axis = self._model_set_axis

            for name in self.param_names:
                # Previously this just used iteritems(params), but we loop over all
                # param_names instead just to ensure some determinism in the
                # ordering behavior
                if name not in params:
                    continue

                value = params[name]
                param_names.append(name)
                # We've already checked that each parameter array is compatible in
                # the model_set_axis dimension, but now we need to check the
                # dimensions excluding that axis
                # Split the array dimensions into the axes before model_set_axis
                # and after model_set_axis
                param_shape = np.shape(value)

                param_ndim = len(param_shape)
                if max_ndim is not None and param_ndim < max_ndim:
                    # All arrays have the same number of dimensions up to the
                    # model_set_axis dimension, but after that they may have a
                    # different number of trailing axes.  The number of trailing
                    # axes must be extended for mutual compatibility.  For example
                    # if max_ndim = 3 and model_set_axis = 0, an array with the
                    # shape (2, 2) must be extended to (2, 1, 2).  However, an
                    # array with shape (2,) is extended to (2, 1).
                    new_axes = (1,) * (max_ndim - param_ndim)

                    if model_set_axis < 0:
                        # Just need to prepend axes to make up the difference
                        broadcast_shape = new_axes + param_shape
                    else:
                        broadcast_shape = (param_shape[:model_set_axis + 1] +
                                           new_axes +
                                           param_shape[model_set_axis + 1:])
                    self._param_metrics[name]['broadcast_shape'] = broadcast_shape
                    all_shapes.append(broadcast_shape)
                else:
                    all_shapes.append(param_shape)

            # Now check mutual broadcastability of all shapes
            try:
                check_broadcast(*all_shapes)
            except IncompatibleShapeError as exc:
                shape_a, shape_a_idx, shape_b, shape_b_idx = exc.args
                param_a = param_names[shape_a_idx]
                param_b = param_names[shape_b_idx]

                raise InputParameterError(
                    "Parameter {0!r} of shape {1!r} cannot be broadcast with "
                    "parameter {2!r} of shape {3!r}.  All parameter arrays "
                    "must have shapes that are mutually compatible according "
                    "to the broadcasting rules.".format(param_a, shape_a,
                                                        param_b, shape_b))

        def _param_sets(self, raw=False):
            """
            Implementation of the Model.param_sets property.

            This internal implementation has a ``raw`` argument which controls
            whether or not to return the raw parameter values (i.e. the values that
            are actually stored in the ._parameters array, as opposed to the values
            displayed to users.  In most cases these are one in the same but there
            are currently a few exceptions.

            Note: This is notably an overcomplicated device and may be removed
            entirely in the near future.
            """

            param_metrics = self._param_metrics
            values = []
            for name in self.param_names:
                if raw:
                    value = getattr(self, name)._raw_value
                else:
                    value = getattr(self, name).value

                broadcast_shape = param_metrics[name].get('broadcast_shape')
                if broadcast_shape is not None:
                    value = value.reshape(broadcast_shape)

                values.append(value)

            shapes = [np.shape(value) for value in values]

            if len(self) == 1:
                # Add a single param set axis to the parameter's value (thus
                # converting scalars to shape (1,) array values) for consistency
                values = [np.array([value]) for value in values]

            if len(set(shapes)) != 1:
                # If the parameters are not all the same shape, converting to an
                # array is going to produce an object array
                # However the way Numpy creates object arrays is tricky in that it
                # will recurse into array objects in the list and break them up
                # into separate objects.  Doing things this way ensures a 1-D
                # object array the elements of which are the individual parameter
                # arrays.  There's not much reason to do this over returning a list
                # except for consistency
                psets = np.empty(len(values), dtype=object)
                psets[:] = values
                return psets

            return np.array(values)

        def _format_repr(self, args=[], kwargs={}, defaults={}):
            """
            Internal implementation of ``__repr__``.

            This is separated out for ease of use by subclasses that wish to
            override the default ``__repr__`` while keeping the same basic
            formatting.
            """

            # TODO: I think this could be reworked to preset model sets better

            parts = [repr(a) for a in args]

            parts.extend(
                "{0}={1}".format(name,
                                 array_repr_oneline(getattr(self, name).value))
                for name in self.param_names)

            if self.name is not None:
                parts.append('name={0!r}'.format(self.name))

            for kwarg, value in kwargs.items():
                if kwarg in defaults and defaults[kwarg] != value:
                    continue
                parts.append('{0}={1!r}'.format(kwarg, value))

            if len(self) > 1:
                parts.append("n_models={0}".format(len(self)))

            return '<{0}({1})>'.format(self.__class__.__name__, ', '.join(parts))

        def _format_str(self, keywords=[]):
            """
            Internal implementation of ``__str__``.

            This is separated out for ease of use by subclasses that wish to
            override the default ``__str__`` while keeping the same basic
            formatting.
            """

            default_keywords = [
                ('Model', self.__class__.__name__),
                ('Name', self.name),
                ('Inputs', self.inputs),
                ('Outputs', self.outputs),
                ('Model set size', len(self))
            ]

            parts = ['{0}: {1}'.format(keyword, value)
                     for keyword, value in default_keywords + keywords
                     if value is not None]

            parts.append('Parameters:')

            if len(self) == 1:
                columns = [[getattr(self, name).value]
                           for name in self.param_names]
            else:
                columns = [getattr(self, name).value
                           for name in self.param_names]

            if columns:
                param_table = Table(columns, names=self.param_names)
                parts.append(indent(str(param_table), width=4))

            return '\n'.join(parts)

    class FittableModel(Model):
        """
        Base class for models that can be fitted using the built-in fitting
        algorithms.
        """

        linear = False
        # derivative with respect to parameters
        fit_deriv = None
        """
        Function (similar to the model's `~Model.evaluate`) to compute the
        derivatives of the model with respect to its parameters, for use by fitting
        algorithms.  In other words, this computes the Jacobian matrix with respect
        to the model's parameters.
        """
        # Flag that indicates if the model derivatives with respect to parameters
        # are given in columns or rows
        col_fit_deriv = True
        fittable = True

    class Fittable1DModel(FittableModel):
        """
        Base class for one-dimensional fittable models.

