/
osl_inverse_mne_weights.m
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osl_inverse_mne_weights.m
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function [W, W_nonorm,lf] = osl_inverse_mne_weights(SensorData, LeadFields, Noise, Options)
%OSL_INVERSE_MNE_WEIGHTS creates weights for source reconstruction using MNE
%
% W = OSL_INVERSE_MNE_WEIGHTS(SensorData, LeadFields, NOISE, OPTIONS)
% calculates cell array of weights, W, for source reconstruction. Sources
% at dipole i are estimated with W{i}*B.
%
% [W, W_NONORM, LF] = OSL_MNE_WEIGHTS(...) also returns the un-normalised
% weights W_NONORM and a cell array of lead fields for each dipole, LF.
%
% The inputs should be formatted as a set of data structures:
%
% SensorData.cov - covariance of the sensor data (nChans x nChans)
% .nSamples - number of time samples used to compute covariance
% matrix.
%
% LeadFields.nSources - number of sources to estimate
% .nDims - dimensionality of each source (expected to be 3 at
% the moment)
% .lf - matrix of lead fields, nChans x (nDims x
% nSources). Fields for x, y and z components of the
% same dipole should appear in consecutive columns.
%
% Noise.model - determines specification of the noise covariance.
% Choices are {'empirical', 'white',
% 'diag_datacov'}. For the first option, a noise
% covariance matrix must be passed in. For the
% second two, a scaling factor lambda should be
% provided, or is assumed unity.
% .lambda - scaling factor on the noise covariance
% .cov - empirical noise covariance (nChans x nChans)
%
% Further algorithm choices are set in the subfields of the Options
% structure.
%
% Options.ReduceRank.weights - Boolean [true] converts to scalar weights
% .leadFields - Integer [3] dimensionality of lead fields
% (ignored at the moment)
%
% .sourceModel - Choose the estimation method using a string, from:
% 'Wens' (based on paper referenced below),
% 'MNE' (which uses gamma-MAP estimation of a white covariance matrix),
% 'MNE-INA' (integrated nested approximation for
% averaging over the prior on gamma)
% 'MNE-scaled-noise' (also estimates a parameter
% rho for the scaling between sensor data and
% noise)
% 'sparse-Bayes' (gamma-MAP estimation of a
% sparse Bayes solution)
% 'rvm-beamformer' - beamformer solution using
% the regularised data covariance estimated using
% the sparse Bayes model.
%
% .Normalise.weights - sLoreta, norm, noise-proj, none: normalises
% reconstruction weights using a variety of
% methods. sLoreta uses normalisation from the
% Wens paper. 'norm' normalised the 2-norm of
% the weights. 'noise-proj' normalises by the
% projection of the noise:
% sqrt(tr[W*noiseCov*W]), e.g. for constructing
% pseudo-z-stats (see Vrba and Robinson 2001).
% 'none' applies no normalisation
% .Normalise.leadFields - [true] or false normalises lead fields as
% a compensation for depth bias.
%
% .gammaPrior - form of prior on source variance parameters. Can
% be 'uniform-sd - flat on sqrt(gamma) [Default].
% 'uniform-variance' - flat on gamma.
% 'log-uniform' - flat on log-gamma (this is the
% Jeffrey's prior but produces an
% improper posterior).
% 'cauchy' - weakly informative prior with a
% peak at zero, flat centre and
% tail-off.
% 'normal' - on gamma
% 'log-normal' - on gamma
% or a handle to a function
% lp(gamma) = sum_i (log p(gamma_i))
% which returns the log of a normalised prior density
% for any scalar or vector gamma >= 0.
%
% References:
% Wipf and Nagarajan. A unified Bayesian framework for MEG/EEG source imaging. Neuroimage (2009) vol. 44 (3) pp. 947-66
%
% Dale, A.M. & Sereno, M.I.(1993) "Improved localization of cortical
% activity by combining EEG and MEG with MRI cortical surface
% reconstruction: a linear approach," J. Cogn. Neurosci 5, pp. 162--176.
%
% Pascual-Marqui, R.D. (2002) "Standardized low-resolution brain
% electromagnetic tomography (sLORETA): technical details," Methods Find.
% Exp. Clin. Pharmacol. 24 (Suppl D) pp. 5--12.
%
% Hamalainen, M.S., Lin, F. & Mosher, J.C. (2010) "Anatomically and
% functionally constrained minimum-norm estimates." In Hansen, P.C.,
% Kringelbach, M.L. & Salmelin, R. (ed.) "MEG: an introduction to
% methods," Oxford University Press, Oxford, UK, pp. 186--215.
%
% Wipf and Nagarajan (2007). Beamforming using the relevance vector
% machine. Proc. 24th Int. Conf. on Machine Learning
%
% Vrba and Robinson (2001). Signal processing in magnetoencephalography.
