Source code for MDAnalysis.analysis.encore.similarity

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"""=================================================================================
Ensemble Similarity Calculations --- :mod:`MDAnalysis.analysis.encore.similarity`
=================================================================================

:Author: Matteo Tiberti, Wouter Boomsma, Tone Bengtsen

.. versionadded:: 0.16.0

The module contains implementations of similarity measures between protein
ensembles described in [Lindorff-Larsen2009]_. The implementation and examples
are described in [Tiberti2015]_.

The module includes facilities for handling ensembles and trajectories through
the :class:`Universe` class, performing clustering or dimensionality reduction
of the ensemble space, estimating multivariate probability distributions from
the input data, and more. ENCORE can be used to compare experimental and
simulation-derived ensembles, as well as estimate the convergence of
trajectories from time-dependent simulations.

ENCORE includes three different methods for calculations of similarity measures
between ensembles implemented in individual functions:

+ **Harmonic Ensemble Similarity** : :func:`hes`
+ **Clustering Ensemble Similarity** : :func:`ces`
+ **Dimensional Reduction Ensemble Similarity** : :func:`dres`

as well as two methods to evaluate the convergence of trajectories:

+ **Clustering based convergence evaluation** : :func:`ces_convergence`
+ **Dimensionality-reduction based convergence evaluation** : :func:`dres_convergence`

When using this module in published work please cite [Tiberti2015]_.

References
==========

.. [Lindorff-Larsen2009] Similarity Measures for Protein
   Ensembles. Lindorff-Larsen, K. Ferkinghoff-Borg, J. PLoS ONE 2008, 4, e4203.

.. [Tiberti2015] ENCORE: Software for Quantitative Ensemble Comparison. Matteo
   Tiberti, Elena Papaleo, Tone Bengtsen, Wouter Boomsma, Kresten
   Lindorff-Larsen. PLoS Comput Biol. 2015, 11, e1004415.

.. _Examples:
Examples
========

The examples show how to use ENCORE to calculate a similarity measurement
of two simple ensembles. The ensembles are obtained from the MDAnalysis
test suite for two different simulations of the protein AdK. To run the
examples first execute: ::

    >>> from MDAnalysis import Universe
    >>> import MDAnalysis.analysis.encore as encore
    >>> from MDAnalysis.tests.datafiles import PSF, DCD, DCD2

To calculate the Harmonic Ensemble Similarity (:func:`hes`)
two ensemble objects are first created and then used for calculation: ::

    >>> ens1 = Universe(PSF, DCD)
    >>> ens2 = Universe(PSF, DCD2)
    >>> HES, details = encore.hes([ens1, ens2])
    >>> print(HES)
    [[        0.        ,  38279683.95892926],
    [ 38279683.95892926,         0.        ]]

HES can assume any non-negative value, i.e. no upper bound exists and the
measurement can therefore be used as an absolute scale.

The calculation of the Clustering Ensemble Similarity (:func:`ces`)
is computationally more expensive. It is based on clustering algorithms that in
turn require a similarity matrix between the frames the ensembles are made
of. The similarity matrix is derived from a distance matrix (By default a RMSD
matrix; a full RMSD matrix between each pairs of elements needs to be computed).
The RMSD matrix is automatically calculated. ::

    >>> ens1 = Universe(PSF, DCD)
    >>> ens2 = Universe(PSF, DCD2)
    >>> CES, details = encore.ces([ens1, ens2])
    >>> print(CES)
    [[ 0.          0.68070702]
    [ 0.68070702  0.        ]]

However, we may want to reuse the RMSD matrix in other calculations e.g.
running CES with different parameters or running DRES. In this
case we first compute the RMSD matrix alone:

    >>> rmsd_matrix = encore.get_distance_matrix(
                                    encore.utils.merge_universes([ens1, ens2]),
                                    save_matrix="rmsd.npz")

In the above example the RMSD matrix was also saved in rmsd.npz on disk, and
so can be loaded and re-used at later times, instead of being recomputed:

    >>> rmsd_matrix = encore.get_distance_matrix(
                                    encore.utils.merge_universes([ens1, ens2]),
                                    load_matrix="rmsd.npz")


For instance, the rmsd_matrix object can be re-used as input for the
Dimensional Reduction Ensemble Similarity (:func:`dres`) method.
DRES is based on the estimation of the probability density in
a dimensionally-reduced conformational space of the ensembles, obtained from
the original space using either the Stochastic Proximity Embedding algorithm or
the Principal Component Analysis.
As the algorithms require the distance matrix calculated on the original space,
we can reuse the previously-calculated RMSD matrix.
In the following example the dimensions are reduced to 3 using the
saved RMSD matrix and the default SPE dimensional reduction method.   : ::

    >>> DRES,details = encore.dres([ens1, ens2],
                                   distance_matrix = rmsd_matrix)
    >>> print(DRES)
    [[ 0.        ,   0.67453198]
     [  0.67453198,  0.        ]]


In addition to the quantitative similarity estimate, the dimensional reduction
can easily be visualized, see the ``Example`` section in
:mod:`MDAnalysis.analysis.encore.dimensionality_reduction.reduce_dimensionality`.
Due to the stochastic nature of SPE, two identical ensembles will not
necessarily result in an exactly 0 estimate of the similarity, but will be very
close. For the same reason, calculating the similarity with the :func:`dres`
twice will not result in necessarily identical values but rather two very close
values.

It should be noted that both in :func:`ces` and :func:`dres` the similarity is
evaluated using the Jensen-Shannon divergence resulting in an upper bound of
ln(2), which indicates no similarity between the ensembles and a lower bound
of 0.0 signifying two identical ensembles. In contrast, the :func:`hes` function uses
a symmetrized version of the Kullback-Leibler divergence, which is unbounded.


Functions for ensemble comparisons
==================================

.. autofunction:: hes

.. autofunction:: ces

.. autofunction:: dres


Function reference
==================

.. All functions are included via automodule :members:.

