## Inverse Solution¶

### How should I regularize the covariance matrix?¶

The estimated covariance can be numerically unstable and tends to induce correlations between estimated source amplitudes and the number of samples available. The MNE manual therefore suggests to regularize the noise covariance matrix (see Regularization of the noise-covariance matrix), especially if only few samples are available. Unfortunately it is not easy to tell the effective number of samples, hence, to chose the appropriate regularization. In MNE-Python, regularization is done using advanced regularization methods described in [1]. For this the ‘auto’ option can be used. With this option cross-validation will be used to learn the optimal regularization:

>>> import mne
>>> cov = mne.compute_covariance(epochs, tmax=0., method='auto')


This procedure evaluates the noise covariance quantitatively by how well it whitens the data using the negative log-likelihood of unseen data. The final result can also be visually inspected. Under the assumption that the baseline does not contain a systematic signal (time-locked to the event of interest), the whitened baseline signal should be follow a multivariate Gaussian distribution, i.e., whitened baseline signals should be between -1.96 and 1.96 at a given time sample. Based on the same reasoning, the expected value for the global field power (GFP) is 1 (calculation of the GFP should take into account the true degrees of freedom, e.g. ddof=3 with 2 active SSP vectors):

>>> evoked = epochs.average()
>>> evoked.plot_white(cov)


This plot displays both, the whitened evoked signals for each channels and the whitened GFP. The numbers in the GFP panel represent the estimated rank of the data, which amounts to the effective degrees of freedom by which the squared sum across sensors is divided when computing the whitened GFP. The whitened GFP also helps detecting spurious late evoked components which can be the consequence of over- or under-regularization.

Note that if data have been processed using signal space separation (SSS) [2], gradiometers and magnetometers will be displayed jointly because both are reconstructed from the same SSS basis vectors with the same numerical rank. This also implies that both sensor types are not any longer linearly independent.

These methods for evaluation can be used to assess model violations. Additional introductory materials can be found [here](https://speakerdeck.com/dengemann/eeg-sensor-covariance-using-cross-validation).

For expert use cases or debugging the alternative estimators can also be compared:

>>> covs = mne.compute_covariance(epochs, tmax=0., method='auto', return_estimators=True)
>>> evoked = epochs.average()
>>> evoked.plot_white(covs)


This will plot the whitened evoked for the optimal estimator and display the GFPs for all estimators as separate lines in the related panel.

### References¶

 [1] Engemann D. and Gramfort A. (2015) Automated model selection in covariance estimation and spatial whitening of MEG and EEG signals, vol. 108, 328-342, NeuroImage.
 [2] Taulu, S., Simola, J., Kajola, M., 2005. Applications of the signal space separation method. IEEE Trans. Signal Proc. 53, 3359–3372.