Dear Christian,
Your answer is much more precise. Thank you.
But i have a question about this formula:
X' = X- t_mean * ones(1,V) - ones (T,1) * v_mean
If i use this formula, the matrix X' is neither centered for the rows,
neither centered for the columns.
Maybe i should first center the matrix for the rows to obtain a matrix
X' and then center this matrix X' for the columns to obtain a matrix X'' ?
Or the inverse ? That is:
first center the matrix for the columns to obtain a matrix X' and then
center this matrix X' for the rows to obtain a matrix X''
What do you think about this point?
Best regards,
Pierre
Christian Beckmann a écrit :
> 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
>>>> 38040 GRENOBLE Cedex 09
>>>> FRANCE
>>>>
>>>> Tél.: 04.76.82.58.73 / Fax: 04.76.82.56.65
>>>> [log in to unmask]
>>>> 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
> €
--
Pierre Lafaye de Micheaux
Bureau 210, bâtiment BSHM
1251 avenue centrale BP 47
38040 GRENOBLE Cedex 09
FRANCE
Tél.: 04.76.82.58.73 / Fax: 04.76.82.56.65
[log in to unmask]
http://www.biostatisticien.eu
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