Fixing errors

  • Found previously that errors in background of frame are not poisson - wanted to investigate how points fluctuate in peaks as well as background.
  • Plotted the spectrum of Helium lamp at micrometer position 6.5mm, exposure time 0.5s, to find pixel positions of background and peaks

spectrum.png

  • Chose 3 random pixel positions of background: 150, 350, 550
  • Chose 3 pixel positions where the two peaks is present: 245 (1st peak), 255 (1st peak), 655 (2nd peak)
  • Analysed the point at these pixel positions to see how they fluctuated in each of the 20 frames

errorback.png

errorpeak.png

  • For each frame, calculated the mean of the 20 data points $\mu$, the calculated standard deviation of the data points $\sigma$, and the expected poisson error of the data points, $\sigma_{\mathrm{Poisson}} = \sqrt{\mu}$
Pixel Position $\mu$ $\sigma$ $\sigma_{\mathrm{Poisson}}$ $\sigma / \sigma_{\mathrm{Poisson}}$
150 (back) 520.96 2.34 1.43 1.64
350 (back) 538.51 2.09 1.45 1.44
550 (back) 560.17 2.42 1.48 1.64
245 (peak) 1928.15 18.34 2.75 6.67
255 (peak) 4016.91 31.86 3.97 8.03
655 (peak) 3964.86 28.23 3.94 7.16

Assigning error bars?

  • Found that for calibration lamps, standard deviation $\sigma$ not equivalent to $\sigma_{\mathrm{Poisson}}$. Wanted to find a way to assign error bars
  • Using Helium lamp, micrometer position at 4.0mm, took 10 frames at each exposure time, starting at 0.05s up to 1s in 0.05s increments. For each of the 10 frames of each exposure time, chose 5 points close together that were all in background (pixel numbers: 170,180,190,200,210) and 5 points close together in the peak (pixel numbers: 185,190,195,200,205). Found the $\sigma$ of the 10 frames for each of these points, then found $<\sigma>$ for that exposure time
  • Plotted $<\sigma>$ as a function of exposure time. Expected to be able to fit a function to these points, such that for a specific exposure time, can find the appropriate error.
  • However, the plot was not what we expected - the points are random, a function cannot be fitted.
  • Unsure what is the best method in assigning error bars?
sigma.png

sigmapeak.png

Skewed Gaussian Fit

  • Firstly a Gaussian convoluted with a circular function was fitted to a spectral line
  • Then as towards the edges of the frame the line appeared to become skewed therefore a skewed Gaussian was convoluted with the circular function and fitted
The skewed Gaussian has the equation:

\begin{equation} f(y;\mu,\sigma) = \frac{1}{\sqrt{2\pi} \sigma^2} e^{\frac{-(y-\mu)^2}{2 \sigma^2}} \end{equation}

where $ y = a_{0} + a_{1} x + a_{2} x^2 $

  • Finally a quadratic term was fitted for the background of the image, which has the equation:
\begin{equation} f(x) = b_{0} + b_{1} x + b_{2} x^2 \end{equation}
  • These are plotted below
3Fits.png

  • The quadratic can be seen to improve the fit to the background of the data as shown below:
3Fit_Back.png

  • The three different fits produced parameter values of:
Parameters Gaussian + circle Skewed Gaussian + Circle Skewed Gaussian + circle + Quadratic background
Gaussian amplitude 369 (7.8e6) 417(7.7e05) 447(4.7e06)
Gaussian mean 33.340.001 14.2(5e04) 13.3(1e04)
Gaussian Sigma 1.990.001 2.19(2e04) 3.14(2.06e04)
Circle radius 4.160.01 4.460.01 4.430.00001
Circle mean 33.340.001 58.60.02 58.40.01
Circle amplitude 365(7.7e6) 422(4.2e06) 532(5e06)
Background 624.70.2 6180.2  
$a_{0}$   8.18(5e04) 4.23(3e04)
$a_{1}$   0.034308 0.2641742
$a_{2}$   0.077705 0.008581
$b_{0}$     6120.3
$b_{1}$     -3.540.003
$b_{2}$     -0.0690.001
$\chi^{2}$ 191158 21259 14404
$N_{\mathrm{DOF}}$ 64 60 58
$\chi^{2}$ per DOF 2987 354 248

