0
$\begingroup$

0

I have a set of data points. The first coordinate is time and the second coordinate is energy. I am trying to figure out how the energy is decaying over time. Particularly, I have to find if it is decaying over time exponentially or as a power law. I used Mathematica FindFit to model my points as both an exponential decay and a power law decay. It turned out that the exponential decay describes my data points better. But I am not sure if I am doing the right thing. I also plotted my data points in a ListLogPlot and ListLogLogPlot. In both cases, I got a straight line. So, I am a little confused about the actual behavior of my data points. Could anyone help me with this issue? I am copying my data points here. Note that I am only interested in the late-time behavior of the function, not the entire time axis. Thank you!

Data1={{5,0.0210796},{7,0.0293022},{9,0.0302858},{11,0.0257149},{13,0.0182589},{15,0.0106745},{17,0.00473577},{19,0.00101295},{21,-0.000754187},{23,-0.00117344},{25,-0.000860244},{27,-0.000278088},{29,0.000293337},{31,0.00072545},{33,0.000988823},{35,0.00110603},{37,0.00111822},{39,0.00106582},{41,0.000980234},{43,0.000882181},{45,0.000783367},{47,0.000689278},{49,0.0006018},{51,0.000521108},{53,0.000446822},{55,0.000378596},{57,0.000316303}, {59, 0.000259989190761133}}

Improve this question
$\endgroup$
9
  • 1
    $\begingroup$ what counts as late time? If you have a specific time, why give data before then? $\endgroup$
    – Glen_b
    Commented Apr 12, 2023 at 1:26
  • 2
    $\begingroup$ The problem with choosing 35 is that there's a mode (a local maximum) around time 38, so that can't be where you count "late" and be debating between two monotonic functions; the relationship is not monotonic there. It looks to me like your first step should be to actually look at the data. If you're just going to choose it arbitrarily like that, at least take t>40 so it's monotonic. In that case I suggest comparing a plot of log(y) vs t with a plot of sqrt(y) vs t. You may find that captures most of the relevant issues. $\endgroup$
    – Glen_b
    Commented Apr 12, 2023 at 1:53
  • 1
    $\begingroup$ Indeed, it looks like using the MASS::boxcox function in R suggests the same conclusion. However, I'd strongly suggest reviewing some of the answers that mention the blog post by Shalizi and the paper by Clauset, Shalizi and Newman in relation to power laws (arxiv version here: arxiv.org/abs/0706.1062), as well as reading those two links directly. $\endgroup$
    – Glen_b
    Commented Apr 12, 2023 at 2:00
  • 1
    $\begingroup$ I should clarify, though. While Shalizi (and the paper) are mostly discussing power-law distributions rather than power law relationships between two variables (which would impact things like suitable analyses, displays etc), some of the cautionary comments are still pretty relevant. $\endgroup$
    – Glen_b
    Commented Apr 12, 2023 at 2:16
  • 1
    $\begingroup$ Exponential decay means on a log-linear plot the right tail of the points looks linear. It's not -- it decays a little faster than linearly -- but you can still approximate the tail with a linear function. Its slope depends on how many points you take. Somewhere between 2 and 11 will work. Draw the plot and use your judgment. $\endgroup$
    – whuber ♦
    Commented Apr 12, 2023 at 3:41

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.