Source code for MDAnalysis.analysis.pca

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r"""Principal Component Analysis (PCA) --- :mod:`MDAnalysis.analysis.pca`
=====================================================================

:Authors: John Detlefs
:Year: 2016
:Copyright: GNU Public License v3

.. versionadded:: 0.16.0

This module contains the linear dimensions reduction method Principal Component
Analysis (PCA). PCA sorts a simulation into 3N directions of descending
variance, with N being the number of atoms. These directions are called
the principal components. The dimensions to be analyzed are reduced by only
looking at a few projections of the first principal components. To learn how to
run a Principal Component Analysis, please refer to the :ref:`PCA-tutorial`.

The PCA problem is solved by solving the eigenvalue problem of the covariance
matrix, a :math:`3N \times 3N` matrix where the element :math:`(i, j)` is the
covariance between coordinates :math:`i` and :math:`j`. The principal
components are the eigenvectors of this matrix.

For each eigenvector, its eigenvalue is the variance that the eigenvector
explains. Stored in :attr:`PCA.results.cumulated_variance`, a ratio for each
number of eigenvectors up to index :math:`i` is provided to quickly find out
how many principal components are needed to explain the amount of variance
reflected by those :math:`i` eigenvectors. For most data,
:attr:`PCA.results.cumulated_variance`
will be approximately equal to one for some :math:`n` that is significantly
smaller than the total number of components. These are the components of
interest given by Principal Component Analysis.

From here, we can project a trajectory onto these principal components and
attempt to retrieve some structure from our high dimensional data.

For a basic introduction to the module, the :ref:`PCA-tutorial` shows how
to perform Principal Component Analysis.

.. _PCA-tutorial:

PCA 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
PCA class.

First load all modules and test data::

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


Given a universe containing trajectory data we can perform Principal Component
Analysis by using the class :class:`PCA` and retrieving the principal
components.::

    u = mda.Universe(PSF, DCD)
    PSF_pca = pca.PCA(u, select='backbone')
    PSF_pca.run()


Inspect the components to determine the principal components you would like
to retain. The choice is arbitrary, but I will stop when 95 percent of the
variance is explained by the components. This cumulated variance by the
components is conveniently stored in the one-dimensional array attribute
:attr:`PCA.results.cumulated_variance`. The value at the ith index of
:attr:`PCA.results.cumulated_variance` is the sum of the variances from 0 to
i.::

    n_pcs = np.where(PSF_pca.results.cumulated_variance > 0.95)[0][0]
    atomgroup = u.select_atoms('backbone')
    pca_space = PSF_pca.transform(atomgroup, n_components=n_pcs)


From here, inspection of the ``pca_space`` and conclusions to be drawn from the
data are left to the user.

Classes and Functions
---------------------

.. autoclass:: PCA
   :members:
   :inherited-members:

.. autofunction:: cosine_content

.. autofunction:: rmsip

.. autofunction:: cumulative_overlap

"""
import warnings

import numpy as np
import scipy.integrate

from MDAnalysis import Universe
from MDAnalysis.analysis.align import _fit_to
from MDAnalysis.lib.log import ProgressBar

