Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Uncertainty in parameters -> Gauss-Newton

  1. Jan 7, 2010 #1
    Uncertainty in parameters --> Gauss-Newton

    Hi guys!

    I have a set of datapoints, and i´m about to use Gauss-Newton to fit a model function (Lorentzian) to these points. So we´re talking abut a nonlinear least squares regression.

    How do I estimate error in function parameters given errors in data points?

    Thanks.
     
  2. jcsd
  3. Jan 8, 2010 #2

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    Re: Uncertainty in parameters --> Gauss-Newton

    Simulation using pseudo-random number generation?
     
  4. Jan 8, 2010 #3
    Re: Uncertainty in parameters --> Gauss-Newton


    Nothing I'm familiar with. Could you develop?

    Isn´t there any method equivalent to that using the covariance matrix in weighted linear least squares?
     
  5. Jan 8, 2010 #4

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    Last edited: Jan 8, 2010
  6. Jan 9, 2010 #5
    Re: Uncertainty in parameters --> Gauss-Newton

    Allright, have read it through. Thanks
    One thing though, where does the errors from my original data come in?
     
  7. Jan 10, 2010 #6

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    Re: Uncertainty in parameters --> Gauss-Newton

    I will write vectors in bold, so for example y = {y(1), ..., y(i), ..., y(n)}.

    In a proper bootstrap, you are just "mixing and matching" the errors you've already computed: y*(i) = y(i) + u(j), where j is "almost surely" different from i, u(.) are the computed errors, and y* is a convolution of y. This "remix" algorithm is repeated many times (say 100 times), so you have 100 different estimates of your model parameters coming from convoluted vectors y1*, ..., y100*.

    In the approximate bootstrap (monte carlo?), you use the computed errors to derive the approximate "population distribution." Suppose the errors "look like" they are distributed normally with mean = 0 and standard deviation = s, i.e. u*(i) ~ N(0,s2). Then, you can use a pseudo-random generator to produce repeated draws from Normal(0,s2) and define y**(i) = y(i) + u*(i). Again, if you run this 100 times, you will have 100 parameter estimates from y1**, ..., y100**.
     
  8. Jan 12, 2010 #7
    Re: Uncertainty in parameters --> Gauss-Newton

    Amazing.. I was experimenting with a a method just like that one when I saw your reply.
    Interesting method that can be applied to any linear/nonlinear method.

    By the way since gauss-newton linearizes the problem in each iteration I should get pretty decent results by making a weighted regression and taking the covariance matrix in the last step. (Like in linear least squares)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook