images: neatly renamed all the images

notes/sections/1.md

@@ -9,7 +9,7 @@ $$
   f(x) = \int \limits_{0}^{+ \infty} dt \, e^{-t \log(t) -xt} \sin (\pi t)
 $$
 
-![Landau distribution.](images/landau-small.pdf){width=50%}
+![Landau distribution.](images/1-landau-small.pdf){width=50%}
 
 The GNU Scientific Library (GSL) provides a number of functions for generating
 random variates following tens of probability distributions. Thus, the function
@@ -20,7 +20,7 @@ an histogram and plotted with matplotlib. The result is shown in @fig:landau.
 
 ![Example of N = 10'000 points generated with the `gsl_ran_landau()`
 function and plotted in a 100-bins histogram ranging from -20 to
-80.](images/landau-histo.pdf){#fig:landau}
+80.](images/1-landau-histo.pdf){#fig:landau}
 
 
 ## Randomness testing of the generated sample
@@ -102,7 +102,7 @@ PDF, very few parameters can be easily checked: mode, median and full width at
 half maximum (FWHM).
 
 ![Landau distribution with emphatized mode $m_e$ and
-  FWHM = ($x_+ - x_-$).](images/landau.pdf)
+  FWHM = ($x_+ - x_-$).](images/1-landau.pdf)
 
 \begin{figure}
 \hypertarget{fig:parameters}{%
@@ -304,7 +304,7 @@ $$
 
 ![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
-  compensate for the otherwise peak height reduction.](images/landau-kde.pdf)
+  compensate for the otherwise peak height reduction.](images/1-landau-kde.pdf)
 
 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

notes/sections/2.md

@@ -26,7 +26,7 @@ numbers, packed in series, partial sums or infinite products. Thus, the
 efficiency of the methods lies on how quickly they converge to their limit.
 
 ![The area of the blue region converges to the Euler–Mascheroni
-constant.](images/gamma-area.png){#fig:gamma width=7cm}
+constant.](images/2-gamma-area.png){#fig:gamma width=7cm}
 
 
 ## Computing the constant

notes/sections/3.md

@@ -45,14 +45,14 @@ easily generated by the GSL function `gsl_rng_uniform()`.
 
 <div id="fig:compare">
   ![Uniformly distributed points with $\theta$ evenly distributed between
-  0 and $\pi$.](images/histo-i-u.pdf){width=50%}
+  0 and $\pi$.](images/3-histo-i-u.pdf){width=50%}
   ![Points uniformly distributed on a spherical
-  surface.](images/histo-p-u.pdf){width=50%}
+  surface.](images/3-histo-p-u.pdf){width=50%}
 
   ![Sample generated according to $F$ with $\theta$ evenly distributed between
-  0 and $\pi$.](images/histo-i-F.pdf){width=50%}
+  0 and $\pi$.](images/3-histo-i-F.pdf){width=50%}
   ![Sample generated according to $F$ with $\theta$ properly
-  distributed.](images/histo-p-F.pdf){width=50%}
+  distributed.](images/3-histo-p-F.pdf){width=50%}
 
 Examples of samples. On the left, points with $\theta$ evenly distributed
 between 0 and $\pi$; on the right, points with $\theta$ properly distributed.

notes/sections/4.md

@@ -150,7 +150,7 @@ $$ {#eq:dip}
 Namely:
 
 ![Plot of the expected distribution with
-  $P_{\text{max}} = 10$.](images/expected.pdf){#fig:plot}
+  $P_{\text{max}} = 10$.](images/4-expected.pdf){#fig:plot}
 
 
 ## Monte Carlo simulation
@@ -194,7 +194,7 @@ following. At first $S_j = 0 \wedge \text{num}_j = 0 \, \forall \, j$, then:
 
 For $P_{\text{max}} = 10$ and $n = 50$, the following result was obtained:
 
-![Sampled points histogram.](images/dip.pdf)
+![Sampled points histogram.](images/4-dip.pdf)
 
 In order to check whether the expected distribution (@eq:dip) properly matches
 the produced histogram, a chi-squared minimization was applied. Being a simple
@@ -232,4 +232,4 @@ distribution. In @fig:fit, the fit function superimposed on the histogram is
 shown.
 
