field.py

# NIFTy
# Copyright (C) 2017  Theo Steininger
#
# Author: Theo Steininger
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

from __future__ import division
import numpy as np

from keepers import Versionable,\
                    Loggable

from d2o import distributed_data_object,\
    STRATEGIES as DISTRIBUTION_STRATEGIES

from nifty.config import nifty_configuration as gc

from nifty.domain_object import DomainObject

from nifty.spaces.power_space import PowerSpace

import nifty.nifty_utilities as utilities
from nifty.random import Random


class Field(Loggable, Versionable, object):
    # ---Initialization methods---

    def __init__(self, domain=None, val=None, dtype=None,
                 distribution_strategy=None, copy=False):

        self.domain = self._parse_domain(domain=domain, val=val)
        self.domain_axes = self._get_axes_tuple(self.domain)

        self.dtype = self._infer_dtype(dtype=dtype,
                                       val=val)

        self.distribution_strategy = self._parse_distribution_strategy(
                                distribution_strategy=distribution_strategy,
                                val=val)

        if val is None:
            self._val = None
        else:
            self.set_val(new_val=val, copy=copy)

    def _parse_domain(self, domain, val=None):
        if domain is None:
            if isinstance(val, Field):
                domain = val.domain
            else:
                domain = ()
        elif isinstance(domain, DomainObject):
            domain = (domain,)
        elif not isinstance(domain, tuple):
            domain = tuple(domain)

        for d in domain:
            if not isinstance(d, DomainObject):
                raise TypeError(
                    "Given domain contains something that is not a "
                    "DomainObject instance.")
        return domain

    def _get_axes_tuple(self, things_with_shape, start=0):
        i = start
        axes_list = []
        for thing in things_with_shape:
            l = []
            for j in range(len(thing.shape)):
                l += [i]
                i += 1
            axes_list += [tuple(l)]
        return tuple(axes_list)

    def _infer_dtype(self, dtype, val):
        if dtype is None:
            try:
                dtype = val.dtype
            except AttributeError:
                if val is not None:
                    dtype = np.result_type(val)
        else:
                    dtype = np.dtype(gc['default_field_dtype'])
        else:
            dtype = np.dtype(dtype)

        return dtype

    def _parse_distribution_strategy(self, distribution_strategy, val):
        if distribution_strategy is None:
            if isinstance(val, distributed_data_object):
                distribution_strategy = val.distribution_strategy
            elif isinstance(val, Field):
                distribution_strategy = val.distribution_strategy
            else:
                self.logger.debug("distribution_strategy set to default!")
                distribution_strategy = gc['default_distribution_strategy']
        elif distribution_strategy not in DISTRIBUTION_STRATEGIES['global']:
            raise ValueError(
                    "distribution_strategy must be a global-type "
                    "strategy.")
        return distribution_strategy

    # ---Factory methods---

    @classmethod
    def from_random(cls, random_type, domain=None, dtype=None,
                    distribution_strategy=None, **kwargs):
        # create a initially empty field
        f = cls(domain=domain, dtype=dtype,
                distribution_strategy=distribution_strategy)

        # now use the processed input in terms of f in order to parse the
        # random arguments
        random_arguments = cls._parse_random_arguments(random_type=random_type,
                                                       f=f,
                                                       **kwargs)

        # extract the distributed_data_object from f and apply the appropriate
        # random number generator to it
        sample = f.get_val(copy=False)
        generator_function = getattr(Random, random_type)
        sample.apply_generator(
            lambda shape: generator_function(dtype=f.dtype,
                                             shape=shape,
                                             **random_arguments))
        return f

    @staticmethod
    def _parse_random_arguments(random_type, f, **kwargs):

        if random_type == "pm1":
            random_arguments = {}

        elif random_type == "normal":
            mean = kwargs.get('mean', 0)
            std = kwargs.get('std', 1)
            random_arguments = {'mean': mean,
                                'std': std}

        elif random_type == "uniform":
            low = kwargs.get('low', 0)
            high = kwargs.get('high', 1)
            random_arguments = {'low': low,
                                'high': high}

        else:
            raise KeyError(
                "unsupported random key '" + str(random_type) + "'.")

        return random_arguments

    # ---Powerspectral methods---

    def power_analyze(self, spaces=None, log=False, nbin=None, binbounds=None,
                      real_signal=True):
        # check if all spaces in `self.domain` are either harmonic or
        # power_space instances
        for sp in self.domain:
            if not sp.harmonic and not isinstance(sp, PowerSpace):
                self.logger.info(
                    "Field has a space in `domain` which is neither "
                    "harmonic nor a PowerSpace.")

