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Higher-Order Covariance Minimum Variance

Solver ID: HOCMV

Usage

from invert import Solver

# fwd = ...    (mne.Forward object)
# evoked = ... (mne.Evoked object)

solver = Solver("HOCMV")
solver.make_inverse_operator(fwd)
stc = solver.apply_inverse_operator(evoked)
stc.plot()

Overview

Minimum-variance beamformer using higher-order covariance statistics (as implemented here).

References

  1. Jonmohamadi, Y., Poudel, G., Innes, C., Weiss, D., Krueger, R., & Jones, R. (2014). Comparison of beamformers for EEG source signal reconstruction. Biomedical Signal Processing and Control, 14, 175-188.

API Reference

Bases: BaseSolver

Class for the Higher-Order Covariance Minimum Variance (HOCMV) Beamformer inverse solution [1].

References

[1] Jonmohamadi, Y., Poudel, G., Innes, C., Weiss, D., Krueger, R., & Jones, R. (2014). Comparison of beamformers for EEG source signal reconstruction. Biomedical Signal Processing and Control, 14, 175-188.

Source code in invert/solvers/beamformers/hocmv.py 
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class SolverHOCMV(BaseSolver):
    """Class for the Higher-Order Covariance Minimum Variance (HOCMV)
        Beamformer inverse solution [1].

    References
    ----------
    [1] Jonmohamadi, Y., Poudel, G., Innes, C., Weiss, D., Krueger, R., & Jones,
    R. (2014). Comparison of beamformers for EEG source signal reconstruction.
    Biomedical Signal Processing and Control, 14, 175-188.

    """

    meta = SolverMeta(
        slug="hocmv",
        full_name="Higher-Order Covariance Minimum Variance",
        category="Beamformers",
        description=(
            "Minimum-variance beamformer using higher-order covariance statistics "
            "(as implemented here)."
        ),
        references=[
            "Jonmohamadi, Y., Poudel, G., Innes, C., Weiss, D., Krueger, R., & Jones, R. "
            "(2014). Comparison of beamformers for EEG source signal reconstruction. "
            "Biomedical Signal Processing and Control, 14, 175-188.",
        ],
    )

    def __init__(
        self, name="HOCMV Beamformer", reduce_rank=True, rank="auto", **kwargs
    ):
        self.name = name
        return super().__init__(reduce_rank=reduce_rank, rank=rank, **kwargs)

    def make_inverse_operator(
        self,
        forward,
        mne_obj,
        *args,
        weight_norm=True,
        alpha="auto",
        order=3,
        verbose=0,
        **kwargs,
    ):
        """Calculate inverse operator.

        Parameters
        ----------
        forward : mne.Forward
            The mne-python Forward model instance.
        mne_obj : [mne.Evoked, mne.Epochs, mne.io.Raw]
            The MNE data object.
        weight_norm : bool
            Normalize the filter weight matrix W to unit length of the columns.
        alpha : float
            The regularization parameter.
        order : int
            The order of the covariance matrix. Should be a positive integer not
            evenly divisible by two {3, 5, 7, ...}

        Return
        ------
        self : object returns itself for convenience
        """
        super().make_inverse_operator(forward, *args, alpha=alpha, **kwargs)
        data = self.unpack_data_obj(mne_obj)

        leadfield = self.leadfield
        leadfield /= np.linalg.norm(leadfield, axis=0)
        n_chans, n_dipoles = self.leadfield.shape

        self.weight_norm = weight_norm

        y = data
        I = np.identity(n_chans)