        This class provides an easier interface to defining new models.
        Examples can be found in `astropy.modeling.functional_models`.
        """

        inputs = ('x',)
        outputs = ('y',)

    class Fittable2DModel(FittableModel):
        """
        Base class for two-dimensional fittable models.

        This class provides an easier interface to defining new models.
        Examples can be found in `astropy.modeling.functional_models`.
        """

        inputs = ('x', 'y')
        outputs = ('z',)

    def _make_arithmetic_operator(oper):
        # We don't bother with tuple unpacking here for efficiency's sake, but for
        # documentation purposes:
        #
        #     f_eval, f_n_inputs, f_n_outputs = f
        #
        # and similarly for g
        def op(f, g):
            return (make_binary_operator_eval(oper, f[0], g[0]), f[1], f[2])

        return op

    def _composition_operator(f, g):
        # We don't bother with tuple unpacking here for efficiency's sake, but for
        # documentation purposes:
        #
        #     f_eval, f_n_inputs, f_n_outputs = f
        #
        # and similarly for g
        return (lambda inputs, params: g[0](f[0](inputs, params), params),
                f[1], g[2])

    def _join_operator(f, g):
        # We don't bother with tuple unpacking here for efficiency's sake, but for
        # documentation purposes:
        #
        #     f_eval, f_n_inputs, f_n_outputs = f
        #
        # and similarly for g
        return (lambda inputs, params: (f[0](inputs[:f[1]], params) +
                                        g[0](inputs[f[1]:], params)),
                f[1] + g[1], f[2] + g[2])

    # TODO: Support a couple unary operators--at least negation?
    BINARY_OPERATORS = {
        '+': _make_arithmetic_operator(operator.add),
        '-': _make_arithmetic_operator(operator.sub),
        '*': _make_arithmetic_operator(operator.mul),
        '/': _make_arithmetic_operator(operator.truediv),
        '**': _make_arithmetic_operator(operator.pow),
        '|': _composition_operator,
        '&': _join_operator
    }

    _ORDER_OF_OPERATORS = [('|',), ('&',), ('+', '-'), ('*', '/'), ('**',)]
    OPERATOR_PRECEDENCE = {}
    for idx, ops in enumerate(_ORDER_OF_OPERATORS):
        for op in ops:
            OPERATOR_PRECEDENCE[op] = idx
    del idx, op, ops

    class _CompoundModelMeta(_ModelMeta):
        _tree = None
        _submodels = None
        _submodel_names = None
        _nextid = 0

        _param_names = None
        # _param_map is a mapping of the compound model's generated param names to
        # the parameters of submodels they are associated with.  The values in this
        # mapping are (idx, name) tuples were idx is the index of the submodel this
        # parameter is associated with, and name is the same parameter's name on
        # the submodel
        # In principle this will allow compound models to give entirely new names
        # to parameters that don't have to be the same as their original names on
        # the submodels, but right now that isn't taken advantage of
        _param_map = None

        _slice_offset = 0
        # When taking slices of a compound model, this keeps track of how offset
        # the first model in the slice is from the first model in the original
        # compound model it was taken from

        # This just inverts _param_map, swapping keys with values.  This is also
        # useful to have.
        _param_map_inverse = None
        _fittable = None

        _evaluate = None

        def __getitem__(cls, index):
            index = cls._normalize_index(index)

            if isinstance(index, int):
                return cls._get_submodels()[index]
            else:
                return cls._get_slice(index.start, index.stop)

        def __getattr__(cls, attr):
            # Make sure the _tree attribute is set; otherwise we are not looking up
            # an attribute on a concrete compound model class and should just raise
            # the AttributeError
            if cls._tree is not None and attr in cls.param_names:
                cls._init_param_descriptors()
                return getattr(cls, attr)

            raise AttributeError(attr)

        def __repr__(cls):
            if cls._tree is None:
                # This case is mostly for debugging purposes
                return cls._format_cls_repr()

            expression = cls._format_expression()
            components = '\n\n'.join('[{0}]: {1!r}'.format(idx, m)
                                     for idx, m in enumerate(cls._get_submodels()))
            keywords = [
                ('Expression', expression),
                ('Components', '\n' + indent(components))
            ]

            return cls._format_cls_repr(keywords=keywords)

        def __dir__(cls, *args):
            """
            Returns a list of attributes defined on a compound model, including
            all of its parameters.
            """

            # The *args is to address a bug (?) on Python 2.6 where the dir()
            # builtin calls __dir__ with an additional (unused) argument

            try:
                # Annoyingly, this will only work for Python 3.3+
                basedir = super(_CompoundModelMeta, cls).__dir__()
            except AttributeError:
                basedir = list(set((dir(type(cls)) + list(cls.__dict__))))

            if cls._tree is not None:
                for name in cls.param_names:
                    basedir.append(name)

                basedir.sort()

            return basedir

        def __reduce__(cls):
            rv = super(_CompoundModelMeta, cls).__reduce__()

            if isinstance(rv, tuple):
                # Delete _evaluate from the members dict
                with ignored(KeyError):
                    del rv[1][2]['_evaluate']

            return rv

        @property
        def submodel_names(cls):
            if cls._submodel_names is not None:
                return cls._submodel_names

            by_name = defaultdict(list)

            for idx, submodel in enumerate(cls._get_submodels()):
                # Keep track of the original sort order of the submodels
                by_name[submodel.name].append(idx)

            names = []
            for basename, indices in six.iteritems(by_name):
                if len(indices) == 1:
                    # There is only one model with this name, so it doesn't need an
                    # index appended to its name
                    names.append((basename, indices[0]))
                else:
                    for idx in indices:
                        names.append(('{0}_{1}'.format(basename, idx), idx))

