# Source code for MDAnalysis.analysis.diffusionmap

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    """Diffusion map --- :mod:`MDAnalysis.analysis.diffusionmap`
    =====================================================================

    :Authors: Eugen Hruska, John Detlefs
    :Year: 2016
    :Copyright: Lesser GNU Public License v2.1+

    This module contains the non-linear dimension reduction method diffusion map.
    The eigenvectors of a diffusion matrix represent the 'collective coordinates'
    of a molecule; the largest eigenvalues are the more dominant collective
    coordinates. Assigning physical meaning to the 'collective coordinates' is a
    fundamentally difficult problem. The time complexity of the diffusion map is
    :math:`O(N^3)`, where N is the number of frames in the trajectory, and the in-memory
    storage complexity is :math:`O(N^2)`. Instead of a single trajectory a sample of
    protein structures can be used. The sample should be equilibrated, at least
    locally. The order of the sampled structures in the trajectory is irrelevant.

    The :ref:`Diffusion-Map-tutorial` shows how to use diffusion map for dimension
    reduction.

    More details about diffusion maps are in
    :footcite:p:`deLaPorte2008,Coifman-Lafon,Ferguson2011,Clementi2011`.

    .. _Diffusion-Map-tutorial:

    Diffusion Map tutorial
    ----------------------

    The example uses files provided as part of the MDAnalysis test suite
    (in the variables :data:`~MDAnalysis.tests.datafiles.PSF` and
    :data:`~MDAnalysis.tests.datafiles.DCD`). This tutorial shows how to use the
    Diffusion Map class.

    First load all modules and test data

    .. code-block:: python

       import MDAnalysis as mda
       import MDAnalysis.analysis.diffusionmap as diffusionmap
       from MDAnalysis.tests.datafiles import PSF, DCD

    Given a universe or atom group, we can create and eigenvalue decompose
    the Diffusion Matrix from that trajectory using :class:`DiffusionMap` and get
    the corresponding eigenvalues and eigenvectors.

    .. code-block:: python

       u = mda.Universe(PSF,DCD)

    We leave determination of the appropriate scale parameter epsilon to the user,
    :footcite:p:`Clementi2011` uses a complex method involving the k-nearest-neighbors
    of a trajectory frame, whereas others simple use a trial-and-error approach
    with a constant epsilon. Currently, the constant epsilon method is implemented
    by MDAnalysis.

    .. code-block:: python

       dmap = diffusionmap.DiffusionMap(u, select='backbone', epsilon=2)
       dmap.run()

    From here we can perform an embedding onto the k dominant eigenvectors. The
    non-linearity of the map means there is no explicit relationship between the
    lower dimensional space and our original trajectory. However, this is an
    isometry (distance preserving map), which means that points close in the lower
    dimensional space are close in the higher-dimensional space and vice versa.
    In order to embed into the most relevant low-dimensional space, there should
    exist some number of dominant eigenvectors, whose corresponding eigenvalues
    diminish at a constant rate until falling off, this is referred to as a
    spectral gap and should be somewhat apparent for a system at equilibrium with a
    high number of frames.

    .. code-block:: python

       import matplotlib.pyplot as plt
       f, ax = plt.subplots()
       upper_limit = # some reasonably high number less than the n_eigenvectors
       ax.plot(dmap.eigenvalues[:upper_limit])
       ax.set(xlabel ='eigenvalue index', ylabel='eigenvalue')
       plt.tight_layout()

    From here we can transform into the diffusion space

    .. code-block:: python

       num_eigenvectors = # some number less than the number of frames after
       # inspecting for the spectral gap
       fit = dmap.transform(num_eigenvectors, time=1)

    It can be difficult to interpret the data, and is left as a task
    for the user. The `diffusion distance` between frames i and j is best
    approximated by the euclidean distance  between rows i and j of
    self.diffusion_space.

    Classes
    -------

    .. autoclass:: DiffusionMap
    .. autoclass:: DistanceMatrix

    References
    ----------

    If you use this Dimension Reduction method in a publication, please
    cite :footcite:p:`Coifman-Lafon`.

    If you choose the default metric, this module uses the fast QCP algorithm
    :footcite:p:`Theobald2005` to calculate the root mean square distance (RMSD)
    between two coordinate sets (as implemented in
    :func:`MDAnalysis.lib.qcprot.CalcRMSDRotationalMatrix`).  When using this
    module in published work please :footcite:p:`Theobald2005`.

    .. footbibliography::
    """
    import logging
    import warnings

    import numpy as np

    from MDAnalysis.core.universe import Universe
    from MDAnalysis.core.groups import AtomGroup, UpdatingAtomGroup
    from .rms import rmsd
    from .base import AnalysisBase

    logger = logging.getLogger("MDAnalysis.analysis.diffusionmap")

    class DistanceMatrix(AnalysisBase):
        """Calculate the pairwise distance between each frame in a trajectory
        using a given metric

        A distance matrix can be initialized on its own and used as an
        initialization argument in :class:`DiffusionMap`.

