# Source code for MDAnalysis.analysis.diffusionmap

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

:Authors: Eugen Hruska, John Detlefs
:Year: 2016

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

.. 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,
[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.

.. 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 [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)

"""
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. Parameters ---------- 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. 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. """ 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.trajectory, **kwargs) 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._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._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.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._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
[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.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
[docs] 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 [Ferguson1]_. """ return (self._eigenvectors[1:n_eigenvectors+1,].T * (self.eigenvalues[1:n_eigenvectors+1]**time))