Error on the Gaussian mean

[kelly0303]

Hello! If I have N points (x,y) which I know they are described by a Gaussian i.e. y(x) is a Gaussian of unknown mean and standard deviation, and each y has an associated error of $\sqrt{y}$, is there a general formula for the uncertainty on the mean of this function? Thank you!

 

[hutchphd]

It is not clear to me what you are asking. In particular what does "each y has an associated error of $\sqrt y$" mean? Perhaps you could further describe the system ?

 

[kelly0303]

It is not clear to me what you are asking. In particular what does "each y has an associated error of $\sqrt y$" mean? Perhaps you could further describe the system ?

I meant that the uncertainty on y is $\sqrt{y}$, which is usually the case with counting experiments. An example of such a system is the measurement of the transition between 2 levels of a system, where the lineshape is Gaussian. One sets the laser at a given frequency for a while and measures the number of counts (e.g. fluorescent photons from the induced transition), then the frequency is changed and the number of counts are measured again. In the end, one ends up with a plot of counts (or rate) vs frequency, where the uncertainty on the number of counts, N is $\sqrt{N}$. Then, in order to get the central value of that transition, one needs to fit a Gaussian to N vs frequency. My question is, is there a general formula for the uncertainty on the mean of this gaussian under these circumstances?

 

[hutchphd]

My question is, is there a general formula for the uncertainty on the mean of this gaussian under these circumstances?

I fear you are conflating two ideas here.
In your example, the lineshape I believe will be Laurentzian (not Gaussian) and the width arises from the intrinsic physics. Given that shape, the best fit of a Laurentzian to the data will provide an estimate of the transition frequency and width. It is not asymmetric distribution.
The uncertainty in the measurement at each frequency, as driven by by signal strength or integration time, will indeed go as $\sqrt N$ and the fitting procedure to a parameterized curve can be weighted accordingly. One can certainly derive a relationship for the goodness of this fit in terms of the uncertianty in the data. These depend upon slopes and curvatures of the fitted curve near the data points in question and the uncertainties.

 

[Twigg]

In your example, the lineshape I believe will be Laurentzian (not Gaussian) and the width arises from the intrinsic physics. Given that shape, the best fit of a Laurentzian to the data will provide an estimate of the transition frequency and width. It is not asymmetric distribution.

Atomic lines aren't always Lorentzian. They are often Gaussian, and more generally they are described by a Voigt profile (a convolution of the two). One example is an atomic transition that is broadened by the Doppler shifts of all the atoms flying around with thermal velocities. This will produce a Gaussian lineshape, reflecting the Maxwell-Boltzmann velocity distribution.

Hello! If I have N points (x,y) which I know they are described by a Gaussian i.e. y(x) is a Gaussian of unknown mean and standard deviation, and each y has an associated error of y, is there a general formula for the uncertainty on the mean of this function? Thank you!

What you want is to do a Levenburg-Marquardt (or whatever other fitting algorithm you prefer) to your data, and extract the covariance matrix. Matlab's "nlinfit" function does this, and I believe scipy.optimize.least_sqaures does the same when you set method='lm'. (I'm not very familiar with scipy, sorry!) For the scipy method, it only returns the jacobian, which you can use to generate the covariance matrix from the error bars on your data points via usual error propagation formulas. Essentially, the Jacobian is the derivative of the fit parameters versus each of your data points $y_i$. Using those derivatives and the error bars on each $y_i$, you can calculate the error bars on the fit parameters. Matlab does calculates the covariance matrix for you (it's the 4th output of nlinfit).

Once you have the covariance matrix, you'll want to look at the diagonal component that corresponds to the gaussian width. This will be the variance on that width, so if you take the square root you will get the standard deviation. I hope that's somewhat helpful!

 

[Twigg]

I realize my description wasn't very good. Here's some example code for matlab. I wasn't able to check it because I don't have MATLAB at home. I believe nlinfit requires the statistics and machine learning toolbox. It's very useful though!

% Make some data
x = 0:0.1:5;
y = 5*exp(-((x-3)/0.71).^2) + 1.7;
yerr = 0.1 + 0.2*rand(size(y));

% Plot the data
figure(1); clf;
errorbar(x,y,yerr,'ks')

% define a gaussian function to fit to
fitfun = @(b,x) b(1)*exp(-((x-b(2))/b(3)).^2)+b(4);
% b(1) is the amplitude of the gaussian line
% b(2) is the center value
% b(3) is the linewidth
% b(4) is the background

% define a vector of inverse-variance weights
w = yerr.^(-2);

% define a guess for the parameters of the gaussian
b0 = [1,0,1,0];

% use nlinfit to fit the data to a gaussian
[b,~,~,covB] = nlinfit(x,y,fitfun,b0,'weights',w);

% plot the result
hold on
plot(x,fitfun(b,x),'k--')
hold off

% print the uncertainty on the linewidth to the command window
display(['The linewidth uncertainty is: ',num2str(sqrt(covB(3,3)))])