% Methods 25(2), 249--271.
%
% Wens, V. et al. (2015) "A geometric correction scheme for spatial leakage effects in MEG/EEG seed-based functional connectivity mapping," Hum. Brain. Mapp. (In review)
% Copyright 2015 OHBA
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% $LastChangedBy$
% $Revision$
% $LastChangedDate$
% Contact: giles.colclough@magd.ox.ac.uk
% Originally written on: GLNXA64 by Giles Colclough, 19-Jan-2015 14:40:17
global DEBUG
DEBUG = true;
%% Input Checking
% Sensor data covariance
SensorData.nSensors = ROInets.rows(SensorData.cov);
validateattributes(SensorData.cov, {'numeric'}, ...
{'2d', 'real', 'ncols', SensorData.nSensors}, ...
mfilename, 'SensorData.cov', 1);
% Check for positive definiteness? No - sensor data may not be full rank
% assert(ROInets.isposdef(SensorData.cov), ...
% [mfilename ':SensorDataCovarianceNotPosDef'], ...
% 'Data covariance matrix SensorData.cov should be positive definite. \n');
% lead fields
D = 3; % dimensions in space
% at the moment, expecting 3D lead fields only
assert(isequal(LeadFields.nDims, D), ...
[mfilename ':LeadFieldsWrongDimensionality'], ...
'Lead field dimensionality should be for 3D space. \n');
% check matrix size matches dimension fields and correct number of sensors
validateattributes(LeadFields.lf, {'numeric'}, ...
{'2d', 'real', 'nrows', SensorData.nSensors, ...
'ncols', LeadFields.nSources * LeadFields.nDims}, ...
mfilename, 'LeadFields.lf');
% set default options
Options = parse_options(Options);
%% Parse noise matrix
% noise covariance can be specified as the identity, the diagonal of the
% data covariance, or passed in as a measured or estimated quantity.
% For the first two cases, a global scaling is possible with parameter
% lambda.
Noise = parse_noise(Noise, diag(SensorData.cov));
%% Lead field normalisation
% normalisation of the leadfields is often performed to compensate to some
% extent for the depth bias.
LeadFields = normalise_leadfields(LeadFields, Options);
%% Source model and weights
fprintf('%s: computing weights using model %s. \n', ...
mfilename, Options.sourceModel);
switch lower(Options.sourceModel)
case 'wens'
W_3d = wens_estimate(SensorData, Noise, LeadFields);
case 'mne'
W_3d = mne_estimate(SensorData, Noise, LeadFields, ...
Options.gammaPrior.fn);
case 'mne-ina'
W_3d = mne_estimate_ina(SensorData, Noise, LeadFields, ...
Options.gammaPrior.fn);
case 'mne-scaled-noise'
W_3d = mne_estimate_scale_noise(SensorData, Noise, LeadFields, ...
Options.gammaPrior.fn);
case 'sparse-bayes'
W_3d = sparse_bayes_estimate(SensorData, Noise, LeadFields, ...
Options.gammaPrior);
case 'rvm-beamformer'
W_3d = rvm_beamformer(SensorData, Noise, LeadFields, ...
Options.gammaPrior);
otherwise
error([mfilename ':InvalidSourceMethod'], ...
'The chosen source method %s is not recognised. \n', ...
Options.method);
end%switch
%% Parse into cell array and normalise weights
for iVox = LeadFields.nSources:-1:1, % initialise matrices by looping backwards
% Partition weights for this dipole
dipInd = (D*iVox - (D-1)):(D*iVox); % relevant indices for this dipole. LeadFields.nDims and D should be identical (checked above)
Ws = W_3d(dipInd,:); % single-dipole weights
if Options.ReduceRank.weights,
% project onto direction of maximum variance using PCA.
dipCov = Ws * SensorData.cov * Ws.'; % single-dipole covariance 3x3
[ns,~] = eigs(dipCov, [], 1); % ns is direction of maximum source variance. This code seems about as fast as [ns,~] = fast_svds(dipCov, 1);
W_nonorm{iVox} = ns.' * Ws; % weights for this dipole projected onto direction of maximum variance
lf{iVox} = LeadFields.lf(:,dipInd) * ns; % lead fields for this dipole projected onto direction of maximum variance
else
W_nonorm{iVox} = Ws;
lf{iVox} = LeadFields.lf(:,dipInd);
end%if
% apply weights normalisation
switch lower(Options.Normalise.weights)
case 'sloreta'
% normalising constant for depth bias using sLORETA
% Wens et al. sec 2.4
lambda_s = sqrt(trace(W_nonorm{iVox} ...
* empirical_bayes_cov(Noise.cov, ...
sourceCov, ...
