# 4.9.2. Principal Component Analysis (PCA) — MDAnalysis.analysis.pca¶

Authors: John Detlefs 2016 GNU Public License v3

New in version 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 PCA Tutorial.

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

For each eigenvector, its eigenvalue is the variance that the eigenvector explains. Stored in PCA.results.cumulated_variance, a ratio for each number of eigenvectors up to index $$i$$ is provided to quickly find out how many principal components are needed to explain the amount of variance reflected by those $$i$$ eigenvectors. For most data, PCA.results.cumulated_variance will be approximately equal to one for some $$n$$ 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 PCA Tutorial shows how to perform Principal Component Analysis.

## 4.9.2.1. PCA Tutorial¶

The example uses files provided as part of the MDAnalysis test suite (in the variables PSF and 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 Analyis by using the 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 PCA.results.cumulated_variance. The value at the ith index of 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.

## 4.9.2.2. Classes and Functions¶

class MDAnalysis.analysis.pca.PCA(universe, select='all', align=False, mean=None, n_components=None, **kwargs)[source]

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 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.
results.p_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].

New in version 2.0.0.

Type: array, (n_atoms * 3, n_components)
p_components

Alias to the results.p_components.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.p_components instead.

Type: array, (n_atoms * 3, n_components)
results.variance

Raw variance explained by each eigenvector of the covariance matrix.

New in version 2.0.0.

Type: array (n_components, )
variance

Alias to the results.variance.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.variance instead.

Type: array (n_components, )
results.cumulated_variance

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.

New in version 2.0.0.

Type: array, (n_components, )
cumulated_variance

Alias to the results.cumulated_variance.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.cumulated_variance instead.

Type: array, (n_components, )
transform(atomgroup, n_components=None)[source]

Take an atomgroup or universe with the same number of atoms as was used for the calculation in PCA.run(), and project it onto the principal components.

Notes

Computation can be sped up by supplying precalculated mean positions.

Changed in version 0.19.0: The start frame is used when performing selections and calculating mean positions. Previously the 0th frame was always used.

Changed in version 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.

Changed in version 2.0.0: mean_atoms removed, as this did not reliably contain the mean positions. mean input now accepts coordinate arrays instead of atomgroup. p_components, variance and cumulated_variance are now stored in a MDAnalysis.analysis.base.Results instance.

cumulative_overlap(other, i=0, n_components=None)[source]

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 (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. 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. float

New in version 1.0.0.

rmsip(other, n_components=None)[source]

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.

Parameters: other (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. Root mean square inner product of the selected subspaces. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical. float

New in version 1.0.0.

run(start=None, stop=None, step=None, verbose=None)

Perform the calculation

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 verbose (bool, optional) – Turn on verbosity
transform(atomgroup, n_components=None, start=None, stop=None, step=None)[source]

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). pca_space array, shape (n_frames, n_components)

Changed in version 0.19.0: Transform now requires that run() has been called before, otherwise a ValueError is raised.

MDAnalysis.analysis.pca.cosine_content(pca_space, i)[source]

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 [BerkHess1].

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. 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

 [BerkHess1] Berk Hess. Convergence of sampling in protein simulations. Phys. Rev. E 65, 031910 (2002).
MDAnalysis.analysis.pca.rmsip(a, b, n_components=None)[source]

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.

Parameters: a (array, shape (n_components, n_features)) – The first subspace. 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. n_components (int or tuple of ints, optional) – number of components to compute for the inner products. None computes all of them. Root mean square inner product of the selected subspaces. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical. float

New in version 1.0.0.

MDAnalysis.analysis.pca.cumulative_overlap(a, b, i=0, n_components=None)[source]

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. 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. float

New in version 1.0.0.