Michael and/or Pierre,
I think the easiest way of understanding this is to realise that both
de-meaning in time and space can be written as follows:
X = t_mean * ones(1,V) + ( ones (T,1) * v_mean + X')
Here, t_mean denotes the mean time course and v_mean denotes the mean
image. X is the original data a,d X' is the data de-meaned both
across the spatial and temporal domain. This I think makes it quite
transparent how any decomposition for X' relates to a decomposition
for the original data.
The reason for de-meaning across the temporal domain (i.e. removing
the mean image first) is that the mean image is not interesting at
all - depending on the exact settings of your EPI acquisition you get
different mean intensities across time - what we're interested in is
the variations around the mean. (if you were to add a fixed offset to
each time series you would not want to generate a different
decomposition i.e. you'd want the method to be insensitive to the
actual mean image. Therefore the mean for each time course (i.e. the
mean image overall) is removed. This is analogous to the way that in
the GLM each time series gets de-meaned. As a result, the estimated
mixing matrix will always have zero mean time courses.
Similarly, the mean time course is not very informative - the mean
time course is generated by averaging across space, weighting each
voxel similarly. As I said before this basically means that the
corresponding spatial map does not have any spatial specificity. Wrt
estimating the IC maps the removel of the mean time course from all
voxels' time series' means that the IC maps are assumed to be mean
zero - noting more.
You are right in saying that this is not generally done in PCA,
though it should be quite clear that with PCA you only ever want to
model those variations in the data that are of interest. In our case
this is variations within X'.
Hope this helps
Christian
On 7 Jun 2007, at 09:39, Michel Dojat wrote:
> Christian,
> I'm not sure I understand correctly your answer. So may be I was
> not clear enough.
>
> For SICA, I have an observation matrix (xij) of size TxV (T=time,
> V=voxels).
> I calculate the mean for each row (i) and remove it to xij for
> centering the data.
> This corresponds to a referential (with T axis) shift and is
> commonly used in PCA.
> BUT, generally we do not do the same for the column (j).
>
> If I read the Melodic code the data centering is performed for each
> i and each j.
> and I do not understand why.
>
> Thanks for your answer
>
> Best
> Pierre Lafaye de Michaux.
>
>> Hi,
>>
>> You're partly right, in melodic the temporal mean is first removed
>> and re-introduced after the decomposition as outlined in the
>> technical report (see first few lines in section 'Maximum
>> Likelihood estimation' of the technical report available at our
>> website for the technical treatment).
>> The mean time course in effect corresponds to a spatial map where
>> all voxels contribute equally, i.e. it is a signal which does not
>> contain any spatial specificity - as such, the mean time course
>> itself carries no useful information wrt the final (spatially
>> specific, i.e. not everywhere the same) maps, particularly not
>> during the estimation stage. The part which lies in the space of
>> the modelled time courses is only useful in that it re-sahpes the
>> corresponding time courses while leaving the spatial maps in tact.
>> hope this helps
>> best
>> christian
>>
>> On 6 Jun 2007, at 15:35, Pierre Lafaye de Micheaux wrote:
>>
>>> Dear members,
>>>
>>> When i do a PCA in melodic, i get the following message:
>>>
>>>> Excluding voxels with constant value
>>>> Data size : 120 x 158227
>>>> Removing mean image ... done
>>>> Estimating data covariance ... done
>>>> Removing mean time course ... done
>>>> Starting PCA ... done
>>>
>>> So, it seems that you center the data BOTH for the lines and the
>>> columns.
>>> I thought that if you make a spatial PCA, you should center the
>>> data only for the variables (time) and not for the statistical
>>> units (space): removing mean tim course only.
>>> In a temporal PCA i thought you should also center only for the
>>> variables (space in this case): removing mean image only.
>>>
>>> Am i right? Do you have a theoretical justification for the
>>> procedure used in Melodic?
>>>
>>> Best regards,
>>>
>>> Pierre Lafaye de Micheaux, Ph.D.
>>>
>>> --Pierre Lafaye de Micheaux
>>> Bureau 210, bâtiment BSHM
>>> 1251 avenue centrale BP 47
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>>>
>>> Tél.: 04.76.82.58.73 / Fax: 04.76.82.56.65
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>>> http://www.biostatisticien.eu
>>
>> ____
>> Christian F. Beckmann
>> University Research Lecturer
>> Oxford University Centre for Functional MRI of the Brain (FMRIB)
>> John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK.
>> [log in to unmask] http://www.fmrib.ox.ac.uk/~beckmann
>> tel: +44 1865 222551 fax: +44 1865 222717
>
>
> --
> Michel Dojat
>
> Grenoble Institut des Neurosciences (GIN)
> Centre de Recherche Inserm U 836-UJF-CEA-CHU
> Equipe : Neuroimagerie Fonctionnelle et Métabolique
> CHU - Pavillon B
> BP 217
> 38043 Grenoble Cedex 9
>
> Tél: (direct) 33 (0)4 76 76 88 97 (secr.) 33 (0)4 76 76 57 48
> Fax: 33 (0)4 76 76 58 96
>
> [log in to unmask]
> http://nifm.ujf-grenoble.fr/~dojatm/index.html
____
Christian F. Beckmann
University Research Lecturer
Oxford University Centre for Functional MRI of the Brain (FMRIB)
John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK.
[log in to unmask] http://www.fmrib.ox.ac.uk/~beckmann
tel: +44 1865 222551 fax: +44 1865 222717
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