"""
import warnings
import logging

import numpy as np
import scipy.stats

import MDAnalysis as mda

from ...coordinates.memory import MemoryReader
from .confdistmatrix import get_distance_matrix
from .bootstrap import (get_distance_matrix_bootstrap_samples,
                        get_ensemble_bootstrap_samples)
from .clustering.cluster import cluster
from .clustering.ClusteringMethod import AffinityPropagationNative
from .dimensionality_reduction.DimensionalityReductionMethod import (
    StochasticProximityEmbeddingNative)
from .dimensionality_reduction.reduce_dimensionality import (
    reduce_dimensionality)
from .covariance import (
    covariance_matrix, ml_covariance_estimator, shrinkage_covariance_estimator)
from .utils import merge_universes
from .utils import trm_indices_diag, trm_indices_nodiag

# Low boundary value for log() argument - ensure no nans
EPSILON = 1E-15

xlogy = np.vectorize(
    lambda x, y: 0.0 if (x <= EPSILON and y <= EPSILON) else x * np.log(y))


[docs]def discrete_kullback_leibler_divergence(pA, pB): """Kullback-Leibler divergence between discrete probability distribution. Notice that since this measure is not symmetric :: :math:`d_{KL}(p_A,p_B) != d_{KL}(p_B,p_A)` Parameters ---------- pA : iterable of floats First discrete probability density function pB : iterable of floats Second discrete probability density function Returns ------- dkl : float Discrete Kullback-Liebler divergence """ return np.sum(xlogy(pA, pA / pB))
# discrete dJS
[docs]def discrete_jensen_shannon_divergence(pA, pB): """Jensen-Shannon divergence between discrete probability distributions. Parameters ---------- pA : iterable of floats First discrete probability density function pB : iterable of floats Second discrete probability density function Returns ------- djs : float Discrete Jensen-Shannon divergence """ return 0.5 * (discrete_kullback_leibler_divergence(pA, (pA + pB) * 0.5) + discrete_kullback_leibler_divergence(pB, (pA + pB) * 0.5))
# calculate harmonic similarity
[docs]def harmonic_ensemble_similarity(sigma1, sigma2, x1, x2): """ Calculate the harmonic ensemble similarity measure as defined in [Tiberti2015]_. Parameters ---------- sigma1 : numpy.array Covariance matrix for the first ensemble. sigma2 : numpy.array Covariance matrix for the second ensemble. x1: numpy.array Mean for the estimated normal multivariate distribution of the first ensemble. x2: numpy.array Mean for the estimated normal multivariate distribution of the second ensemble. Returns ------- dhes : float harmonic similarity measure """ # Inverse covariance matrices sigma1_inv = np.linalg.pinv(sigma1) sigma2_inv = np.linalg.pinv(sigma2) # Difference between average vectors d_avg = x1 - x2 # Distance measure trace = np.trace(np.dot(sigma1, sigma2_inv) + np.dot(sigma2, sigma1_inv) - 2 * np.identity(sigma1.shape[0])) d_hes = 0.25 * (np.dot(np.transpose(d_avg), np.dot(sigma1_inv + sigma2_inv, d_avg)) + trace) return d_hes
[docs]def clustering_ensemble_similarity(cc, ens1, ens1_id, ens2, ens2_id, select="name CA"): """Clustering ensemble similarity: calculate the probability densities from the clusters and calculate discrete Jensen-Shannon divergence. Parameters ---------- cc : encore.clustering.ClustersCollection Collection from cluster calculated by a clustering algorithm (e.g. Affinity propagation) ens1 : :class:`~MDAnalysis.core.universe.Universe` First ensemble to be used in comparison ens1_id : int First ensemble id as detailed in the ClustersCollection metadata ens2 : :class:`~MDAnalysis.core.universe.Universe` Second ensemble to be used in comparison ens2_id : int Second ensemble id as detailed in the ClustersCollection metadata select : str Atom selection string in the MDAnalysis format. Default is "name CA". Returns ------- djs : float Jensen-Shannon divergence between the two ensembles, as calculated by the clustering ensemble similarity method """ ens1_coordinates = ens1.trajectory.timeseries(ens1.select_atoms(select), order='fac') ens2_coordinates = ens2.trajectory.timeseries(ens2.select_atoms(select), order='fac') tmpA = np.array([np.where(c.metadata['ensemble_membership'] == ens1_id)[ 0].shape[0] / float(ens1_coordinates.shape[0]) for c in cc]) tmpB = np.array([np.where(c.metadata['ensemble_membership'] == ens2_id)[ 0].shape[0] / float(ens2_coordinates.shape[0]) for c in cc]) # Exclude clusters which have 0 elements in both ensembles pA = tmpA[tmpA + tmpB > EPSILON] pB = tmpB[tmpA + tmpB > EPSILON] return discrete_jensen_shannon_divergence(pA, pB)
[docs]def cumulative_clustering_ensemble_similarity(cc, ens1_id, ens2_id, ens1_id_min=1, ens2_id_min=1): """ Calculate clustering ensemble similarity between joined ensembles. This means that, after clustering has been performed, some ensembles are merged and the dJS is calculated between the probability distributions of the two clusters groups. In particular, the two ensemble groups are defined by their ensembles id: one of the two joined ensembles will comprise all the ensembles with id [ens1_id_min, ens1_id], and the other ensembles will comprise all the ensembles with id [ens2_id_min, ens2_id]. Parameters ---------- cc : encore.ClustersCollection Collection from cluster calculated by a clustering algorithm (e.g. Affinity propagation) ens1_id : int First ensemble id as detailed in the ClustersCollection metadata ens2_id : int Second ensemble id as detailed in the ClustersCollection metadata Returns ------- djs : float Jensen-Shannon divergence between the two ensembles, as calculated by the clustering ensemble similarity method """ ensA = [np.where(np.logical_and( c.metadata['ensemble_membership'] <= ens1_id, c.metadata['ensemble_membership']) >= ens1_id_min)[0].shape[0] for c in cc] ensB = [np.where(np.logical_and( c.metadata['ensemble_membership'] <= ens2_id, c.metadata['ensemble_membership']) >= ens2_id_min)[0].shape[0] for c in cc] sizeA = float(np.sum(ensA)) sizeB = float(np.sum(ensB)) tmpA = np.array(ensA) / sizeA tmpB = np.array(ensB) / sizeB # Exclude clusters which have 0 elements in both ensembles pA = tmpA[tmpA + tmpB > EPSILON] pB = tmpB[tmpA + tmpB > EPSILON] return discrete_jensen_shannon_divergence(pA, pB)
