Class ISFANode

Perform Independent Slow Feature Analysis.

The notation is the same used in the paper by Blaschke et al. Please
refer to the paper for more information.

:Parameters:
  lags
    list of time-lags to generate the time-delayed covariance
    matrices (in the paper this is the set of   au). If
    lags is an integer, time-lags 1,2,...,'lags' are used.
    Note that time-lag == 0 (instantaneous correlation) is
    always implicitly used.

  sfa_ica_coeff
    a list of float with two entries, which defines the
    weights of the SFA and ICA part of the objective
    function. They are called b_{SFA} and b_{ICA} in the
    paper.

  sfaweights
    weighting factors for the covariance matrices relative
    to the SFA part of the objective function (called
    \kappa_{SFA}^{      au} in the paper). Default is
    [1., 0., ..., 0.]
    For possible values see the description of icaweights.

  icaweights
    weighting factors for the cov matrices relative
    to the ICA part of the objective function (called
    \kappa_{ICA}^{      au} in the paper). Default is 1.
    Possible values are:

    - an integer ``n``: all matrices are weighted the same
      (note that it does not make sense to have ``n != 1``)

    - a list or array of floats of ``len == len(lags)``:
      each element of the list is used for weighting the
      corresponding matrix

    - ``None``: use the default values.

  whitened
    ``True`` if input data is already white, ``False``
    otherwise (the data will be whitened internally).

  white_comp
    If whitened is false, you can set ``white_comp`` to the
    number of whitened components to keep during the
    calculation (i.e., the input dimensions are reduced to
    ``white_comp`` by keeping the components of largest variance).
  white_parm
    a dictionary with additional parameters for whitening.
    It is passed directly to the WhiteningNode constructor.
    Ex: white_parm = { 'svd' : True }

  eps_contrast
    Convergence is achieved when the relative
    improvement in the contrast is below this threshold.
    Values in the range [1E-4, 1E-10] are usually
    reasonable.

  max_iter
    If the algorithms does not achieve convergence within
    max_iter iterations raise an Exception. Should be
    larger than 100.

  RP
    Starting rotation-permutation matrix. It is an
    input_dim x input_dim matrix used to initially rotate the
    input components. If not set, the identity matrix is used.
    In the paper this is used to start the algorithm at the
    SFA solution (which is often quite near to the optimum).

  verbose
    print progress information during convergence. This can
    slow down the algorithm, but it's the only way to see
    the rate of improvement and immediately spot if something
    is going wrong.

  output_dim
    sets the number of independent components that have to
    be extracted. Note that if this is not smaller than
    input_dim, the problem is solved linearly and SFA
    would give the same solution only much faster.

Overrides: object.__init__

Process the data contained in `x`.

If the object is still in the training phase, the function
`stop_training` will be called.
`x` is a matrix having different variables on different columns
and observations on the rows.

By default, subclasses should overwrite `_execute` to implement
their execution phase. The docstring of the `_execute` method
overwrites this docstring.

Overrides: Node.execute

Invert `y`.

If the node is invertible, compute the input ``x`` such that
``y = execute(x)``.

By default, subclasses should overwrite `_inverse` to implement
their `inverse` function. The docstring of the `inverse` method
overwrites this docstring.

Overrides: Node.inverse

Stop the training phase.

If the node is used on large datasets it may be wise to first
learn the covariance matrices, and then tune the parameters
until a suitable parameter set has been found (learning the
covariance matrices is the slowest part in this case).  This
could be done for example in the following way (assuming the
data is already white):

>>> covs=[mdp.utils.DelayCovarianceMatrix(dt, dtype=dtype)
...       for dt in lags]
>>> for block in data:
...     [covs[i].update(block) for i in range(len(lags))]

You can then initialize the ISFANode with the desired parameters,
do a fake training with some random data to set the internal
node structure and then call stop_training with the stored covariance
matrices. For example:

>>> isfa = ISFANode(lags, .....)
>>> x = mdp.numx_rand.random((100, input_dim)).astype(dtype)
>>> isfa.train(x)
>>> isfa.stop_training(covs=covs)

This trick has been used in the paper to apply ISFA to surrogate
matrices, i.e. covariance matrices that were not learnt on a
real dataset.

Overrides: Node.stop_training

Update the internal structures according to the input data `x`.

`x` is a matrix having different variables on different columns
and observations on the rows.

By default, subclasses should overwrite `_train` to implement their
training phase. The docstring of the `_train` method overwrites this
docstring.

Note: a subclass supporting multiple training phases should implement
the *same* signature for all the training phases and document the
meaning of the arguments in the `_train` method doc-string. Having
consistent signatures is a requirement to use the node in a flow.

Overrides: Node.train