1

I have the cubic relationship between two variables, x and y, and I need to find the error in x.

y = ax^3 + bx^2 + cx + d

I have the values for the coefficients and their respective uncertainties. I also have the y values and their associated uncertainties. I know how to propagate errors for simpler relationships, but this one is a bit trickier.

This cubic equation defines the relationship between y and x (it's a cubic polynomial fit to a graph between the two variables). I have thousands of x values and their corresponding y values. I also have the y-errors for each of the y-values. What I need to find are the errors for each x-value. I'm trying to find a function that might help me do this, as I have thousands of data points.

I'd appreciate any help, thanks :)

Thanks

Check the comments for more info.

eshbee
  • 11
  • 3
  • 1
    As a general proposition, the error in $x$ is a complicated, potentially trimodal function. Thus, any information you can provide to simplify your problem can be helpful. – whuber Sep 06 '21 at 18:45
  • I was hoping that there might've been an easy way to do this. Thanks for replying. Would it be useful to provide the coefficients and their respective uncertainties? This cubic equation defines the relationship between y and x (it's a cubic polynomial fit to a graph between the two variables). I have hundreds of x values and their corresponding y values. I also have the y-errors for each of the y-axis values. What I need to find are the errors for each x-value. I'm trying to find a function that might help me do this, as I have hundreds of data points. Any help would be appreciated :) – eshbee Sep 07 '21 at 11:55
  • By y-error, do you mean the residual $y - ax^3-bx^2-cx-d$? If not please clarify the definition. Also, what do you mean by x-error? If you have "figured out a way to propagate the errors on x", please describe that. This will help us understand what kind of errors you are considering. – Kota Mori Sep 07 '21 at 12:10
  • Sorry, I'm not trying to be cryptic. The data that I have access to isn't in a database form and so I can't directly access the individual data points. I can only plot graphs with the data, and the graph plotted of y against x can be fitted with a cubic polynomial fitting. The data contains the uncertainties associated with the y-values (as those values could be quantified). I need to work out the uncertainties associated with the x-values in the data, using the cubic polynomial fitting. The way I'd tried to propagate the errors wouldn't work, as I don't have the individual data points. – eshbee Sep 07 '21 at 12:32
  • The essential qualitative distinction between a tractable problem and one that is going to be very complicated is whether the graphs always suggest the relationship between $x$ and $y$ is *monotonic.* Would that be the case or not? And if it is the case, are there strong theoretical grounds for fitting a cubic, or are you just using that as a convenient way to model a little bit of nonlinearity? – whuber Sep 07 '21 at 15:58
  • The graphs very strongly suggest that the relationship between x and y is monotonic. There aren't strong theoretical grounds for the fitting, though I am searching through files to see if a relationship has been formally defined. I have just used the cubic fitting to model the nonlinear graphs and generate an appropriate relationship. Other more complicated fittings would have worked a bit better, but as the y-axis error bars are extremely large for larger values of y, the cubic fitting works just fine. I didn't want to complicate things further by having fitting equations with x^5 in them. – eshbee Sep 07 '21 at 17:13

0 Answers0