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I have a function of a dozen discrete variables, $$y = f(x_1, ..., x_n)$$ and $k$ samples of $y$ (a lot), $y$ being continuous.

I can use Matlab to analyse the data, with the statistics toolbox. The goal is to analyse the relationship between $y$ and my variables. For example, which variables $x_m$ explain best variations of $y$? (I was thinking of using an Principal Components Analysis, but I can't think of a way to adapt the method to it) To what extent do they have an influence on $y$? Can I find clusters?

My question is: do you have methods I should look into and hints about how to adapt them to my analysis?

Thank you

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    $\begingroup$ Since you want to find the effect of the predictors - $x_1, ..., x_n$ on a univariate response $y$, this seems like a classical multiple regression modeling problem (not multivariate, which would imply you have multiple dependent variables). Is there any reason multiple regression doesn't do what you need? $\endgroup$
    – Macro
    Jul 9, 2012 at 13:58
  • $\begingroup$ Well the main problem is that I have no idea of the form of f, and it's quite difficult to observe so many variables. Also, I am mostly trying to see which variables are relevant to explain changes in y instead of having an explicit formula that is very likely to be wrong. Lastly, my variables are discrete, which doesn't seem to be very suited for this kind of analysis. $\endgroup$
    – Flavian
    Jul 9, 2012 at 14:01
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    $\begingroup$ OK but since your predictor variables are categorical, the most general function form (the "saturated" model) is just to have a different function value for each combination of the predictor values, which can be accomplished with a regression model with interactions between all of the variables. This is probably overkill though - pairing down the model would have to involve some substantive theory which is impossible to comment on without more information about what the data actually is - can you provide some background? $\endgroup$
    – Macro
    Jul 9, 2012 at 14:07
  • $\begingroup$ Well for example you can take y: sales for x_1: date, x_2: product type, x_3: product size, x_4: store size $\endgroup$
    – Flavian
    Jul 9, 2012 at 14:11
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    $\begingroup$ Have you looked into regression trees? They seem like a good fit to your question. $\endgroup$
    – Peter Flom
    Jul 9, 2012 at 14:47

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