            # Sort according to the models' original sort orders
            names.sort(key=lambda k: k[1])

            names = tuple(k[0] for k in names)

            cls._submodels_names = names
            return names

        @property
        def param_names(cls):
            if cls._param_names is None:
                cls._init_param_names()

            return cls._param_names

        @property
        def fittable(cls):
            if cls._fittable is None:
                cls._fittable = all(m.fittable for m in cls._get_submodels())

            return cls._fittable

        # TODO: Maybe we could use make_function_with_signature for evaluate, but
        # it's probably not worth it (and I'm not sure what the limit is on number
        # of function arguments/local variables but we could break that limit for
        # complicated compound models...
        def evaluate(cls, *args):
            if cls._evaluate is None:
                func = cls._tree.evaluate(BINARY_OPERATORS,
                                          getter=cls._model_evaluate_getter)[0]
                # Making this a staticmethod isn't strictly necessary for Python 3,
                # but it is necessary on Python 2 since looking up cls._evaluate
                # will return an unbound method otherwise
                cls._evaluate = staticmethod(func)

            inputs = args[:cls.n_inputs]
            params = iter(args[cls.n_inputs:])
            result = cls._evaluate(inputs, params)

            if cls.n_outputs == 1:
                return result[0]
            else:
                return result

        # TODO: This supports creating a new compound model from two existing
        # compound models (or normal models) and a single operator.  However, it
        # ought also to be possible to create a new model from an *entire*
        # expression, represented as a sequence of operators and their operands (or
        # an exiting ExpressionTree) and build that into a compound model without
        # creating an intermediate _CompoundModel class for every single operator
        # in the expression.  This will prove to be a useful optimization in many
        # cases
        @classmethod
        def _from_operator(mcls, operator, left, right, additional_members={}):
            """
            Given a Python operator (represented by a string, such as ``'+'``
            or ``'*'``, and two model classes or instances, return a new compound
            model that evaluates the given operator on the outputs of the left and
            right input models.

            If either of the input models are a model *class* (i.e. a subclass of
            `~astropy.modeling.Model`) then the returned model is a new subclass of
            `~astropy.modeling.Model` that may be instantiated with any parameter
            values.  If both input models are *instances* of a model, a new class
            is still created, but this method returns an *instance* of that class,
            taking the parameter values from the parameters of the input model
            instances.

            If given, the ``additional_members`` `dict` may provide additional
            class members that should be added to the generated
            `~astropy.modeling.Model` subclass.  Some members that are generated by
            this method should not be provided by ``additional_members``.  These
            include ``_tree``, ``inputs``, ``outputs``, ``linear``,
            ``standard_broadcasting``, and ``__module__`.  This is currently for
            internal use only.
            """

            # Note, currently this only supports binary operators, but could be
            # easily extended to support unary operators (namely '-') if/when
            # needed
            children = []
            for child in (left, right):
                if isinstance(child, (_CompoundModelMeta, _CompoundModel)):
                    children.append(child._tree)
                else:
                    children.append(ExpressionTree(child))

            tree = ExpressionTree(operator, left=children[0], right=children[1])

            name = str('CompoundModel{0}'.format(_CompoundModelMeta._nextid))
            _CompoundModelMeta._nextid += 1

            mod = find_current_module(3)
            if mod:
                modname = mod.__name__
            else:
                modname = '__main__'

            # TODO: These aren't the full rules for handling inputs and outputs, but
            # this will handle most basic cases correctly
            if operator == '|':
                inputs = left.inputs
                outputs = right.outputs

                if left.n_outputs != right.n_inputs:
                    raise ModelDefinitionError(
                        "Unsupported operands for |: {0} (n_inputs={1}, "
                        "n_outputs={2}) and {3} (n_inputs={4}, n_outputs={5}); "
                        "n_outputs for the left-hand model must match n_inputs "
                        "for the right-hand model.".format(
                            left.name, left.n_inputs, left.n_outputs, right.name,
                            right.n_inputs, right.n_outputs))
            elif operator == '&':
                inputs = combine_labels(left.inputs, right.inputs)
                outputs = combine_labels(left.outputs, right.outputs)
            else:
                # Without loss of generality
                inputs = left.inputs
                outputs = left.outputs

                if (left.n_inputs != right.n_inputs or
                        left.n_outputs != right.n_outputs):
                    raise ModelDefinitionError(
                        "Unsupported operands for {0}: {1} (n_inputs={2}, "
                        "n_outputs={3}) and {4} (n_inputs={5}, n_outputs={6}); "
                        "models must have the same n_inputs and the same "
                        "n_outputs for this operator".format(
                            operator, left.name, left.n_inputs, left.n_outputs,
                            right.name, right.n_inputs, right.n_outputs))

            if operator in ('|', '+', '-'):
                linear = left.linear and right.linear
            else:
                # Which is not to say it is *definitely* not linear but it would be
                # trickier to determine
                linear = False

            standard_broadcasting = \
                    left.standard_broadcasting and right.standard_broadcasting

            # Note: If any other members are added here, make sure to mention them
            # in the docstring of this method.
            members = additional_members
            members.update({
                '_tree': tree,
                '_is_dynamic': True,  # See docs for _ModelMeta._is_dynamic
                'inputs': inputs,
                'outputs': outputs,
                'linear': linear,
                'standard_broadcasting': standard_broadcasting,
                '__module__': str(modname)})

            new_cls = mcls(name, (_CompoundModel,), members)

            if isinstance(left, Model) and isinstance(right, Model):
                # Both models used in the operator were already instantiated models,
                # not model *classes*.  As such it's not particularly useful to return
                # the class itself, but to instead produce a new instance:
                instance = new_cls()

                # Workaround for https://github.com/astropy/astropy/issues/3542
                # TODO: Any effort to restructure the tree-like data structure for
                # compound models should try to obviate this workaround--if
                # intermediate compound models are stored in the tree as well then
                # we can immediately check for custom inverses on sub-models when
                # computing the inverse
                instance._user_inverse = mcls._make_user_inverse(
                        operator, left, right)

                return instance

            # Otherwise return the new uninstantiated class itself
            return new_cls

        @classmethod
        def _handle_backwards_compat(mcls, name, members):
            # Override _handle_backwards_compat from _ModelMeta to be a no-op; it
            # is not needed since compound models did not exist before version 1.0
            # anyways.

