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Error propagation when dividing by fitted model

  1. May 15, 2014 #1
    1. The problem statement, all variables and given/known data

    I am taking a dataset of intensity vs. frequency (which I'll call dataset I_1), and fitting it with a linear model (I_mod). I want to divide another intensity vs. wavelength data (I_2) by this fitted model to get fractional changes in the second data set compared to the model of the first (I_result).

    The problem is that I cannot for the life of me work out the correct way to propagate/combine the uncertainty on the fitted parameters from I_mod with the uncertainty on the intensity measurements in dataset I_2 to get the final uncertainty estimate on the relative intensity values in I_result.

    2. Relevant equations

    The fit equation is just the standard form for a straight line: I = m*nu + b, with I = model intensity at frequency nu, m the gradient, and b the y-offset. The fitting is done via a Monte Carlo Markov method, which leaves me with Gaussian likelihood functions for the parameters m and b from which I can estimate their standard deviation - i.e. the errors on the slope and the y-offset.

    The data set I_2 has errors associated with each of the data points, estimated during the measurement process.

    3. The attempt at a solution

    No idea really, and I've been looking around for ages.

    I was thinking that I could take the uncertainty in the I_mod slope as estimated from the standard deviation of its likelihood distribution obtained from the fitting process (call it d.m, say), then calculate d.m*nu to obtain the uncertainty on the model data points due to the slope uncertainty. Then take the uncertainty in the y-offset as estimated from the standard deviation of its likelihood distribution (d.b), and add these in quadrature to get the total error on each of the model data points. I.e.:

    d.I_mod = sqrt( (d.m*nu)^2 + (d.b)^2 )

    This gives me errors on the model data points - d.I_mod - and I already have estimates for the errors on the data points for my second data set which I'll call d.I_2. Because I'm dividing the data points in I_2 by I_mod to get I_result, I was then thinking I would add the fractional errors in each of these points - i.e.:

    total fractional error on I_result datapoint = sqrt( (d.I_2 / I_2)^2 + (d.I_mod / I_mod)^2 )

    I have a feeling that this is a terrible way to do things, but it's the best I've come up with :(

    Any help much appreciated, even if it is just pointing we to a website with some basic stats covering this problem.
     
  2. jcsd
  3. May 16, 2014 #2

    Simon Bridge

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    Welcome to PF;
    The secret is to be careful about describing what you want to know rather than what you are trying to do.

    i.e. if you want to know how close your new dataset is to the model, you would find the difference between the data and the model.

    You could compare the two datasets by getting a new model for the second dataset and seeing how the fitted parameters are different...

    Dividing a dataset by a model would, graphically, involve plotting the the intensity data against that predicted by the model. The resulting curve will tell you about the relationship between the new data and the model.

    In each case you should be able to see how the errors propagate.

    But what I am saying is that you need to think more about what you hope to find out.
     
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