 ![Fitted sampled data. $P^{\text{oss}}_{\text{max}} = 10.005
-  \pm 0.018$, $\chi_r^2 = 0.071$.](images/fit.pdf){#fig:fit}
+  \pm 0.018$, $\chi_r^2 = 0.071$.](images/4-fit.pdf){#fig:fit}

notes/sections/5.md

@@ -77,7 +77,7 @@ given.
 ![Estimated values of $I$ obatined by Plain MC technique with different
 number of function calls; logarithmic scale; errorbars showing their
 estimated uncertainties. As can be seen, the process does a sort o seesaw
-around the correct value.](images/MC_MC.pdf){#fig:MC}
+around the correct value.](images/5-MC_MC.pdf){#fig:MC}
 
 ---------------------------------------------------------------------------
 calls               $I^{\text{oss}}$        $\sigma$            diff
@@ -220,7 +220,7 @@ to give an overall result and an estimate of its error [@sayah19].
 
 ![Estimations $I^{\text{oss}}$ of the integral $I$ obtained for the three
   implemented method for different values of function calls. Errorbars
-  showing their estimated uncertainties.](images/MC_MC_MI.pdf){#fig:MI}
+  showing their estimated uncertainties.](images/5-MC_MC_MI.pdf){#fig:MI}
 
 Results for this particular sample are shown in black in @fig:MI and some of
 them are listed in @tbl:MI. Except for the first very little number of calls,
@@ -400,7 +400,7 @@ calls, the difference is smaller than \SI{1e-10}{}.
 
 ![Only the most accurate results are shown in order to stress the
   differences between VEGAS (in gray) and MISER (in black) methods
-  results.](images/MC_MI_VE.pdf){#fig:MI_VE}
+  results.](images/5-MC_MI_VE.pdf){#fig:MI_VE}
 
 In conclusion, between a plain Monte Carlo technique, stratified sampling and
 importance sampling, the last one turned out to be the most powerful mean to

notes/sections/6.md

@@ -87,13 +87,14 @@ The so obtained sample was binned and stored in a histogram with a customizable
 number $n$ of bins (default set $n = 150$) ranging from $\theta = 0$ to $\theta
 = \pi/2$ bacause of the system symmetry. In @fig:original an example is shown.
 
-![Example of intensity histogram.](images/original.pdf){#fig:original}
+![Example of intensity histogram.](images/6-original.pdf){#fig:original}
 
 
 ## Gaussian convolution {#sec:convolution}
 
-The sample has then to be smeared with a Gaussian function with the aim to
+In order to simulate the instrumentation response, the sample was then to be
-recover the original sample afterwards, implementing a deconvolution routine.  
+smeared with a Gaussian function with the aim to recover the original sample
+afterwards, implementing a deconvolution routine.  
 For this purpose, a 'kernel' histogram with an even number $m$ of bins and the
 same bin width of the previous one, but a smaller number of them ($m \sim 6\%
 \, n$), was created according to a Gaussian distribution with mean $\mu$,
@@ -102,10 +103,10 @@ descussed lately.
 Then, the original histogram was convolved with the kernel in order to obtain
 the smeared signal. As an example, the result obtained for $\sigma = \Delta
 \theta$, where $\Delta \theta$ is the bin width, is shown in @fig:convolved.
-As expected, the smeared signal looks smoother with respect to the original
+The smeared signal looks smoother with respect to the original one: the higher
-one.
+$\sigma$, the greater the smoothness. 
 