        # check if the `spaces` input is valid
        spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))
        if spaces is None:
            if len(self.domain) == 1:
                spaces = (0,)
            else:
                raise ValueError(
                    "Field has multiple spaces as domain "
                    "but `spaces` is None.")

        if len(spaces) == 0:
            raise ValueError(
                "No space for analysis specified.")
        elif len(spaces) > 1:
            raise ValueError(
                "Conversion of only one space at a time is allowed.")

        space_index = spaces[0]

        if not self.domain[space_index].harmonic:
            raise ValueError(
                "The analyzed space must be harmonic.")

        # Create the target PowerSpace instance:
        # If the associated signal-space field was real, we extract the
        # hermitian and anti-hermitian parts of `self` and put them
        # into the real and imaginary parts of the power spectrum.
        # If it was complex, all the power is put into a real power spectrum.

        distribution_strategy = \
            self.val.get_axes_local_distribution_strategy(
                self.domain_axes[space_index])

        if real_signal:
            power_dtype = np.dtype('complex')
        else:
            power_dtype = np.dtype('float')

        harmonic_domain = self.domain[space_index]
        power_domain = PowerSpace(harmonic_domain=harmonic_domain,
                                  distribution_strategy=distribution_strategy,
                                  log=log, nbin=nbin, binbounds=binbounds,
                                  dtype=power_dtype)

        # extract pindex and rho from power_domain
        pindex = power_domain.pindex
        rho = power_domain.rho

        if real_signal:
            hermitian_part, anti_hermitian_part = \
                harmonic_domain.hermitian_decomposition(
                                            self.val,
                                            axes=self.domain_axes[space_index])

            [hermitian_power, anti_hermitian_power] = \
                [self._calculate_power_spectrum(
                                            x=part,
                                            pindex=pindex,
                                            rho=rho,
                                            axes=self.domain_axes[space_index])
                 for part in [hermitian_part, anti_hermitian_part]]

            power_spectrum = hermitian_power + 1j * anti_hermitian_power
        else:
            power_spectrum = self._calculate_power_spectrum(
                                            x=self.val,
                                            pindex=pindex,
                                            rho=rho,
                                            axes=self.domain_axes[space_index])

        # create the result field and put power_spectrum into it
        result_domain = list(self.domain)
        result_domain[space_index] = power_domain

        result_field = self.copy_empty(
                   domain=result_domain,
                   distribution_strategy=power_spectrum.distribution_strategy)
        result_field.set_val(new_val=power_spectrum, copy=False)

        return result_field

    def _calculate_power_spectrum(self, x, pindex, rho, axes=None):
        fieldabs = abs(x)
        fieldabs **= 2

        if axes is not None:
            pindex = self._shape_up_pindex(
                                    pindex=pindex,
                                    target_shape=x.shape,
                                    target_strategy=x.distribution_strategy,
                                    axes=axes)
        power_spectrum = pindex.bincount(weights=fieldabs,
                                         axis=axes)
        if axes is not None:
            new_rho_shape = [1, ] * len(power_spectrum.shape)
            new_rho_shape[axes[0]] = len(rho)
            rho = rho.reshape(new_rho_shape)
        power_spectrum /= rho

        power_spectrum **= 0.5
        return power_spectrum

    def _shape_up_pindex(self, pindex, target_shape, target_strategy, axes):
        if pindex.distribution_strategy not in \
                DISTRIBUTION_STRATEGIES['global']:
            raise ValueError("pindex's distribution strategy must be "
                             "global-type")

        if pindex.distribution_strategy in DISTRIBUTION_STRATEGIES['slicing']:
            if ((0 not in axes) or
                    (target_strategy is not pindex.distribution_strategy)):
                raise ValueError(
                    "A slicing distributor shall not be reshaped to "
                    "something non-sliced.")

