# Regression with incomplete basis [closed]

Suppose that $$f(x) = \frac{1}{2}\text{max}(x+10,0) + \frac{1}{3}\text{max}(x+20,0) + \frac{1}{4}\text{max}(x+110,0) + \frac{1}{5}\text{max}(x+120,0)$$

If I randomly simulate values for $$x$$ such that $$\text{max}(x+10,0), \text{max}(x+20,0), \text{max}(x+110,0), \text{max}(x+120,0)$$ are linearly independent and make a regression of $$f(x)$$ on these, I am quite convinced that regression coefficients will be close to $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$$ and will converge to this as I simulate more values for regression. (am I right?)

What interests me is what happens if I omit one of the functions and make a regression:

$$f(x) = \beta_1max(x+10,0) + \beta_2max(x+110,0) + \beta_3max(x+120,0)$$

Can we say that since $$\text{max}(x+10,0)$$ is closer to $$\text{max}(x+20,0)$$ than $$\text{max}(x+110,0)$$ or $$\text{max}(x+120,0)$$ the coefficient of $$\text{max}(x+10,0)$$ will grow more than coefficients of the last two.

Intuitively I feel like the coefficient of $$\text{max}(x+10,0)$$ will absorb the most of the coefficient of $$\text{max}(x+20,0)$$ and something will be left for other two.

Of course my question right now is not mathematically precise. I didn't define what I mean by the closeness of functions for example.

Was this question studied before? Ideally I would like to know what happens if $$f(x)$$ has infinite dimensional basis:

$$f(x) = \alpha_1g_1(x) + \alpha_2g_2(x)+...+\alpha_ng_n(x)+...$$

and we make a regression on some finite set of functions $$g_i(x), g_{i+1}(x),...g_{i+k}(x)$$

I understand that it is a projection on the space spanned by these functions, but what happens to the coefficients? Will they somehow absorb the coefficients of the functions that are close to them (in some metric )

## closed as off-topic by Alexey Ustinov, user44191, Chris Godsil, LSpice, StoppleMay 17 at 16:50

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