Uncertanty in a non-linear regression with least squares method

[VictorH]

Homework Statement

Ok, so I'm trying to fit a set of data (21000 points to be exact) to a sine function.

Homework Equations

Y = A*sin(ωt)

The Attempt at a Solution

I used NumPy to get the parameters A and ω with the least squares method. So far, so good. However, i appear to have reached an impass, this values don't have an uncertainty that accompanies them.

My question is: how do i propagate the error in the amplitude and the frecuency? In a quick web search I have not found any helpful inside in anything different of uncertainty of slopes in linear regressions. Can you recommend literature or websites that cover this topic? Bear in mind a freshman undergrad level of computer expertise and experimentation skill.

Thanks in advance

 

[maajdl]

You might get some inspiration by reading Numerical Recipes.

http://www.nr.com

http://apps.nrbook.com/c/index.html chap 15

Don't forget also to check for practical problems.
For example: aliasing, several frequencies can fit the same data, ...

 

[BvU]

Don't give a list of 21000 points, but:
Tell us a bit more about what was measured, and how.
Approximate frequency, sampling frequency,
Then write out what you did with numpy.
Did you get a $\chi^2$ out?

 

[VictorH]

Alright. So, i measured the movement of a vibrating table with a LVDT sensor, the approximate frecuency is around 30 Hz and a sampling frecuency of 1000 samples per second. Hopefully i can show it behaves like a harmonic oscillator. I looked at tha data, the graph is pretty nice and definitive to a sine function, so $\chi^2$ is a given, plus Python says so. I did least squares with Numpy (don't have the exact code at hand) and got some values for amplitude and frecuency.
Now, this numbers come very close of what i can see and measured in the lab. But, as in any experiment, this numbers mean nothing without uncertanties. My question is: how do i do error propagation in a non-linear regression using least squares?

 

[HalfLostAndSearching]

Uncertainty in non-linear regression can be more complex compared to linear regression, as it involves finding the best fit curve for the data. In this case, you can use the least squares method to find the parameters A and ω, but it is important to also consider the uncertainty in these values. There are a few approaches you can take to propagate the error in the amplitude and frequency. One option is to use the bootstrap method, which involves repeatedly sampling from your data and calculating the parameters to get a distribution of values. Another approach is to use a Monte Carlo simulation, where you randomly generate values within a certain range for the parameters and calculate the best fit curve for each set of parameters. This will give you a distribution of possible curves and you can use this to estimate the uncertainty in the amplitude and frequency.

As for literature or websites that cover this topic, I recommend checking out statistical textbooks or online resources that discuss non-linear regression and error propagation. Some examples include "Data Analysis: A Bayesian Tutorial" by Sivia and Skilling, "An Introduction to Error Analysis" by Taylor, or the website "Stat Trek" which has a section on non-linear regression and error propagation. It may also be helpful to consult with a professor or teaching assistant for guidance on this topic. Good luck with your analysis!