Improving the fit to the line
- Initially a gaussian convoluted with a circular distribution was fitted to the spectral line, with the equations of:
- Gaussian:
- Then for the background term a quadratic was used of the form:
After this a skewed Gaussian was fitted to the spectral line:
- where
This gave a plot of:
- Finally the quadratic background was fitted with the skewed gaussian:
- The parameters for each of the its is listed below:
Parameters |
Conv |
Conv+Quadratic back |
Skew Guass |
Skew Gauss + Quadratic back |
Gauss Amp |
364.6 ±(1.8e7) |
347.8 ±(4.8e6) |
(3.4±160569)e05 |
(6.23±1090)e4 |
Mean |
59.41±0.0003 |
33.4±0.001 |
777±1729 |
-8.52±14871 |
Gauss |
1.998±0.001 |
1.785±0.002 |
75.1±3489 |
1.354±2369 |
Circle radius |
4.15±0.004 |
4.516±0.002 |
|
|
Circle Amplitude |
369.8±(1.8e7) |
388.2±(5.4e6) |
|
|
background |
624.7±0.2 |
|
612±0.3 |
|
|
|
|
1295±931461 |
0.9306±679 |
|
|
|
-5.69±2644 |
-0.108±19.1 |
|
|
|
0.31±1407 |
-0.005±9.4 |
|
|
681±0.41 |
|
63.2±920 |
|
|
-0.366±0.009 |
|
0.126±480 |
|
|
0.096±0.001 |
|
-0.036±0.006 |
|
191158 |
138587 |
131805 |
127977 |
|
70 |
70 |
70 |
70 |
per DOF |
2730 |
1979 |
1882 |
1828 |
- it was noticed that towards the edges of the frame the distributions were becoming skewed, this was symmetric on both sides of the chip as the plots below show:
- This shows a skew on the left side of the chip:
- This shows a skew to the right of the chip:
- This shows the same peak in the center
Halogen Lamp
- To look at the response of the camera a halogen lamp was observed across its spectrum, this gave a plot of:
Are
Poisson?
- Wanted to check whether
actually followed Poisson distribution, as the errors seemed too small
- Took 20 frames of the same spectral line. Then studied the same point for each of the 20 frames to see how it fluctuated in each frame.
- Plotted
against the pixel position #350 for each of the 20 frames, and calculated the
and
for the 20 points
Mean |
Standard Deviation |
138928.0 |
623.85 |
- For a Poisson distribution, the standard deviation is equal to the square root of the mean,
. For 138928 photoelectrons this corresponds to
= 372.73
- This indicates that
is NOT in fact Poisson distributed, as we had been assuming before?
Mapping wavelength with pixel position and micrometer position
- First to visualise the fit to the 3d plot the residuals of the points were plotted, this is done by:
- The error bar for
was calculated by propagating the errors in the pixel position and the micrometer position.
- This was done using the equation:
- where
,micrometer position and
, the position on the camera.
- The first equation that was fitted was:
- This gave a residual plot of:
- This process was then repeated with a quadratic fit with the equation:
- This gave a residual histogram of:
- This was then repeated one more time with a cubic term within the fit:
- This have a histogram of residuals as:
- this gave parameters for each of the fits as:
Parameters |
Linear |
Squared |
Cubed |
|
275 ± 4 |
276±9 |
314.9±23.6 |
|
0.1069±0.0057 |
0.118±0.028 |
0.098±0.092 |
|
55.26±0.66 |
53.76±3.57 |
26.1±15.1 |
|
|
-(2±3)e-05 |
(-1.3±2)e-04 |
|
|
0.11±0.34 |
5.53±3.22 |
|
|
0.0014±0.003 |
0.003±0.002 |
|
|
|
(1.17±1.62)e-07 |
|
|
|
-0.321±0.219 |
|
|
|
(-3±20)e-05 |
|
|
|
(-2.75±1.67)e-03 |
|
7305 |
7220 |
6772 |
|
72 |
72 |
72 |
per DOF |
101 |
100 |
94 |
* Ask about errors that go into the least squares fit, as currently set to one?
---+ Standard Deviation vs Exposure time
- Took frames of one spectral line of Helium at different exposure times starting at 0.1s, increasing in 0.3s increments up to 3.0s.
- This was to see how the
of the line changed with exposure time
- Did fit an exponential distribution, however this is redundant without any error bars, so we need to sort out our errors before fitting it.
Vega
- The plot below shows vega at 3.5mm grating setting,
- 120s frame
--
JosephBayley - 22 Nov 2015