        semiscaled_shape = [1, ] * len(target_shape)
        for i in axes:
            semiscaled_shape[i] = target_shape[i]
        local_data = pindex.get_local_data(copy=False)
        semiscaled_local_data = local_data.reshape(semiscaled_shape)
        result_obj = pindex.copy_empty(global_shape=target_shape,
                                       distribution_strategy=target_strategy)
        result_obj.set_full_data(semiscaled_local_data, copy=False)

        return result_obj

    def power_synthesize(self, spaces=None, real_power=True, real_signal=True,
                         mean=None, std=None):

        # check if the `spaces` input is valid
        spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))

        for power_space_index in spaces:
            power_space = self.domain[power_space_index]
            if not isinstance(power_space, PowerSpace):
                raise ValueError("A PowerSpace is needed for field "
                                 "synthetization.")

        # create the result domain
        result_domain = list(self.domain)
        for power_space_index in spaces:
            power_space = self.domain[power_space_index]
            harmonic_domain = power_space.harmonic_domain
            result_domain[power_space_index] = harmonic_domain

        # create random samples: one or two, depending on whether the
        # power spectrum is real or complex
        if real_power:
            result_list = [None]
        else:
            result_list = [None, None]

        result_list = [self.__class__.from_random(
                             'normal',
                             mean=mean,
                             std=std,
                             domain=result_domain,
                             dtype=np.complex,
                             distribution_strategy=self.distribution_strategy)
                       for x in result_list]

        # from now on extract the values from the random fields for further
        # processing without killing the fields.
        # if the signal-space field should be real, hermitianize the field
        # components

        spec = self.val.get_full_data()
        for power_space_index in spaces:
            spec = self._spec_to_rescaler(spec, result_list, power_space_index)
        local_rescaler = spec

        result_val_list = [x.val for x in result_list]

        # apply the rescaler to the random fields
        result_val_list[0].apply_scalar_function(
                                            lambda x: x * local_rescaler.real,
                                            inplace=True)

        if not real_power:
            result_val_list[1].apply_scalar_function(
                                            lambda x: x * local_rescaler.imag,
                                            inplace=True)

        if real_signal:
            for power_space_index in spaces:
                harmonic_domain = result_domain[power_space_index]
                result_val_list = [harmonic_domain.hermitian_decomposition(
                                    result_val,
                                    axes=result.domain_axes[power_space_index],
                                    preserve_gaussian_variance=True)[0]
                                   for (result, result_val)
                                   in zip(result_list, result_val_list)]

        # store the result into the fields
        [x.set_val(new_val=y, copy=False) for x, y in
            zip(result_list, result_val_list)]

        if real_power:
            result = result_list[0]
        else:
            result = result_list[0] + 1j*result_list[1]

        return result

    def _spec_to_rescaler(self, spec, result_list, power_space_index):
        power_space = self.domain[power_space_index]

        # weight the random fields with the power spectrum
        # therefore get the pindex from the power space
        pindex = power_space.pindex
        # take the local data from pindex. This data must be compatible to the
        # local data of the field given the slice of the PowerSpace
        local_distribution_strategy = \
            result_list[0].val.get_axes_local_distribution_strategy(
                result_list[0].domain_axes[power_space_index])

        if pindex.distribution_strategy is not local_distribution_strategy:
            self.logger.warn(
                "The distribution_stragey of pindex does not fit the "
                "slice_local distribution strategy of the synthesized field.")

        # Now use numpy advanced indexing in order to put the entries of the
        # power spectrum into the appropriate places of the pindex array.
        # Do this for every 'pindex-slice' in parallel using the 'slice(None)'s
        local_pindex = pindex.get_local_data(copy=False)

        local_blow_up = [slice(None)]*len(self.shape)
        local_blow_up[self.domain_axes[power_space_index][0]] = local_pindex
        # here, the power_spectrum is distributed into the new shape
        local_rescaler = spec[local_blow_up]
        return local_rescaler