            # TODO: Remove this at the same time as removing
            # _ModelMeta._handle_backwards_compat
            return

        @classmethod
        def _make_user_inverse(mcls, operator, left, right):
            """
            Generates an inverse `Model` for this `_CompoundModel` when either
            model in the operation has a *custom inverse* that was manually
            assigned by the user.

            If either model has a custom inverse, and in particular if another
            `_CompoundModel` has a custom inverse, then none of that model's
            sub-models should be considered at all when computing the inverse.
            So in that case we just compute the inverse ahead of time and set
            it as the new compound model's custom inverse.

            Note, this use case only applies when combining model instances,
            since model classes don't currently have a notion of a "custom
            inverse" (though it could probably be supported by overriding the
            class's inverse property).

            TODO: Consider fixing things so the aforementioned class-based case
            works as well.  However, for the present purposes this is good enough.
            """

            if not (operator in ('&', '|') and
                    (left._user_inverse or right._user_inverse)):
                # These are the only operators that support an inverse right now
                return None

            try:
                left_inv = left.inverse
                right_inv = right.inverse
            except NotImplementedError:
                # If either inverse is undefined then just return False; this
                # means the normal _CompoundModel.inverse routine will fail
                # naturally anyways, since it requires all sub-models to have
                # an inverse defined
                return None

            if operator == '&':
                return left_inv & right_inv
            else:
                return right_inv | left_inv

        # TODO: Perhaps, just perhaps, the post-order (or ???-order) ordering of
        # leaf nodes is something the ExpressionTree class itself could just know
        def _get_submodels(cls):
            # Would make this a lazyproperty but those don't currently work with
            # type objects
            if cls._submodels is not None:
                return cls._submodels

            submodels = [c.value for c in cls._tree.traverse_postorder()
                         if c.isleaf]
            cls._submodels = submodels
            return submodels

        def _init_param_descriptors(cls):
            """
            This routine sets up the names for all the parameters on a compound
            model, including figuring out unique names for those parameters and
            also mapping them back to their associated parameters of the underlying
            submodels.

            Setting this all up is costly, and only necessary for compound models
            that a user will directly interact with.  For example when building an
            expression like::

                >>> M = (Model1 + Model2) * Model3  # doctest: +SKIP

            the user will generally never interact directly with the temporary
            result of the subexpression ``(Model1 + Model2)``.  So there's no need
            to setup all the parameters for that temporary throwaway.  Only once
            the full expression is built and the user initializes or introspects
            ``M`` is it necessary to determine its full parameterization.
            """

            # Accessing cls.param_names will implicitly call _init_param_names if
            # needed and thus also set up the _param_map; I'm not crazy about that
            # design but it stands for now
            for param_name in cls.param_names:
                submodel_idx, submodel_param = cls._param_map[param_name]
                submodel = cls[submodel_idx]

                orig_param = getattr(submodel, submodel_param, None)
                if not isinstance(orig_param, Parameter):
                    # This is just a pathological case that is only really needed
                    # to support the deprecated _CompositeModel--composite models
                    # claim to have some parameters, but don't actually implement
                    # the parameter descriptors, so we just make one up basically,
                    # with a default value of zero.  This value will just be thrown
                    # away, basically.
                    # TODO: Remove this special case once the legacy interfaces
                    # have been removed (basically this entire if statement--keep
                    # only the parts in the else: clause.
                    new_param = Parameter(name=param_name, default=0)
                else:
                    if isinstance(submodel, Model):
                        # Take the parameter's default from the model's value for that
                        # parameter
                        default = orig_param.value
                    else:
                        default = orig_param.default

                    # Copy constraints
                    constraints = dict((key, getattr(orig_param, key))
                                       for key in Model.parameter_constraints)

                    # Note: Parameter.copy() returns a new unbound Parameter, never
                    # a bound Parameter even if submodel is a Model instance (as
                    # opposed to a Model subclass)
                    new_param = orig_param.copy(name=param_name, default=default,
                                                **constraints)

                setattr(cls, param_name, new_param)

        def _init_param_names(cls):
            """
            This subroutine is solely for setting up the ``param_names`` attribute
            itself.

