One variable (black) in presumably independent and likely affects the other two variables, nonetheless cannot rule out a possible effect of the other variables on the first.
The two other variables (green and maroon) seem dependent on the first but also interact between them and may affect each other (and possibly the first variable too)
My question is what is the best way to explore these non-linear relationships in order to determine whether the effects are significant and how much of the variation in any given variable in explained by the other two variables?
The attached graph show how the three variables change with time.
Hi Ohad Peleg
One suggestion to explore the relationship is to plot black vs green and black vs maroon. Such plots may help highlight any systematic trend of green or maroon with respect to black. If no systematic trend can be seen, may be the dataset is too small or too noisy.
A dataset such as this may require specification of a linear/nonlinear state-space model in which these three variables represent the inter-dependent states of the system. Such a system of equations can be trained and cross-validated on a large dataset. Significance testing will have to wait till a suitable model is at hand which predicts specific hypotheses about the data.
Thanks Usman Rashid
The attached graph is derived from averaged data.
I used the pre-averaged data to plot the relationships like you've suggested using geom_smooth (method = "loess"). It showed non-linear interactions with maroon and green that were almost 'mirror imaged' (see attachment).
How do you suggest I proceed exploring these interdependecies?
I assume the graph is between black and the other two. I don't see any relationship between black and maroon. There is some variability and a few outliers but overall it seems the relationship is either linear or may be an offset difference: Maroon = Black + constant + Gaussian measurement error.
There may be some relationship between black and green. Perhaps an exponential decay: Green = constant1 * exp(-Black * constant2) + constant3 + Gaussian measurement error.
Can you please also plot maroon vs green and green vs maroon? The difficulty is that a large dataset is required to first hypothesise these relationships on first half of the data and then test them on the second half. In R, you can perform nonlinear regression with package "nlme" (https://cran.r-project.org/web/packages/nlme/nlme.pdf)
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