    # ---Properties---

    def set_val(self, new_val=None, copy=False):
        new_val = self.cast(new_val)
        if copy:
            new_val = new_val.copy()
        self._val = new_val
        return self

    def get_val(self, copy=False):
        if self._val is None:
            self.set_val(None)

        if copy:
            return self._val.copy()
        else:
            return self._val

    @property
    def val(self):
        return self.get_val(copy=False)

    @val.setter
    def val(self, new_val):
        self.set_val(new_val=new_val, copy=False)

    @property
    def shape(self):
        shape_tuple = tuple(sp.shape for sp in self.domain)
        try:
            global_shape = reduce(lambda x, y: x + y, shape_tuple)
        except TypeError:
            global_shape = ()

        return global_shape

    @property
    def dim(self):
        dim_tuple = tuple(sp.dim for sp in self.domain)
        try:
            return reduce(lambda x, y: x * y, dim_tuple)
        except TypeError:
            return 0

    @property
    def dof(self):
        dof = self.dim
        if issubclass(self.dtype.type, np.complexfloating):
            dof *= 2
        return dof

    @property
    def total_volume(self):
        volume_tuple = tuple(sp.total_volume for sp in self.domain)
        try:
            return reduce(lambda x, y: x * y, volume_tuple)
        except TypeError:
            return 0.

    # ---Special unary/binary operations---

    def cast(self, x=None, dtype=None):
        if dtype is None:
            dtype = self.dtype
        else:
            dtype = np.dtype(dtype)

        casted_x = x

        for ind, sp in enumerate(self.domain):
            casted_x = sp.pre_cast(casted_x,
                                   axes=self.domain_axes[ind])

        casted_x = self._actual_cast(casted_x, dtype=dtype)

        for ind, sp in enumerate(self.domain):
            casted_x = sp.post_cast(casted_x,
                                    axes=self.domain_axes[ind])

        return casted_x

    def _actual_cast(self, x, dtype=None):
        if isinstance(x, Field):
            x = x.get_val()

        if dtype is None:
            dtype = self.dtype

        return_x = distributed_data_object(
                            global_shape=self.shape,
                            dtype=dtype,
                            distribution_strategy=self.distribution_strategy)
        return_x.set_full_data(x, copy=False)
        return return_x

    def copy(self, domain=None, dtype=None, distribution_strategy=None):
        copied_val = self.get_val(copy=True)
        new_field = self.copy_empty(
                                domain=domain,
                                dtype=dtype,
                                distribution_strategy=distribution_strategy)
        new_field.set_val(new_val=copied_val, copy=False)
        return new_field

    def copy_empty(self, domain=None, dtype=None, distribution_strategy=None):
        if domain is None:
            domain = self.domain
        else:
            domain = self._parse_domain(domain)

        if dtype is None:
            dtype = self.dtype
        else:
            dtype = np.dtype(dtype)

        if distribution_strategy is None:
            distribution_strategy = self.distribution_strategy

        fast_copyable = True
        try:
            for i in xrange(len(self.domain)):
                if self.domain[i] is not domain[i]:
                    fast_copyable = False
                    break
        except IndexError:
            fast_copyable = False

        if (fast_copyable and dtype == self.dtype and
                distribution_strategy == self.distribution_strategy):
            new_field = self._fast_copy_empty()
        else:
            new_field = Field(domain=domain,
                              dtype=dtype,
                              distribution_strategy=distribution_strategy)
        return new_field

    def _fast_copy_empty(self):
        # make an empty field
        new_field = EmptyField()
        # repair its class
        new_field.__class__ = self.__class__
        # copy domain, codomain and val
        for key, value in self.__dict__.items():
            if key != '_val':
                new_field.__dict__[key] = value
            else:
                new_field.__dict__[key] = self.val.copy_empty()
        return new_field

    def weight(self, power=1, inplace=False, spaces=None):
        if inplace:
            new_field = self
        else:
            new_field = self.copy_empty()

        new_val = self.get_val(copy=False)

        spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))
        if spaces is None:
            spaces = range(len(self.domain))

        for ind, sp in enumerate(self.domain):
            if ind in spaces:
                new_val = sp.weight(new_val,
                                    power=power,
                                    axes=self.domain_axes[ind],
                                    inplace=inplace)

        new_field.set_val(new_val=new_val, copy=False)
        return new_field

    def dot(self, x=None, spaces=None, bare=False):

        if not isinstance(x, Field):
            raise ValueError("The dot-partner must be an instance of " +
                             "the NIFTy field class")

