Hi Christian,
Hi Hans,Under the null assumption these two approaches are identical, the only difference being that in the case of the more complicated variance normalisation the variance is based on an estimate from a truncated Gaussian (i.e. you normalise to e.g. 0.95 globally rather than 1 globally but this is irrelavant given the later Z-transform). Under the alternative hypothesis of there being signal somewhere, the PCA based variance normalisation is basically a robust estimate for the the noise standard deviation.
Given the parameterisation we can flexibly move between both extremes (no variance normalisation to full variance normalisation). The details are not in the paper, the overall concept of having to do such a normalising step is.Given typical sparseness and low SNR of BOLD signals of interest the difference between the two cases is largely negligible - where it really matters is wrt non-BOLD signal variations, e.g. slice dropouts or spiking, which can induce changes in variance _much_ above the level you get from signals due to activation.The reason why it matters is that in such extreme cases the noise standard deviation does not correspond to an isotropic case which then messes with the mixture model fit used for inference in that voxels which do not contain any signal (the background noise class) cannot simply be modelled using a single Gaussian (as the corresponding PDF for the background noise will be a mixture of Gaussians just as in the case where no variance normalisation is carried out) - it is for this reason that we now use a robust estimator of the voxel-wise variances in the software.The thresholding simply is used temporarily to get a robust estimate of the noise which is then applied to the non-thresholded data.Practically, apart from the extreme cases of scanner-indiced variation, this normalisation deals with the very simple problem of tissue-dependent noise variance. Below is a plot of this (on log-scale!) highlighting that if no variance normalisation is carried out then the a single CSF voxel can easily weigh as much as 10 GM voxels in the calculation of the data covariance - one reason for not doing what you suggest (i.e. only use this normalisation in the dim-estimation step)hthChristianOn 20 Aug 2009, at 17:02, Hans Tissot wrote:Hi Christian,Sorry for jumping into this thread but I am very confused. I have 2 questions:(1) I found your description in the previous e-mail different from that described in your paper. Here's the text from your IEEE paper:"An immediate consequence of the fact that we are operatingunder an isotropic noise model is that as an initial preprocessingstep we will modify the original data time courses to be normalizedto zero mean and unit variance. This appears to be a sensiblestep in that on the one hand we know that the voxel-wise standarddeviation of resting state data varies significantly over thebrain but on the other hand, all voxels’ time courses are assumedto be generated from the same noise process. This variance-normalizationpreconditions the data under the “null hypotheses” ofpurely Gaussian noise, i.e., in the absence of any signal: the datamatrix X is identical up to second-order statistics to a simple setof realizations from a N(0,1) noise process. Any signal componentcontained in X will have to reveal itself via its deviationfrom Gaussianity. This will turn out to be of prime importanceboth for the estimation of the number of sources and the finalinferential steps.After a voxel-wise normalization of variance, two voxels withcomparable noise level that are modulated by the same signaltime course, aj, say, but by different amounts will have the sameregression coefficient upon regression against aj. The differencein the original amount of modulation is, therefore, containedin the standard deviation of the residual noise. Formingvoxel-wise statistics, i.e., dividing the PICA maps by the estimatedstandard deviation of eta, thus, is invariant under the initialvariance-normalization."Based on this description it seems that each time course is indeed normalized to unit variance (and this is a critical step as per the description above). Could you please clarify.
(2)A different approach that you described in the previous e-mail seems to divide each time course x_i by hat_sigma_i to equalize noise variance across voxels as assumed in the model. I could not find anything in the paper about thresholding the loadings at p .05 and doing the division by hat_sigma_i only on those voxels which pass the threshold. If I missed it, could you please direct me to the correct section? Why is this thresholding required anyways, why not just divide all voxels by their sigma_hat_i, once the latent dimensionality is estimated?Thanks,Hans Tissot.On Wed, Aug 19, 2009 at 4:37 PM, Christian F. Beckmann <[log in to unmask]> wrote:
HiNeither - Melodic variance normalisation attempts to normalise the residual _noise_ and therefore scales by the inverse of a robust estimate of the noise, rather than normalise by signal + noise (which, as you say, would remove important amplitude information). More specifically, melodic runs a full PCA on the data, thresholds loadings at a p .05 level (uncorrected) in order to indentify signals which potentially will later on pass thresholding. Melodic then regresses this out of the data and uses the residuals from this regression to identify voxel-wise standard deviations. Each voxel then gets normalised by the inverse of this estimate. This then renders the data closer to the model assumptions of the probabilistic ICA model (which assumes isotropic noise) and improves signal detection and gives a much improved mixture model fit for inference - see Beckmann and Smith (2004) IEEE TMI for details.hthChristianOn 19 Aug 2009, at 21:17, MCLAREN, Donald wrote:I've been trying to figure out how the variance normalization works in FSL.Is the variance of each voxel set to 1 or is the overall variance of all voxels set to 1?In the context of resting fMRI, if the former is true, isn't information about the amplitude of the flucuations being removed? This would seem to be important if your comparing data from different scans were done in two different states (e.g. eyes open versus eyes closed) since the amplitude of the flucuations how been shown to be different in these states.Any thoughts would be appreciated.
--
Best Regards, Donald McLaren
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Christian F. Beckmann, DPhil
Senior Lecturer, Clinical Neuroscience Department
Division of Neuroscience and Mental Health
Imperial College London, Hammersmith Campus
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Tel.: +44 (0)20 7594 6685 --- Fax: +44 (0)20 7594 6548
Email: [log in to unmask]
http://www.imperial.ac.uk/medicine/people/c.beckmann/
Senior Research Fellow, FMRIB Centre
University of Oxford
JR Hospital - Oxford OX3 9DU
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Email: [log in to unmask]
http://www.fmrib.ox.ac.uk/~beckmann
_______________________________________________
Christian F. Beckmann, DPhil
Senior Lecturer, Clinical Neuroscience Department
Division of Neuroscience and Mental Health
Imperial College London, Hammersmith Campus
Rm 419, Burlington Danes Bldg, Du Cane Road, London W12 0NN, UK
Tel.: +44 (0)20 7594 6685 --- Fax: +44 (0)20 7594 6548
Email: [log in to unmask]
http://www.imperial.ac.uk/medicine/people/c.beckmann/
Senior Research Fellow, FMRIB Centre
University of Oxford
JR Hospital - Oxford OX3 9DU
Tel.: +44 (0)1865 222551 --- Fax: +44 (0)1865 222717
Email: [log in to unmask]
http://www.fmrib.ox.ac.uk/~beckmann