I want to find the principal “parameters” driving the relative location of a set of $K$ discrete curves (observables, dependent variables).
The data shape is as follows:
n n c c o
e1: a1 b1 m1 n1 x1=[x1_1,...,x1_n]
e2: a2 b2 m2 n2 x2=[x2_1,...,x2_n]
...
eK: aK bK mK nK xK=[xk_1,...,xk_n]
The parameters (independent variables) are either numerical ($a_k$, $b_k$) or categorical ($m_k$, $n_k$). I would like to explain or predict the relative location of the curves with a model based on those parameters.
In a simplification of the above problem for illustration, in the case the categories are the same for all data, and only the parameter $a_k$ varies (with values $1,2,4,8, 16,32, 64$ here), the curves may look as follows:
In the central part of the plot ($[0.1,0.6]$), the curves lay with an order which is monotonous with $a_k$ ($x_1$ is above $x_2$, etc.). It becomes very 1D, and can be handled relatively easily. Yet, curves may cross at some locations, and other numercial and categorical variables may come into play. I welcome suggestions in the following directions, from the most generic to simplified cases.
- What are names of techniques and tools (if any) to address the prediction of the observed curves (on their whole range)?
- Are there more interesting techniques if I restrict to a range (like $[0.1,0.6]$) where curves behave nicer (are more of less above each other)?
- Should the curves be converted to simpler numbers (like their hierarchical rank, an area between curves)?