Editing Gaussian function (section)

=== Parameter precision ===

Once one has an algorithm for estimating the Gaussian function parameters, it is also important to know how [[Accuracy and precision|precise]] those estimates are. Any [[least squares]] estimation algorithm can provide numerical estimates for the variance of each parameter (i.e., the variance of the estimated height, position, and width of the function). One can also use [[Cramér–Rao bound]] theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data.<ref name="Hagen1">[https://dx.doi.org/10.1364/AO.46.005374 N. Hagen, M. Kupinski, and E. L. Dereniak, "Gaussian profile estimation in one dimension," Appl. Opt. 46:5374–5383 (2007)]</ref><ref name="Hagen2">[https://dx.doi.org/10.1364/AO.47.006842 N. Hagen and E. L. Dereniak, "Gaussian profile estimation in two dimensions," Appl. Opt. 47:6842–6851 (2008)]</ref>
# The noise in the measured profile is either [[Independent and identically-distributed random variables|i.i.d.]] Gaussian, or the noise is [[Poisson distribution|Poisson-distributed]].
# The spacing between each sampling (i.e. the distance between pixels measuring the data) is uniform.
# The peak is "well-sampled", so that less than 10% of the area or volume under the peak (area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region.
# The width of the peak is much larger than the distance between sample locations (i.e. the detector pixels must be at least 5 times smaller than the Gaussian FWHM).
When these assumptions are satisfied, the following [[covariance matrix]] '''K''' applies for the 1D profile parameters <math>a</math>, <math>b</math>, and <math>c</math> under i.i.d. Gaussian noise and under Poisson noise:<ref name="Hagen1" />
<math display="block"> \mathbf{K}_{\text{Gauss}} = \frac{\sigma^2}{\sqrt{\pi} \delta_X Q^2} \begin{pmatrix} \frac{3}{2c} &0 &\frac{-1}{a} \\ 0 &\frac{2c}{a^2} &0 \\ \frac{-1}{a} &0 &\frac{2c}{a^2} \end{pmatrix} \ ,
\qquad
\mathbf{K}_\text{Poiss} = \frac{1}{\sqrt{2 \pi}} \begin{pmatrix} \frac{3a}{2c} &0 &-\frac{1}{2} \\ 0 &\frac{c}{a} &0 \\ -\frac{1}{2} &0 &\frac{c}{2a} \end{pmatrix} \ ,</math>
where <math>\delta_X</math> is the width of the pixels used to sample the function, <math>Q</math> is the quantum efficiency of the detector, and <math>\sigma</math> indicates the standard deviation of the measurement noise. Thus, the individual variances for the parameters are, in the Gaussian noise case,
<math display="block">\begin{align}
\operatorname{var} (a) &= \frac{3 \sigma^2}{2 \sqrt{\pi} \, \delta_X Q^2 c} \\
\operatorname{var} (b) &= \frac{2 \sigma^2 c}{\delta_X \sqrt{\pi} \, Q^2 a^2} \\
\operatorname{var} (c) &= \frac{2 \sigma^2 c}{\delta_X \sqrt{\pi} \, Q^2 a^2}
\end{align}</math>

and in the Poisson noise case,
<math display="block">\begin{align}
\operatorname{var} (a) &= \frac{3a}{2 \sqrt{2 \pi} \, c} \\
\operatorname{var} (b) &= \frac{c}{\sqrt{2 \pi} \, a} \\
\operatorname{var} (c) &= \frac{c}{2 \sqrt{2 \pi} \, a}.
\end{align} </math>

For the 2D profile parameters giving the amplitude <math>A</math>, position <math>(x_0,y_0)</math>, and width <math>(\sigma_X,\sigma_Y)</math> of the profile, the following covariance matrices apply:<ref name="Hagen2" />

<math display="block">\begin{align}
\mathbf{K}_\text{Gauss} = \frac{\sigma^2}{\pi \delta_X \delta_Y Q^2} & \begin{pmatrix}
\frac{2}{\sigma_X \sigma_Y} &0 &0 &\frac{-1}{A \sigma_Y} &\frac{-1}{A \sigma_X} \\
 0 &\frac{2 \sigma_X}{A^2 \sigma_Y} &0 &0 &0 \\
 0 &0 &\frac{2 \sigma_Y}{A^2 \sigma_X} &0 &0 \\
 \frac{-1}{A \sigma_y} &0 &0 &\frac{2 \sigma_X}{A^2 \sigma_y} &0 \\
 \frac{-1}{A \sigma_X} &0 &0 &0 &\frac{2 \sigma_Y}{A^2 \sigma_X}
\end{pmatrix} \\[6pt]
\mathbf{K}_{\operatorname{Poisson}} = \frac{1}{2 \pi} & \begin{pmatrix}
 \frac{3A}{\sigma_X \sigma_Y} &0 &0 &\frac{-1}{\sigma_Y} &\frac{-1}{\sigma_X} \\
 0 & \frac{\sigma_X}{A \sigma_Y} &0 &0 &0 \\ 0 &0 &\frac{\sigma_Y}{A \sigma_X} &0 &0 \\
 \frac{-1}{\sigma_Y} &0 &0 &\frac{2 \sigma_X}{3A \sigma_Y} &\frac{1}{3A} \\
 \frac{-1}{\sigma_X} &0 &0 &\frac{1}{3A} &\frac{2 \sigma_Y}{3A \sigma_X}
 \end{pmatrix}.
\end{align}</math>
where the individual parameter variances are given by the diagonal elements of the covariance matrix.