[docs]def gen_kde_pdfs(embedded_space, ensemble_assignment, nensembles, nsamples): """ Generate Kernel Density Estimates (KDE) from embedded spaces and elaborate the coordinates for later use. Parameters ---------- embedded_space : numpy.array Array containing the coordinates of the embedded space ensemble_assignment : numpy.array Array containing one int per ensemble conformation. These allow to distinguish, in the complete embedded space, which conformations belong to each ensemble. For instance if ensemble_assignment is [1,1,1,1,2,2], it means that the first four conformations belong to ensemble 1 and the last two to ensemble 2 nensembles : int Number of ensembles nsamples : int samples to be drawn from the ensembles. Will be required in a later stage in order to calculate dJS. Returns ------- kdes : scipy.stats.gaussian_kde KDEs calculated from ensembles resamples : list of numpy.array For each KDE, draw samples according to the probability distribution of the KDE mixture model embedded_ensembles : list of numpy.array List of numpy.array containing, each one, the elements of the embedded space belonging to a certain ensemble """ kdes = [] embedded_ensembles = [] resamples = [] for i in range(1, nensembles + 1): this_embedded = embedded_space.transpose()[ np.where(np.array(ensemble_assignment) == i)].transpose() embedded_ensembles.append(this_embedded) kdes.append(scipy.stats.gaussian_kde(this_embedded)) # # Set number of samples # if not nsamples: # nsamples = this_embedded.shape[1] * 10 # Resample according to probability distributions for this_kde in kdes: resamples.append(this_kde.resample(nsamples)) return (kdes, resamples, embedded_ensembles)
[docs]def dimred_ensemble_similarity(kde1, resamples1, kde2, resamples2, ln_P1_exp_P1=None, ln_P2_exp_P2=None, ln_P1P2_exp_P1=None, ln_P1P2_exp_P2=None): """ Calculate the Jensen-Shannon divergence according the the Dimensionality reduction method. In this case, we have continuous probability densities, this we need to integrate over the measurable space. The aim is to first calculate the Kullback-Liebler divergence, which is defined as: .. math:: D_{KL}(P(x) || Q(x)) = \\int_{-\\infty}^{\\infty}P(x_i) ln(P(x_i)/Q(x_i)) = \\langle{}ln(P(x))\\rangle{}_P - \\langle{}ln(Q(x))\\rangle{}_P where the :math:`\\langle{}.\\rangle{}_P` denotes an expectation calculated under the distribution P. We can, thus, just estimate the expectation values of the components to get an estimate of dKL. Since the Jensen-Shannon distance is actually more complex, we need to estimate four expectation values: .. math:: \\langle{}log(P(x))\\rangle{}_P \\langle{}log(Q(x))\\rangle{}_Q \\langle{}log(0.5*(P(x)+Q(x)))\\rangle{}_P \\langle{}log(0.5*(P(x)+Q(x)))\\rangle{}_Q Parameters ---------- kde1 : scipy.stats.gaussian_kde Kernel density estimation for ensemble 1 resamples1 : numpy.array Samples drawn according do kde1. Will be used as samples to calculate the expected values according to 'P' as detailed before. kde2 : scipy.stats.gaussian_kde Kernel density estimation for ensemble 2 resamples2 : numpy.array Samples drawn according do kde2. Will be used as sample to calculate the expected values according to 'Q' as detailed before. ln_P1_exp_P1 : float or None Use this value for :math:`\\langle{}log(P(x))\\rangle{}_P`; if None, calculate it instead ln_P2_exp_P2 : float or None Use this value for :math:`\\langle{}log(Q(x))\\rangle{}_Q`; if None, calculate it instead ln_P1P2_exp_P1 : float or None Use this value for :math:`\\langle{}log(0.5*(P(x)+Q(x)))\\rangle{}_P`; if None, calculate it instead ln_P1P2_exp_P2 : float or None Use this value for :math:`\\langle{}log(0.5*(P(x)+Q(x)))\\rangle{}_Q`; if None, calculate it instead Returns ------- djs : float Jensen-Shannon divergence calculated according to the dimensionality reduction method """ if not ln_P1_exp_P1 and not ln_P2_exp_P2 and not ln_P1P2_exp_P1 and not \ ln_P1P2_exp_P2: ln_P1_exp_P1 = np.average(np.log(kde1.evaluate(resamples1))) ln_P2_exp_P2 = np.average(np.log(kde2.evaluate(resamples2))) ln_P1P2_exp_P1 = np.average(np.log( 0.5 * (kde1.evaluate(resamples1) + kde2.evaluate(resamples1)))) ln_P1P2_exp_P2 = np.average(np.log( 0.5 * (kde1.evaluate(resamples2) + kde2.evaluate(resamples2)))) return 0.5 * ( ln_P1_exp_P1 - ln_P1P2_exp_P1 + ln_P2_exp_P2 - ln_P1P2_exp_P2)
[docs]def cumulative_gen_kde_pdfs(embedded_space, ensemble_assignment, nensembles, nsamples, ens_id_min=1, ens_id_max=None): """ Generate Kernel Density Estimates (KDE) from embedded spaces and elaborate the coordinates for later use. However, consider more than one ensemble as the space on which the KDE will be generated. In particular, will use ensembles with ID [ens_id_min, ens_id_max]. Parameters ---------- embedded_space : numpy.array Array containing the coordinates of the embedded space ensemble_assignment : numpy.array array containing one int per ensemble conformation. These allow to distinguish, in the complete embedded space, which conformations belong to each ensemble. For instance if ensemble_assignment is [1,1,1,1,2,2], it means that the first four conformations belong to ensemble 1 and the last two to ensemble 2 nensembles : int Number of ensembles nsamples : int Samples to be drawn from the ensembles. Will be required in a later stage in order to calculate dJS. ens_id_min : int Minimum ID of the ensemble to be considered; see description ens_id_max : int Maximum ID of the ensemble to be considered; see description. If None, it will be set to the maximum possible value given the number of ensembles. Returns ------- kdes : scipy.stats.gaussian_kde KDEs calculated from ensembles resamples : list of numpy.array For each KDE, draw samples according to the probability distribution of the kde mixture model embedded_ensembles : list of numpy.array List of numpy.array containing, each one, the elements of the embedded space belonging to a certain ensemble """ kdes = [] embedded_ensembles = [] resamples = [] if not ens_id_max: ens_id_max = nensembles + 1 for i in range(ens_id_min, ens_id_max): this_embedded = embedded_space.transpose()[np.where( np.logical_and(ensemble_assignment >= ens_id_min, ensemble_assignment <= i))].transpose() embedded_ensembles.append(this_embedded) kdes.append(scipy.stats.gaussian_kde(this_embedded)) # Resample according to probability distributions for this_kde in kdes: resamples.append(this_kde.resample(nsamples)) return (kdes, resamples, embedded_ensembles)
