# TD2¶

TD2 module performs temporal decorrelation of second order on the spatially whitened data

TD2 module removes dominant second order temporal correlations by computing a time-delayed covariance matrix and performing PCA.

This code makes use of AMUSE algorithm which can be found using the link

http://docs.markovmodel.org/lecture_tica.html

An eigen value decomposition is performed over time delayed covariance matrix and the data is projected onto the dominant eigen vectors to obtain spatially whitened and temporally decorrelated data.

Problem:

TD2 fails to capture insights obtained from the inherent anharmonicity in atomic fluctuations and it’s long tail behavior.

Parameters

Y - an mxT spatially whitened matrix (m dimensionality of subspace, T snapshots). May be a numpy array or a matrix where,

m - dimensionality of the subspace we are interested in; Default value is None, in which case m=n

T - number of snapshots of MD trajectory

U - whitening matrix obtained after doing the PCA analysis on m components of real data

lag - lag time in the form of an integer denoting the time steps

verbose - print information on progress. Default is true.


Returns

V      - An n x m matrix V (NumPy matrix type) is a separating matrix such that V = Btd2 x U
(U is obtained from SD2 of data matrix and Btd2 is obtained from time-delayed covariance of matrix Y)

Z      - B2td2 * Y is spatially whitened and temporally decorrelated (2nd order) source extracted from
the m x T spatially whitened matrix Y.

Dstd2  - has eigen values sorted by increasing variance

PCstd2 -  holds the index for m most significant principal components by decreasing variance
R = Dstd2[PCstd2]

R      - Eigen values of the time-delayed covariance matrix of Y

Btd2   - Eigen vectors of the time-delayed covariance matrix of Y


Note

• Tic is the time delayed covariance matrix computed with time lag = lag. To ensure symmetricity of the covariance matrix a mathematical computation is made: Tic = 0.5 * [Tic + Tic.T]
• Eigen value decomposition of this time delayed symmetrized covariance matrix is performed to obtain eigen vector matrix
• Z is obtained by projecting the trajectory obtained from Y onto the dominant eigen vectors. Z is a matrix of spatially whitened and temporally uncorrelated components in the 2nd order