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I have a discrete function $f$ which lies over a certain domain $X$. My goal is to find the value of $X$, $x_{max}$, for which the function is maximum. I have opted for a simply search: using numpy python library, I have simply searched the argmax of the function, i.e. I found idx=numpy.argmax(f); x_max=f[idx]. Things work well. But, since the function is known with uncertainties, I'm wondering what is the method with which I can propagate errors of $f$ on $x_{max}$.

EDIT Follow @whuber suggestion, I will add some information about the funtion $f$ and its uncertainties. My function is computed from observed data. I started with the observed spectrum, i.e. the flux measured for each frequency within a range (~4000 - 8000 Angstron). Since these fluxes are measured, I have an error for each flux. Fluxes and their errors are provided by the instrument. Then, I take a spectra template, it is a model of how the flux varying along a wavelength range. I assume for this template no errors. I map the template at various redshift ($z$). This is a simple process, based on which I moved the wavelength by a factor $1+z$ for $z \in [z_{min}, z_{max}]$. The redshift range is imposed as hyperparameter, together with the sampling in $z$, i.e. the number of $z$ point in the $z$-range. Finally, I cross-correlate my observed spectra with each template mapped at different $z$, finding a cross-correlation function, i.e. the cross-correlation value estimated for each $z$. Finding the maximum value, I get the redshift. This is a common and widely-used process. Indeed it works well.

Finally, since the observed data have errors, I have propagated these errors through the cross-correlation process, computing the various derivatives and obtaining uncertainties on $f$ for each $z$. Thus, my question is: how can I deduce the error on $z$, knowing the error on $f$? Hoping that this can clarify my problem.

Giuseppe Angora
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  • Could you explain what you mean by a "discrete" function? – whuber Mar 23 '20 at 15:13
  • @whuber : My function is digitized, i.e. I don't know what is its analytic form, I have a value of f, for each x. Is it clear? – Giuseppe Angora Mar 23 '20 at 15:15
  • Not really. What does "digitized" mean? Rounded to some very coarse unit? If so, could you explain what "known with uncertainties" means? Does it mean, for instance, that a "noisy" version of $f$ is rounded or perhaps it means the rounded version of $f$ is randomly altered to another rounded value? – whuber Mar 23 '20 at 15:19
  • @whuber Ok It's clear that I was not able to explain my self. I simply meant that I have a function, computed from data. So each each point of this function has an error. – Giuseppe Angora Mar 24 '20 at 10:17
  • The answer, then, depends on (a) how the function was computed and (b) what you assume about the nature of the error. For some indication of how such problems are approached and what you need to assume, see (for example) https://stats.stackexchange.com/a/446205/919 (about finding zeros of a function, which is a similar problem) and https://stats.stackexchange.com/a/370201/919 (which is about testing whether a point is an extremum of a function--a procedure that has evident application to your kind of problem). Maybe they will help you provide more specific information. – whuber Mar 24 '20 at 14:03
  • @whuber I ahve read the post you have linked, the problem is that i don't have an analytic form for the function. Maybe I can apply a bootstrap method, extracting may functions within errors range and taking the maximum for each one of them. In this way I should be able to estimate the variance of x, respect to f variability. What do you think? Is acceptable? – Giuseppe Angora Mar 24 '20 at 14:27
  • As I have written, *you must provide the details.* If you don't have an analytic form for the function, then you need to explain what form you do have and how you derived this form from the data. Your questions do not have a general answer that will apply in all cases. The difficulty with your edits lies in their vagueness: what exactly is a "template", what does "sampled within a certain range" mean, and how have you "propagated these errors"? – whuber Mar 24 '20 at 15:18

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