[docs]def write_output(matrix, base_fname=None, header="", suffix="", extension="dat"): """ Write output matrix with a nice format, to stdout and optionally a file. Parameters ---------- matrix : encore.utils.TriangularMatrix Matrix containing the values to be printed base_fname : str Basic filename for output. If None, no files will be written, and the matrix will be just printed on standard output header : str Text to be written just before the matrix suffix : str String to be concatenated to basename, in order to get the final file name extension : str Extension for the output file """ if base_fname is not None: fname = base_fname + "-" + suffix + "." + extension else: fname = None matrix.square_print(header=header, fname=fname)
[docs]def prepare_ensembles_for_convergence_increasing_window(ensemble, window_size, select="name CA"): """ Generate ensembles to be fed to ces_convergence or dres_convergence from a single ensemble. Basically, the different slices the algorithm needs are generated here. Parameters ---------- ensemble : :class:`~MDAnalysis.core.universe.Universe` object Input ensemble window_size : int size of the window (in number of frames) to be used select : str Atom selection string in the MDAnalysis format. Default is "name CA" Returns ------- tmp_ensembles : The original ensemble is divided into different ensembles, each being a window_size-long slice of the original ensemble. The last ensemble will be bigger if the length of the input ensemble is not exactly divisible by window_size. """ ens_size = ensemble.trajectory.timeseries(ensemble.select_atoms(select), order='fac').shape[0] rest_slices = ens_size // window_size residuals = ens_size % window_size slices_n = [0] tmp_ensembles = [] for rs in range(rest_slices - 1): slices_n.append(slices_n[-1] + window_size) slices_n.append(slices_n[-1] + residuals + window_size) for s,sl in enumerate(slices_n[:-1]): tmp_ensembles.append(mda.Universe( ensemble.filename, ensemble.trajectory.timeseries(order='fac') [slices_n[s]:slices_n[s + 1], :, :], format=MemoryReader)) return tmp_ensembles
[docs]def hes(ensembles, select="name CA", cov_estimator="shrinkage", weights='mass', align=False, estimate_error=False, bootstrapping_samples=100, calc_diagonal=False): """ Calculates the Harmonic Ensemble Similarity (HES) between ensembles using the symmetrized version of Kullback-Leibler divergence as described in [Tiberti2015]_. Parameters ---------- ensembles : list List of Universe objects for similarity measurements. select : str, optional Atom selection string in the MDAnalysis format. Default is "name CA" cov_estimator : str, optional Covariance matrix estimator method, either shrinkage, `shrinkage`, or Maximum Likelyhood, `ml`. Default is shrinkage. weights : str/array_like, optional specify optional weights. If ``mass`` then chose masses of ensemble atoms align : bool, optional Whether to align the ensembles before calculating their similarity. Note: this changes the ensembles in-place, and will thus leave your ensembles in an altered state. (default is False) estimate_error : bool, optional Whether to perform error estimation (default is False). bootstrapping_samples : int, optional Number of times the similarity matrix will be bootstrapped (default is 100), only if estimate_error is True. calc_diagonal : bool, optional Whether to calculate the diagonal of the similarity scores (i.e. the similarities of every ensemble against itself). If this is False (default), 0.0 will be used instead. Returns ------- hes, details : numpy.array, dictionary Harmonic similarity measurements between each pair of ensembles, and dict containing mean and covariance matrix for each ensemble Notes ----- The method assumes that each ensemble is derived from a multivariate normal distribution. The mean and covariance matrix are, thus, estimatated from the distribution of each ensemble and used for comparision by the symmetrized version of Kullback-Leibler divergence defined as: .. math:: D_{KL}(P(x) || Q(x)) = \\int_{-\\infty}^{\\infty}P(x_i) ln(P(x_i)/Q(x_i)) = \\langle{}ln(P(x))\\rangle{}_P - \\langle{}ln(Q(x))\\rangle{}_P where the :math:`\\langle{}.\\rangle{}_P` denotes an expectation calculated under the distribution P. For each ensemble, the mean conformation is estimated as the average over the ensemble, and the covariance matrix is calculated by default using a shrinkage estimation method (or by a maximum-likelihood method, optionally). Note that the symmetrized version of the Kullback-Leibler divergence has no upper bound (unlike the Jensen-Shannon divergence used by for instance CES and DRES). When using this similarity measure, consider whether you want to align the ensembles first (see example below). Example ------- To calculate the Harmonic Ensemble similarity, two ensembles are created as Universe objects from a topology file and two trajectories. The topology- and trajectory files used are obtained from the MDAnalysis test suite for two different simulations of the protein AdK. To run the examples see the module `Examples`_ for how to import the files: :: >>> ens1 = Universe(PSF, DCD) >>> ens2 = Universe(PSF, DCD2) >>> HES, details = encore.hes([ens1, ens2]) >>> print(HES) [[ 0. 38279683.95892926] [ 38279683.95892926 0. ]] You can use the align=True option to align the ensembles first. This will align everything to the current timestep in the first ensemble. Note that this changes the ens1 and ens2 objects: >>> print(encore.hes([ens1, ens2], align=True)[0]) [[ 0. 6880.34140106] [ 6880.34140106 0. ]] Alternatively, for greater flexibility in how the alignment should be done you can call use an AlignTraj object manually: >>> from MDAnalysis.analysis import align >>> align.AlignTraj(ens1, ens1, select="name CA", in_memory=True).run() >>> align.AlignTraj(ens2, ens1, select="name CA", in_memory=True).run() >>> print(encore.hes([ens1, ens2])[0]) [[ 0. 