            See ``_init_param_descriptors`` for the full parameter setup.
            """

            # Currently this skips over Model *instances* in the expression tree;
            # basically these are treated as constants and do not add
            # fittable/tunable parameters to the compound model.
            # TODO: I'm not 100% happy with this design, and maybe we need some
            # interface for distinguishing fittable/settable parameters with
            # *constant* parameters (which would be distinct from parameters with
            # fixed constraints since they're permanently locked in place). But I'm
            # not sure if this is really the best way to treat the issue.

            names = []
            param_map = {}

            # Start counting the suffix indices to put on parameter names from the
            # slice_offset.  Usually this will just be zero, but for compound
            # models that were sliced from another compound model this may be > 0
            param_suffix = cls._slice_offset

            for idx, model in enumerate(cls._get_submodels()):
                if not model.param_names:
                    # Skip models that don't have parameters in the numbering
                    # TODO: Reevaluate this if it turns out to be confusing, though
                    # parameter-less models are not very common in practice (there
                    # are a few projections that don't take parameters)
                    continue

                for param_name in model.param_names:
                    # This is sort of heuristic, but we want to check that
                    # model.param_name *actually* returns a Parameter descriptor,
                    # and that the model isn't some inconsistent type that happens
                    # to have a param_names attribute but does not actually
                    # implement settable parameters.
                    # In the future we can probably remove this check, but this is
                    # here specifically to support the legacy compat
                    # _CompositeModel which can be considered a pathological case
                    # in the context of the new framework
                    #if not isinstance(getattr(model, param_name, None),
                    #                  Parameter):
                    #    break
                    name = '{0}_{1}'.format(param_name, param_suffix + idx)
                    names.append(name)
                    param_map[name] = (idx, param_name)

            cls._param_names = tuple(names)
            cls._param_map = param_map
            cls._param_map_inverse = dict((v, k) for k, v in param_map.items())

        def _format_expression(cls):
            # TODO: At some point might be useful to make a public version of this,
            # albeit with more formatting options
            return cls._tree.format_expression(OPERATOR_PRECEDENCE)

        def _normalize_index(cls, index):
            """
            Converts an index given to __getitem__ to either an integer, or
            a slice with integer start and stop values.

            If the length of the slice is exactly 1 this converts the index to a
            simple integer lookup.

            Negative integers are converted to positive integers.
            """

            def get_index_from_name(name):
                try:
                    return cls.submodel_names.index(name)
                except ValueError:
                    raise IndexError(
                        'Compound model {0} does not have a component named '
                        '{1}'.format(cls.name, name))

            def check_for_negative_index(index):
                if index < 0:
                    new_index = len(cls.submodel_names) + index
                    if new_index < 0:
                        # If still < 0 then this is an invalid index
                        raise IndexError(
                                "Model index {0} out of range.".format(index))
                    else:
                        index = new_index

                return index

            if isinstance(index, six.string_types):
                return get_index_from_name(index)
            elif isinstance(index, slice):
                if index.step not in (1, None):
                    # In principle it could be but I can scarcely imagine a case
                    # where it would be useful.  If someone can think of one then
                    # we can enable it.
                    raise ValueError(
                        "Step not supported for compound model slicing.")
                start = index.start if index.start is not None else 0
                stop = (index.stop
                        if index.stop is not None else len(cls.submodel_names))
                if isinstance(start, int):
                    start = check_for_negative_index(start)
                if isinstance(stop, int):
                    stop = check_for_negative_index(stop)
                if isinstance(start, six.string_types):
                    start = get_index_from_name(start)
                if isinstance(stop, six.string_types):
                    stop = get_index_from_name(stop) + 1
                length = stop - start

                if length == 1:
                    return start
                elif length <= 0:
                    raise ValueError("Empty slice of a compound model.")

                return slice(start, stop)
            elif isinstance(index, int):
                if index >= len(cls.submodel_names):
                    raise IndexError(
                            "Model index {0} out of range.".format(index))

                return check_for_negative_index(index)

            raise TypeError(
                'Submodels can be indexed either by their integer order or '
                'their name (got {0!r}).'.format(index))

        def _get_slice(cls, start, stop):
            """
            Return a new model build from a sub-expression of the expression
            represented by this model.

            Right now this is highly inefficient, as it creates a new temporary
            model for each operator that appears in the sub-expression.  It would
            be better if this just built a new expression tree, and the new model
            instantiated directly from that tree.

            Once tree -> model instantiation is possible this should be fixed to
            use that instead.
            """

            members = {'_slice_offset': cls._slice_offset + start}
            operators = dict((oper, _model_oper(oper, additional_members=members))
                             for oper in BINARY_OPERATORS)

            return cls._tree.evaluate(operators, start=start, stop=stop)

        @staticmethod
        def _model_evaluate_getter(idx, model):
            n_params = len(model.param_names)
            n_inputs = model.n_inputs
            n_outputs = model.n_outputs

            # There is currently an unfortunate inconsistency in some models, which
            # requires them to be instantiated for their evaluate to work.  I think
            # that needs to be reconsidered and fixed somehow, but in the meantime
            # we need to check for that case
            if (not isinstance(model, Model) and
                    isinstancemethod(model, model.evaluate)):
                if n_outputs == 1:
                    # Where previously model was a class, now make an instance
                    def f(inputs, params):
                        param_values = tuple(islice(params, n_params))
                        return (model(*param_values).evaluate(
                            *chain(inputs, param_values)),)
                else:
                    def f(inputs, params):
                        param_values = tuple(islice(params, n_params))
                        return model(*param_values).evaluate(
                            *chain(inputs, param_values))
            else:
                evaluate = model.evaluate
                if n_outputs == 1:
                    f = lambda inputs, params: \
                        (evaluate(*chain(inputs, islice(params, n_params))),)
                else:
                    f = lambda inputs, params: \
                        evaluate(*chain(inputs, islice(params, n_params)))

            return (f, n_inputs, n_outputs)

    @six.add_metaclass(_CompoundModelMeta)
    class _CompoundModel(Model):
        fit_deriv = None
        col_fit_deriv = False