from ..lib import util
from ..due import due, Doi
from .base import AnalysisBase


[docs] class PCA(AnalysisBase): """Principal component analysis on an MD trajectory. After initializing and calling method with a universe or an atom group, principal components ordering the atom coordinate data by decreasing variance will be available for analysis. As an example::: pca = PCA(universe, select='backbone').run() pca_space = pca.transform(universe.select_atoms('backbone'), 3) generates the principal components of the backbone of the atomgroup and then transforms those atomgroup coordinates by the direction of those variances. Please refer to the :ref:`PCA-tutorial` for more detailed instructions. Parameters ---------- universe : Universe Universe select : string, optional A valid selection statement for choosing a subset of atoms from the atomgroup. align : boolean, optional If True, the trajectory will be aligned to a reference structure. mean : array_like, optional Optional reference positions to be be used as the mean of the covariance matrix. n_components : int, optional The number of principal components to be saved, default saves all principal components verbose : bool (optional) Show detailed progress of the calculation if set to ``True``. Attributes ---------- results.p_components: array, (n_atoms * 3, n_components) Principal components of the feature space, representing the directions of maximum variance in the data. The column vector p_components[:, i] is the eigenvector corresponding to the variance[i]. .. versionadded:: 2.0.0 p_components: array, (n_atoms * 3, n_components) Alias to the :attr:`results.p_components`. .. deprecated:: 2.0.0 Will be removed in MDAnalysis 3.0.0. Please use :attr:`results.p_components` instead. results.variance : array (n_components, ) Raw variance explained by each eigenvector of the covariance matrix. .. versionadded:: 2.0.0 variance : array (n_components, ) Alias to the :attr:`results.variance`. .. deprecated:: 2.0.0 Will be removed in MDAnalysis 3.0.0. Please use :attr:`results.variance` instead. results.cumulated_variance : array, (n_components, ) Percentage of variance explained by the selected components and the sum of the components preceding it. If a subset of components is not chosen then all components are stored and the cumulated variance will converge to 1. .. versionadded:: 2.0.0 cumulated_variance : array, (n_components, ) Alias to the :attr:`results.cumulated_variance`. .. deprecated:: 2.0.0 Will be removed in MDAnalysis 3.0.0. Please use :attr:`results.cumulated_variance` instead. Notes ----- Computation can be sped up by supplying precalculated mean positions. .. versionchanged:: 0.19.0 The start frame is used when performing selections and calculating mean positions. Previously the 0th frame was always used. .. versionchanged:: 1.0.0 ``n_components`` now limits the correct axis of ``p_components``. ``cumulated_variance`` now accurately represents the contribution of each principal component and does not change when ``n_components`` is given. If ``n_components`` is not None or is less than the number of ``p_components``, ``cumulated_variance`` will not sum to 1. ``align=True`` now correctly aligns the trajectory and computes the correct means and covariance matrix. .. versionchanged:: 2.0.0 ``mean_atoms`` removed, as this did not reliably contain the mean positions. ``mean`` input now accepts coordinate arrays instead of atomgroup. :attr:`p_components`, :attr:`variance` and :attr:`cumulated_variance` are now stored in a :class:`MDAnalysis.analysis.base.Results` instance. """ def __init__(self, universe, select='all', align=False, mean=None, n_components=None, **kwargs): super(PCA, self).__init__(universe.trajectory, **kwargs) self._u = universe # for transform function self.align = align self._calculated = False self._n_components = n_components self._select = select self._mean = mean def _prepare(self): # access start index self._u.trajectory[self.start] # reference will be start index self._reference = self._u.select_atoms(self._select) self._atoms = self._u.select_atoms(self._select) self._n_atoms = self._atoms.n_atoms if self._mean is None: self.mean = np.zeros((self._n_atoms, 3)) self._calc_mean = True else: self.mean = np.asarray(self._mean) if self.mean.shape[0] != self._n_atoms: raise ValueError('Number of atoms in reference ({}) does ' 'not match number of atoms in the ' 'selection ({})'.format(self._n_atoms, self.mean.shape[0])) self._calc_mean = False if self.n_frames == 1: raise ValueError('No covariance information can be gathered from a' 'single trajectory frame.