7032.19607004] [ 7032.19607004 0. ]] .. versionchanged:: 1.0.0 hes doesn't accept the *details* argument anymore, it always returns the details of the calculation instead, in the form of a dictionary """ if not isinstance(weights, (list, tuple, np.ndarray)) and weights == 'mass': weights = ['mass' for _ in range(len(ensembles))] elif weights is not None: if len(weights) != len(ensembles): raise ValueError("need weights for every ensemble") else: weights = [None for _ in range(len(ensembles))] # Ensure in-memory trajectories either by calling align # with in_memory=True or by directly calling transfer_to_memory # on the universe. if align: for e, w in zip(ensembles, weights): mda.analysis.align.AlignTraj(e, ensembles[0], select=select, weights=w, in_memory=True).run() else: for ensemble in ensembles: ensemble.transfer_to_memory() if calc_diagonal: pairs_indices = list(trm_indices_diag(len(ensembles))) else: pairs_indices = list(trm_indices_nodiag(len(ensembles))) logging.info("Chosen metric: Harmonic similarity") if cov_estimator == "shrinkage": covariance_estimator = shrinkage_covariance_estimator logging.info(" Covariance matrix estimator: Shrinkage") elif cov_estimator == "ml": covariance_estimator = ml_covariance_estimator logging.info(" Covariance matrix estimator: Maximum Likelihood") else: logging.error( "Covariance estimator {0} is not supported. " "Choose between 'shrinkage' and 'ml'.".format(cov_estimator)) return None out_matrix_eln = len(ensembles) xs = [] sigmas = [] if estimate_error: data = [] ensembles_list = [] for i, ensemble in enumerate(ensembles): ensembles_list.append( get_ensemble_bootstrap_samples( ensemble, samples=bootstrapping_samples)) for t in range(bootstrapping_samples): logging.info("The coordinates will be bootstrapped.") xs = [] sigmas = [] values = np.zeros((out_matrix_eln, out_matrix_eln)) for i, e_orig in enumerate(ensembles): xs.append(np.average( ensembles_list[i][t].trajectory.timeseries( e_orig.select_atoms(select), order=('fac')), axis=0).flatten()) sigmas.append(covariance_matrix(ensembles_list[i][t], weights=weights[i], estimator=covariance_estimator, select=select)) for pair in pairs_indices: value = harmonic_ensemble_similarity(x1=xs[pair[0]], x2=xs[pair[1]], sigma1=sigmas[pair[0]], sigma2=sigmas[pair[1]]) values[pair[0], pair[1]] = value values[pair[1], pair[0]] = value data.append(values) avgs = np.average(data, axis=0) stds = np.std(data, axis=0) return (avgs, stds) # Calculate the parameters for the multivariate normal distribution # of each ensemble values = np.zeros((out_matrix_eln, out_matrix_eln)) for e, w in zip(ensembles, weights): # Extract coordinates from each ensemble coordinates_system = e.trajectory.timeseries(e.select_atoms(select), order='fac') # Average coordinates in each system xs.append(np.average(coordinates_system, axis=0).flatten()) # Covariance matrices in each system sigmas.append(covariance_matrix(e, weights=w, estimator=covariance_estimator, select=select)) for i, j in pairs_indices: value = harmonic_ensemble_similarity(x1=xs[i], x2=xs[j], sigma1=sigmas[i], sigma2=sigmas[j]) values[i, j] = value values[j, i] = value # Save details as required details = {} for i in range(out_matrix_eln): details['ensemble{0:d}_mean'.format(i + 1)] = xs[i] details['ensemble{0:d}_covariance_matrix'.format(i + 1)] = sigmas[i] return values, details
[docs]def ces(ensembles, select="name CA", clustering_method=AffinityPropagationNative( preference=-1.0, max_iter=500, convergence_iter=50, damping=0.9, add_noise=True), distance_matrix=None, estimate_error=False, bootstrapping_samples=10, ncores=1, calc_diagonal=False, allow_collapsed_result=True): """ Calculates the Clustering Ensemble Similarity (CES) between ensembles using the Jensen-Shannon divergence as described in [Tiberti2015]_. Parameters ---------- ensembles : list List of ensemble objects for similarity measurements select : str, optional Atom selection string in the MDAnalysis format. Default is "name CA" clustering_method : A single or a list of instances of the :class:`MDAnalysis.analysis.encore.clustering.ClusteringMethod` classes from the clustering module. Different parameters for the same clustering method can be explored by adding different instances of the same clustering class. Clustering methods options are the Affinity Propagation (default), the DBSCAN and the KMeans. The latter two methods need the sklearn python module installed. distance_matrix : encore.utils.TriangularMatrix Distance matrix clustering methods. If this parameter is not supplied the matrix will be calculated on the fly. estimate_error : bool, optional Whether to perform error estimation (default is False). Only bootstrapping mode is supported. bootstrapping_samples : int, optional number of samples to be used for estimating error. ncores : int, optional Maximum number of cores to be used (default is 1). calc_diagonal : bool, optional Whether to calculate the diagonal of the similarity scores (i.e. the similarities of every ensemble against itself). If this is False (default), 0.0 will be used instead. allow_collapsed_result: bool, optional Whether a return value of a list of one value should be collapsed into just the value. Returns ------- ces, details : numpy.array, numpy.array ces contains the similarity values, arranged in a numpy.array. If only one clustering_method is provided the output will be a 2-dimensional square symmetrical numpy.array. The order of the matrix elements depends on the order of the input ensembles: for instance, if ensemble = [ens1, ens2, ens3] the matrix elements [0,2] and [2,0] will both contain the similarity value between ensembles ens1 and ens3. Elaborating on the previous example, if *n* ensembles are given and *m* clustering_methods are provided the output will be a list of *m* arrays ordered by the input sequence of methods, each with a *n*x*n* symmetrical similarity matrix. details contains information on the clustering: the individual size of each cluster, the centroids and the frames associated with each cluster. Notes ----- In the Jensen-Shannon divergence the upper bound of ln(2) signifies no similarity between the two ensembles, the lower bound, 0.0, signifies identical ensembles. To calculate the CES, the affinity propagation method (or others, if specified) is used to partition the whole space of conformations. The population of each ensemble in each cluster is then taken as a probability density function. Different probability density functions from each ensemble are finally compared using the Jensen-Shannon divergence measure. Examples -------- To calculate the Clustering Ensemble similarity, two ensembles are created as Universe object using a topology file and two trajectories. The topology- and trajectory files used are obtained from the MDAnalysis test suite for two different simulations of the protein AdK. To run the examples see the module `Examples`_ for how to import the files. Here the simplest case of just two instances of :class:`Universe` is illustrated: :: >>> ens1 = Universe(PSF, DCD) >>> ens2 = Universe(PSF, DCD2) >>> CES, details = encore.ces([ens1,ens2]) >>> print(CES) [[ 0. 