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In the image below, the orange curve is the true function, the dots are samples from the true function plus a unit variance gaussian, while the blue curve are the estimated function using linear regression with polynomial basis.

Graph

I want to find a meaningful way to compute the true error between the blue and orange curves. I used $$ \epsilon=\|f(x)-\hat f(x)\|_2 $$ where $f(x), \hat f(x)$ are the vector of all the outputs in the interval and $x$ consists of small increments (points) in the X-axis. The problem is that the true error increases as I let the increments become small to smoothen the plot of the curve. What shall I divide with my true error function to make them more meaningful like staying between 0 and 1 for example?

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  • $\begingroup$ Are you multiplying your point-estimates by the "dx" you are using? If not, that would certainly be a reason your error is diverging with decreasing stepsize. $\endgroup$
    – costrom
    Jul 19, 2022 at 15:39
  • $\begingroup$ @costrom unfortunately no, I am simply using the function values obtained on those many x points in my error function. May I ask where should I place those dx in my function? $\endgroup$
    – wd violet
    Jul 19, 2022 at 15:40
  • $\begingroup$ This question may be useful: math.stackexchange.com/questions/675833/… $\endgroup$
    – costrom
    Jul 19, 2022 at 20:32
  • $\begingroup$ Could you please explain what you mean by "true error"? After all, you already have an estimate of the error between your polynomial fit and the data. $\endgroup$
    – whuber
    Jul 19, 2022 at 20:34
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    $\begingroup$ That "true error" is not defined until (1) you specify a compact domain for the function and (2) you specify the metric you are using to compare two functions. $\endgroup$
    – whuber
    Jul 20, 2022 at 12:50

1 Answer 1

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The "total error" you are looking for may be something like the distance between functions.

You are currently computing:

$$error = \Sigma (f(x_i) -g(x_i))^2 $$

This will, as you have found, not converge to a single answer when you vary your sampling 'increment'.

What you should be computing is something more like:

$$error = \Sigma (f(x_i) -g(x_i))^2 \Delta x $$

where $\Delta x$ is the 'increment' in $x$ you use to estimate the error.

This should produce the converging results that you expected.

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  • $\begingroup$ That error function approaches 0 as I let $\Delta x$ approach 0, is that behavior expected? $\endgroup$
    – wd violet
    Jul 20, 2022 at 3:31
  • $\begingroup$ I would not expect it to go to zero, no. Can you provide the functions that produce the blue and orange curves? $\endgroup$
    – costrom
    Jul 22, 2022 at 20:05

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