\n') n_dim = self._n_atoms * 3 self.cov = np.zeros((n_dim, n_dim)) self._ref_atom_positions = self._reference.positions self._ref_cog = self._reference.center_of_geometry() self._ref_atom_positions -= self._ref_cog if self._calc_mean: for ts in ProgressBar(self._u.trajectory[self.start:self.stop:self.step], verbose=self._verbose, desc="Mean Calculation"): if self.align: mobile_cog = self._atoms.center_of_geometry() mobile_atoms, old_rmsd = _fit_to(self._atoms.positions - mobile_cog, self._ref_atom_positions, self._atoms, mobile_com=mobile_cog, ref_com=self._ref_cog) self.mean += self._atoms.positions self.mean /= self.n_frames self._xmean = np.ravel(self.mean) def _single_frame(self): if self.align: mobile_cog = self._atoms.center_of_geometry() mobile_atoms, old_rmsd = _fit_to(self._atoms.positions - mobile_cog, self._ref_atom_positions, self._atoms, mobile_com=mobile_cog, ref_com=self._ref_cog) # now all structures are aligned to reference x = mobile_atoms.positions.ravel() else: x = self._atoms.positions.ravel() x -= self._xmean self.cov += np.dot(x[:, np.newaxis], x[:, np.newaxis].T) def _conclude(self): self.cov /= self.n_frames - 1 e_vals, e_vects = np.linalg.eig(self.cov) sort_idx = np.argsort(e_vals)[::-1] self._variance = e_vals[sort_idx] self._p_components = e_vects[:, sort_idx] self._calculated = True self.n_components = self._n_components @property def p_components(self): wmsg = ("The `p_components` attribute was deprecated in " "MDAnalysis 2.0.0 and will be removed in MDAnalysis 3.0.0. " "Please use `results.p_components` instead.") warnings.warn(wmsg, DeprecationWarning) return self.results.p_components @property def variance(self): wmsg = ("The `variance` attribute was deprecated in " "MDAnalysis 2.0.0 and will be removed in MDAnalysis 3.0.0. " "Please use `results.variance` instead.") warnings.warn(wmsg, DeprecationWarning) return self.results.variance @property def cumulated_variance(self): wmsg = ("The `cumulated_variance` attribute was deprecated in " "MDAnalysis 2.0.0 and will be removed in MDAnalysis 3.0.0. " "Please use `results.cumulated_variance` instead.") warnings.warn(wmsg, DeprecationWarning) return self.results.cumulated_variance @property def n_components(self): return self._n_components @n_components.setter def n_components(self, n): if self._calculated: if n is None: n = len(self._variance) self.results.variance = self._variance[:n] self.results.cumulated_variance = (np.cumsum(self._variance) / np.sum(self._variance))[:n] self.results.p_components = self._p_components[:, :n] self._n_components = n
[docs] def transform(self, atomgroup, n_components=None, start=None, stop=None, step=None): """Apply the dimensionality reduction on a trajectory Parameters ---------- atomgroup : AtomGroup or Universe The AtomGroup or Universe containing atoms to be PCA transformed. n_components : int, optional The number of components to be projected onto. The default ``None`` maps onto all components. start : int, optional The frame to start on for the PCA transform. The default ``None`` becomes 0, the first frame index. stop : int, optional Frame index to stop PCA transform. The default ``None`` becomes the total number of frames in the trajectory. Iteration stops *before* this frame number, which means that the trajectory would be read until the end. step : int, optional Include every `step` frames in the PCA transform. If set to ``None`` (the default) then every frame is analyzed (i.e., same as ``step=1``). Returns ------- pca_space : array, shape (n_frames, n_components) .. versionchanged:: 0.19.0 Transform now requires that :meth:`run` has been called before, otherwise a :exc:`ValueError` is raised. """ if not self._calculated: raise ValueError('Call run() on the PCA before using transform') if isinstance(atomgroup, Universe): atomgroup = atomgroup.atoms if(self._n_atoms != atomgroup.n_atoms): raise ValueError('PCA has been fit for' '{} atoms. Your atomgroup' 'has {} atoms'.format(self._n_atoms, atomgroup.n_atoms)) if not (self._atoms.types == atomgroup.types).all(): warnings.warn('Atom types do not match with types used to fit PCA') traj = atomgroup.universe.trajectory start, stop, step = traj.check_slice_indices(start, stop, step) n_frames = len(range(start, stop, step)) dim = (n_components if n_components is not None else self.results.p_components.shape[1]) dot = np.zeros((n_frames, dim)) for i, ts in enumerate(traj[start:stop:step]): xyz = atomgroup.positions.ravel() - self._xmean dot[i] = np.dot(xyz, self._p_components[:, :dim]) return dot