0.68070702] [ 0.68070702 0. ]] To use a different clustering method, set the parameter clustering_method (Note that the sklearn module must be installed). Likewise, different parameters for the same clustering method can be explored by adding different instances of the same clustering class: :: >>> CES, details = encore.ces([ens1,ens2], clustering_method = [encore.DBSCAN(eps=0.45), encore.DBSCAN(eps=0.50)]) >>> print("eps=0.45: ", CES[0]) eps=0.45: [[ 0. 0.20447236] [ 0.20447236 0. ]] >>> print("eps=0.5: ", CES[1]) eps=0.5: [[ 0. 0.25331629] [ 0.25331629 0. ]]" """ for ensemble in ensembles: ensemble.transfer_to_memory() if calc_diagonal: pairs_indices = list(trm_indices_diag(len(ensembles))) else: pairs_indices = list(trm_indices_nodiag(len(ensembles))) clustering_methods = clustering_method if not hasattr(clustering_method, '__iter__'): clustering_methods = [clustering_method] any_method_accept_distance_matrix = \ np.any([method.accepts_distance_matrix for method in clustering_methods]) all_methods_accept_distance_matrix = \ np.all([method.accepts_distance_matrix for method in clustering_methods]) # Register which ensembles the samples belong to ensemble_assignment = [] for i, ensemble in enumerate(ensembles): ensemble_assignment += [i+1]*len(ensemble.trajectory) # Calculate distance matrix if not provided if any_method_accept_distance_matrix and not distance_matrix: distance_matrix = get_distance_matrix(merge_universes(ensembles), select=select, ncores=ncores) if estimate_error: if any_method_accept_distance_matrix: distance_matrix = \ get_distance_matrix_bootstrap_samples( distance_matrix, ensemble_assignment, samples=bootstrapping_samples, ncores=ncores) if not all_methods_accept_distance_matrix: ensembles_list = [] for i, ensemble in enumerate(ensembles): ensembles_list.append( get_ensemble_bootstrap_samples( ensemble, samples=bootstrapping_samples)) ensembles = [] for j in range(bootstrapping_samples): ensembles.append([]) for i, e in enumerate(ensembles_list): ensembles[-1].append(e[j]) else: # if all methods accept distances matrices, duplicate # ensemble so that it matches size of distance matrices # (no need to resample them since they will not be used) ensembles = [ensembles]*bootstrapping_samples # Call clustering procedure ccs = cluster(ensembles, method= clustering_methods, select=select, distance_matrix = distance_matrix, ncores = ncores, allow_collapsed_result=False) # Do error analysis if estimate_error: k = 0 values = {} avgs = [] stds = [] for i, p in enumerate(clustering_methods): failed_runs = 0 values[i] = [] for j in range(bootstrapping_samples): if ccs[k].clusters is None: failed_runs += 1 k += 1 continue values[i].append(np.zeros((len(ensembles[j]), len(ensembles[j])))) for pair in pairs_indices: # Calculate dJS this_djs = \ clustering_ensemble_similarity(ccs[k], ensembles[j][ pair[0]], pair[0] + 1, ensembles[j][ pair[1]], pair[1] + 1, select=select) values[i][-1][pair[0], pair[1]] = this_djs values[i][-1][pair[1], pair[0]] = this_djs k += 1 outs = np.array(values[i]) avgs.append(np.average(outs, axis=0)) stds.append(np.std(outs, axis=0)) if hasattr(clustering_method, '__iter__'): pass else: avgs = avgs[0] stds = stds[0] return avgs, stds values = [] details = {} for i, p in enumerate(clustering_methods): if ccs[i].clusters is None: continue else: values.append(np.zeros((len(ensembles), len(ensembles)))) for pair in pairs_indices: # Calculate dJS this_val = \ clustering_ensemble_similarity(ccs[i], ensembles[pair[0]], pair[0] + 1, ensembles[pair[1]], pair[1] + 1, select=select) values[-1][pair[0], pair[1]] = this_val values[-1][pair[1], pair[0]] = this_val details['clustering'] = ccs if allow_collapsed_result and not hasattr(clustering_method, '__iter__'): values = values[0] return values, details
[docs]def dres(ensembles, select="name CA", dimensionality_reduction_method = StochasticProximityEmbeddingNative( dimension=3, distance_cutoff = 1.5, min_lam=0.1, max_lam=2.0, ncycle=100, nstep=10000), distance_matrix=None, nsamples=1000, estimate_error=False, bootstrapping_samples=100, ncores=1, calc_diagonal=False, allow_collapsed_result=True): """ Calculates the Dimensional Reduction Ensemble Similarity (DRES) between ensembles using the Jensen-Shannon divergence as described in [Tiberti2015]_. Parameters ---------- ensembles : list List of ensemble objects for similarity measurements select : str, optional Atom selection string in the MDAnalysis format. Default is "name CA" dimensionality_reduction_method : A single or a list of instances of the DimensionalityReductionMethod classes from the dimensionality_reduction module. Different parameters for the same method can be explored by adding different instances of the same dimensionality reduction class. Provided methods are the Stochastic Proximity Embedding (default) and the Principal Component Analysis. distance_matrix : encore.utils.TriangularMatrix conformational distance matrix, It will be calculated on the fly from the ensemble data if it is not provided. nsamples : int, optional Number of samples to be drawn from the ensembles (default is 1000). This is used to resample the density estimates and calculate the Jensen-Shannon divergence between ensembles. estimate_error : bool, optional Whether to perform error estimation (default is False) bootstrapping_samples : int, optional number of samples to be used for estimating error. ncores : int, optional Maximum number of cores to be used (default is 1). calc_diagonal : bool, optional Whether to calculate the diagonal of the similarity scores (i.e. the simlarities of every ensemble against itself). If this is False (default), 0.0 will be used instead. allow_collapsed_result: bool, optional Whether a return value of a list of one value should