        Parameters
        ----------
        universe : `~MDAnalysis.core.universe.Universe` or `~MDAnalysis.core.groups.AtomGroup`
            The MD Trajectory for dimension reduction, remember that
            computational cost of eigenvalue decomposition
            scales at O(N^3) where N is the number of frames.
            Cost can be reduced by increasing step interval or specifying a
            start and stop value when calling :meth:`DistanceMatrix.run`.
        select : str, optional
            Any valid selection string for
            :meth:`~MDAnalysis.core.groups.AtomGroup.select_atoms`
            This selection of atoms is used to calculate the RMSD between
            different frames. Water should be excluded.
        metric : function, optional
            Maps two numpy arrays to a float, is positive definite and
            symmetric. The API for a metric requires that the arrays must have
            equal length, and that the function should have weights as an
            optional argument. Weights give each index value its own weight for
            the metric calculation over the entire arrays. Default: metric is
            set to rms.rmsd().
        cutoff : float, optional
            Specify a given cutoff for metric values to be considered equal,
            Default: 1EO-5
        weights : array, optional
            Weights to be given to coordinates for metric calculation
        verbose : bool, optional
                Show detailed progress of the calculation if set to ``True``; the
                default is ``False``.

        Attributes
        ----------
        atoms : `~MDAnalysis.core.groups.AtomGroup`
            Selected atoms in trajectory subject to dimension reduction
        results.dist_matrix : numpy.ndarray, (n_frames, n_frames)
            Array of all possible ij metric distances between frames in trajectory.
            This matrix is symmetric with zeros on the diagonal.

            .. versionadded:: 2.0.0

        dist_matrix : numpy.ndarray, (n_frames, n_frames)

            .. deprecated:: 2.0.0
                     Will be removed in MDAnalysis 3.0.0. Please use
                     :attr:`results.dist_matrix` instead.

        Example
        -------
        Often, a custom distance matrix could be useful for local
        epsilon determination or other manipulations on the diffusion
        map method. The :class:`DistanceMatrix` exists in
        :mod:`~MDAnalysis.analysis.diffusionmap` and can be passed
        as an initialization argument for :class:`DiffusionMap`.

        .. code-block:: python

            import MDAnalysis as mda
            import MDAnalysis.analysis.diffusionmap as diffusionmap
            from MDAnalysis.tests.datafiles import PSF, DCD

        Now create the distance matrix and pass it as an argument to
        :class:`DiffusionMap`.

            u = mda.Universe(PSF,DCD)
            dist_matrix = diffusionmap.DistanceMatrix(u, select='all')
            dist_matrix.run()
            dmap = diffusionmap.DiffusionMap(dist_matrix)
            dmap.run()

        .. versionchanged:: 1.0.0
           ``save()`` method has been removed. You can use ``np.save()`` on
           :attr:`DistanceMatrix.results.dist_matrix` instead.
        .. versionchanged:: 2.0.0
             :attr:`dist_matrix` is now stored in a
             :class:`MDAnalysis.analysis.base.Results` instance.
        .. versionchanged:: 2.2.0
             :class:`DistanceMatrix` now also accepts `AtomGroup`.
        .. versionchanged:: 2.8.0
             :class:`DistanceMatrix` is now correctly works with `frames=...`
             parameter (#4432) by iterating over `self._sliced_trajectory`
        """

        def __init__(
            self,
            universe,
            select="all",
            metric=rmsd,
            cutoff=1e0 - 5,
            weights=None,
            **kwargs,
        ):
            # remember that this must be called before referencing self.n_frames
            super(DistanceMatrix, self).__init__(
                universe.universe.trajectory, **kwargs
            )

            if isinstance(universe, UpdatingAtomGroup):
                wmsg = (
                    "U must be a static AtomGroup. Parsing an updating AtomGroup "
                    "will result in a static AtomGroup with the current frame "
                    "atom selection."
                )
                warnings.warn(wmsg)

            self.atoms = universe.select_atoms(select)
            self._metric = metric
            self._cutoff = cutoff
            self._weights = weights
            self._calculated = False

        def _prepare(self):
            self.results.dist_matrix = np.zeros((self.n_frames, self.n_frames))

        def _single_frame(self):
            iframe = self._frame_index
            i_ref = self.atoms.positions
            # diagonal entries need not be calculated due to metric(x,x) == 0 in
            # theory, _ts not updated properly. Possible savings by setting a
            # cutoff for significant decimal places to sparsify matrix
            for j, ts in enumerate(self._sliced_trajectory[iframe:]):
                self._ts = ts
                j_ref = self.atoms.positions
                dist = self._metric(i_ref, j_ref, weights=self._weights)
                self.results.dist_matrix[
                    self._frame_index, j + self._frame_index
                ] = (dist if dist > self._cutoff else 0)
                self.results.dist_matrix[
                    j + self._frame_index, self._frame_index
                ] = self.results.dist_matrix[
                    self._frame_index, j + self._frame_index
                ]
            self._ts = self._sliced_trajectory[iframe]

