.. _td4:

*TD4*
=====

**TD4 module performs temporal decorrelation of fourth order on time-delayed cumulant matrices**  

In the interest of resolving spatial and temporal anharmonic dependencies in the molecular simulation trajectories, we have designed the TD4 module which performs joint diagonalization of time-delayed cumulant matrices (a tensor of fourth-order time-delayed statistics signifying kurtosis). TD4 is the counterpart of :ref:`SD4 <sd4>`, where fourth-order spatial correlations are minimized, implying zero time lag.

Conceptually, the assumption we make is that a molecular simulation trajectory is a linear combination of independent, anharmonically fluctuating protein motions. To discover these anharmonic motions, we borrow a technique from signal processing literature, called Blind Source Separation (BSS), which attempts to extract or unmix independent non-Gaussian sources from signal mixtures with Gaussian noise. To facilitate the extraction of anharmonic modes of motion of the fourth-order, the trajectory data :math:`X_{orig} \in \mathbb{R}^{3N \times t}`, where *3N* represents *(x,y,z)* coordinates from individual atom selections and *t* represents conformations is decorrelated for second-order dependencies both spatially and temporally by transforming it through the modules of :ref:`SD2 <sd2>` and :ref:`TD2 <td2>`. :ref:`SD2 <sd2>` module removes dominant second order spatial correlations by computing a spatial covariance matrix and performing principal component analysis (PCA). The function diagonalizes covariance matrix to obtain the projection matrix :math:`Y = B^TX_{orig} (m \times t)`, where *m* is subspace dimensionality and :math:`B (3N \times m)` are the dominant eigenvectors. Consequently, :ref:`TD2 <td2>` module removes dominant second order temporal correlations by computing a time-delayed (specified by a lag time :math:`\tau`) covariance matrix and performing PCA. A matrix *Z* is obtained by projecting the spatially resolved data matrix *Y* onto the dominant eigenvectors :math:`B_{TD2}`. The matrix *Z* then undergoes transformations to retrieve mutually independent signals by obtaining a separating matrix :math:`W \in \mathbb{R}^{m \times 3N}`.

Algorithmically, the method of unmixing temporally correlated signals of fourth-order can be viewed as a symmetric eigenvalue problem of a generalized cumulant matrix :math:`Q_{ij}`. As a measure of statistical independence, we will consider the 'diagonality' of a set of cumulant matrices. The cumulant matrices are generated in a low-dimensional subspace denoted by *m*, which is the best guess for the most compact summary of the fourth-order statistics.
The subspace dimensionality can be adjusted by examining the inflection points in the cumulative variance plots generated from :ref:`SD2 <sd2>` module. 

In order to generate the cumulant matrices, a time-lagged covariance matrix is defined by:

.. math:: R_z{(\tau)} = E\left\{ZZ_{\tau}^T\right\},

where :math:`Z \in \mathbb{R}^{m \times t}` is second-order spatially and temporally resolved molecular simulation data, :math:`\tau` is time delay and :math:`Z_{\tau} = Z(t-\tau)` is the time-lagged version of *Z*. A fourth-order cumulant matrix :math:`Q_{ij}` of this data matrix *Z* is defined by:

.. math:: Q_{ij} = E\left\{ZZ^TZ_{\tau}^TZ_{\tau}\right\} - E\left\{ZZ^T\right\} \textrm{tr}\, E\left\{Z_{\tau}Z_{\tau}^T\right\} -2E\left\{ZZ_{\tau}^T\right\}E\left\{Z_{\tau}Z^T\right\},

where :math:`Q_{ij} \in \mathbb{R}^{m \times m}` computes a time-lagged cumulant matrix.
The possibility of computational errors, such as round-off errors, can destroy the symmetricity of the cumulant matrix which is restored by performing:

.. math:: Q_{ij} = \frac{1}{2} \left[Q_{ij} + Q_{ij}^T\right].

A time-lagged cumulant tensor :math:`\mathbb{Q} \in \mathbb{R}^{m \times (m \times k)}`, where *k* = :math:`[{m \times (m+1)}]/2` is defined for the storage of cumulant matrices computed by the symmetric :math:`Q_{ij}` matrix. 
Joint diagonalization of these time-lagged cumulant matrices reduces fourth-order temporal dependencies leading to anharmonic modes of motion of the trajectory data. This is done through Jacobi's iterative method of finding solution to a system of linear equations. In particular, the method uses successive transformations to calculate diagonal elements of the cumulant tensor by decimating off-diagonal elements with each iteration.
The spatio-temporally decorrelated matrix of fourth-order is computed by:

.. math:: {Z_{TD4}} = W X_{orig},  

where *W* attempts to separate sources from signal mixture :math:`X_{orig}` by finding directions, such that projections onto these directions have maximum statistical independence. The computed parameter :math:`Z_{TD4}` is fourth-order spatially and temporally resolved matrix.


**Parameters** ::
 
                 Z       - an mxT spatially uncorrelated of order 2 and temporally uncorrelated of order 2 matrix (m subspaces, T samples). May be a numpyarray or matrix where,

                 m       - dimensionality of the subspace we are interested in. Defaults to None, in which case m=n. 

                 T       - number of snapshots of MD trajectory

                 V       - separating matrix obtained after doing the PCA analysis on m components of real data followed temporal decorrelation of the spatially whitened data

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

                 verbose - print information on progress. Default is true.

**Returns** ::
 
                 W - a separating matrix obtained from resolving fourth order temporal correlations


**Reference** ::

1. Georgiev, P., & Cichocki, A. (2003). Robust independent component analysis via time-delayed cumulant functions. IEICE Transactions on Fundamentals of
   Electronics, Communications and Computer Sciences, 86(3), 573-579.