[docs] def project_single_frame(self, components=None, group=None, anchor=None): r"""Computes a function to project structures onto selected PCs Applies Inverse-PCA transform to the PCA atomgroup. Optionally, calculates one displacement vector per residue to extrapolate the transform to atoms not in the PCA atomgroup. Parameters ---------- components : int, array, optional Components to be projected onto. The default ``None`` maps onto all components. group : AtomGroup, optional The AtomGroup containing atoms to be projected. The projection applies to whole residues in ``group``. The atoms in the PCA class are not affected by this argument. The default ``None`` does not extrapolate the projection to non-PCA atoms. anchor : string, optional The string to select the PCA atom whose displacement vector is applied to non-PCA atoms in a residue. The ``anchor`` selection is applied to ``group``.The resulting atomselection must have exactly one PCA atom in each residue of ``group``. The default ``None`` does not extrapolate the projection to non-PCA atoms. Returns ------- function The resulting function f(ts) takes as input a :class:`~MDAnalysis.coordinates.timestep.Timestep` ts, and returns ts with the projected structure .. warning:: The transformation function takes a :class:`Timestep` as input because this is required for :ref:`transformations`. However, the inverse-PCA transformation is applied on the atoms of the Universe that was used for the PCA. It is *expected* that the `ts` is from the same Universe but this is currently not checked. Notes ----- When the PCA class is run for an atomgroup, the principal components are cached. The trajectory can then be projected onto one or more of these principal components. Since the principal components are sorted in the order of decreasing explained variance, the first few components capture the essential molecular motion. If N is the number of atoms in the PCA group, each component has the length 3N. A PCA score :math:`w\_i`, along component :math:`u\_i`, is calculated for a set of coordinates :math:`(r(t))` of the same atoms. The PCA scores are then used to transform the structure, :math:`(r(t))` at a timestep, back to the original space. .. math:: w_{i}(t) = ({\textbf r}(t) - \bar{{\textbf r}}) \cdot {\textbf u}_i \\ {\textbf r'}(t) = (w_{i}(t) \cdot {\textbf u}_i^T) + \bar{{\textbf r}} For each residue, the projection can be extended to atoms that were not part of PCA by applying the displacement vector of a PCA atom to all the atoms in the residue. This could be useful to preserve the bond distance between a PCA atom and other non-PCA atoms in a residue. If there are r residues and n non-PCA atoms in total, the displacement vector has the size 3r. This needs to be broadcasted to a size 3n. An extrapolation trick is used to shape the array, since going over each residue for each frame can be expensive. Non-PCA atoms' displacement vector is calculated with fancy indexing on the anchors' displacement vector. `index_extrapolate` saves which atoms belong to which anchors. If there are two non-PCA atoms in the first anchor's residue and three in the second anchor's residue, `index_extrapolate` is [0, 0, 1, 1, 1] Examples -------- Run PCA class before using this function. For backbone PCA, run:: pca = PCA(universe, select='backbone').run() Obtain a transformation function to project the backbone trajectory onto the first principal component:: project = pca.project_single_frame(components=0) To project onto the first two components, run:: project = pca.project_single_frame(components=[0,1]) Alternatively, the transformation can be applied to PCA atoms and extrapolated to other atoms according to the CA atom's translation in each residue:: all = u.select_atoms('all') project = pca.project_single_frame(components=0, group=all, anchor='name CA') Finally, apply the transformation function to a timestep:: project(u.trajectory.ts) or apply the projection to the universe:: u.trajectory.add_transformations(project) .. versionadded:: 2.2.0 """ if not self._calculated: raise ValueError('Call run() on the PCA before projecting') if group is not None: if anchor is None: raise ValueError("'anchor' cannot be 'None'" + " if 'group' is not 'None'") anchors = group.select_atoms(anchor) anchors_res_ids = anchors.resindices if np.unique(anchors_res_ids).size != anchors_res_ids.size: raise ValueError("More than one 'anchor' found in residues") if not np.isin(group.resindices, anchors_res_ids).all(): raise ValueError("Some residues in 'group'" + " do not have an 'anchor'") if not anchors.issubset(self._atoms): raise ValueError("Some 'anchors' are not part of PCA class") # non_pca has "all" the atoms in residues of `group`. This makes # sure that extrapolation works on residues, not random atoms. non_pca = group.residues.atoms - self._atoms pca_res_indices, pca_res_counts = np.unique( self._atoms.resindices, return_counts=True) non_pca_atoms = np.array([], dtype=int) for res in group.residues: # n_common is the number of pca atoms in a residue n_common = pca_res_counts[np.where( pca_res_indices == res.resindex)][0] non_pca_atoms = np.append(non_pca_atoms, res.atoms.n_atoms - n_common) # index_extrapolate records the anchor number for each non-PCA atom index_extrapolate = np.repeat(np.arange(anchors.atoms.n_atoms), non_pca_atoms) if components is None: components = np.arange(self.results.p_components.shape[1]) def wrapped(ts): """Projects a timestep""" if group is not None: anchors_coords_old = anchors.positions xyz = self._atoms.positions.ravel() - self._xmean self._atoms.positions = np.reshape( (np.dot(np.dot(xyz, self._p_components[:, components]), self._p_components[:, components].T) + self._xmean), (-1, 3) ) if group is not None: non_pca.positions += (anchors.positions - anchors_coords_old)[index_extrapolate] return ts return wrapped
[docs] @due.dcite( Doi('10.1002/(SICI)1097-0134(19990901)36:4<419::AID-PROT5>3.0.CO;2-U'), Doi('10.1529/biophysj.104.052449'), description="RMSIP", path='MDAnalysis.analysis.pca', ) def rmsip(self, other, n_components=None): """Compute the root mean square inner product between subspaces. This is only symmetric if the number of components is the same for both instances. The RMSIP effectively measures how correlated the vectors of this instance are to those of ``other``. Please cite [Amadei1999]_ and [Leo-Macias2004]_ if you use this function. Parameters ---------- other : :class:`~MDAnalysis.analysis.pca.PCA` Another PCA class. This must have already been run. n_components : int or tuple of ints, optional number of components to compute for the inner products. ``None`` computes all of them. Returns ------- float: Root mean square inner product of the selected subspaces. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical. Examples -------- .. testsetup:: >>> import MDAnalysis as mda >>> import MDAnalysis.analysis.pca as pca >>> from MDAnalysis.tests.datafiles import PSF, DCD >>> u = mda.Universe(PSF, DCD) You can compare the RMSIP between different intervals of the same trajectory. For example, to compare similarity within the top three principal components: .. doctest:: >>> first_interval = pca.PCA(u, select="backbone").run(start=0, stop=25) >>> second_interval = pca.PCA(u, select="backbone").run(start=25, stop=50) >>> last_interval = pca.PCA(u, select="backbone").run(start=75) >>> first_interval.rmsip(second_interval, n_components=3) 0.38147609331128324 >>> first_interval.rmsip(last_interval, n_components=3) 0.17478244043688052 See also -------- :func:`~MDAnalysis.analysis.pca.rmsip` .. versionadded:: 1.0.0 """ try: a = self.results.p_components except AttributeError: raise ValueError('Call run() on the PCA before using rmsip') try: b = other.results.p_components except AttributeError: if isinstance(other, type(self)): raise ValueError( 'Call run() on the other PCA before using rmsip') else: raise ValueError('other must be another PCA class') return rmsip(a.T, b.T, n_components=n_components)
[docs] @due.dcite( Doi('10.1016/j.str.2007.12.011'), description="Cumulative overlap", path='MDAnalysis.analysis.pca', ) def cumulative_overlap(self, other, i=0, n_components=None): """Compute the cumulative overlap of a vector in a subspace. This is not symmetric. The cumulative overlap measures the overlap of the chosen vector in this instance, in the ``other`` subspace. Please cite [Yang2008]_ if you use this function. Parameters ---------- other : :class:`~MDAnalysis.analysis.pca.PCA` Another PCA class. This must have already been run. i : int, optional The index of eigenvector to be analysed. n_components : int, optional number of components in ``other`` to compute for the cumulative overlap. ``None`` computes all of them. Returns ------- float: Cumulative overlap of the chosen vector in this instance to the ``other`` subspace. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical. See also -------- :func:`~MDAnalysis.analysis.pca.cumulative_overlap` .. versionadded:: 1.0.0 """ try: a = self.results.p_components except AttributeError: raise ValueError( 'Call run() on the PCA before using cumulative_overlap') try: b = other.results.p_components except AttributeError: if isinstance(other, type(self)): raise ValueError( 'Call run() on the other PCA before using cumulative_overlap') else: raise ValueError('other must be another PCA class') return cumulative_overlap(a.T, b.T, i=i, n_components=n_components)