LeadFields.lf) ...
* W_nonorm{iVox}.'));
case 'norm'
% Normalise by norm of weights
lambda_s = norm(W_nonorm{iVox});
case 'noise-proj'
% Normalise by projected noise power.
lambda_s = sqrt(trace(W_nonorm{iVox} * Noise.cov * W_nonorm{iVox}.'));
case 'none'
% apply no normalisation
lambda_s = 1.0;
otherwise
error([mfilename ':InvalidNormalisationMethod'], ...
'Normalisation method %s is invalid. \n', ...
Options.Normalise);
end%switch
W{iVox} = W_nonorm{iVox} ./ lambda_s; % normalised scalar weights vector for this dipole
end%loop over voxels
end%osl_inverse_mne_weights
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function sensorCov = empirical_bayes_cov(noiseCov, sourceCov, lf3d)
%EMPIRICAL_BAYES_COV regularised estimate of data covariance in sensors
%
% SENSORCOV = EMPIRICAL_BAYES_COV(NOISECOV, SOURCECOV, 3D_LEADFIELDS)
% returns the empirical bayes estimate of the sensor covariances based on
% the noise covariance matrix, the modelled source covariance matrix and
% the lead field matrix in 3D.
sensorCov = noiseCov + lf3d * sourceCov * lf3d.';
% ensure real - sometimes goes a bit off.
% sensorCov = real(sensorCov);
sensorCov = (sensorCov + sensorCov')./2.0;
end%empirical_bayes_cov
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W_3d = estimate_weights(noiseCov, sourceCov, lf3d, rho)
%ESTIMATE_WEIGHTS for reconstruction of 3d sources
%
% W = ESTIMATE_WEIGHTS(NOISECOV, SOURCECOV, 3D_LEADFIELDS) estimates
% weights W for estimating sources S = W*B in 3d.
%
% W = ESTIMATE_WEIGHTS(NOISECOV, SOURCECOV, 3D_LEADFIELDS, RHO) estimates
% weights W for estimating sources S = W*B in 3d when there is an
% additional scaling factor rho between the measured noise and measured
% data.
%
% equation 13 from Wipf and Nagarajan (2009).
% given B = (LS + e)./rho, and S ~ N(0,sourceCov) then S is normally
% distributed with mean
% W * B = rho sourceCov L' (noiseCov + L sourceCov L')^{-1} B
% set default rho as identity
if nargin < 4 || ~exist('rho', 'var') || isempty(rho),
rho = 1;
end%if
W_3d = rho .* sourceCov * lf3d.' * ...
osl_cholinv(empirical_bayes_cov(noiseCov, sourceCov, lf3d));
end%estimate_weights
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W_3d = beamformer_weights(noiseCov, sourceCov, lf3d, rho)
%ESTIMATE_WEIGHTS for reconstruction of 3d sources
%
% W = BEAMFORMER_WEIGHTS(NOISECOV, SOURCECOV, 3D_LEADFIELDS) estimates
% weights W for estimating sources S = W*B in 3d.
%
% W = BEAMFORMER_WEIGHTS(NOISECOV, SOURCECOV, 3D_LEADFIELDS, RHO) estimates
% weights W for estimating sources S = W*B in 3d when there is an
% additional scaling factor rho between the measured noise and measured
% data.
%
% Vrba and Robinson 2001
% set default rho as identity
if nargin < 4 || ~exist('rho', 'var') || isempty(rho),
rho = 1;
end%if
invDataCov = osl_cholinv(empirical_bayes_cov(noiseCov, sourceCov, lf3d));
W_3d = rho .* lf3d.' * invDataCov ./ trace(lf3d.' * invDataCov * lf3d);
end%estimate_weights
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W = wens_estimate(SensorData, Noise, LeadFields)
%WENS_ESTIMATE reconstruction weights by Vincent Wens' method
% regularization parameter from Wens et al. Sec 2.4
% use property sum(eig(B, A)) = trace(inv(A) * B)
k = sum(eig(LeadFields.lf * LeadFields.lf.', Noise.cov)) ./ ...
(sum(eig(SensorData.cov, Noise.cov)) - SensorData.nSensors);
% In our formulation, gamma = 1/k and C = I
sourceCov = (1.0 ./ real(k)) * speye(LeadFields.nDims * LeadFields.nSources);
% Extract weights for 3d sources
W = estimate_weights(Noise.cov, sourceCov, LeadFields.lf);
end%wens_estimate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [W, logGamma] = mne_estimate(Data, Noise, LeadFields, prior)
%MNE_ESIMATE regularized source reconstruction weights
%
% Estimates source covariance for MNE, using a white covariance matrix,
% with a single scaling parameter.
