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I Best fit of a function containing costants with error

  1. May 16, 2017 #1
    Hi.

    Suppose I have a function [itex]y=f(a,b;p;x)[/itex] where a, b are known with some uncertainty and x, y were measured multiple times.
    I want to find p through best fit.

    I consider a, b to be gaussian variable.

    I can imagine multiple ways to do this:
    * I consider a, b parameters like p. I don't like this, moreover with the function I'm working on, I get uncertainties orders of magnitude greater than the value of the parameters.
    * I ignore the uncertainties for a, b and proceed to do a numerical fit for parameter p like I would normally do.
    * I thought I could make a large number of fits where a, b get gaussian-random values and obtain the probability distribution of p (I ignore the std dev given by the fits). Supposing this makes sense, how do I get the uncertainty for p from his probability distribution if this is not a gaussian function?

    The function I'm working on is [tex] v(t) = \frac{F_T}{ \lambda} \ln \left( \frac{M_0 }{M_0 - \lambda t } \right)- gt [/tex] where [itex]F_T[/itex] is known exactly and [itex]M_0[/itex] is the parameter.
    [itex]\lambda[/itex] and [itex]g[/itex] are known with uncertainties.


    Thank you very much for any advice.
    I have another problem, again with uncertainties, that arises from the same exercise. I will make another post in the future about that.
     
  2. jcsd
  3. May 17, 2017 #2

    BvU

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    Hello general,

    You should be more specific. Not clear to me what T is in ##F_T##. Is ##F## a constant or a function ?

    And a bit clearer: Why start with ##f(a,b,p,x)##, and then continue with a function ##v(\lambda, g, M_0, t)## (did I make the right reverse translation ?) ?

    Do you have a heap of measurement series ##v(t)## with different ##\lambda## ? What is v, what is ##\lambda## ?

    Is it feasible to develop the ln as a taylor series ##{\lambda\over M_0} t + ... \ ##? That way you get rid of ##\lambda## -- if ##M_0 >> \lambda##.

    What are the dimensions ? ##M_0## looks like a mass but is a length ?
     
  4. May 17, 2017 #3
    [itex]f(a,b;p;x) [/itex] was generic to describe the problem; in my case the function is [itex]v(\lambda,g;M_0;t)[/itex]. [itex]F_T=45000[/itex] N (newtons) exactly.

    I have a bunch of t measured with error, and the respective v measure with error. The parameter to fit is [itex]M_0[/itex].

    This function describes the velocity in (function of) time considering a thrust of 45000N that reduces the mass by [itex]\lambda t[/itex]. [itex]\lambda[/itex] has the dimension of Kg/s.

    [itex]\lambda[/itex] and g were estimated with uncertainties by other means.


    I could use taylor, but I'd like to have also the exact result. Moreover, if I use taylor I still have g, which has uncertainty, which is the problem I'm posing.

    Yesterday I tried gaussian generating [itex]\lambda[/itex] 10000 times (keeping g fixed to his best estimate) and making an histogram of the best fits of [itex]M_0[/itex]. I got a gaussian shape and the sigma was negligible compared to the sigma given by the fits. I'm not surprised because [itex]\lambda[/itex] disappears in taylor.
    I tried the same thing with g. This time the sigma was not negligible.

    I thought I could sum the two sigmas in quadrature and consider this the solution of my problem. But I'm not sure it's correct.

    I also gaussian generated both [itex]\lambda[/itex] and g, 10000 times at the same time. Maybe this is better than keeping one fixed and generating the other.
     
  5. May 18, 2017 #4

    BvU

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    Hats off for the thorough work !
    If all is well, there should be no big difference and both ways come with the error in g ?

    So you analyze a series of experiments where ##M_0## is to be determined ? One single ##M_0## or one per launch ?
    Pretty hefty ##F_T## -- how do you know it's determined perfectly and the same each time ?
     
  6. May 18, 2017 #5

    BvU

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    And: about the formula. It starts off with ##\Bigl( {F_T\over M_0}-g\Bigr)t ## but goes to infinity for ##\lambda t \rightarrow M_0## which prompts me to ask if it is sufficiently valid for the range of ##t## in your experiments. Surely there is the mass of the hull to consider too ?
     
  7. May 18, 2017 #6
    My ispiration for the problem comes from here:
    https://phet.colorado.edu/sims/lunar-lander/lunar-lander_en.html

    Considering the free falling ship, I want to find the height (or the time) at which I can turn on the full thrust and let it land at zero speed. The algebrical problem is not difficult (altough the solution is found numerically) but I also know that saying [itex]\tau=6.74[/itex]s is meaningless without any indication of uncertainties.

    [itex]M_0[/itex] comprises the mass of the ship and the initial mass of the fuel. I also assume (and verify) that for the time intervals considered the fuel won't end.
    From that minigame there are missing some parameters that I need in order to solve the problem. I got them through "experiments": [itex]g, \lambda, M_0[/itex].

    My subproblem is now to propagate [itex]g, \lambda[/itex] errors to get an uncertainty for [itex]M_0[/itex].

    The next thing I need to do is to propagate the errors of these three parameters when solving the following system of equations:
    [tex]
    y(T,\tau)=0 \\
    \dot{y}(T,\tau)=0
    [/tex]

    where T is the total time and [itex]\tau[/itex] the time when the thrust is turned on.


    This was the second problem I was going to pose. But I see I could use the same technique: gaussian generating the parameters, solving numerically, and plotting the histogram.

    My friend also had another idea: because in this particular system I can separate T and [itex]\tau[/itex] for each function like this:
    [tex]
    y(T,\tau)=f(T) - g(\tau)=0 \\
    \dot{y}(T,\tau)=h(T) - p(\tau) =0
    [/tex]
    [tex]
    f(T)=g(\tau) \\
    h(t)=p(\tau)
    [/tex]

    So, he says, the uncertainty on the evaluation of f is the same as the uncertainty on g.
    Expanding both function I obtain a system of equations with the unkown [itex]\sigma_T,\sigma_\tau[/itex].
     
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