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I have seen regression models with a continuous predictor fitted as a spline written like this:

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What is the meaning of the little "addition symbol" subscript that I have circled in red? Is it simply a "pointer" to the next equation below, where extra information is given regarding the particular values we assign to x depending on the conditions? I wasn't sure if it was intended to mean any "addition" because it is a plus symbol. It doesn't look that way because the sigma symbol already implies this? It just seems a bit unnecessary to me, but I might be mistaken.

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    $\begingroup$ The usual reading of this notation is "the positive part of." It can be more succinctly defined as $(y)_+=\max(0,y)=(y+|y|)/2.$ Of course it's necessary, because whenever $y\lt 0,$ $y\ne(y)_+.$ $\endgroup$
    – whuber
    Dec 3, 2023 at 15:51

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The second part of your question answers the first part. The + subscript says "Evaluate this and, if it is negative, substitute 0".

This is pretty common with splines, where effects are piecewise.

But the third part of your question is slightly off. This is not a value to assign to x, but to the whole expression that has the subscript.

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