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You are here: Frontpage : Introduction » Tutorials » Granger causality analysis of EEG data » Class: var

Table of Contents

`var`

. For a detailed description of the class properties and methods
see `help var.var`

in MATLAB.

The `var`

class implements the machinery necessary for generating observations
from a Vector Autoregressive (VAR) process:

where is the realization at time of the _VAR observation vector_, is the coefficient matrix for lag and is the realization at time of the _innovation process_. We will often refer to the elements of vector as the _nodes_ of the model.

Typically, the innovation process is considered to be
a sequence of independent and identically distributed (i.i.d.) Gaussian random
variables. However, class `var`

more generally models the innovation process as
a Generalized Normal Distribution.

Class `var`

also implements a linear _measurement model_ in which measurements
are obtained by linearly mixing the observations of the VAR process:

where is the _measurement vector_ and is a _mixing matrix_. Note that the dimensionality of the measurement vector () does not need to be the same as the dimensionality of the underlying VAR process (). A noisy version of the measurement model is also possible:

where the noise process is i.i.d. Gaussian.

Finally, class `var`

also incorporates a pre-processing model that that can
be used to implement standard pre-processing filters. That is, if the
pre-processing model is specified, the measurements
are obtained as:

where is a digital filter. Digital filters are defined as
`filter.dfilt`

objects. See `help filter.dfilt`

for more information.

In the following I will assume that the `dynamics`

package (which contains
class `var`

) has been imported via the command:

import dynamics.*

Objects of class `var`

are constructed as:

obj = var('propKey', propValue)

See `help dynamics.var`

for a detailed description of the set
of accepted property key/values. A default `var`

object can be constructed
using:

obj = var

The default constructor produces a noisless mix-free model with Gaussian innovations.

Class `var`

` implements the interface `

`data`

which defines the
following operation to generate realizations from the model described above:

[pset, obj] = generate(obj, nSamples, nTrials)

where `pset`

` is a `

`pointset`

, `nSamples`

is the number of samples
to be generated, and `nTrials`

is the number of \\trials\\. The following code
can be used to plot a single trial of 1000 samples, from a 3-dimensional
VAR model of order 5:

varObj = var('NbDims', 3, 'Order', 5); [pset, varObj] = generate(varObj, 1000); plot(varObj.Observations{1}');

Class `var`

implements various methods for modifying the state
of `var`

objects. For instance, method `set_noisecov()`

can be used for
modifying the covariance matrix of the noise:

obj = var('NbDims', 3); % This is a noiseless model obj = set_noisecov(obj, eye(3)); % Now it is a noisy one

A useful modifier method is `randomize()`

, which can be used to randomize
different properties of a `var`

object. For instance, the following command
can be used to randomize the VAR coefficients and the noise covariance matrix:

obj = randomize(obj, 'varcoeffs', true, 'noisecov', true)

It is important to note that modifying the state of a `var`

object
does not automatically trigger the re-generation of the model observations,
i.e. method `generate()`

is not automatically called after such modification
of the object state.

**sigma = acov(obj, lag)**

If the innovation process of the model is Gaussian, then the covariance of the
measurement vector can be obtained analytically. This functionality is provided
by method `acov`

:

obj = var('NbDims', 3); % A 3-dimensional var object lag = 3; % The covariance lag Sigma = acov(obj, lag); % The 3x3 covariance matrix

**value = pte(obj, gamma, phi, theta)**

Using the analytical covariance, the (partial) transfer entropy or the
(partial) mutual information between any set of elements of the measurement
vector can also be obtained analytically (see ref. [1] below). For instance,
given the 3-dimensional `var`

object that we created above, the exact value of
the transfer entropy from the second component towards the first, partialized
with respect to the third is:

pteValue = pte(obj, 1, 2, 3);

Method `pmi`

works similarly to `pte`

and provides the exact value of the
partial mutual information.

A useful property of a `var`

object is its `Topology`

. The topology of a `var`

object describes the way the nodes of the model are connected to each other.
The topology of a `var`

object can be specified during construction:

obj = var('NbDims', 3, 'Topology', 'tree');

of using the modifier method `set_topology()`

:

obj = var('NbDims', 5); % Default: 'random' topology obj = set_topology(obj, 'sparse'); % Now it has 'sparse' topology

The figures below describe the topologies that class `var`

implements:

**[1] Barnett et al., 2009**, *Granger Causality and Transfer Entropy Are Equivalent for
Gaussian Variables*, Physical Review Letters 103, 238701
doi: 10.1103/PhysRevLett.103.238701