% Use Eq 18 from Wipf and Nagarajan (2009), optimising w.r.t scalar gamma
global DEBUG
sourceCovFn = @(logGamma) exp(real(logGamma)) .* ...
speye(LeadFields.nDims * LeadFields.nSources);
% variance of data can change by factor of e^20 relative to noise in search
% space
% note that for priors flat in log-space, which create improper posteriors,
% the inference can be quite sensitive to the lower bound used here.
noiseToLFScale = mean(log(diag(Noise.cov))) - ...
mean(log(sum(LeadFields.lf.^2)));
logGammaBound = noiseToLFScale + [-20 50];
% if the prior has a scale, set it to be about e^7 above noise
scale = exp(noiseToLFScale + 7);
optimise_target = @(logGamma) optimise_target_single_prior(Data, Noise, ...
LeadFields, logGamma, prior, scale, sourceCovFn);
% optimise using a golden section search and parabolic interpolation
logGamma = fminbnd(optimise_target, logGammaBound(1), logGammaBound(2));
sourceCov = sourceCovFn(logGamma);
% Extract weights for 3d sources
W = estimate_weights(Noise.cov, sourceCov, LeadFields.lf);
if DEBUG,
% plot
lg = log(logspace(log10(exp(logGammaBound(1))), log10(exp(logGammaBound(2))), 50));
L = arrayfun(optimise_target, lg);
figure('Color', 'w', 'Name', 'Optimisation target for log(gamma)');
semilogy(lg, L);
xlabel('Log (\gamma)');
ylabel('L');
% display result
fprintf('\nBounds on logGamma: %0.2g, %0.2g. \n', logGammaBound);
fprintf('Chosen logGamma: %0.6g. \n', logGamma);
end%if DEBUG
end%mne_Jeffreys_prior_estimate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W = mne_estimate_scale_noise(Data, Noise, LeadFields, priorFn)
%MNE_DOUBLE_SEARCH_ESIMATE regularized source estimates with noise scaling
%
% Estimates source covariance for MNE, using a white covariance matrix,
% with a single scaling parameter.
% Allows an additional parameter to control scaling between noise and data
% as rho*B = LS + e
global DEBUG
% It seems as though this functionality will not work well. No minimum for
% rho, unless forced by prior?
warning([mfilename ':scale_noise'], ...
['The scale-noise mne estimate is unstable. ', ...
'It may not find a minimum for rho. \n']);
sourceCovFn = @(logGamma) exp(real(logGamma)) .* ...
speye(LeadFields.nDims * LeadFields.nSources);
% put noise and sources are on same scale
noiseToLFScale = mean(log(diag(Noise.cov))) - ...
mean(log(sum(LeadFields.lf.^2)));
% put data and noise on scale set by mean eigenvalue
dataEigVals = eig(Data.cov);
noiseEigVals = eig(Noise.cov);
NoiseToDataScale = log(mean(noiseEigVals)) - log(mean(dataEigVals));
% set initial values
logGammaInit = noiseToLFScale;
logRhoInit = NoiseToDataScale./2;
paramsInit = [logGammaInit; logRhoInit];
% priors forms are the same for rho and gamma
priors = {priorFn; priorFn};
% If there is a scale on the priors, we want to constrain the space to
% moderately sensible values
% If we reckon our initial guesses are any good, let's constrain the
% variance parameters to be within a factor of exp(7).
scales = exp(paramsInit + 7);
optimise_target = @(params) optimise_target_double_prior(Data, Noise, ...
LeadFields, params, ...
priors, scales, ...
sourceCovFn);
% set optimisation params
maxIter = 4000;
% optimise using a multivariate nonlinear Nelder-Mead minimization
[params, fval, success] = fminsearch(optimise_target, paramsInit, ...
optimset('MaxFunEvals', maxIter, ...
'MaxIter', maxIter));
logGamma = params(1);
sourceCov = sourceCovFn(logGamma);
rho = exp(params(2));
% Extract weights for 3d sources
W = estimate_weights(Noise.cov, sourceCov, LeadFields.lf, rho);
if DEBUG || ~success,
% use a grid search
lrBound = real(log10(rho)) + [-7 7];
lgBound = real(log10(exp(logGamma))) + [-10 10];
nR = 20;
nG = 30;
lr = log(logspace(lrBound(1), lrBound(2), nR));
lg = log(logspace(lgBound(1), lgBound(2), nG));
for iR = length(lr):-1:1,
for iG = length(lg):-1:1,
L(iG,iR) = optimise_target([lg(iG), lr(iR)]);
end%for
end%for
figure('Color', 'w', 'Name', 'Optimisation target for log(gamma)');
surfl(real(lr), real(lg), real(L));
colormap pink
xlabel('Log (\rho)');
ylabel('Log (\gamma)');
zlabel('L');
% best gamma and rho?
