On enter I'm having a sequence of pairs $(X_n, Y_n)$. Consider the following two graphics of two possible sequences.
in first case $Y_n$ can be modeled as $f(X_n)$ while in second case it obviously has no sense. The question is how can I determine if trying to represent $Y_n$ as $f(X_n)$ makes sense? Probable there is some criterium or something like that?
What can be done in multivariate situation (i.e. when we're trying to represent $Y_n$ as $f(X_1^{(n)}, X_2^{(n)}, ..., X_k^{(n)})$)?
Please note that trying to fit points with something graphicaly (e.g. like on first graph) and seeing if it fits the data is not OK. I'm looking for numerical criterium.
UPD. the variant that I'm thinking of is trying to look at variation of sequence. I.e. we order $(X_n, Y_n)$ in such a way that $X_{k-1} < X_k < X_{k+1}$ and then look at $\sum_{i=1}^N \frac{|Y_i - Y_{i-1}|}{X_i - X_{i-1}}$. And there can even be an analog for multivariate case.
You are looking for correlation measures from descriptive statistics.
There are different correlation measures for metric variables. For a linear function, there is the correlation coefficient. For a monotonic function, there is the rank correlation coefficient.
There are formulas for bivariable distributions and for multivariable distributions. For multivariable distributions, there are also the partial correlation measures.
Before applying one of the correlation measures, you can dissect your data into monotonic branches. You can do this by using autocorrelation coefficients.