-![Convolved signal.](images/smoothed.pdf){#fig:convolved}
+![Convolved signal.](images/6-smoothed.pdf){#fig:convolved}
 
 The convolution was implemented as follow. Consider the definition of
 convolution of two functions $f(x)$ and $g(x)$:
@@ -331,13 +332,22 @@ variable $\xi$ when the available measure is a sample {$x_i$} of points
 dronwn not by $f(x)$ but by another function $\phi(x)$ such that:
 $$
   \phi(x) = \int d\xi \, f(\xi) P(x | \xi)
-$$
+$$ {#eq:conv}
 
 where $P(x | \xi) \, d\xi$ is the probability (presumed known) that $x$ falls
-in the interval $(x, x + dx)$ when $\xi = \xi$. An example of this problem is
+in the interval $(x, x + dx)$ when $\xi = \xi$. If the so-colled point spread
-precisely that of correcting an observed distribution $\phi(x)$ for the effect
+function $P(x | \xi)$ follows a normal distribution with variance $\sigma$,
-of observational errors, which are represented by the function $P (x | \xi)$,
+namely:
-called point spread function.
+$$
+  P(x | \xi) = \frac{1}{\sqrt{2 \pi} \sigma}
+  \exp \left( - \frac{(x - \xi)^2}{2 \sigma^2} \right)
+$$
+
+then, @eq:conv becomes a convolution and finding $f(\xi)$ turns out to be a
+deconvolution.  
+An example of this problem is precisely that of correcting an observed
+distribution $\phi(x)$ for the effect of observational errors, which are
+represented by the function $P (x | \xi)$.
 
 Let $Q(\xi | x) d\xi$ be the probability that $\xi$ comes from the interval
 $(\xi, \xi + d\xi)$ when the measured quantity is $x = x$. The probability that
@@ -382,22 +392,18 @@ $$
   P(x | \xi)
 $$ {#eq:solution}
 
-If the spread function $P(x | \xi)$ follows a normal distribution with variance
+When the spread function $P(x | \xi)$ is Gaussian, @eq:solution can be
-$\sigma$, namely:
+rewritten in terms of convolutions:
-$$
-  P(x | \xi) = \frac{1}{\sqrt{2 \pi} \sigma}
-  \exp \left( - \frac{(x - \xi)^2}{2 \sigma^2} \right)
-$$
-
-then, @eq:solution can be rewritten in terms of convolutions:
 $$
    f^{t + 1} = f^{t}\left( \frac{\phi}{{f^{t}} * P} * P^{\star} \right)
 $$
+
 where $P^{\star}$ is the flipped point spread function [@lucy74].
 
-In this special case, the Gaussian kernel stands for the point spread function
+In this special case, the Gaussian kernel which was convolved with the original
-and, dealing with discrete values, the division and multiplication are element
+histogram stands for the point spread function and, dealing with discrete
-wise and the convolution is to be carried out as described in @sec:convolution.
+values, the division and multiplication are element wise and the convolution is
+to be carried out as described in @sec:convolution.  
 When implemented, this method results in an easy step-wise routine:
 
   - create a flipped copy of the kernel;
@@ -405,53 +411,16 @@ When implemented, this method results in an easy step-wise routine:
   - compute the convolutions, the product and the division at each step;
   - proceed until a given number of reiterations is achieved.
 
-In this case, the zero-order was set $f(\xi) = 0.5 \, \forall \, \xi$.
+In this case, the zero-order was set $f(\xi) = 0.5 \, \forall \, \xi$. Different
+number of iterations where tested. Results are discussed in
+@sec:conv_results.
 