        _submodels = None

        def __getattr__(self, attr):
            value = getattr(self.__class__, attr)
            if hasattr(value, '__get__'):
                # Object is a descriptor, so we should really return the result of
                # its __get__
                value = value.__get__(self, self.__class__)
            return value

        def __getitem__(self, index):
            index = self.__class__._normalize_index(index)
            model = self.__class__[index]

            if isinstance(index, slice):
                param_names = model.param_names
            else:
                param_map = self.__class__._param_map_inverse
                param_names = tuple(param_map[index, name]
                                    for name in model.param_names)

            return model._from_existing(self, param_names)

        if sys.version_info[:3] < (2, 7, 3):
            def __reduce__(self):
                # _CompoundModel classes have a generated evaluate() that is cached
                # off in the _evaluate attribute.  This can't be pickled, and so
                # should be regenerated after unpickling (alas)
                if find_current_module(2) is not copy:
                    # The copy module also uses __reduce__, but there's no problem
                    # there.
                    raise RuntimeError(
                        "Pickling of compound models is not possible using Python "
                        "versions less than 2.7.3 due to a bug in Python.  See "
                        "http://docs.astropy.org/en/v1.0.4/known_issues.html#"
                        "pickling-error-on-compound-models for more information ("
                        "tried to pickle {0!r}).".format(self))
                else:
                    return super(_CompoundModel, self).__reduce__()

        @property
        def submodel_names(self):
            return self.__class__.submodel_names

        @property
        def param_names(self):
            return self.__class__.param_names

        @property
        def fittable(self):
            return self.__class__.fittable

        @sharedmethod
        def evaluate(self, *args):
            return self.__class__.evaluate(*args)

        # TODO: The way this works is highly inefficient--the inverse is created by
        # making a new model for each operator in the compound model, which could
        # potentially mean creating a large number of temporary throwaway model
        # classes.  This can definitely be optimized in the future by implementing
        # a way to construct a single model class from an existing tree
        @property
        def inverse(self):
            def _not_implemented(oper):
                def _raise(x, y):
                    raise NotImplementedError(
                        "The inverse is not currently defined for compound "
                        "models created using the {0} operator.".format(oper))
                return _raise

            operators = dict((oper, _not_implemented(oper))
                             for oper in ('+', '-', '*', '/', '**'))
            operators['&'] = operator.and_
            # Reverse the order of compositions
            operators['|'] = lambda x, y: operator.or_(y, x)

            leaf_idx = -1

            def getter(idx, model):
                try:
                    # By indexing on self[] this will return an instance of the
                    # model, with all the appropriate parameters set, which is
                    # currently required to return an inverse
                    return self[idx].inverse
                except NotImplementedError:
                    raise NotImplementedError(
                        "All models in a composite model must have an inverse "
                        "defined in order for the composite model to have an "
                        "inverse.  {0!r} does not have an inverse.".format(model))

            return self._tree.evaluate(operators, getter=getter)

        @sharedmethod
        def _get_submodels(self):
            return self.__class__._get_submodels()

    def custom_model(*args, **kwargs):
        """
        Create a model from a user defined function. The inputs and parameters of
        the model will be inferred from the arguments of the function.

        This can be used either as a function or as a decorator.  See below for
        examples of both usages.

        .. note::

            All model parameters have to be defined as keyword arguments with
            default values in the model function.  Use `None` as a default argument
            value if you do not want to have a default value for that parameter.

        Parameters
        ----------
        func : function
            Function which defines the model.  It should take N positional
            arguments where ``N`` is dimensions of the model (the number of
            independent variable in the model), and any number of keyword arguments
            (the parameters).  It must return the value of the model (typically as
            an array, but can also be a scalar for scalar inputs).  This
            corresponds to the `~astropy.modeling.Model.evaluate` method.
        fit_deriv : function, optional
            Function which defines the Jacobian derivative of the model. I.e., the
            derivative with respect to the *parameters* of the model.  It should
            have the same argument signature as ``func``, but should return a
            sequence where each element of the sequence is the derivative
            with respect to the corresponding argument. This corresponds to the
            :meth:`~astropy.modeling.FittableModel.fit_deriv` method.

        Examples
        --------
        Define a sinusoidal model function as a custom 1D model::

            >>> from astropy.modeling.models import custom_model
            >>> import numpy as np
            >>> def sine_model(x, amplitude=1., frequency=1.):
            ...     return amplitude * np.sin(2 * np.pi * frequency * x)
            >>> def sine_deriv(x, amplitude=1., frequency=1.):
            ...     return 2 * np.pi * amplitude * np.cos(2 * np.pi * frequency * x)
            >>> SineModel = custom_model(sine_model, fit_deriv=sine_deriv)

        Create an instance of the custom model and evaluate it::

            >>> model = SineModel()
            >>> model(0.25)
            1.0

        This model instance can now be used like a usual astropy model.

        The next example demonstrates a 2D Moffat function model, and also
        demonstrates the support for docstrings (this example could also include
        a derivative, but it has been omitted for simplicity)::

            >>> @custom_model
            ... def Moffat2D(x, y, amplitude=1.0, x_0=0.0, y_0=0.0, gamma=1.0,
            ...            alpha=1.0):
            ...     \"\"\"Two dimensional Moffat function.\"\"\"
            ...     rr_gg = ((x - x_0) ** 2 + (y - y_0) ** 2) / gamma ** 2
            ...     return amplitude * (1 + rr_gg) ** (-alpha)
            ...
            >>> print(Moffat2D.__doc__)
            Two dimensional Moffat function.
            >>> model = Moffat2D()
            >>> model(1, 1)  # doctest: +FLOAT_CMP
            0.3333333333333333
        """

        fit_deriv = kwargs.get('fit_deriv', None)

        if len(args) == 1 and six.callable(args[0]):
            return _custom_model_wrapper(args[0], fit_deriv=fit_deriv)
        elif not args:
            return functools.partial(_custom_model_wrapper, fit_deriv=fit_deriv)
        else:
            raise TypeError(
                "{0} takes at most one positional argument (the callable/"
                "function to be turned into a model.  When used as a decorator "
                "it should be passed keyword arguments only (if "
                "any).".format(__name__))

    def _custom_model_wrapper(func, fit_deriv=None):
        """
        Internal implementation `custom_model`.

