pyemma.coordinates.pca

pyemma.coordinates.pca(data=None, dim=-1, var_cutoff=0.95, stride=1, mean=None)

Principal Component Analysis (PCA).

PCA is a linear transformation method that finds coordinates of maximal variance. A linear projection onto the principal components thus makes a minimal error in terms of variation in the data. Note, however, that this method is not optimal for Markov model construction because for that purpose the main objective is to preserve the slow processes which can sometimes be associated with small variance.

It estimates a PCA transformation from data. When input data is given as an argument, the estimation will be carried out right away, and the resulting object can be used to obtain eigenvalues, eigenvectors or project input data onto the principal components. If data is not given, this object is an empty estimator and can be put into a pipeline() in order to use PCA in streaming mode.

Parameters:
  • data (ndarray (T, d) or list of ndarray (T_i, d) or a reader created by) – source function data array or list of data arrays. T or T_i are the number of time steps in a trajectory. When data is given, the PCA is immediately parametrized by estimating the covariance matrix and computing its eigenvectors.
  • dim (int, optional, default -1) – the number of dimensions (principal components) to project onto. A call to the map function reduces the d-dimensional input to only dim dimensions such that the data preserves the maximum possible variance amongst dim-dimensional linear projections. -1 means all numerically available dimensions will be used unless reduced by var_cutoff. Setting dim to a positive value is exclusive with var_cutoff.
  • var_cutoff (float in the range [0,1], optional, default 0.95) – Determines the number of output dimensions by including dimensions until their cumulative kinetic variance exceeds the fraction subspace_variance. var_cutoff=1.0 means all numerically available dimensions (see epsilon) will be used, unless set by dim. Setting var_cutoff smaller than 1.0 is exclusive with dim
  • stride (int, optional, default = 1) – If set to 1, all input data will be used for estimation. Note that this could cause this calculation to be very slow for large data sets. Since molecular dynamics data is usually correlated at short timescales, it is often sufficient to estimate transformations at a longer stride. Note that the stride option in the get_output() function of the returned object is independent, so you can parametrize at a long stride, and still map all frames through the transformer.
  • mean (ndarray, optional, default None) – Optionally pass pre-calculated means to avoid their re-computation. The shape has to match the input dimension.
Returns:

pca – Object for Principle component analysis (PCA) analysis. It contains PCA eigenvalues and eigenvectors, and the projection of input data to the dominant PCA

Return type:

a PCA transformation object

Notes

Given a sequence of multivariate data \(X_t\), computes the mean-free covariance matrix.

\[C = (X - \mu)^T (X - \mu)\]

and solves the eigenvalue problem

\[C r_i = \sigma_i r_i,\]

where \(r_i\) are the principal components and \(\sigma_i\) are their respective variances.

When used as a dimension reduction method, the input data is projected onto the dominant principal components.

See Wiki page for more theory and references. for more theory and references.

Examples

Create some input data:

>>> import numpy as np
>>> from pyemma.coordinates import pca
>>> data = np.ones((1000, 2))
>>> data[0, -1] = 0

Project all input data on the first principal component:

>>> pca_obj = pca(data, dim=1)
>>> pca_obj.get_output() 
[array([[-0.99900001],
       [ 0.001     ],
       [ 0.001     ],...

See also

PCA : pca object

tica : for time-lagged independent component analysis

class pyemma.coordinates.transform.pca.PCA(dim=-1, var_cutoff=0.95, mean=None)

Principal component analysis.

Methods

describe(*args, **kwargs) Get a descriptive string representation of this class.
dimension() output dimension
fit(X, **kwargs) For compatibility with sklearn
fit_transform(X, **kwargs) For compatibility with sklearn
get_output([dimensions, stride]) Maps all input data of this transformer and returns it as an array or list of arrays.
iterator([stride, lag]) Returns an iterator that allows to access the transformed data.
map(X) Deprecated: use transform(X)
n_frames_total([stride]) Returns total number of frames.
number_of_trajectories() Returns the number of trajectories.
output_type() By default transformers return single precision floats.
parametrize([stride]) Parametrize this Transformer
register_progress_callback(call_back[, stage]) Registers the progress reporter.
trajectory_length(itraj[, stride]) Returns the length of trajectory of the requested index.
trajectory_lengths([stride]) Returns the length of each trajectory.
transform(X) Maps the input data through the transformer to correspondingly shaped output data array/list.

