Data sets, when structured, can be put in vector form (v(1)...v(n)), adding time dependency it's v(i, t) for i=1...n and t=1...T.
Then we have a matrix of terms v(i, j)...
Matrices are important, they can represent linear operators in finite dimension. Composing such operators f, g, as fog translates into matrix product FxG with obvious notations.
Now a classical matrix M is a table of lines and columns, containing numbers or variables. Precisely at line i and column j of such table, we store term m(i, j), usually belonging to real number set R, or complex number set C, or more generally to a group G.
What about generalising such matrix of numbers into a matrix of set (in any field of science, this could mean storing all data collected for a particular parameter "m(i, j)", which is a set M(i, j) of data?
What can we observe, say, define on such matrices of sets?
If you are as curious as me, in your own field of science or engineering, please follow the link below, and more importantly, feedback here with comments, thoughts, advice on how to take this further.
Ref:
https://www.researchgate.net/publication/348786944_Matrices_of_Sets_MoS_Why_and_How
Dear Prof. Di Francesco
Thank you for sharing this excellent question.
kind regards
@Radoslaw Walkowiak, many thanks for your views. I am indeed looking at products of matrices of sets A. B, with A containing for instance resource factors (customer or company willingness to pay for certain needs and demands) and market availability factors (sets of products, features, services, in the market), and interpreting the product of the two, as compound scenarios.
This fairly neutral model (just sets) should allow to sort the possible (demand, supply) configurations without entering the details of cost, price, quantity, too early in the game, thus keeping "options" and "degrees of freedom". I will be more specific in a next paper, with use cases to illustrate (economic scenarios, in particular)
The link is very nice generalization from points or vectors to arrays of matrices that may be further extend to generalized to tensors. My own research is also based on matrix valued probability distribution functions with manifold valued data. A very good library on such work via matrices as data points is here : https://www.manopt.org/
Please check the following link
https://www.researchgate.net/publication/338004118_A_literature_survey_of_matrix_methods_for_data_science
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