Annalisa, I am also not sure. But let's summarise: The PCA yielded a kind of projection of several variables (measurements, brain activities) observed over time instances. The trajectories you have drawed (in a plane  or space if they are based on 2 or 3 principal components) -- how did you do it? probably by linking temporal points indicated by rows of your data matrix serving for PCA calculations. If so, you have a visualisation of changes in time of your variables  which happen jointly for these variables in the multivariate unobservable data space.   If something undergoes some changes in time, then we we may refer to it as manifesting some dynamics.  And just the illustration of this dynamics (changes in time of your variables) is illustrated in your 2D or 3D pca graphs. It seems to make sense.

 Good night. We have in Wroclaw -11C frost now.

Anna

Good morning Anna and Annalisa

I believe I can explain what PCA analysis is doing with your datasets and how and why it reveals new information. The method is closely related to an analytical method I use which I call Grey Scale Tracking (or GST), and both techniques seek to isolate and quantify component dynamic (aka scalar) relationships. My particular research interest is the analysis and resolution of multivariate cross dependency relationships in psychophysical systems such as the human colour sense.

The  relationship between PCA and my  GST analysis is intriguing , so here is a brief account ( see also my paper ‘The control of colour using measurement and feedback’ and my Thesis ‘Selected Papers on Colorimetric Theory an Colour Modelling’ both pasted as full texts on RG).

Any given multi-variate data set may be regarded as a vector space with a set of basis vectors that represent component case and effect variables. Within this space a unique subset of values exists whose members lie on a ‘grey scale axis’ where all the causal variables of the observed phenomenon have precisely equated added value. In effect, all members of this unique subset of measurements share a strictly constant and precisely equated definition of observed added value. It follows therefore that all possible differences between set members must by definition be scalar differences of basis vector value . These scalar relationships are by definition strictly intra-dimensional and in consequence they are available for analysis and quantification using GST experiments where only a single variable is isolated and changed. Note that PCA attempts to achieve the same isolation and quantification, but if additive difference is not held constant at null point equivalence the iterative solution must inevitably be multidimensional; whereas in the GST approach it is possible to iteratively refine cause and effect relationships dimension by dimension.

This is all very well when analysing the colour sense with its clear visual neutrality axis as discussed in my 2010 Thesis, but I am now  fascinated with the possibility of applying GST to the analysis of multivariate Psychological phenomena in general.

It is after all simply a set of experimental constraints and analytical criteria which in principle enable the classic principles of scientific method  to be applied to the field of psychophysics  and psychology  where cause and effect cannot be measured directly

Do you think it could work in your area of specialty?

Regards and best wishes

David.

Dear Annalisa, 

are you talking to use affine connections and homotopies, in order to reduce the dimensionality of the 3D brain into a 2D manifold (through  vectors or tensors)? in this case, you may look at the methodology (the first section) in this paper:

https://www.researchgate.net/publication/287583633_THE_NEUROSCIENTIFIC_MECHANISMS_OF_KNOWLEDGE_A_TOPOLOGICAL_APPROACH 

Preprint THE NEUROSCIENTIFIC MECHANISMS OF KNOWLEDGE: A TOPOLOGICAL APPROACH