        @property
        def dist_matrix(self):
            wmsg = (
                "The `dist_matrix` attribute was deprecated in "
                "MDAnalysis 2.0.0 and will be removed in MDAnalysis 3.0.0. "
                "Please use `results.dist_matrix` instead."
            )
            warnings.warn(wmsg, DeprecationWarning)
            return self.results.dist_matrix

        def _conclude(self):
            self._calculated = True

    class DiffusionMap(object):
        """Non-linear dimension reduction method

        Dimension reduction with diffusion mapping of selected structures in a
        trajectory.

        Attributes
        ----------
        eigenvalues: array (n_frames,)
            Eigenvalues of the diffusion map

        Methods
        -------
        run()
            Constructs an anisotropic diffusion kernel and performs eigenvalue
            decomposition on it.
        transform(n_eigenvectors, time)
            Perform an embedding of a frame into the eigenvectors representing
            the collective coordinates.

        .. versionchanged:: 2.2.0
             :class:`DiffusionMap` now also accepts `AtomGroup`.
        """

        def __init__(self, u, epsilon=1, **kwargs):
            """
            Parameters
            -------------
            u : MDAnalysis Universe or AtomGroup or DistanceMatrix object.
                Can be a Universe or AtomGroup, in which case one must supply kwargs for the
                initialization of a DistanceMatrix. Otherwise, this can be a
                DistanceMatrix already initialized. Either way, this will be made
                into a diffusion kernel.
            epsilon : Float
                Specifies the method used for the choice of scale parameter in the
                diffusion map. More information in
                :footcite:p:`Coifman-Lafon,Ferguson2011,Clementi2011`, Default: 1.
            **kwargs
                Parameters to be passed for the initialization of a
                :class:`DistanceMatrix`.
            """
            if isinstance(u, AtomGroup) or isinstance(u, Universe):
                self._dist_matrix = DistanceMatrix(u, **kwargs)
            elif isinstance(u, DistanceMatrix):
                self._dist_matrix = u
            else:
                raise ValueError(
                    "U is not a Universe or AtomGroup or DistanceMatrix and"
                    " so the DiffusionMap has no data to work with."
                )
            self._epsilon = epsilon

        def run(self, start=None, stop=None, step=None):
            """Create and decompose the diffusion matrix in preparation
            for a diffusion map.

            Parameters
            ----------
            start : int, optional
                start frame of analysis
            stop : int, optional
                stop frame of analysis
            step : int, optional
                number of frames to skip between each analysed frame

            .. versionchanged:: 0.19.0
               Added start/stop/step kwargs
            """
            # run only if distance matrix not already calculated
            if not self._dist_matrix._calculated:
                self._dist_matrix.run(start=start, stop=stop, step=step)
            # important for transform function and length of .run() method
            self._n_frames = self._dist_matrix.n_frames
            if self._n_frames > 5000:
                warnings.warn(
                    "The distance matrix is very large, and can "
                    "be very slow to compute. Consider picking a larger "
                    "step size in distance matrix initialization."
                )
            self._scaled_matrix = (
                self._dist_matrix.results.dist_matrix**2 / self._epsilon
            )
            # take negative exponent of scaled matrix to create Isotropic kernel
            self._kernel = np.exp(-self._scaled_matrix)
            D_inv = np.diag(1 / self._kernel.sum(1))
            self._diff = np.dot(D_inv, self._kernel)
            self._eigenvals, self._eigenvectors = np.linalg.eig(self._diff)
            sort_idx = np.argsort(self._eigenvals)[::-1]
            self.eigenvalues = self._eigenvals[sort_idx]
            self._eigenvectors = self._eigenvectors[sort_idx]
            self._calculated = True
            return self

        def transform(self, n_eigenvectors, time):
            """Embeds a trajectory via the diffusion map

            Parameters
            ---------
            n_eigenvectors : int
                The number of dominant eigenvectors to be used for
                diffusion mapping
            time : float
                Exponent that eigenvalues are raised to for embedding, for large
                values, more dominant eigenvectors determine diffusion distance.
            Return
            ------
            diffusion_space : array (n_frames, n_eigenvectors)
                The diffusion map embedding as defined by :footcite:p:`Ferguson2011`.
            """
            return self._eigenvectors[1 : n_eigenvectors + 1,].T * (
                self.eigenvalues[1 : n_eigenvectors + 1] ** time
            )