305 lines
11 KiB
Markdown
305 lines
11 KiB
Markdown
# Exercise 1
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## Random numbers following the Landau distribution
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The Landau distribution is a probability density function which can be defined
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as follows:
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$$
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f(x) = \int \limits_{0}^{+ \infty} dt \, e^{-t log(t) -xt} \sin (\pi t)
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$$
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{width=50%}
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The GNU Scientific Library (GSL) provides a number of functions for generating
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random variates following tens of probability distributions. Thus, the function
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for generating numbers from the Landau distribution, namely `gsl_ran_landau()`,
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was used.
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For the purpose of visualizing the resulting sample, the data was put into
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an histogram and plotted with matplotlib. The result is shown in @fig:landau.
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{#fig:landau}
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## Randomness testing of the generated sample
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### Kolmogorov-Smirnov test
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In order to compare the sample with the Landau distribution, the
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Kolmogorov-Smirnov (KS) test was applied. This test statistically quantifies the
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distance between the cumulative distribution function of the Landau distribution
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and the one of the sample. The null hypothesis is that the sample was
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drawn from the reference distribution.
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The KS statistic for a given cumulative distribution function $F(x)$ is:
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$$
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D_N = \text{sup}_x |F_N(x) - F(x)|
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$$
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where:
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- $x$ runs over the sample,
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- $F(x)$ is the Landau cumulative distribution and function
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- $F_N(x)$ is the empirical cumulative distribution function of the sample.
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If $N$ numbers have been generated, for every point $x$,
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$F_N(x)$ is simply given by the number of points preceding the point (itself
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included) normalized by $N$, once the sample is sorted in ascending order.
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$F(x)$ was computed numerically from the Landau distribution with a maximum
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relative error of $10^{-6}$, using the function `gsl_integration_qagiu()`,
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found in GSL.
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Under the null hypothesis, the distribution of $D_N$ is expected to
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asymptotically approach a Kolmogorov distribution:
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$$
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\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K
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$$
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where $K$ is the Kolmogorov variable, with cumulative
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distribution function given by:
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$$
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P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
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\sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
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$$
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Plugging the observed value $\sqrt{N}D_N$ in $K_0$, the $p$-value can be
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computed. At 95% confidence level (which is the probability of confirming the
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null hypothesis when correct) the compatibility with the Landau distribution
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cannot be disproved if $p > α = 0.05$.
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To approximate the series, the convergence was accelerated using the Levin
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$u$-transform with the `gsl_sum_levin_utrunc_accel()` function. The algorithm
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terminates when the difference between two successive extrapolations reaches a
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minimum.
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For $N = 1000$, the following results were obtained:
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- $D = 0.020$
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- $p = 0.79$
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Hence, the data was reasonably sampled from a Landau distribution.
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**Note**:
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Contrary to what one would expect, the $\chi^2$ test on a histogram is not very
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useful in this case. For the test to be significant, the data has to be binned
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such that at least several points fall in each bin. However, it can be seen
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(@fig:landau) that many bins are empty both in the right and left side of the
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distribution, so it would be necessary to fit only the region where the points
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cluster or use very large bins in the others, making the $\chi^2$ test
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unpractical.
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## Parameters comparison
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When a sample of points is generated in a given range, different tests can be
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applied in order to check whether they follow an even distribution or not. The
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idea which lies beneath most of them is to measure how far the parameters of
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the distribution are from the ones measured in the sample.
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The same principle can be used to verify if the generated sample effectively
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follows the Landau distribution. Since it turns out to be a very pathological
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PDF, very few parameters can be easily checked: mode, median and
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full-width-half-maximum (FWHM).
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### Mode
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\begin{figure}
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\hypertarget{fig:parameters}{%
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\begin{tikzpicture}[overlay]
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\begin{scope}[shift={(0,0.4)}]
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% Mode
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\draw [thick, dashed] (7.57,3.1) -- (7.57,8.55);
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\draw [thick, dashed] (1.9,8.55) -- (7.57,8.55);
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\node [above right] at (7.6,3.1) {$m_e$};
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\node [below right] at (1.9,8.55) {$f(m_e)$};
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% FWHM
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\draw [thick, dashed] (1.9,5.95) -- (9.05,5.95);
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\draw [thick, dashed] (6.85,5.83) -- (6.85,3.1);
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\draw [thick, dashed] (8.95,5.83) -- (8.95,3.1);
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\node [below right] at (1.9,5.95) {$\frac{f(m_e)}{2}$};
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\node [above right] at (6.85,3.1) {$x_-$};
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\node [above right] at (8.95,3.1) {$x_+$};
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\end{scope}
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\end{tikzpicture}}
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\end{figure}
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The mode of a set of data values is defined as the value that appears most
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often, namely: it is the maximum of the PDF. Since there is no closed form
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for the mode of the Landau PDF, it was computed numerically by the *golden
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section* method (`gsl_min_fminimizer_goldensection` in GSL), applied to $-f$
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with an arbitrary error of $10^{-2}$, giving:
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$$
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\text{expected mode: } m_e = \SI{-0.2227830 \pm 0.0000001}{}
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$$
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This is a minimization algorithm that begins with a bounded region known to
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contain a minimum. The region is described by a lower bound $x_\text{min}$ and
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an upper bound $x_\text{max}$, with an estimate of the location of the minimum
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$x_e$. The value of the function at $x_e$ must be less than the value of the
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function at the ends of the interval, in order to guarantee that a minimum is
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contained somewhere within the interval.
