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ex-1: reviewed

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Michele Guerini Rocco 2020-05-19 20:51:13 +02:00
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notes/sections/1.md
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@ -167,7 +167,7 @@ or *Robertson-Cryer* estimator was used. This estimator was chosen because makes
no assumptions on the underlying distribution and is not computationally no assumptions on the underlying distribution and is not computationally
expensive. The HSM is obtained by iteratively identifying the half modal expensive. The HSM is obtained by iteratively identifying the half modal
interval, which is the smallest interval containing half of the observation. interval, which is the smallest interval containing half of the observation.
Once the sample is reduced to less that three points the mode is computed as the Once the sample is reduced to less than three points the mode is computed as the
average. The special case $n=3$ is dealt with by averaging the two closer points average. The special case $n=3$ is dealt with by averaging the two closer points
[@robertson74]. [@robertson74].
@ -191,8 +191,8 @@ where $\sigma_e$ and $\sigma_o$ are the absolute errors of $m_e$ and $m_o$
respectively. At 95% confidence level, the values are compatible if $p > 0.05$. respectively. At 95% confidence level, the values are compatible if $p > 0.05$.
In this case: In this case:
- t = 1.012 - $t = 1.012$
- p = 0.311 - $p = 0.311$
Thus, the observed mode is compatible with the mode of the Landau distribution, Thus, the observed mode is compatible with the mode of the Landau distribution,
although the result is quite imprecise. although the result is quite imprecise.
@ -246,7 +246,7 @@ $$
for x, given a probability value $p_0$, where $p(x)$ is the CDF. The (unique) for x, given a probability value $p_0$, where $p(x)$ is the CDF. The (unique)
root of this equation was found by a root-finding routine root of this equation was found by a root-finding routine
(`gsl_root_fsolver_brent` in GSL) based on the Brent-Dekker method it too. (`gsl_root_fsolver_brent` in GSL) based on the Brent-Dekker method.
The following condition was checked for convergence: The following condition was checked for convergence:
$$ $$
|a - b| < \varepsilon_\text{abs} + \varepsilon_\text{rel} \min(|a|, |b|) |a - b| < \varepsilon_\text{abs} + \varepsilon_\text{rel} \min(|a|, |b|)
@ -290,7 +290,7 @@ $$
The function derivative $f'(x)$ was minimized using the same minimization method The function derivative $f'(x)$ was minimized using the same minimization method
used for finding $m_e$. Once $f_\text{max}$ was known, the equation: used for finding $m_e$. Once $f_\text{max}$ was known, the equation:
$$ $$
f'(x) = \frac{f_\text{max}}{2} f(x) = \frac{f_\text{max}}{2}
$$ $$
was solved by performing the Brent-Dekker method (described before) in the was solved by performing the Brent-Dekker method (described before) in the
@ -303,8 +303,8 @@ $$
\vspace{-1em} \vspace{-1em}
![Example of a Moyal distribution density obtained by the KDE method. The rug ![Example of a Moyal distribution density obtained by the KDE method. The rug
plot shows the original sample used in the reconstruction. The 0.6 factor plot shows the original sample used in the reconstruction.
compensate for the otherwise peak height reduction.](images/1-landau-kde.pdf) ](images/landau-kde.pdf)
On the other hand, obtaining a good estimate of the FWHM from a sample is much On the other hand, obtaining a good estimate of the FWHM from a sample is much
more difficult. In principle, it could be measured by binning the data and more difficult. In principle, it could be measured by binning the data and