 
-## Results comparison {#sec:conv_Results}
+## The earth mover's distance
 
-In [@fig:results1; @fig:results2; @fig:results3] the results obtained for three
+With the aim of comparing the two deconvolution methods, the similarity of a
-different $\sigma$s are shown. The tested values are $\Delta \theta$, $0.5 \,
+deconvolved outcome with the original signal was quantified using the earth
-\Delta \theta$ and $0.05 \, \Delta \theta$, where $\Delta \theta$ is the bin
+mover's distance.
-width of the original histogram, which is the one previously introduced in
-@fig:original. In each figure, the convolved signal is shown above, the
-histogram deconvolved with the FFT method is in the middle and the one
-deconvolved with RL is located below.
-
-As can be seen, increasing the value of $\sigma$ implies a stronger smoothing of
-the curve. The FFT deconvolution process seems not to be affected by $\sigma$
-amplitude changes: it always gives the same outcome, which is exactly the
-original signal. In fact, the FFT is the analitical result of the deconvolution.
-In the real world, it is unpratical, since signals are inevitably blurred by
-noise.  
-The same can't be said about the RL deconvolution, which, on the other hand,
-looks heavily influenced by the variance magnitude: the greater $\sigma$, the
-worse the deconvolved result. In fact, given the same number of steps, the
-deconvolved signal is always the same 'distance' far form the convolved one:
-if it very smooth, the deconvolved signal is very smooth too and if the
-convolved is less smooth, it is less smooth too.
-
-The original signal is shown below for convenience.
-
-
-
-It was also implemented the possibility to add a Poisson noise to the
-convolved histogram to check weather the deconvolution is affected or not by
-this kind of interference. It was took as an example the case with $\sigma =
-\Delta \theta$. In @fig:poisson the results are shown for both methods when a
-Poisson noise with mean $\mu = 50$ is employed.  
-In both cases, the addition of the noise seems to partially affect the
-deconvolution. When the FFT method is applied, it adds little spikes nearly
-everywhere on the curve and it is particularly evident on the edges, where the
-expected data are very small. On the other hand, the Richardson-Lucy routine is
-less affected by this further complication.
-
-In order to quantify the similarity of a deconvolution outcome with the original
-signal, a null hypotesis test was made up.  
-Likewise in @sec:Landau, the original sample was treated as a population from
-which other samples of the same size were sampled with replacements. For each
-new sample, the earth mover's distance with respect to the original signal was
-computed.
 
 In statistics, the earth mover's distance (EMD) is the measure of distance
 between two probability distributions [@cock41]. Informally, the distributions
@@ -460,22 +429,36 @@ a region and the EMD is the minimum cost of turning one pile into the other,
 where the cost is the amount of dirt moved times the distance by which it is
 moved. It is valid only if the two distributions have the same integral, that
 is if the two piles have the same amount of dirt.  
-Computing the EMD is based on a solution to the well-known transportation
+Computing the EMD is based on a solution to the transportation problem, which
-problem, which can be formalized as follows.
+can be formalized as follows.
 
-Consider two vectors:
+Consider two vectors $P$ and $Q$ which represent the two probability
+distributions whose EMD has to be measured:
 
 $$
-  P = \{ (p_1, w_{p1}) \dots (p_n, w_{pm}) \} \et
+  P = \{ (p_1, w_{p1}) \dots (p_m, w_{pm}) \} \et
   Q = \{ (q_1, w_{q1}) \dots (q_n, w_{qn}) \}
 $$
 
-where $p_i$ and $q_i$ are the 'values' and $w_{pi}$ and $w_{qi}$ are their
+L'istogramma P deve essere distrutto in modo tale da ottenere l'istogramma Q,
-weights. The entries $d_{ij}$ of the ground distance matrix $D_{ij}$ are
+che in partenza è vuoto ma so che vorrò avere w_qj in ogni bin che sta alla
-defined as the distances between $p_i$ and $q_j$.  
+posizione qj.
-The aim is to find the flow $F =$ {$f_{ij}$}, where $f_{ij}$ is the flow
+- sposto solo da P a Q
-between $p_i$ and $p_j$ (which would be the quantity of moved dirt), which
+- sposto non più di ogni ingresso di P
-minimizes the cost $W$:
+- ottengo non più di ogni ingreddo di Q
+- sposto tutto quello che posso: o ottengo tutto Q o ho finito P
+
+e non devono venire uguali, quindi!
+
+
+
+
+where $p_i$ and $q_i$ are the 'values' (that is, the location of the dirt) and
+$w_{pi}$ and $w_{qi}$ are the 'weights' (that is, the quantity of dirt). A
+ground distance matrix $D_{ij}$ is defined such as its entries $d_{ij}$ are the
+distances between $p_i$ and $q_j$. The aim is to find the flow matrix $F_{ij}$,
+where each entry $f_{ij}$ is the flow from $p_i$ to $q_j$ (which would be
+the quantity of moved dirt), which minimizes the cost $W$:
 
