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The main difference is the choice of knots and penalization. To start, let's focus on cubic splines for x, y data.* I find it useful to start from the extreme situation, an interpolating spline that connects all data points with a smooth curve. Then each x-axis value is a knot at which two cubic functions interpolating on either side of the knot agree in ...

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All you've really told us is that the relationships are nonlinear, and you are asking us to tell you how to get the best model. It's not really possible to do that, since "nonlinear" is a huge class of relationships. Nevertheless, here are some things to think about: Ideally, there would be some aspect of the variables that would suggest a class ...

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You could train a non-linear model using only $x_c$ to predict $y$, i.e. $f(x_c) = \hat{y}$. Then compute the residual $\epsilon = \hat{y} - y$. The residual $\epsilon$ will have been controlled for with regards to $x_c$ as $f$ has captured the relationship between $x_c$ and $y$ as well as it possibly could. You can now use the residual $\epsilon$ as input ...

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When bivariate data present this shape — here, with the points falling nearly vertically around x = 0 and x = 1, and then nearly horizontally when x > 2 — a continuous model is unlikely to produce satisfactory results. Sometimes a segmented model, such as a linear-plateau or quadratic plateau, will work well enough. There is, however, another approach ...

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