[searchFval, iMin] = min(L(:));
if searchFval < fval,
fprintf('%s: Using best parameters from grid search. \n', mfilename);
[iG,iR] = ind2sub(size(L), iMin);
logGamma = lg(iG);
sourceCov = sourceCovFn(logGamma);
rho = exp(lr(iR));
W = estimate_weights(Noise.cov, sourceCov, LeadFields.lf, rho);
end%if
end%if DEBUG
end%mne_estimate_scale_noise
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W = mne_estimate_ina(Data, Noise, LeadFields, prior)
%MNE_ESTIMATE_INA regularized source estimates
%
% Estimates source distribution using mean of marginal distribution on
% sources. Marginal distribution is approximated by averaging over several
% discrete points in gamma-space, rather than taking the MAP estimate of
% gamma. This is the integrated nested approximation.
% Returns weight W to estimate sources as S = W*B.
global DEBUG
% Find or estimate peak of marginal gamma distribution
% either:
[~, logGammaPeak] = mne_estimate(Data, Noise, LeadFields, prior);
gammaPeak = exp(logGammaPeak);
% or:
noiseToLFScale = mean(log(diag(Noise.cov))) - ...
mean(log(sum(LeadFields.lf.^2)));
% gammaPeak = exp(noiseToLFScale);
% define inverse weights function
nSourceElements = LeadFields.nDims * LeadFields.nSources;
weights = @(g) estimate_weights(Noise.cov, ...
g.*speye(nSourceElements), ...
LeadFields.lf);
% if the prior has a scale, set it to be about e^7 above noise
scale = exp(noiseToLFScale + 7);
sourceCovFn = @(logGamma) exp(real(logGamma)) .* speye(nSourceElements);
% integrate with clenshaw-curtis integration in 3 bands
nLower = 2^7 + 1;
nInner = 2^12 + 1;
nUpper = 2^6 + 1;
gammaBounds = [gammaPeak./1e40, gammaPeak./1e9, gammaPeak*1e6, gammaPeak*1e30];
[g1, w1] = clen_curt_points(nLower, gammaBounds(1), gammaBounds(2)*0.9999);
[g2, w2] = clen_curt_points(nInner, gammaBounds(2), gammaBounds(3)*0.9999);
[g3, w3] = clen_curt_points(nUpper, gammaBounds(3), gammaBounds(4));
g = cat(1, g1, g2, g3);
w = cat(1, w1, w2, w3);
[g,is] = sort(g);
w = w(is);
% we want to avoid exact zeros in x when working in log-space
exactZeros = 0 == g;
g(exactZeros) = [];
w(exactZeros) = [];
N = length(g);
% set up tracking plot
if DEBUG,
fprintf('Finding values of logp(g|B). \n');
flowBound = eps(min(g));
logPrior = arrayfun(@(lx) prior(lx, scale), log(g+flowBound));
figure('Name', 'p(g)', 'Color', 'w');
plot(log(g), exp(logPrior - log_sum_exp(logPrior)), 'r');
xlabel('Log(\gamma)');
hold on;
end%if
% declare memory
W = zeros(nSourceElements, Data.nSensors);
log_pg = zeros(N,1);
% find coefficients for expected weights at each x
for i = 1:N,
if ~mod(i, 100),
fprintf('%s: marginal contribution %d of %d. \n', ...
mfilename, i, N);
end%if
log_pg(i) = -0.5*optimise_target_single_prior(Data, Noise, ...
LeadFields, log(g(i)), prior, scale, sourceCovFn); %logp_g_on_B(x(i));
end%for
log_wpg = log_pg + log(w);
log_Exp_pg = log_sum_exp(log_wpg);
if DEBUG,
figure('Name', 'p(g|B)', 'Color', 'w');
plot(log(g), log_pg - log_Exp_pg, 'k');
xlabel('Log(\gamma)');
end%if
% loop over integration points, accumulating result
for iInt = 1:N,
W = W + exp(log_wpg(iInt) - log_Exp_pg) .* weights(g(iInt));
% write out progress
if DEBUG
if ~mod(iInt, 100),
fprintf('%s: INA contribution %d of %d. \n', ...
mfilename, iInt, N);
end%if
end%if
end%for
if DEBUG,
fprintf('%s: normalisation check: sum of weighting factors: %0.3g. \n', ...
mfilename, exp(log_sum_exp(log_wpg - log_Exp_pg)));
gammaMapW = weights(gammaPeak);
fractionalImprovement = norm(W - gammaMapW)./norm(W) * 100;
gammaMean = sum(exp(log_wpg - log_Exp_pg) .* g);
fprintf('%s: difference between INA estimate and gamma-MAP: %0.2g%%. \n', ...
mfilename, fractionalImprovement);
fprintf(['%s: gamma-MAP = %0.4g,\tE[gamma] = %0.4g,', ...