        When `custom_model` is called as a function its arguments are passed to
        this function, and the result of this function is returned.

        When `custom_model` is used as a decorator a partial evaluation of this
        function is returned by `custom_model`.
        """

        if not six.callable(func):
            raise ModelDefinitionError(
                "func is not callable; it must be a function or other callable "
                "object")

        if fit_deriv is not None and not six.callable(fit_deriv):
            raise ModelDefinitionError(
                "fit_deriv not callable; it must be a function or other "
                "callable object")

        model_name = func.__name__

        inputs, params = get_inputs_and_params(func)

        if (fit_deriv is not None and
                len(six.get_function_defaults(fit_deriv)) != len(params)):
            raise ModelDefinitionError("derivative function should accept "
                                       "same number of parameters as func.")

        # TODO: Maybe have a clever scheme for default output name?
        if inputs:
            output_names = (inputs[0].name,)
        else:
            output_names = ('x',)

        params = dict((param.name, Parameter(param.name, default=param.default))
                      for param in params)

        mod = find_current_module(2)
        if mod:
            modname = mod.__name__
        else:
            modname = '__main__'

        members = {
            '__module__': str(modname),
            '__doc__': func.__doc__,
            'inputs': tuple(x.name for x in inputs),
            'outputs': output_names,
            'evaluate': staticmethod(func),
        }

        if fit_deriv is not None:
            members['fit_deriv'] = staticmethod(fit_deriv)

        members.update(params)

        return type(model_name, (FittableModel,), members)

    def render_model(model, arr=None, coords=None):
        """
        Evaluates a model on an input array. Evaluation is limited to
        a bounding box if the `Model.bounding_box` attribute is set.

        Parameters
        ----------
        model : `Model`
            Model to be evaluated.
        arr : `numpy.ndarray`, optional
            Array on which the model is evaluated.
        coords : array-like, optional
            Coordinate arrays mapping to ``arr``, such that
            ``arr[coords] == arr``.

        Returns
        -------
        array : `numpy.ndarray`
            The model evaluated on the input ``arr`` or a new array from ``coords``.
            If ``arr`` and ``coords`` are both `None`, the returned array is
            limited to the `Model.bounding_box` limits. If
            `Model.bounding_box` is `None`, ``arr`` or ``coords`` must be passed.

        Examples
        --------
        :ref:`bounding-boxes`
        """

        bbox = model.bounding_box

        if (coords is None) & (arr is None) & (bbox is None):
            raise AssertionError('If no bounding_box is set, coords or arr must be input.')

        # for consistent indexing
        if model.n_inputs == 1:
            if coords is not None:
                coords = [coords]
            if bbox is not None:
                bbox = [bbox]

        if arr is not None:
            arr = arr.copy()
            # Check dimensions match model
            assert arr.ndim == model.n_inputs

        if coords is not None:
            # Check dimensions match arr and model
            coords = np.array(coords)
            assert len(coords) == model.n_inputs
            if arr is not None:
                assert coords[0].shape == arr.shape
            else:
                arr = np.zeros(coords[0].shape)

        if bbox is not None:
            # assures position is at center pixel, important when using add_array
            pd = pos, delta = np.array([(np.mean(bb), np.ceil((bb[1] - bb[0]) / 2))
                                        for bb in bbox]).astype(int).T

            if coords is not None:
                sub_shape = tuple(delta * 2 + 1)
                sub_coords = np.array([extract_array(c, sub_shape, pos) for c in coords])
            else:
                limits = [slice(p - d, p + d + 1, 1) for p, d in pd.T]
                sub_coords = np.mgrid[limits]

            sub_coords = sub_coords[::-1]

            if arr is None:
                arr = model(*sub_coords)
            else:
                try:
                    arr = add_array(arr, model(*sub_coords), pos)
                except ValueError:
                    raise ValueError('The `bounding_box` is larger than the input'
                                    ' arr in one or more dimensions. Set '
                                    '`model.bounding_box = None`.')
        else:

            if coords is None:
                im_shape = arr.shape
                limits = [slice(i) for i in im_shape]
                coords = np.mgrid[limits]

            arr += model(*coords[::-1])

        return arr

    def _prepare_inputs_single_model(model, params, inputs, **kwargs):
        broadcasts = []

        for idx, _input in enumerate(inputs):
            input_shape = _input.shape

            # Ensure that array scalars are always upgrade to 1-D arrays for the
            # sake of consistency with how parameters work.  They will be cast back
            # to scalars at the end
            if not input_shape:
                inputs[idx] = _input.reshape((1,))

            if not params:
                max_broadcast = input_shape
            else:
                max_broadcast = ()

            for param in params:
                try:
                    if model.standard_broadcasting:
                        broadcast = check_broadcast(input_shape, param.shape)
                    else:
                        broadcast = input_shape
                except IncompatibleShapeError:
                    raise ValueError(
                        "Model input argument {0!r} of shape {1!r} cannot be "
                        "broadcast with parameter {2!r} of shape "
                        "{3!r}.".format(model.inputs[idx], input_shape,
                                        param.name, param.shape))

                if len(broadcast) > len(max_broadcast):
                    max_broadcast = broadcast
                elif len(broadcast) == len(max_broadcast):
                    max_broadcast = max(max_broadcast, broadcast)

            broadcasts.append(max_broadcast)

        if model.n_outputs > model.n_inputs:
            if len(set(broadcasts)) > 1:
                raise ValueError(
                    "For models with n_outputs > n_inputs, the combination of "
                    "all inputs and parameters must broadcast to the same shape, "
                    "which will be used as the shape of all outputs.  In this "
                    "case some of the inputs had different shapes, so it is "
                    "ambiguous how to format outputs for this model.  Try using "
                    "inputs that are all the same size and shape.")
            else:
                # Extend the broadcasts list to include shapes for all outputs
                extra_outputs = model.n_outputs - model.n_inputs
                if not broadcasts:
                    # If there were no inputs then the broadcasts list is empty
                    # just add a None since there is no broadcasting of outputs and
                    # inputs necessary (see _prepare_outputs_single_model)
                    broadcasts.append(None)
                broadcasts.extend([broadcasts[0]] * extra_outputs)

