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I'm looking to interpret the output from my SVR model.

I know that with SVM you can't directly interpret the coefficients of the model but that you first have to take a dot product

With that said, how would one interpret the coefficients of a linear kernel SVR model? Hence, once I've obtained the coefficients vector, what transformation do I need to apply and how would I go about interpreting the results?

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Assuming you mean that you've trained an SVR model in the dual using a linear kernel, so that your predictive function is $$ f(x) = \sum_i \alpha_i k(X_i, x) + b $$ and $$ k(x, y) = x^\top y ,$$ then you can find the coefficients in each dimension like so: \begin{align} f(x) &= \sum_i \alpha_i X_i^\top x + b = \left( \sum_i \alpha_i X_i \right)^\top x + b \end{align} so that $$ f(x) = w^\top x + b \qquad w = \sum_i \alpha_i X_i .$$

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