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"""Diffusion map --- :mod:`MDAnalysis.analysis.diffusionmap`
=====================================================================
:Authors: Eugen Hruska, John Detlefs
:Year: 2016
:Copyright: GNU Public License v2
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 phyiscal 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 equiblibrated, 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 [deLaPorte1]_, [Lafon1]_ ,
[Ferguson1]_, and [Clementi1]_.
.. _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 ::
>>> 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.
>>> u = mda.Universe(PSF,DCD)
We leave determination of the appropriate scale parameter epsilon to the user,
[Clementi1]_ 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.
>>> 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.
>>> # first cell of a jupyter notebook should contain: %matplotlib inline
>>> 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
>>> 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.
.. _Distance-Matrix-tutorial:
Distance Matrix tutorial
------------------------
Often a, 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`.
>>> 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()
Classes
-------
.. autoclass:: DiffusionMap
.. autoclass:: DistanceMatrix
References
----------
If you use this Dimension Reduction method in a publication, please
cite [Lafon1]_.
If you choose the default metric, this module uses the fast QCP algorithm
[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 cite [Theobald2005]_.
.. [Lafon1] Coifman, Ronald R., Lafon, Stephane. Diffusion
maps. Appl. Comput. Harmon. Anal. 21, 5–30 (2006).
.. [deLaPorte1] J. de la Porte, B. M. Herbst, W. Hereman, S. J. van der Walt.
An Introduction to Diffusion Maps. In: The 19th Symposium of the
Pattern Recognition Association of South Africa (2008).
.. [Clementi1] Rohrdanz, M. A, Zheng, W, Maggioni, M, & Clementi, C.
Determination of reaction coordinates via locally scaled diffusion
map. J. Chem. Phys. 134, 124116 (2011).
.. [Ferguson1] Ferguson, A. L.; Panagiotopoulos, A. Z.; Kevrekidis, I. G.
Debenedetti, P. G. Nonlinear dimensionality reduction in molecular
simulation: The diffusion map approach Chem. Phys. Lett. 509, 1−11
(2011)
"""
from __future__ import division, absolute_import
from six.moves import range
import logging
import warnings
import numpy as np
from MDAnalysis.core.universe import Universe
from .rms import rmsd
from .base import AnalysisBase
logger = logging.getLogger("MDAnalysis.analysis.diffusionmap")
[docs]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`. Refer to the
:ref:`Distance-Matrix-tutorial` for a demonstration.
Attributes
----------
atoms : AtomGroup
Selected atoms in trajectory subject to dimension reduction
dist_matrix : array, (n_frames, n_frames)
Array of all possible ij metric distances between frames in trajectory.
This matrix is symmetric with zeros on the diagonal.
.. versionchanged:: 1.0.0
``save()`` method has been removed. You can use ``np.save()`` on
:attr:`DistanceMatrix.dist_matrix` instead.
"""
def __init__(self, u, select='all', metric=rmsd, cutoff=1E0-5,
weights=None, **kwargs):
"""
Parameters
----------
u : universe `~MDAnalysis.core.universe.Universe`
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``.
"""
self._u = u
traj = self._u.trajectory
# remember that this must be called before referencing self.n_frames
super(DistanceMatrix, self).__init__(self._u.trajectory, **kwargs)
self.atoms = self._u.select_atoms(select)
self._metric = metric
self._cutoff = cutoff
self._weights = weights
self._calculated = False
def _prepare(self):
self.dist_matrix = np.zeros((self.n_frames, self.n_frames))
def _single_frame(self):
iframe = self._ts.frame
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._u.trajectory[iframe:self.stop:self.step]):
self._ts = ts
j_ref = self.atoms.positions
dist = self._metric(i_ref, j_ref, weights=self._weights)
self.dist_matrix[self._frame_index, j+self._frame_index] = (
dist if dist > self._cutoff else 0)
self.dist_matrix[j+self._frame_index, self._frame_index] = (
self.dist_matrix[self._frame_index, j+self._frame_index])
self._ts = self._u.trajectory[iframe]
def _conclude(self):
self._calculated = True
[docs]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.
"""
def __init__(self, u, epsilon=1, **kwargs):
"""
Parameters
-------------
u : MDAnalysis Universe or DistanceMatrix object
Can be a Universe, 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 [Lafon1]_, [Ferguson1]_ and
[Clementi1]_, Default: 1.
**kwargs
Parameters to be passed for the initialization of a
:class:`DistanceMatrix`.
"""
if 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 DistanceMatrix and"
" so the DiffusionMap has no data to work with.")
self._epsilon = epsilon
[docs] 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.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