'\tdifference: %0.3g%%. \n'], ...
mfilename, gammaPeak, gammaMean, ...
(gammaMean - gammaPeak)./gammaMean * 100);
end%if
end%mne_estimate_ina
%{
% % Previously used a nested function:
% function logp = logp_g_on_B(g)
% %LOGP_G_ON_B
% weightsCov = weights(g)'*weights(g);
% IminusLW = (speye(Data.nSensors) - LeadFields.lf*weights(g));
% logp = - 0.5 * Data.nSamples * Data.nSensors * log(2*pi) ...
% - 0.5 * Data.nSamples * logDetNoise ...
% - 0.5 * trace(IminusLW * Data.cov * IminusLW.' * noiseInv) * Data.nSamples ...
% - 0.5 * Data.nSamples * nSourceElements * log(2*pi*g) ...
% - 0.5 * sum(sum(weightsCov .* Data.cov)) * Data.nSamples ./ g ...
% + prior(log(g),scale);
% end%logp_g_on_B
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W = sparse_bayes_estimate(Data, Noise, LeadFields, GammaPrior)
%SPARSE_BAYES_ESIMATE weights for sparse bayes source reconstruction
%
% Estimates source covariance using sparse Bayes estimate: allow for
% source variance at each location and in each orientation in space.
%
% To use this algorithm, -log p(gamma) MUST BE CONCAVE for each gamma.
%
% Ref: Wipf and Nagarajan, 2009
% estimate regularized covariance
sourceCov = sparse_bayes_covariance(Data, Noise, LeadFields, GammaPrior);
% Extract weights for 3d sources
W = estimate_weights(Noise.cov, sourceCov, LeadFields.lf);
end%sparse_bayes_estimate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function W = rvm_beamformer(Data, Noise, LeadFields, GammaPrior)
%RVM_BEAMFORMER weights for relevance vector machine beamforming
%
% Estimates source covariance using sparse Bayes estimate: allow for
% source variance at each location and in each orientation in space,
% according to Wipf and Nagarajan 2007, but updated to use a full noise
% covariance estimate.
%
% MAP optimisation of source variances occurs using formulae from Wipf and
% Nagarajan, 2009.
%
% To use this algorithm, -log p(gamma) MUST BE CONCAVE for each gamma.
% estimate regularized covariance
% Wipf and Nagarajan 2007 eq 10
sourceCov = sparse_bayes_covariance(Data, Noise, LeadFields, GammaPrior);
% Extract weights for 3d sources
% Wipf and Nagarajan 2007 Eq 7
W = beamformer_weights(Noise.cov, sourceCov, LeadFields.lf);
end%rvm_beamformer
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function sourceCov = sparse_bayes_covariance(Data, Noise, LeadFields, ...
GammaPrior)
%SPARSE_BAYES_COVARIANCE regularized source covariance
%
% Estimates source covariance using sparse Bayes estimate: allow for
% source variance at each location and in each orientation in space.
%
% To use this algorithm, -log p(gamma) MUST BE CONCAVE for each gamma.
%
% Ref: Wipf and Nagarajan, 2009
global DEBUG
nSourceElements = LeadFields.nDims * LeadFields.nSources;
sourceCovFn = @(gamma) spdiags(real(gamma), ...
0, nSourceElements, nSourceElements);
logGammaInit = (mean(log(diag(Noise.cov))) ...
- mean(log(sum(LeadFields.lf.^2)))) ...
.* ones(nSourceElements, 1);
% set scale for priors on gamma
scale = exp(logGammaInit + 5);
% use updates from gamma-MAP in Wipf and Nagarajan (2009)
oldGamma = exp(logGammaInit);
nIterMax = 1000;
iIter = 0;
deltaGamma = NaN(nIterMax,1);
gamma = zeros(size(oldGamma));
L = NaN(nIterMax,1);
HALT_CONDITION = 1e-4; %fractional change in norm(gamma) to class as convergence
while iIter <= nIterMax,
% some monitoring behaviour
if DEBUG,
% monitor cost function Eq 18 in Wipf and Nagarajan
if iIter > 0,
L(iIter) = optimise_target_single_prior(Data, Noise, LeadFields, ...
log(oldGamma), GammaPrior.fn, ...
scale, @(lg) sourceCovFn(exp(lg)), ...