[docs] def cosine_content(pca_space, i): """Measure the cosine content of the PCA projection. The cosine content of pca projections can be used as an indicator if a simulation is converged. Values close to 1 are an indicator that the simulation isn't converged. For values below 0.7 no statement can be made. If you use this function please cite :footcite:p:`BerkHess2002`. Parameters ---------- pca_space : array, shape (number of frames, number of components) The PCA space to be analyzed. i : int The index of the pca_component projection to be analyzed. Returns ------- A float reflecting the cosine content of the ith projection in the PCA space. The output is bounded by 0 and 1, with 1 reflecting an agreement with cosine while 0 reflects complete disagreement. References ---------- .. footbibliography:: """ t = np.arange(len(pca_space)) T = len(pca_space) cos = np.cos(np.pi * t * (i + 1) / T) return ((2.0 / T) * (scipy.integrate.simpson(cos*pca_space[:, i])) ** 2 / scipy.integrate.simpson(pca_space[:, i] ** 2))
[docs] @due.dcite( Doi('10.1002/(SICI)1097-0134(19990901)36:4<419::AID-PROT5>3.0.CO;2-U'), Doi('10.1529/biophysj.104.052449'), description="RMSIP", path='MDAnalysis.analysis.pca', ) def rmsip(a, b, n_components=None): """Compute the root mean square inner product between subspaces. This is only symmetric if the number of components is the same for ``a`` and ``b``. The RMSIP effectively measures how correlated the vectors of ``a`` are to those of ``b``. Please cite [Amadei1999]_ and [Leo-Macias2004]_ if you use this function. Parameters ---------- a : array, shape (n_components, n_features) The first subspace. Must have the same number of features as ``b``. If you are using the results of :class:`~MDAnalysis.analysis.pca.PCA`, this is the TRANSPOSE of ``p_components`` (i.e. ``p_components.T``). b : array, shape (n_components, n_features) The second subspace. Must have the same number of features as ``a``. If you are using the results of :class:`~MDAnalysis.analysis.pca.PCA`, this is the TRANSPOSE of ``p_components`` (i.e. ``p_components.T``). n_components : int or tuple of ints, optional number of components to compute for the inner products. ``None`` computes all of them. Returns ------- float: Root mean square inner product of the selected subspaces. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical. Examples -------- .. testsetup:: >>> import MDAnalysis as mda >>> import MDAnalysis.analysis.pca as pca >>> from MDAnalysis.tests.datafiles import PSF, DCD >>> u = mda.Universe(PSF, DCD) You can compare the RMSIP between different intervals of the same trajectory. For example, to compare similarity within the top three principal components: .. doctest:: >>> first_interval = pca.PCA(u, select="backbone").run(start=0, stop=25) >>> second_interval = pca.PCA(u, select="backbone").run(start=25, stop=50) >>> last_interval = pca.PCA(u, select="backbone").run(start=75) >>> pca.rmsip(first_interval.results.p_components.T, ... second_interval.results.p_components.T, ... n_components=3) 0.38147609331128324 >>> pca.rmsip(first_interval.results.p_components.T, ... last_interval.results.p_components.T, ... n_components=3) 0.17478244043688052 .. versionadded:: 1.0.0 """ n_components = util.asiterable(n_components) if len(n_components) == 1: n_a = n_b = n_components[0] elif len(n_components) == 2: n_a, n_b = n_components else: raise ValueError('Too many values provided for n_components') if n_a is None: n_a = len(a) if n_b is None: n_b = len(b) sip = np.matmul(a[:n_a], b[:n_b].T) ** 2 msip = sip.sum()/n_a return msip**0.5
[docs] @due.dcite( Doi('10.1016/j.str.2007.12.011'), description="Cumulative overlap", path='MDAnalysis.analysis.pca', ) def cumulative_overlap(a, b, i=0, n_components=None): """Compute the cumulative overlap of a vector in a subspace. This is not symmetric. The cumulative overlap measures the overlap of the chosen vector in ``a``, in the ``b`` subspace. Please cite [Yang2008]_ if you use this function. Parameters ---------- a : array, shape (n_components, n_features) or vector, length n_features The first subspace containing the vector of interest. Alternatively, the actual vector. Must have the same number of features as ``b``. b : array, shape (n_components, n_features) The second subspace. Must have the same number of features as ``a``. i : int, optional The index of eigenvector to be analysed. n_components : int, optional number of components in ``b`` to compute for the cumulative overlap. ``None`` computes all of them. Returns ------- float: Cumulative overlap of the chosen vector in ``a`` to the ``b`` subspace. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical. .. versionadded:: 1.0.0 """ if len(a.shape) < len(b.shape): a = a[np.newaxis, :] vec = a[i][np.newaxis, :] vec_norm = (vec**2).sum() ** 0.5 if n_components is None: n_components = len(b) b = b[:n_components] b_norms = (b**2).sum(axis=1) ** 0.5 o = np.abs(np.matmul(vec, b.T)) / (b_norms*vec_norm) return (o**2).sum() ** 0.5