        # Compute the dot respecting the fact of discrete/continuous spaces
        if bare:
            y = self
        else:
            y = self.weight(power=1)

        if spaces is None:
            x_val = x.get_val(copy=False)
            y_val = y.get_val(copy=False)
            result = (x_val.conjugate() * y_val).sum()
            return result
        else:
            # create a diagonal operator which is capable of taking care of the
            # axes-matching
            from nifty.operators.diagonal_operator import DiagonalOperator
            diagonal = y.val.conjugate()
            diagonalOperator = DiagonalOperator(domain=y.domain,
                                                diagonal=diagonal,
                                                copy=False)
            dotted = diagonalOperator(x, spaces=spaces)
            return dotted.sum(spaces=spaces)

    def norm(self, q=2):
        """
            Computes the Lq-norm of the field values.

            Parameters
            ----------
            q : scalar
                Parameter q of the Lq-norm (default: 2).

            Returns
            -------
            norm : scalar
                The Lq-norm of the field values.

        """
        if q == 2:
            return (self.dot(x=self)) ** (1 / 2)
        else:
            return self.dot(x=self ** (q - 1)) ** (1 / q)

    def conjugate(self, inplace=False):
        """
            Computes the complex conjugate of the field.

            Returns
            -------
            cc : field
                The complex conjugated field.

        """
        if inplace:
            work_field = self
        else:
            work_field = self.copy_empty()

        new_val = self.get_val(copy=False)
        new_val = new_val.conjugate()
        work_field.set_val(new_val=new_val, copy=False)

        return work_field

    # ---General unary/contraction methods---

    def __pos__(self):
        return self.copy()

    def __neg__(self):
        return_field = self.copy_empty()
        new_val = -self.get_val(copy=False)
        return_field.set_val(new_val, copy=False)
        return return_field

    def __abs__(self):
        return_field = self.copy_empty()
        new_val = abs(self.get_val(copy=False))
        return_field.set_val(new_val, copy=False)
        return return_field

    def _contraction_helper(self, op, spaces):
        # build a list of all axes
        if spaces is None:
            spaces = xrange(len(self.domain))
        else:
            spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))

        axes_list = tuple(self.domain_axes[sp_index] for sp_index in spaces)

        try:
            axes_list = reduce(lambda x, y: x+y, axes_list)
        except TypeError:
            axes_list = ()

        # perform the contraction on the d2o
        data = self.get_val(copy=False)
        data = getattr(data, op)(axis=axes_list)

        # check if the result is scalar or if a result_field must be constr.
        if np.isscalar(data):
            return data
        else:
            return_domain = tuple(self.domain[i]
                                  for i in xrange(len(self.domain))
                                  if i not in spaces)

            return_field = Field(domain=return_domain,
                                 val=data,
                                 copy=False)
            return return_field

    def sum(self, spaces=None):
        return self._contraction_helper('sum', spaces)

    def prod(self, spaces=None):
        return self._contraction_helper('prod', spaces)

    def all(self, spaces=None):
        return self._contraction_helper('all', spaces)

    def any(self, spaces=None):
        return self._contraction_helper('any', spaces)

    def min(self, spaces=None):
        return self._contraction_helper('min', spaces)

    def nanmin(self, spaces=None):
        return self._contraction_helper('nanmin', spaces)

    def max(self, spaces=None):
        return self._contraction_helper('max', spaces)

    def nanmax(self, spaces=None):
        return self._contraction_helper('nanmax', spaces)

    def mean(self, spaces=None):
        return self._contraction_helper('mean', spaces)

    def var(self, spaces=None):
        return self._contraction_helper('var', spaces)

    def std(self, spaces=None):
        return self._contraction_helper('std', spaces)