Attributes

chunksize chunksize defines how much data is being processed at once.
covariance_matrix
data_producer where the transformer obtains its data.
in_memory are results stored in memory?
logger The logger for this class instance
mean
name The name of this instance
ntraj
chunksize

chunksize defines how much data is being processed at once.

covariance_matrix
data_producer

where the transformer obtains its data.

describe(*args, **kwargs)

Get a descriptive string representation of this class.

dimension()

output dimension

fit(X, **kwargs)

For compatibility with sklearn

fit_transform(X, **kwargs)

For compatibility with sklearn

get_output(dimensions=slice(0, None, None), stride=1)

Maps all input data of this transformer and returns it as an array or list of arrays.

Parameters:
  • dimensions (list-like of indexes or slice) – indices of dimensions you like to keep, default = all
  • stride (int) – only take every n’th frame, default = 1
Returns:

output – the mapped data, where T is the number of time steps of the input data, or if stride > 1, floor(T_in / stride). d is the output dimension of this transformer. If the input consists of a list of trajectories, Y will also be a corresponding list of trajectories

Return type:

ndarray(T, d) or list of ndarray(T_i, d)

Notes

  • This function may be RAM intensive if stride is too large or too many dimensions are selected.
  • if in_memory attribute is True, then results of this methods are cached.

Example

plotting trajectories

>>> import pyemma.coordinates as coor 
>>> import matplotlib.pyplot as plt 

Fill with some actual data!

>>> tica = coor.tica() 
>>> trajs = tica.get_output(dimensions=(0,), stride=100) 
>>> for traj in trajs: 
...     plt.figure() 
...     plt.plot(traj[:, 0]) 
in_memory

are results stored in memory?

iterator(stride=1, lag=0)

Returns an iterator that allows to access the transformed data.

Parameters:
  • stride (int) – Only transform every N’th frame, default = 1
  • lag (int) – Configure the iterator such that it will return time-lagged data with a lag time of lag. If lag is used together with stride the operation will work as if the striding operation is applied before the time-lagged trajectory is shifted by lag steps. Therefore the effective lag time will be stride*lag.
Returns:

iterator – If lag = 0, a call to the .next() method of this iterator will return the pair (itraj, X) : (int, ndarray(n, m)), where itraj corresponds to input sequence number (eg. trajectory index) and X is the transformed data, n = chunksize or n < chunksize at end of input.

If lag > 0, a call to the .next() method of this iterator will return the tuple (itraj, X, Y) : (int, ndarray(n, m), ndarray(p, m)) where itraj and X are the same as above and Y contain the time-lagged data.

Return type:

a TransformerIterator

logger

The logger for this class instance

map(X)

Deprecated: use transform(X)

Maps the input data through the transformer to correspondingly shaped output data array/list.

mean
n_frames_total(stride=1)

Returns total number of frames.

Parameters:stride (int) – return value is the number of frames in trajectories when running through them with a step size of stride.
Returns:int
Return type:n_frames_total
name

The name of this instance

ntraj
number_of_trajectories()

Returns the number of trajectories.

Returns:int
Return type:number of trajectories
output_type()

By default transformers return single precision floats.

parametrize(stride=1)

Parametrize this Transformer

register_progress_callback(call_back, stage=0)

Registers the progress reporter.

Parameters:
  • call_back (function) –

    This function will be called with the following arguments:

    1. stage (int)
    2. instance of pyemma.utils.progressbar.ProgressBar
    3. optional *args and named keywords (**kw), for future changes
  • stage (int, optional, default=0) – The stage you want the given call back function to be fired.
trajectory_length(itraj, stride=1)

Returns the length of trajectory of the requested index.

Parameters:
  • itraj (int) – trajectory index
  • stride (int) – return value is the number of frames in the trajectory when running through it with a step size of stride.
Returns:

int

Return type:

length of trajectory

trajectory_lengths(stride=1)

Returns the length of each trajectory.

Parameters:stride (int) – return value is the number of frames of the trajectories when running through them with a step size of stride.
Returns:array(dtype=int)
Return type:containing length of each trajectory
transform(X)

Maps the input data through the transformer to correspondingly shaped output data array/list.

Parameters:X (ndarray(T, n) or list of ndarray(T_i, n)) – The input data, where T is the number of time steps and n is the number of dimensions. If a list is provided, the number of time steps is allowed to vary, but the number of dimensions are required to be to be consistent.
Returns:Y – The mapped data, where T is the number of time steps of the input data and d is the output dimension of this transformer. If called with a list of trajectories, Y will also be a corresponding list of trajectories
Return type:ndarray(T, d) or list of ndarray(T_i, d)

References

[1]Pearson, K. 1901 On Lines and Planes of Closest Fit to Systems of Points in Space Phil. Mag. 2, 559–572
[2]Hotelling, H. 1933. Analysis of a complex of statistical variables into principal components. J. Edu. Psych. 24, 417-441 and 498-520.