The Package: PTAk Version: 1.1-4 performs Principal Tensor Analysis on k modes
as the main multiway method of the package.
last released in R in CRAN contributed packages.
examples using PTAk
PTA-k modes is a multiway method to decompose a tensor (array) of any order,
as a generalisation of SVD also supporting non-identity metrics and penalisations.
2-way SVD with these extensions is also available. The package includes also some other multiway
methods: PCAn (Tucker-n) and PARAFAC/CANDECOMP with these extensions.
A comparison between a canonical SVD and a penalised SVD
forcing components ($u and $v) to be smooth...
Now an example with fMRI data (brain x time x subjects)
for a multi-subject verbal experiment without metrics or penalisations .
The time series (for each voxel and each subject) were detrended, and each subject was scaled to variance one.
The first Principal Tensor i.e. singular value vs111 (figure of the triple of components):
The same analysis using a non-identity metric for time
-the inverse of a robust estimate of the within "group"-(without knowing the group structure) computed on the (brain x time) mean over subjects.
The same analysis using a identity metric but
penalisation for time.
The same analysis using a identity metric but
penalisation for time and space.
The same previous analysis using non-identity metric (the inverse of the within group as above) and penalisation for time and space.
Now the same data example with the full brain masked for only grey matter voxels. Smoothing operating only on time component (with the same smoother as before) and just identity metrics.
summary of the decomposition
The correlation is improved and we still get the "same" spatial and subject component for the first Principal Tensor.
In the previous examples of fMRI analysis using PTAk one focused on Principal Tensor where the time component well correlated to the paradigm, but one could also look PTs where the subject component shows some group differences, or look for PTs where a specific region would be activated (using appropriate inference).
One subject analysis using PCA, FCA FCA 4modes.
Correspondence Analysis is an established multidimensional method looking at the of lack of indepence in a bivariate distribution usually represented using a contingency table. The raw fMRI data can seen as sptio-temporal distribution of BOLD signal but after detrending the time series one has either to separate positive and negative values or shift them all to be able to consider it as "count". Note on the "raw detrended" data the rows margins are zeros.
The spatio-temporal distribution can also be looked as 4-dimensional distribution instead of bivariate. In this case indpendence is also inforced in the space directions. A generalisation of Correspondence Analysis to multiple contingency table has to be used
( Leibovici(2000a)). Theoretically it would be possible to perform the analysis in the continous case ( Leibovici and El Maach(1997))
We show now some results with fMRI using FCA, and FCA4-modes.
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Didier Leibovici
Last modified: Thu Jul 5 10:21:34 BST 2006