    # ---General binary methods---

    def _binary_helper(self, other, op, inplace=False):
        # if other is a field, make sure that the domains match
        if isinstance(other, Field):
            try:
                assert len(other.domain) == len(self.domain)
                for index in xrange(len(self.domain)):
                    assert other.domain[index] == self.domain[index]
            except AssertionError:
                raise ValueError(
                    "domains are incompatible.")
            other = other.get_val(copy=False)

        self_val = self.get_val(copy=False)
        return_val = getattr(self_val, op)(other)

        if inplace:
            working_field = self
        else:
            working_field = self.copy_empty(dtype=return_val.dtype)

        working_field.set_val(return_val, copy=False)
        return working_field

    def __add__(self, other):
        return self._binary_helper(other, op='__add__')

    def __radd__(self, other):
        return self._binary_helper(other, op='__radd__')

    def __iadd__(self, other):
        return self._binary_helper(other, op='__iadd__', inplace=True)

    def __sub__(self, other):
        return self._binary_helper(other, op='__sub__')

    def __rsub__(self, other):
        return self._binary_helper(other, op='__rsub__')

    def __isub__(self, other):
        return self._binary_helper(other, op='__isub__', inplace=True)

    def __mul__(self, other):
        return self._binary_helper(other, op='__mul__')

    def __rmul__(self, other):
        return self._binary_helper(other, op='__rmul__')

    def __imul__(self, other):
        return self._binary_helper(other, op='__imul__', inplace=True)

    def __div__(self, other):
        return self._binary_helper(other, op='__div__')

    def __rdiv__(self, other):
        return self._binary_helper(other, op='__rdiv__')

    def __idiv__(self, other):
        return self._binary_helper(other, op='__idiv__', inplace=True)

    def __pow__(self, other):
        return self._binary_helper(other, op='__pow__')

    def __rpow__(self, other):
        return self._binary_helper(other, op='__rpow__')

    def __ipow__(self, other):
        return self._binary_helper(other, op='__ipow__', inplace=True)

    def __lt__(self, other):
        return self._binary_helper(other, op='__lt__')

    def __le__(self, other):
        return self._binary_helper(other, op='__le__')

    def __ne__(self, other):
        if other is None:
            return True
        else:
            return self._binary_helper(other, op='__ne__')

    def __eq__(self, other):
        if other is None:
            return False
        else:
            return self._binary_helper(other, op='__eq__')

    def __ge__(self, other):
        return self._binary_helper(other, op='__ge__')

    def __gt__(self, other):
        return self._binary_helper(other, op='__gt__')

    def __repr__(self):
        return "<nifty_core.field>"

    def __str__(self):
        minmax = [self.min(), self.max()]
        mean = self.mean()
        return "nifty_core.field instance\n- domain      = " + \
               repr(self.domain) + \
               "\n- val         = " + repr(self.get_val()) + \
               "\n  - min.,max. = " + str(minmax) + \
               "\n  - mean = " + str(mean)

    # ---Serialization---

    def _to_hdf5(self, hdf5_group):
        hdf5_group.attrs['dtype'] = self.dtype.name
        hdf5_group.attrs['distribution_strategy'] = self.distribution_strategy
        hdf5_group.attrs['domain_axes'] = str(self.domain_axes)
        hdf5_group['num_domain'] = len(self.domain)

        if self._val is None:
            ret_dict = {}
        else:
            ret_dict = {'val': self.val}

        for i in range(len(self.domain)):
            ret_dict['s_' + str(i)] = self.domain[i]

        return ret_dict

    @classmethod
    def _from_hdf5(cls, hdf5_group, repository):
        # create empty field
        new_field = EmptyField()
        # reset class
        new_field.__class__ = cls
        # set values
        temp_domain = []
        for i in range(hdf5_group['num_domain'][()]):
            temp_domain.append(repository.get('s_' + str(i), hdf5_group))
        new_field.domain = tuple(temp_domain)

        exec('new_field.domain_axes = ' + hdf5_group.attrs['domain_axes'])

        try:
            new_field._val = repository.get('val', hdf5_group)
        except(KeyError):
            new_field._val = None

        new_field.dtype = np.dtype(hdf5_group.attrs['dtype'])
        new_field.distribution_strategy =\
            hdf5_group.attrs['distribution_strategy']

        return new_field


class EmptyField(Field):
    def __init__(self):
        pass