be collapsed into just the value. Returns ------- dres, details : numpy.array, numpy.array dres contains the similarity values, arranged in numpy.array. If one number of dimensions is provided as an integer, the output will be a 2-dimensional square symmetrical numpy.array. The order of the matrix elements depends on the order of the input ensemble: for instance, if ensemble = [ens1, ens2, ens3] then the matrix elements [0,2] and [2,0] will both contain the similarity value between ensembles ens1 and ens3. Elaborating on the previous example, if *n* ensembles are given and *m* methods are provided the output will be a list of *m* arrays ordered by the input sequence of methods, each with a *n*x*n* symmetrical similarity matrix. details provide an array of the reduced_coordinates. Notes ----- To calculate the similarity, the method first projects the ensembles into lower dimensions by using the Stochastic Proximity Embedding (or others) algorithm. A gaussian kernel-based density estimation method is then used to estimate the probability density for each ensemble which is then used to compute the Jensen-Shannon divergence between each pair of ensembles. In the Jensen-Shannon divergence the upper bound of ln(2) signifies no similarity between the two ensembles, the lower bound, 0.0, signifies identical ensembles. However, due to the stochastic nature of the dimensional reduction in :func:`dres`, two identical ensembles will not necessarily result in an exact 0.0 estimate of the similarity but will be very close. For the same reason, calculating the similarity with the :func:`dres` twice will not result in two identical numbers; small differences have to be expected. Examples -------- To calculate the Dimensional Reduction Ensemble similarity, two ensembles are created as Universe objects from a topology file and two trajectories. The topology- and trajectory files used are obtained from the MDAnalysis test suite for two different simulations of the protein AdK. To run the examples see the module `Examples`_ for how to import the files. Here the simplest case of comparing just two instances of :class:`Universe` is illustrated: :: >>> ens1 = Universe(PSF,DCD) >>> ens2 = Universe(PSF,DCD2) >>> DRES, details = encore.dres([ens1,ens2]) >>> print(DRES) [[ 0. 0.67996043] [ 0.67996043 0. ]] In addition to the quantitative similarity estimate, the dimensional reduction can easily be visualized, see the ``Example`` section in :mod:`MDAnalysis.analysis.encore.dimensionality_reduction.reduce_dimensionality`` To use a different dimensional reduction methods, simply set the parameter dimensionality_reduction_method. Likewise, different parameters for the same clustering method can be explored by adding different instances of the same method class: :: >>> DRES, details = encore.dres([ens1,ens2], dimensionality_reduction_method = encore.PrincipalComponentAnalysis(dimension=2)) >>> print(DRES) [[ 0. 0.69314718] [ 0.69314718 0. ]] """ for ensemble in ensembles: ensemble.transfer_to_memory() if calc_diagonal: pairs_indices = list(trm_indices_diag(len(ensembles))) else: pairs_indices = list(trm_indices_nodiag(len(ensembles))) dimensionality_reduction_methods = dimensionality_reduction_method if not hasattr(dimensionality_reduction_method, '__iter__'): dimensionality_reduction_methods = [dimensionality_reduction_method] any_method_accept_distance_matrix = \ np.any([method.accepts_distance_matrix for method in dimensionality_reduction_methods]) all_methods_accept_distance_matrix = \ np.all([method.accepts_distance_matrix for method in dimensionality_reduction_methods]) # Register which ensembles the samples belong to ensemble_assignment = [] for i, ensemble in enumerate(ensembles): ensemble_assignment += [i+1]*len(ensemble.trajectory) # Calculate distance matrix if not provided if any_method_accept_distance_matrix and not distance_matrix: distance_matrix = get_distance_matrix(merge_universes(ensembles), select=select, ncores=ncores) if estimate_error: if any_method_accept_distance_matrix: distance_matrix = \ get_distance_matrix_bootstrap_samples( distance_matrix, ensemble_assignment, samples=bootstrapping_samples, ncores=ncores) if not all_methods_accept_distance_matrix: ensembles_list = [] for i, ensemble in enumerate(ensembles): ensembles_list.append( get_ensemble_bootstrap_samples( ensemble, samples=bootstrapping_samples)) ensembles = [] for j in range(bootstrapping_samples): ensembles.append(ensembles_list[i, j] for i in range(ensembles_list.shape[0])) else: # if all methods accept distances matrices, duplicate # ensemble so that it matches size of distance matrices # (no need to resample them since they will not be used) ensembles = [ensembles] * bootstrapping_samples # Call dimensionality reduction procedure coordinates, dim_red_details = reduce_dimensionality( ensembles, method=dimensionality_reduction_methods, select=select, distance_matrix = distance_matrix, ncores = ncores, allow_collapsed_result = False) details = {} details["reduced_coordinates"] = coordinates details["dimensionality_reduction_details"] = dim_red_details if estimate_error: k = 0 values = {} avgs = [] stds = [] for i,method in enumerate(dimensionality_reduction_methods): values[i] = [] for j in range(bootstrapping_samples): values[i].append(np.zeros((len(ensembles[j]), len(ensembles[j])))) kdes, resamples, embedded_ensembles = gen_kde_pdfs( coordinates[k], ensemble_assignment, len(ensembles[j]), nsamples=nsamples) for pair in pairs_indices: this_value = dimred_ensemble_similarity(kdes[pair[0]], resamples[pair[0]], kdes[pair[1]], resamples[pair[1]]) values[i][-1][pair[0], pair[1]] = this_value values[i][-1][pair[1], pair[0]] = this_value k += 1 outs = np.array(values[i]) avgs.append(np.average(outs, axis=0)) stds.append(np.std(outs, axis=0)) if hasattr(dimensionality_reduction_method, '__iter__'): pass else: avgs = avgs[0] stds = stds[0] return avgs, stds values = [] for i,method in enumerate(dimensionality_reduction_methods): values.append(np.zeros((len(ensembles), len(ensembles)))) kdes, resamples, embedded_ensembles = gen_kde_pdfs(coordinates[i], ensemble_assignment, len(ensembles), nsamples=nsamples) for pair in pairs_indices: this_value = dimred_ensemble_similarity(kdes[pair[0]], resamples[pair[0]], kdes[pair[1]], resamples[pair[1]]) values[-1][pair[0], pair[1]] = this_value values[-1][pair[1], pair[0]] = this_value if allow_collapsed_result and not hasattr(dimensionality_reduction_method, '__iter__'): values = values[0] return values, details
