Norm used for scaling CL is neither Frobenius nor L2

[pranaysy]

Problem

In fuse_lfp:L137 the $\text{norm}$ of the leadfield that has been reduced from $sensors\times vertices$ to $sensors\times ROIs$ does not correspond to any known $\text{norm}$.

More precisely, the $\text{norm}$ is estimated in code as: sqrt(trace(CL{m}*CL{m}')/Nchn(m)),
but, for matrices $\text{Frobenius norm}$ is defined as `sqrt(trace(CL{m}CL{m}'))(of many sources, [here's one at MATLAB's docs](https://uk.mathworks.com/help/matlab/ref/norm.html#bvhji30-5)) while $L_2 \text{ norm}$ for a matrix (also called the ***spectral norm***) is defined asmax(svd(CL{m}))` (MATLAB's docs)

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For matrices: $\text{Frobenius norm}$ = norm(CL{m}, 'fro') = `sqrt(trace(CL{m}CL{m}'))=sqrt(sum(CL{m}(:).^2))`

Reproducible example A

x = ones(5);

% Frobenius norms
norm(x, 'fro')      % returns 5
sqrt(trace(x*x'))   % returns 5
sqrt(sum(x(:).^2))  % returns 5

% L2 norms
norm(x, 2)          % returns 5
max(svd(x))         % returns 5

% Implemented norm
sqrt(trace(x*x')/5) % returns 2.2361

Reproducible example B

x = magic(5);

% x =
%
%    17    24     1     8    15
%    23     5     7    14    16
%     4     6    13    20    22
%    10    12    19    21     3
%    11    18    25     2     9

% Frobenius norms
norm(x, 'fro')      % returns 74.330
sqrt(trace(x*x'))   % returns 74.330
sqrt(sum(x(:).^2))  % returns 74.330

% L2 norms
norm(x, 2)          % returns 65
max(svd(x))         % returns 65

% Implemented norm
sqrt(trace(x*x')/5) % returns 33.242

Solution

As the implemented $\text{norm}$ closely matches the $\text{Frobenius norm}$, switch to MATLAB's default implementation with norm(x, 'fro')

[pranaysy]

Lines 77-80 of spm_eeg_invert.m describe how scaling is performed for both the lead field and data.

% Potential scaling differences between the lead-fields are handled by
% scaling L{1} such that trace(L{1}*L{1}') = constant (number of spatial
% modes or channels), while scaling the data such that trace(AY{n}*AY{n}') =
% constant over subjects (and modalities; see below).

L316 of spm_eeg_invert.m performs this normalization, which is exactly the same as fuse_lfp.m#L137

% normalise lead-field
%------------------------------------------------------------------
Scale = sqrt(trace(UL{m}*UL{m}')/Nm(m));
UL{m} = UL{m}/Scale;

[pranaysy]

The normalization issue is addressed in this paper:

Henson, Richard N., Elias Mouchlianitis, and Karl J. Friston. “MEG and EEG Data Fusion: Simultaneous Localisation of Face-Evoked Responses.” NeuroImage 47, no. 2 (August 15, 2009): 581–89. https://doi.org/10.1016/j.neuroimage.2009.04.063.

In section An empirical Bayesian approach to MEG/EEG fusion:

This essentially argues that the relative sensitivity of the lead field may not be identical between different modalities. One way to normalize this is to make sure that the variance/scaling done by each sensor through the lead field is normalized to one. The trace of the covariance matrix of the lead field, $tr(LL^T)$, estimates the total variance/scaling done when $L$ is applied. The square root of of this trace yields the total standard deviation instead of total variance. So by dividing the lead field by $\sqrt{tr(LL^T)}$ or the $\text{Frobenius norm}$ ensures that the scaling achieved by $L$ is normalized to the $\text{Frobenius norm}$ of $L$ which is equivalent to the $\text{RMS gain}$ of $L$ and of course need not be one. However when the trace of the covariance matrix itself is normalized by the number of elements in the trace, $m$ (or on the diagonal), i.e. $\frac{1}{m} tr(LL^T)$, the meaning of this quantity changes from total scaling to the average scaling achieved when $L$ is applied. When $L$ is normalized by the square root of this quantity, it implies that the average scaling done by each row of $L$ is one. If there are $m$ elements in $L$ then the total scaling will therefore be $m$. This normalization procedure therefore acts in such a way that the average scaling obtained by each sensor regardless of modality will be one (exactly one if no sensor noise and iid sources)

Attached PDF: MEG and EEG data fusion - Simultaneous localisation of face-evoked responses - Henson et al [2009][NeuroImage].pdf

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External links for further reading:

• Discussion on r/math
• The accessible text Clifford Algebras - An Introduction by D. J. H. Garling, 2012 has a section on traces, and touches briefly upon normalized traces, example from page 43: