neuropsychology/NeuroKit

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neurokit2/complexity/information_fisher.py

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import numpy as np
import pandas as pd

from .utils_complexity_embedding import complexity_embedding


def fisher_information(signal, delay=1, dimension=2):
    """**Fisher Information (FI)**

    The Fisher information was introduced by R. A. Fisher in 1925, as a measure of "intrinsic
    accuracy" in statistical estimation theory. It is central to many statistical fields far beyond
    that of complexity theory. It measures the amount of information that an observable random
    variable carries about an unknown parameter. In complexity analysis, the amount of information
    that a system carries "about itself" is measured. Similarly to :func:`SVDEn <entropy_svd>`, it
    is based on the Singular Value Decomposition (SVD) of the :func:`time-delay embedded <complexity_embedding>`
    signal. The value of FI is usually anti-correlated with other measures of complexity (the more
    information a system withholds about itself, and the more predictable and thus, less complex it
    is).

    See Also
    --------
    entropy_svd, information_mutual, complexity_embedding, complexity_delay, complexity_dimension

    Parameters
    ----------
    signal : Union[list, np.array, pd.Series]
        The signal (i.e., a time series) in the form of a vector of values.
    delay : int
        Time delay (often denoted *Tau* :math:`\\tau`, sometimes referred to as *lag*) in samples.
        See :func:`complexity_delay` to estimate the optimal value for this parameter.
    dimension : int
        Embedding Dimension (*m*, sometimes referred to as *d* or *order*). See
        :func:`complexity_dimension` to estimate the optimal value for this parameter.

    Returns
    -------
    fi : float
        The computed fisher information measure.
    info : dict
        A dictionary containing additional information regarding the parameters used
        to compute fisher information.

    Examples
    ----------
    .. ipython:: python

      import neurokit2 as nk

      signal = nk.signal_simulate(duration=2, frequency=5)
      fi, info = nk.fisher_information(signal, delay=10, dimension=3)
      fi

    """
    # Sanity checks
    if isinstance(signal, (np.ndarray, pd.DataFrame)) and signal.ndim > 1:
        raise ValueError(
            "Multidimensional inputs (e.g., matrices or multichannel data) are not supported yet."
        )

    embedded = complexity_embedding(signal, delay=delay, dimension=dimension)
    W = np.linalg.svd(embedded, compute_uv=False)
    W /= np.sum(W)  # normalize singular values
    FI_v = (W[1:] - W[:-1]) ** 2 / W[:-1]

    return np.sum(FI_v), {"Dimension": dimension, "Delay": delay, "Values": FI_v}