        return inputs, (broadcasts,)

    def _prepare_outputs_single_model(model, outputs, format_info):
        broadcasts = format_info[0]

        outputs = list(outputs)

        for idx, output in enumerate(outputs):
            broadcast_shape = broadcasts[idx]
            if broadcast_shape is not None:
                if not broadcast_shape:
                    # Shape is (), i.e. a scalar should be returned
                    outputs[idx] = np.asscalar(output)
                else:
                    outputs[idx] = output.reshape(broadcast_shape)

        if model.n_outputs == 1:
            return outputs[0]
        else:
            return tuple(outputs)

    def _prepare_inputs_model_set(model, params, inputs, n_models, model_set_axis,
                                  **kwargs):
        reshaped = []
        pivots = []

        for idx, _input in enumerate(inputs):
            max_param_shape = ()

            if n_models > 1 and model_set_axis is not False:
                # Use the shape of the input *excluding* the model axis
                input_shape = (_input.shape[:model_set_axis] +
                               _input.shape[model_set_axis + 1:])
            else:
                input_shape = _input.shape

            for param in params:
                try:
                    check_broadcast(input_shape, param.shape)
                except IncompatibleShapeError:
                    raise ValueError(
                        "Model input argument {0!r} of shape {1!r} cannot be "
                        "broadcast with parameter {2!r} of shape "
                        "{3!r}.".format(model.inputs[idx], input_shape,
                                        param.name, param.shape))

                if len(param.shape) > len(max_param_shape):
                    max_param_shape = param.shape

            # We've now determined that, excluding the model_set_axis, the
            # input can broadcast with all the parameters
            input_ndim = len(input_shape)
            if model_set_axis is False:
                if len(max_param_shape) > input_ndim:
                    # Just needs to prepend new axes to the input
                    n_new_axes = 1 + len(max_param_shape) - input_ndim
                    new_axes = (1,) * n_new_axes
                    new_shape = new_axes + _input.shape
                    pivot = model.model_set_axis
                else:
                    pivot = input_ndim - len(max_param_shape)
                    new_shape = (_input.shape[:pivot] + (1,) +
                                 _input.shape[pivot:])
                new_input = _input.reshape(new_shape)
            else:
                if len(max_param_shape) >= input_ndim:
                    n_new_axes = len(max_param_shape) - input_ndim
                    pivot = model.model_set_axis
                    new_axes = (1,) * n_new_axes
                    new_shape = (_input.shape[:pivot + 1] + new_axes +
                                 _input.shape[pivot + 1:])
                    new_input = _input.reshape(new_shape)
                else:
                    pivot = _input.ndim - len(max_param_shape) - 1
                    new_input = np.rollaxis(_input, model_set_axis,
                                            pivot + 1)

            pivots.append(pivot)
            reshaped.append(new_input)

        if model.n_inputs < model.n_outputs:
            pivots.extend([model_set_axis] * (model.n_outputs - model.n_inputs))

        return reshaped, (pivots,)

    def _prepare_outputs_model_set(model, outputs, format_info):
        pivots = format_info[0]

        outputs = list(outputs)

        for idx, output in enumerate(outputs):
            pivot = pivots[idx]
            if pivot < output.ndim and pivot != model.model_set_axis:
                outputs[idx] = np.rollaxis(output, pivot,
                                           model.model_set_axis)

        if model.n_outputs == 1:
            return outputs[0]
        else:
            return tuple(outputs)

    def _validate_input_shapes(inputs, argnames, n_models, model_set_axis,
                               validate_broadcasting):
        """
        Perform basic validation of model inputs--that they are mutually
        broadcastable and that they have the minimum dimensions for the given
        model_set_axis.

        If validation succeeds, returns the total shape that will result from
        broadcasting the input arrays with each other.
        """

        check_model_set_axis = n_models > 1 and model_set_axis is not False

        if not (validate_broadcasting or check_model_set_axis):
            # Nothing else needed here
            return

        all_shapes = []

        for idx, _input in enumerate(inputs):
            input_shape = np.shape(_input)
            # Ensure that the input's model_set_axis matches the model's
            # n_models
            if input_shape and check_model_set_axis:
                # Note: Scalar inputs *only* get a pass on this
                if len(input_shape) < model_set_axis + 1:
                    raise ValueError(
                        "For model_set_axis={0}, all inputs must be at "
                        "least {1}-dimensional.".format(
                            model_set_axis, model_set_axis + 1))
                elif input_shape[model_set_axis] != n_models:
                    raise ValueError(
                        "Input argument {0!r} does not have the correct "
                        "dimensions in model_set_axis={1} for a model set with "
                        "n_models={2}.".format(argnames[idx], model_set_axis,
                                               n_models))
            all_shapes.append(input_shape)

        if not validate_broadcasting:
            return

        try:
            input_broadcast = check_broadcast(*all_shapes)
        except IncompatibleShapeError as exc:
            shape_a, shape_a_idx, shape_b, shape_b_idx = exc.args
            arg_a = argnames[shape_a_idx]
            arg_b = argnames[shape_b_idx]

            raise ValueError(
                "Model input argument {0!r} of shape {1!r} cannot "
                "be broadcast with input {2!r} of shape {3!r}".format(
                    arg_a, shape_a, arg_b, shape_b))

        return input_broadcast

    copyreg.pickle(_ModelMeta, _ModelMeta.__reduce__)
    copyreg.pickle(_CompoundModelMeta, _CompoundModelMeta.__reduce__)