Chi Squared

  • Original $\chi^2$ used:
\begin{equation} \chi^2 = \sum_{i = 1}^{N} \frac{(y_i - f(x_i; \bar{\theta}))^2}{\sigma_i^2} \end{equation}

  • Amended $\chi^2$:
\begin{equation} \chi^2 = \sum_{i = 1}^{N} \frac{(y_i - f(x_i; \bar{\theta}))^2}{\sigma_i^2 + \sigma_0^2} \end{equation}
  • This is done such that $\chi^2 = N_{\rm{DOF}} = N-m$, where N is the number of data points and m is the number of fitted parameters

  • Form of $\sigma_0$:
\begin{equation} \sigma_0 = \sigma (1 + \frac{1}{f(x_i;\bar{\theta})}) \end{equation}

  • Sigma0 as a function of exposure time, with an exponential fit:
    Sigma0_exposure_exp.png
  • Not including most lamp data

  • For the Vega data:
Micrometer Position (mm) Original $\chi^2$ New $\chi^2$
0.80 1082766 48873
1.60 1037859 62358
2.30 2877388 177871
2.90 926898 62729
6.00 275787 28712
7.00 1562246 69592

Error propagation

  • Gaussian equation - fitted to lamp data.
\begin{equation} f(p;a,b,\sigma,\mu)=\frac{a}{\sqrt{2\pi}\sigma}e^{\frac{-(p-\mu)^2}{2\sigma^2}} \end{equation}
  • $\mu$ is the peak location. Gaussian fitted only near the peak.
  • An example plot of how well a gaussian function fits:

Helium0600.png
Helium0600_zoomed.png
  • Fit gave back error on the peak location and their error (Peak and error attached).
  • Propagated this error through to wavelength using below equation.
    \begin{equation} \sigma ^2=\sum_{i,j=1}^N\left[\frac{dy}{dx_i}\frac{dy}{dx_j} \right] V_{ij} \end{equation}

  • Error on the lambda is given by
\begin{equation} \Delta \lambda(p,m)^2=(-D_1-E_1(p-p_0)+2D_2(m-m_0))^2(\Delta m)^2+(-C_1-E_1(m-m_0)+2C_2(p-p_0))^2(\Delta p)^2 \end{equation}
  • The micrometer term can be ignored due to $\Delta m=1\times10^{-4}$.
  • Error recieved was in the order of $10^{-3}$.
  • Unconstrained averaging method used. Changed the value of $\sigma_{0}$ until the value of $\chi^2=N_{DOF}$. This happened at $\sigma_{0}=0.5$. Used this as the error on the wavelength.
  • The plot below shows the relation between pixel and wavelength with data and their relative errors.


plot0200.png
plot0200_zoomed.png

plot0500.png
plot0500_zoomed.png

plot0550.png
plot0550_zoomed.png

plot0600.png
plot0600_zoomed.png

plot0800_zoomed.png 		 
  • Coefficents for second order polynomial:
Coeficients value
I 588.5$\pm$0.2
$C_1$ 0.1011$\pm$0.0004
$D_1$ 53.32$\pm$0.07
$C_2$ (-4.5$\pm$2.3)e-6
$D_2$ -0.86$\pm$0.03
$E_1$ (-2.83$\pm$0.25)e-3
$\chi^2/N_{DOF}$ 1.009

-- JamesAngthopo - 14 Feb 2017

Attachments Attachments
Topic attachments
I Attachment History Action Size Date Who Comment
PNGpng Helium0600.png r1 manage 99.9 K 14 Feb 2017 - 23:25 JamesAngthopo Helium0600
PNGpng Helium0600_zoomed.png r1 manage 86.9 K 14 Feb 2017 - 23:26 JamesAngthopo Helium0600zoomed
PNGpng Sigma0_exposure_exp.png r1 manage 27.6 K 15 Feb 2017 - 10:14 WilliamBurrows Sigma0 as a function of exposure time, with an exponential fit
Texttxt calibration_var.txt r1 manage 1.3 K 15 Feb 2017 - 10:55 JamesAngthopo micrometer peak position and wavelength file
PNGpng plot0200.png r1 manage 133.0 K 14 Feb 2017 - 23:18 JamesAngthopo Pixelwavelength plot at 02.00 mm micrometer-1
PNGpng plot0200_zoomed.png r1 manage 123.3 K 14 Feb 2017 - 22:57 JamesAngthopo Pixelwavelength plot at 02.00 mm micrometer zoomed
PNGpng plot0500.png r1 manage 132.8 K 14 Feb 2017 - 22:57 JamesAngthopo Pixelwavelength plot at 05.00 mm micrometer
PNGpng plot0500_zoomed.png r1 manage 120.2 K 14 Feb 2017 - 22:58 JamesAngthopo Pixelwavelength plot at 05.00 mm micrometer zoomed
PNGpng plot0550.png r1 manage 122.2 K 14 Feb 2017 - 22:58 JamesAngthopo Pixelwavelength plot at 05.50 mm micrometer
PNGpng plot0550_zoomed.png r1 manage 114.5 K 14 Feb 2017 - 22:59 JamesAngthopo Pixelwavelength plot at 05.50 mm micrometer
PNGpng plot0600.png r1 manage 129.2 K 14 Feb 2017 - 23:05 JamesAngthopo Pixelwavelength plot at 06.00 mm micrometer
PNGpng plot0600_zoomed.png r1 manage 120.4 K 14 Feb 2017 - 23:06 JamesAngthopo Pixelwavelength plot at 06.00 mm micrometer zoomed
PNGpng plot0800_zoomed.png r1 manage 118.2 K 14 Feb 2017 - 23:04 JamesAngthopo Pixelwavelength plot at 08.00 mm micrometer zoomed
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Topic revision: r5 - 15 Feb 2017 - JamesAngthopo

 
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