true);
end%if
if (~mod(iIter, 5) || 1 == iIter) && 0 ~= iIter,
fprintf('%s: SB g-MAP iteration no %4d, deltaGamma %0.5g, L %0.6g.\n', mfilename, iIter, deltaGamma(iIter), L(iIter));
end%if
if 0 == iIter,
ff = figure('Color', 'w', 'Name', 'Updating gamma estimates');
rmFF = onCleanup(@() close(ff));
hist(log(gamma), 3000);
xlabel('Log(\gamma)');
elseif ~mod(iIter, 5) || (1 == iIter),
nzg = gamma~=0;
hist(log(gamma(nzg)), 3000);
title(sprintf('fraction non-zero: %0.2f%%', ...
sum(nzg)./nSourceElements * 100));
drawnow;
end%if
end%if
% start of loop proper
iIter = iIter + 1;
% use Factorize object to hold inverse of regularized covariance
% without computation
invSigmaB = osl_cholinv(empirical_bayes_cov(Noise.cov, sourceCovFn(oldGamma), LeadFields.lf));
% update rules
% Wipf and Nagarajan 2009 Eq. 31
for i = nSourceElements:-1:1,
gamma(i) = oldGamma(i) ...
* sqrt(trace(LeadFields.lf(:,i).' * invSigmaB * Data.cov * invSigmaB * LeadFields.lf(:,i))) ... % Frobenius norm term
./ sqrt(trace(LeadFields.lf(:,i).' * invSigmaB * LeadFields.lf(:,i) ...
- GammaPrior.diff_fn(oldGamma(i), scale) ./ Data.nSamples)); % Final trace term
end%for
% monitor change
deltaGamma(iIter) = norm(gamma - oldGamma) ./ norm(oldGamma);
% test for convergence
isConverged = deltaGamma(iIter) < HALT_CONDITION;
if isConverged,
break
end%if
% re-assign old gamma
oldGamma = gamma;
end%while
% tidy up
deltaGamma(isnan(deltaGamma)) = [];
% check if hit max iter
if iIter >= nIterMax,
warning([mfilename ':gMAPMaxIter'], ...
'Max number of iterations, %d, reached without convergence. \n', ...
nIterMax);
end%if
% final result
sourceCov = sourceCovFn(gamma);
if DEBUG
delete(rmFF); % delete old figure
figure('Name', 'Sparse-Bayes convergence', 'Color', 'w');
semilogy(deltaGamma, 'r', 'LineWidth', 2);
xlabel('Iteration', 'FontSize', 14);
ylabel('Fractional change in norm(\gamma)', 'FontSize', 14);
figure('Name', 'Gamma estimates', 'Color', 'w');
width = 0.8;
nzg = gamma~=0;
[n,x] = hist(log(gamma(nzg)), 3000);
hh = bar(x, n, width);
set(get(hh, 'Children'), ...
'FaceColor', [120, 120, 120]/255, ... % set to be grey
'FaceAlpha', 0.4, ...
'EdgeColor', 'none');
title(sprintf('fraction non-zero: %0.2f%%', ...
sum(nzg)./nSourceElements * 100));
xlabel('Log(\gamma)');
end%if
end%sparse_bayes_covariance
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function L = optimise_target_single_prior(Data, Noise, LeadFields, ...
logGamma, prior, scale, ...
sourceCovFn, quiet)
%OPTIMISE_TARGET_single_PRIOR
% minimise L to find gamma-MAP solution
global DEBUG
persistent iTARGETCALL
if nargin < 8 || ~exist('quiet', 'var'),
quiet = false;
end%if
Sigma_EB = empirical_bayes_cov(Noise.cov, sourceCovFn(logGamma), ...
LeadFields.lf);
% we want to maximise
% log p(g|B) = -0.5 Tr(BB' Sigma_b^{-1}) - n/2 logdet Sigma_b + log p(g).
try
logDetSigma = ROInets.logdet(Sigma_EB, 'chol');
catch ME
% some errors with non pos def matrices occuring, which is surprising.
if strcmp(ME.identifier, 'MATLAB:posdef'),
warning([mfilename ':OptimTarget:posdef'], ...
'Regularised covariance not positive defninite. \n');
logDetSigma = ROInets.logdet(Sigma_EB);
else
rethrow(ME);
end%if
end%try
% use property sum(eig(B, A)) = trace(inv(A) * B)
% or trace(AB) = sum(sum(A .* B')) (and covariance matrices are symmetric)
L = real((trace(Data.cov * osl_cholinv(Sigma_EB)) ... % faster than elementwise product or sum(eig()).
+ logDetSigma) * Data.nSamples - 2 * prior(logGamma, scale));
if DEBUG && ~quiet,
if ~exist('iTARGETCALL', 'var') || isempty(iTARGETCALL),
iTARGETCALL = 1;
else
iTARGETCALL = iTARGETCALL + 1;
end%if
fprintf(['Call to optim fn %4.0d: L = %0.8G, logGamma = %0.4G, ', ...
