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I have data that is the result of measurements $f(x)$ at points $x$. These measurements fluctuate (call it noise), also differently for each $x$ so we have that $\sigma(x)$ are the fluctuations. By measuring several times at each $x$ I can estimate the fluctuations as well as $f$ (which would be the average). So then I have something that looks like this:

enter image description here

Now my goal is, given one measurement $f$, to tell which value of $x$ produced such measurement, which is probably a typical problem in many disciplines. Due to the shape of $f$ and the fluctuations, then there will be regions in which there will be few doubts about the value of $x$ and regions in which many values of $x$ could be possible, so there is a "prediction power", I hope the following drawing explains what I mean:

enter image description here

I want to quantify this. What are typical measures for this?

Intuitively I think of something like this: If $f(x)$ was sampled at discrete points $x_1,x_2,...$ with equal spacing, i.e. $x_i-x_j=\Delta x ~ \forall ~ i,j$, then a possible measure of this "prediction power" at point $x_i$ would be $\frac{\Delta f_i}{\sigma_i}$. Now, since this is probably a super common problem, I would guess there are already better measures of this (or similar).

user171780
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  • Your "prediction power" has historically been answered with a fiducial interval and that in turn has been interpreted as a confidence interval (or confidence region) using inverse regression. In the case of univariate OLS regression I provide the details and a worked example at https://stats.stackexchange.com/a/206682/919. The technique remains the same for your more complicated model. You can do better if you know or assume a probability distribution for $X,$ for then your problem reduces to estimating the conditional distribution on $X$ given $f.$ – whuber May 12 '23 at 16:37
  • BTW, your proposal makes intuitive sense but omits two crucial factors: the measurement error, as revealed in the variation of the replicates, and the interpolation error between the observed values. Both can play important roles. For instance, where the slope of the fit is low but the standard error of prediction is also low you might have better "prediction power" than where the slope is high but the standard error is high, too. The SE will be high at and beyond the right and left ends of your data and perhaps in intermediate points, too, depending on the model and pattern of replication. – whuber May 12 '23 at 16:42
  • So, it is probably too broad and vague to inquire about "typical measures." Instead, please provide any of the additional information I have indicated and, if possible, give us some contextual information such as why you're doing this and what the potential consequences might be of not estimating the "prediction power" well. – whuber May 12 '23 at 16:44

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