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$$
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f(x_\text{min}) > f(x_e) < f(x_\text{max})
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$$
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On each iteration the interval is divided in a golden section (using the ratio
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($(3 - \sqrt{5})/2 \approx 0.3819660$) and the value of the function at this new
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point $x'$ is calculated. If the new point is a better estimate of the minimum,
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namely if $f(x') < f(x_e)$, then the current estimate of the minimum is
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updated.
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The new point allows the size of the bounded interval to be reduced, by choosing
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the most compact set of points which satisfies the constraint $f(a) > f(x') <
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f(b)$ between $f(x_\text{min})$, $f(x_\text{min})$ and $f(x_e)$. The interval is
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reduced until it encloses the true minimum to a desired tolerance.
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In the sample, on the other hand, once the data were binned, the mode can be
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estimated as the central value of the bin with maximum events and the error
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is the half width of the bins. In this case, with 40 bins between -20 and 20,
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the following result was obtained:
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$$
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\text{observed mode: } m_o = \SI{0 \pm 1}{}
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$$
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In order to compare the values $m_e$ and $x_0$, the following compatibility
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t-test was applied:
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$$
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p = 1 - \text{erf}\left(\frac{t}{\sqrt{2}}\right)\ \with
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t = \frac{|m_e - m_o|}{\sqrt{\sigma_e^2 + \sigma_o^2}}
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$$
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where $\sigma_e$ and $\sigma_o$ are the absolute errors of $m_e$ and $m_o$
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respectively. At 95% confidence level, the values are compatible if $p > 0.05$.
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In this case:
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$$
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p = 0.82
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$$
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Thus, the observed value is compatible with the expected one.
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### Median
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The median is a central tendency statistics that, unlike the mean, is not
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very sensitive to extreme values, albeit less indicative. For this reason
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is well suited as test statistic in a pathological case such as the
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Landau distribution.
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The median of a real probability distribution is defined as the value
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such that its cumulative probability is $1/2$. In other words the median
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partitions the probability in two (connected) halves.
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The median of a sample, once sorted, is given by its middle element if the
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sample size is odd, or the average of the two middle elements otherwise.
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The expected median was derived from the quantile function (QDF) of the Landau
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distribution[^1].
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Once this is know, the median is simply given by $QDF(1/2)$.
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Since both the CDF and QDF have no known closed form they must
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be computed numerically. The comulative probability has been computed by
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quadrature-based numerical integration of the PDF (`gsl_integration_qagiu()`
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function in GSL). The function calculate an approximation of the integral
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$$
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I(x) = \int_x^{+\infty} f(t)dt
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$$
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[^1]: This is neither necessary nor the easiest way: it was chosen simply
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because the quantile had been already implemented and was initially
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used for reverse sampling.
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The $CDF$ is then given by $p(x) = 1 - I(x)$. This was done to avoid the
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left tail of the distribution, where the integration can sometimes fail.
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The integral $I$ is actually mapped beforehand onto $(0, 1]$ by
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the change of variable $t = a + (1-u)/u$, because the integration
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routine works on definite integrals. The result should satisfy the following
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accuracy requirement:
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$$
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|\text{result} - I| \le \max(\varepsilon_\text{abs}, \varepsilon_\text{rel}I)
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$$
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The tolerances have been set to \SI{1e-10}{} and \SI{1e-6}{}, respectively.
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As for the QDF, this was implemented by numerically inverting the CDF. This is
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done by solving the equation
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$$
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p(x) = p_0
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$$
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for x, given a probability value $p_0$, where $p(x)$ is again the CDF.
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The (unique) root of this equation is found by a root-finding routine
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(`gsl_root_fsolver_brent` in GSL) based on the Brent-Dekker method. This
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algorithm consists in a bisection search, similar to the one employed in the
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mode optimisation, but improved by interpolating the function with a parabola
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at each step. The following condition is checked for convergence:
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$$
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|a - b| < \varepsilon_\text{abs} + \varepsilon_\text{rel} \min(|a|, |b|)
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$$
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where $a,b$ are the current interval bounds. The condition immediately gives an
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upper bound on the error of the root as $\varepsilon = |a-b|$. The tolerances
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here have been set to 0 and \SI{1e-3}{}.
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The result of the numerical computation is:
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$$
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\text{expected median: } m_e = \SI{1.3557804 \pm 0.0000091}{}
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$$
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while the sample median was found to be
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$$
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\text{observed median: } m_e = \SI{1.3479314}{}
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$$
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### FWHM
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The same approach was taken as regards the FWHM. This statistic is defined as
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the distance between the two points at which the function assumes half times
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the maximum value. Even in this case, there is not an analytic expression for
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it, thus it was computed numerically ad follow.
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First, some definitions must be given:
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$$
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f_{\text{max}} := f(m_e) \et \text{FWHM} = x_+ - x_- \with
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f(x_{\pm}) = \frac{f_{\text{max}}}{2}
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$$
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then the function $f'(x)$ was minimized using the same minimization method
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used for finding $m_e$, dividing the range into $[x_\text{min}, m_e]$ and
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$[m_e, x_\text{max}]$ (where $x_\text{min}$ and $x_\text{max}$ are the limits
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in which the points have been sampled) in order to be able to find both the
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minima of the function:
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$$
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f'(x) = \left|f(x) - \frac{f_{\text{max}}}{2}\right|
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$$
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resulting in:
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$$
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\text{expected FWHM: } w_e = \SI{4.0186457 \pm 0.0000001}{}
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$$
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On the other hand, the observed FWHM was computed as the difference between
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the center of the bins with the values closer to $\frac{f_{\text{max}}}{2}$
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and the error was taken as twice the width of the bins, obtaining:
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$$
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\text{observed FWHM: } w_o = \SI{4 \pm 2}{}
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$$
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This two values turn out to be compatible too, with:
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$$
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p = 0.99
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$$
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