[docs]def ces_convergence(original_ensemble, window_size, select="name CA", clustering_method=AffinityPropagationNative( preference=-1.0, max_iter=500, convergence_iter=50, damping=0.9, add_noise=True), ncores=1): """ Use the CES to evaluate the convergence of the ensemble/trajectory. CES will be calculated between the whole trajectory contained in an ensemble and windows of such trajectory of increasing sizes, so that the similarity values should gradually drop to zero. The rate at which the value reach zero will be indicative of how much the trajectory keeps on resampling the same regions of the conformational space, and therefore of convergence. Parameters ---------- original_ensemble : :class:`~MDAnalysis.core.universe.Universe` object ensemble containing the trajectory whose convergence has to estimated window_size : int Size of window to be used, in number of frames select : str, optional Atom selection string in the MDAnalysis format. Default is "name CA" clustering_method : MDAnalysis.analysis.encore.clustering.ClusteringMethod A single or a list of instances of the ClusteringMethod classes from the clustering module. Different parameters for the same clustering method can be explored by adding different instances of the same clustering class. ncores : int, optional Maximum number of cores to be used (default is 1). Returns ------- out : np.array array of shape (number_of_frames / window_size, preference_values). Example -------- To calculate the convergence of a trajectory using the clustering ensemble similarity method a Universe object is created from a topology file and the trajectory. The topology- and trajectory files used are obtained from the MDAnalysis test suite for two different simulations of the protein AdK. To run the examples see the module `Examples`_ for how to import the files. Here the simplest case of evaluating the convergence is illustrated by splitting the trajectory into a window_size of 10 frames : :: >>> ens1 = Universe(PSF,DCD) >>> ces_conv = encore.ces_convergence(ens1, 10) >>> print(ces_conv) [[ 0.48194205] [ 0.40284672] [ 0.31699026] [ 0.25220447] [ 0.19829817] [ 0.14642725] [ 0.09911411] [ 0.05667391] [ 0. ]] """ ensembles = prepare_ensembles_for_convergence_increasing_window( original_ensemble, window_size, select=select) ccs = cluster(ensembles, select=select, method=clustering_method, allow_collapsed_result=False, ncores=ncores) out = [] for cc in ccs: if cc.clusters is None: continue out.append(np.zeros(len(ensembles))) for j, ensemble in enumerate(ensembles): out[-1][j] = cumulative_clustering_ensemble_similarity( cc, len(ensembles), j + 1) out = np.array(out).T return out
[docs]def dres_convergence(original_ensemble, window_size, select="name CA", dimensionality_reduction_method = \ StochasticProximityEmbeddingNative( dimension=3, distance_cutoff=1.5, min_lam=0.1, max_lam=2.0, ncycle=100, nstep=10000 ), nsamples=1000, ncores=1): """ Use the DRES to evaluate the convergence of the ensemble/trajectory. DRES will be calculated between the whole trajectory contained in an ensemble and windows of such trajectory of increasing sizes, so that the similarity values should gradually drop to zero. The rate at which the value reach zero will be indicative of how much the trajectory keeps on resampling the same ares of the conformational space, and therefore of convergence. Parameters ---------- original_ensemble : :class:`~MDAnalysis.core.universe.Universe` object ensemble containing the trajectory whose convergence has to estimated window_size : int Size of window to be used, in number of frames select : str, optional Atom selection string in the MDAnalysis format. Default is "name CA" dimensionality_reduction_method : A single or a list of instances of the DimensionalityReductionMethod classes from the dimensionality_reduction module. Different parameters for the same method can be explored by adding different instances of the same dimensionality reduction class. nsamples : int, optional Number of samples to be drawn from the ensembles (default is 1000). This is akin to the nsamples parameter of dres(). ncores : int, optional Maximum number of cores to be used (default is 1). Returns ------- out : np.array array of shape (number_of_frames / window_size, preference_values). Example -------- To calculate the convergence of a trajectory using the DRES method, a Universe object is created from a topology file and the trajectory. The topology- and trajectory files used are obtained from the MDAnalysis test suite for two different simulations of the protein AdK. To run the examples see the module `Examples`_ for how to import the files. Here the simplest case of evaluating the convergence is illustrated by splitting the trajectory into a window_size of 10 frames : :: >>> ens1 = Universe(PSF,DCD) >>> dres_conv = encore.dres_convergence(ens1, 10) >>> print(dres_conv) [[ 0.5295528 ] [ 0.40716539] [ 0.31158669] [ 0.25314041] [ 0.20447271] [ 0.13212364] [ 0.06979114] [ 0.05214759] [ 0. ]] Here, the rate at which the values reach zero will be indicative of how much the trajectory keeps on resampling the same ares of the conformational space, and therefore of convergence. """ ensembles = prepare_ensembles_for_convergence_increasing_window( original_ensemble, window_size, select=select) coordinates, dimred_details = \ reduce_dimensionality( ensembles, select=select, method=dimensionality_reduction_method, allow_collapsed_result=False, ncores=ncores) ensemble_assignment = [] for i, ensemble in enumerate(ensembles): ensemble_assignment += [i+1]*len(ensemble.trajectory) ensemble_assignment = np.array(ensemble_assignment) out = [] for i, _ in enumerate(coordinates): out.append(np.zeros(len(ensembles))) kdes, resamples, embedded_ensembles = \ cumulative_gen_kde_pdfs( coordinates[i], ensemble_assignment=ensemble_assignment, nensembles=len(ensembles), nsamples=nsamples) for j, ensemble in enumerate(ensembles): out[-1][j] = dimred_ensemble_similarity(kdes[-1], resamples[-1], kdes[j], resamples[j]) out = np.array(out).T return out