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I describe here a model to describe how neighbours influence a node.

I wish to implement it to attempt forecasting to values associate nodes; I post here asking for suggestions on mathematical model and machine learning techniques that could have already considered a similar approach, but I am not aware of, and hints for their implementation (python).

Suppose you a have a squared 2D lattice (a grid of 9 squares for simplicity), and:

  • for each time t
  • from each cell in the grid

you have vectors pointing to or from the central square.

Like this example, at time t1 :

enter image description here

I want to "map" the vectors to a parameter, associated to the central square.

I think vectors could be modelled as "a matrices X " ( or a family of vectors), and a resulting value y, but I am actually struggling to correctly represent it in a formal way.

Like in this second image:

enter image description here

I then want to forecast the possible y value in the future. I think I could explore linear or non linear models, like regressions or forecasting applied to times series:

but I want to train on the time-series of vectors, not on the time-series of ys (Hope it s clear).

Could you help to suggest models to look at and possible implementations (python), which may have already approached this dynamics?

Could you also suggest approach to generalise to manifolds structures (3d) ?

As examples for generalisation, I am thinking at:

  • a lattice of 3D ( neighbors cubes influences a central cube);
  • Ys as vectors, not scalars ( see final image to explain what I mean ) where a resulting vector is interpreted as the resulting vector at the next step in the future t + 1

enter image description here

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It really seems like a problem of formulating your probkem. On your 2D exemple you can split your $X_i$ along your axes and juxtapose them to build a data set. Each $X_i$ has two components : along axis 1 ($X_{i1}$) and 2 ($X_{i2}$).

Basically you would just build your data set by juxtaposing the 18 columns :

$X_{11}$ , $X_{12}$ , ..., $X_{i1}$ , $X_{i2}$ ,..., $X_{91}$ ,$X_{92}$

To try to predict your y.

You can augment the dimension of you problem by similar juxstaposition of your features. It would get a bit more complex if you augment the dimension of y, as you would need multiple models.

Trying to predict $y_{t+1}$ would lead to other considerations, like considering $y_t$ as a feature of your model.

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