'logdet(Sigma_EB) = %0.6G. \n'], ...
iTARGETCALL, L, logGamma, logDetSigma);
end%if DEBUG
end%optimise_target_single_prior
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function L = optimise_target_double_prior(Data, Noise, LeadFields, params, ...
priors, priorScales, sourceCovFn)
%OPTIMISE_TARGET_DOUBLE_PRIOR
% minimise L to find gamma-MAP solution for S under B = (LS + e)/r.
global DEBUG
persistent iTARGETCALL
logGamma = real(params(1));
logRho = real(params(2));
pg = priors{1};
pr = priors{2};
Sigma_EB = empirical_bayes_cov(Noise.cov, sourceCovFn(logGamma), ...
LeadFields.lf);
try
logDetSigma = ROInets.logdet(Sigma_EB, 'chol');
catch ME
% some errors with non pos def matrices occuring, which is surprising.
if strcmp(ME.identifier, 'MATLAB:posdef'),
warning([mfilename ':OptimTarget:posdef'], ...
'Regularised covariance not positive defninite. \n');
logDetSigma = ROInets.logdet(Sigma_EB);
else
rethrow(ME);
end%if
end%try
% we want to maximise
% log p(r,g|B) = -0.5 r^2 Tr(BB' Sigma_b^{-1}) - n/2 logdet Sigma_b
% + n*log(r) + log p(r) + log p(g).
%
% use property sum(eig(B, A)) = trace(inv(A) * B)
% or trace(AB) = sum(sum(A .* B')) (and covariance matrices are symmetric)
L = (exp(2*logRho) * trace(Data.cov * osl_cholinv(Sigma_EB)) ... % faster than elementwise product or sum(eig()).
+ logDetSigma - 2*logRho) .* Data.nSamples ...
- 2*pg(logGamma, priorScales(1)) - 2*pr(logRho, priorScales(2));
if DEBUG,
if ~exist('iTARGETCALL', 'var') || isempty(iTARGETCALL),
iTARGETCALL = 1;
else
iTARGETCALL = iTARGETCALL + 1;
end%if
fprintf(['Call to optim fn %4.0d: L = %0.8G, logGamma = %0.4G, ', ...
'logRho = %0.4G, logdet(Sigma_EB) = %0.6G. \n'], ...
iTARGETCALL, L, logGamma, logRho, ...
logDetSigma);
end%if DEBUG
end%optimise_target_double_prior
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [lp, diff_lp] = parse_priors(priorChoice)
%PARSE_PRIORS creates function handle for priors on hierarchical variance
%parameters.
%
% Some priors require scales, so lp takes two arguments, lp(loggamma, scale).
% May also require d(lp)/dg. (= - df(g)/dg in Wipf and Nagarajan)
% pass on any passed-in functions
if isa(priorChoice, 'function_handle'),
lp = @(log_g,~) priorChoice(exp(log_g));
diff_lp = @(~,~) error([mfilename ':NoDiffLP'], ...
'No differential of gamma prior passed in.\n');
return
end%if
switch lower(priorChoice)
case 'uniform-variance'
% p(g) ~ 1
lp = @(log_g,~) 0; % ignore additive constant
diff_lp = @(g,~) 0;
case 'uniform-sd'
%p(sqrt(g)) ~ 1 --> p(g) ~ 0.5 / sqrt(g)
lp = @(log_g,~) - 0.5 * sum(log_g); % ignore additive constant
diff_lp = @(g,~) - 0.5 ./ sum(g);
case 'uniform-log'
% p(g) ~ 1/g
lp = @(log_g,~) - sum(log_g); % ignore additive constants as improper unless bounded
diff_lp = @(g,~) - 1.0 ./ sum(g);
case 'cauchy'
% p(sqrt(g) ~ 1/pi sqrt(A) / (g+A)
% --> p(g) ~ 1/(2 pi) sqrt(A) / ((A+g) sqrt(g))
lp = @(log_g,A) sum(-log(2*pi) + 0.5*log(A) ...
- 0.5*log_g - log(exp(log_g) + A));
diff_lp = @(g,A) sum(-0.5 ./ g - 1.0 ./ (g + A));
case 'log-normal' % scale on same space as g, so s.d. is log(A).
lp = @(log_g,A) - 0.5 * (log_g.' * log_g) ./ log(A)^2 ...
- length(log_g) * 0.5 * log(2*pi*log(A)^2);
diff_lp = @(g,A) sum( - log(g) ./ (log(A)^2 * g));
case 'normal'
lp = @(log_g,A) - 0.5 * (exp(log_g).' * exp(log_g)) ./ A^2 ...
- length(log_g) * 0.5 * log(2*pi*A^2);