Fitting to Vega Line

  • The line chosen was at a micrometer setting of 3.5mm, the exposure was 120s,
  • Initially the convolution of the skewed Gaussian and circle plus the quadratic background was fitted to the Vega line, this had the equations same as above.
  • Then to get a better fit the skewed Gaussian was convoluted with a Lorentz distribution so that a Voigt profile could be fit. The Lorentz had the equation:
 \begin{equation} L(x;\gamma, x_{0}) = \frac{A}{\pi} \frac{\gamma/2}{(x-x_{0})^2 + (\gamma/2)^2} \end{equation}
  • Both of these fits are shown with the data in the plot below:
VegaFit.png

Paramater Skewed Gaussian + Circle + Quadratic Background Voigt + Circle + Quadratic Background
Gauss Amp 1.077 -1.48(1.39e05)
Gauss mean, $\mu$ 130 138(4.2e06)
Gauss $\sigma$ 9.33 19.7(3.1e05)
$a_{0}$ 4.3 -15.6(1.6e06)
$a_{1}$ 0.26 -5.44(3.93e04)
$a_{2}$ 0.088 0.09821226
Lorentz Amp, $A$   -7.43(6.8e05)
Lorentz width, $\gamma$   18.690.29
Lorentz center, $x_{0}$   2041.86
Circle radius 5.98 2.39286
Circle mean 130 128231
Circle Amplitude -21.7 -30.2(7.1e05)
$b_{1}$ 3.64 3.680.19
$b_{2}$ 0.00088 (8.4214.7)e-04
$b_{3}$ -8.49e-07 (-3.180.07)e-06
$\chi^2$ 1369 1115
$N_{\mathrm{dof}}$ 288 285
$\chi^2$ per dof 4.75 3.91

Mapping Wavelength

  • The three different equations that were fitted were:
 \begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m \end{equation} \begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m + a_{2} \mu^{2} + b_{2} m^{2} + c_{2} \mu m \end{equation} \begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m + a_{2} \mu^{2} + b_{2} m^{2} + c_{2} \mu m + a_{3}\mu^{3} + b_{3} m^{3} + c_{3} \mu^{2} m + d_{3} \mu m^{2}\end{equation}
  • These were then plotted for each micrometer setting with the data.
  • Ask about what errors to put in??
  • The parameters were found to be:

Parameter Linear Squared Cubed
$\lambda_{0}$ 2782 2652.8 2757
$a_{1}$ 0.1050.002 0.1220.008 0.0740.028
$b_{1}$ 55.640.26 62.071.08 56.754.45
$a_{2}$   (-1.190.89)e-05 (6.736.16)e-05
$b_{2}$   -0.6890.104 0.080.94
$c_{2}$   (-1.610.98)e-03 0.0110.007
$a_{3}$     (-4.624.81)e-08
$b_{3}$     (-3.356.44)e-02
$c_{3}$     (-7.596.11)e-06
$d_{3}$     (-7.635.07)e-04
$\chi^2$ 109524 63944 60107
$N_{dof}$ 69 66 62
$\chi^2$ per dof 1587 968 969

  • Some of the grating setting are shown below
0.5mm.png
2.0mm.png
5mm.png
7mm.png
8mm.png

Halogen Lamp

  • For the halogen lamp the above fit parameters were used to set the wavelength of each frame, these were then plotted as shown below:
  • The below plot shows the linear and squared plots and it can clearly be seen to improve at larger and smaller wavelengths, however not much change in the center.
LinearSquare.png
  • When the cubed parameters were used, the plot did no seem to improve, as the image shows
Cubed.png
  • This may be due to an incorrect point which the cubed plot fits better to however is further from the true value
-- AshleaKemp - 27 Nov 2015
Topic attachments
I Attachment Action Size Date Who Comment
PNGpng 150.png manage 36.3 K 27 Nov 2015 - 18:09 AshleaKemp Pixel position 150
PNGpng 255.png manage 33.4 K 27 Nov 2015 - 18:11 AshleaKemp Pixel position 255
PNGpng Helium.png manage 20.0 K 27 Nov 2015 - 17:56 AshleaKemp  
PNGpng errorback.png manage 47.3 K 29 Nov 2015 - 17:47 AshleaKemp  
PNGpng errorpeak.png manage 41.0 K 29 Nov 2015 - 17:46 AshleaKemp  
PNGpng sigma.png manage 15.6 K 01 Dec 2015 - 11:23 AshleaKemp  
PNGpng sigmapeak.png manage 17.7 K 01 Dec 2015 - 11:57 AshleaKemp  
PNGpng spectrum.png manage 29.1 K 29 Nov 2015 - 18:01 AshleaKemp  

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Topic revision: r11 - 12 Sep 2016 - AshleaKemp

 
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