        # Recompute regularization based on the max eigenvalue of the Covariance
        # Matrix (opposed to that of the leadfield)
        y -= y.mean(axis=1, keepdims=True)
        C = self.data_covariance(y, center=False, ddof=1)
        self.alphas = self.get_alphas(reference=C)

        inverse_operators = []
        for alpha in self.alphas:
            C_inv = self.robust_inverse(C + alpha * I)
            C_inv_n = np.linalg.matrix_power(C_inv, order)

            upper = C_inv @ leadfield
            lower = np.sqrt(
                abs(np.einsum("ij,jk,ki->i", leadfield.T, C_inv_n, leadfield))
            )
            W = upper / lower

            if self.weight_norm:
                W /= np.linalg.norm(W, axis=0)

            inverse_operator = W.T
            inverse_operators.append(inverse_operator)

        self.inverse_operators = [
            InverseOperator(inverse_operator, self.name)
            for inverse_operator in inverse_operators
        ]
        return self

__init__ 

__init__(
    name="HOCMV Beamformer",
    reduce_rank=True,
    rank="auto",
    **kwargs,
)
Source code in invert/solvers/beamformers/hocmv.py 
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def __init__(
    self, name="HOCMV Beamformer", reduce_rank=True, rank="auto", **kwargs
):
    self.name = name
    return super().__init__(reduce_rank=reduce_rank, rank=rank, **kwargs)

make_inverse_operator 

make_inverse_operator(
    forward,
    mne_obj,
    *args,
    weight_norm=True,
    alpha="auto",
    order=3,
    verbose=0,
    **kwargs,
)

Calculate inverse operator.

Parameters:

Name Type Description Default
forward Forward

The mne-python Forward model instance.

 required
mne_obj [Evoked, Epochs, Raw]

The MNE data object.

 required
weight_norm bool

Normalize the filter weight matrix W to unit length of the columns.

 True
alpha float

The regularization parameter.

 'auto'
order int

The order of the covariance matrix. Should be a positive integer not evenly divisible by two {3, 5, 7, ...}

 3
Return

self : object returns itself for convenience

Source code in invert/solvers/beamformers/hocmv.py 
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def make_inverse_operator(
    self,
    forward,
    mne_obj,
    *args,
    weight_norm=True,
    alpha="auto",
    order=3,
    verbose=0,
    **kwargs,
):
    """Calculate inverse operator.

    Parameters
    ----------
    forward : mne.Forward
        The mne-python Forward model instance.
    mne_obj : [mne.Evoked, mne.Epochs, mne.io.Raw]
        The MNE data object.
    weight_norm : bool
        Normalize the filter weight matrix W to unit length of the columns.
    alpha : float
        The regularization parameter.
    order : int
        The order of the covariance matrix. Should be a positive integer not
        evenly divisible by two {3, 5, 7, ...}

    Return
    ------
    self : object returns itself for convenience
    """
    super().make_inverse_operator(forward, *args, alpha=alpha, **kwargs)
    data = self.unpack_data_obj(mne_obj)

    leadfield = self.leadfield
    leadfield /= np.linalg.norm(leadfield, axis=0)
    n_chans, n_dipoles = self.leadfield.shape

    self.weight_norm = weight_norm

    y = data
    I = np.identity(n_chans)

    # Recompute regularization based on the max eigenvalue of the Covariance
    # Matrix (opposed to that of the leadfield)
    y -= y.mean(axis=1, keepdims=True)
    C = self.data_covariance(y, center=False, ddof=1)
    self.alphas = self.get_alphas(reference=C)

    inverse_operators = []
    for alpha in self.alphas:
        C_inv = self.robust_inverse(C + alpha * I)
        C_inv_n = np.linalg.matrix_power(C_inv, order)

        upper = C_inv @ leadfield
        lower = np.sqrt(
            abs(np.einsum("ij,jk,ki->i", leadfield.T, C_inv_n, leadfield))
        )
        W = upper / lower

        if self.weight_norm:
            W /= np.linalg.norm(W, axis=0)

        inverse_operator = W.T
        inverse_operators.append(inverse_operator)

    self.inverse_operators = [
        InverseOperator(inverse_operator, self.name)
        for inverse_operator in inverse_operators
    ]
    return self