.. _td2:

*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