 $$
   W (P, Q, F) = \sum_{i = 1}^m \sum_{j = 1}^n f_{ij} d_{ij}
@@ -486,7 +469,7 @@ with the constraints:
 \begin{align*}
   &f_{ij} \ge 0 \hspace{15pt}       &1 \le i \le m \wedge 1 \le j \le n \\
   &\sum_{j = 1}^n f_{ij} \le w_{pi} &1 \le i \le m                      \\
-  &\sum_{j = 1}^m f_{ij} \le w_{qj} &1 \le j \le n
+  &\sum_{i = 1}^m f_{ij} \le w_{qj} &1 \le j \le n
 \end{align*}
 $$
   \sum_{j = 1}^n f_{ij} \sum_{j = 1}^m f_{ij} \le w_{qj}
@@ -555,3 +538,35 @@ the original one cannot be disporoved if its distance from the original signal
 is grater than  \textcolor{red}{value}.
 
 \textcolor{red}{counts}
+
+
+## Results comparison {#sec:conv_results}
+
+As can be seen, increasing the value of $\sigma$ implies a stronger smoothing of
+the curve. The FFT deconvolution process seems not to be affected by $\sigma$
+amplitude changes: it always gives the same outcome, which is exactly the
+original signal. In fact, the FFT is the analitical result of the deconvolution.
+In the real world, it is unpratical, since signals are inevitably blurred by
+noise.  
+The same can't be said about the RL deconvolution, which, on the other hand,
+looks heavily influenced by the variance magnitude: the greater $\sigma$, the
+worse the deconvolved result. In fact, given the same number of steps, the
+deconvolved signal is always the same 'distance' far form the convolved one:
+if it very smooth, the deconvolved signal is very smooth too and if the
+convolved is less smooth, it is less smooth too.
+
+The original signal is shown below for convenience.
+
+
+
+It was also implemented the possibility to add a Poisson noise to the
+convolved histogram to check weather the deconvolution is affected or not by
+this kind of interference. It was took as an example the case with $\sigma =
+\Delta \theta$. In @fig:poisson the results are shown for both methods when a
+Poisson noise with mean $\mu = 50$ is employed.  
+In both cases, the addition of the noise seems to partially affect the
+deconvolution. When the FFT method is applied, it adds little spikes nearly
+everywhere on the curve and it is particularly evident on the edges, where the
+expected data are very small. On the other hand, the Richardson-Lucy routine is
+less affected by this further complication.
+

notes/sections/7.md

@@ -40,7 +40,7 @@ An example of the two samples is shown in @fig:points.
 
 ![Example of points sorted according to two Gaussian with
 the given parameters. Noise points in pink and signal points
-in yellow.](images/points.pdf){#fig:points}
+in yellow.](images/7-points.pdf){#fig:points}
 
 Assuming not to know how the points were generated, a model of classification
 must then be implemented in order to assign each point to the right class
@@ -96,7 +96,7 @@ maximization, it can be found that $w \propto (m_2 − m_1)$.
 resulting from projection onto the line joining the class means: note that
 there is considerable overlap in the projected space. The right plot shows the
 corresponding projection based on the Fisher linear discriminant, showing the
-greatly improved classes separation.](images/fisher.png){#fig:overlap}
+greatly improved classes separation.](images/7-fisher.png){#fig:overlap}
 
 There is still a problem with this approach, however, as illustrated in
 @fig:overlap: the two classes are well separated in the original 2D space but
@@ -195,9 +195,9 @@ The projection of the points was accomplished by the use of the function
 this case were the weight vector and the position of the point to be projected.
 
 <div id="fig:fisher_proj">
-![View from above of the samples.](images/fisher-plane.pdf){height=5.7cm}
+![View from above of the samples.](images/7-fisher-plane.pdf){height=5.7cm}
 ![Gaussian of the samples on the projection
-  line.](images/fisher-proj.pdf){height=5.7cm}
+  line.](images/7-fisher-proj.pdf){height=5.7cm}
 
 Aerial and lateral views of the projection direction, in blue, and the cut, in
 red.
@@ -265,9 +265,9 @@ $$
 $$
 
 <div id="fig:percep_proj">
-![View from above of the samples.](images/percep-plane.pdf){height=5.7cm}
+![View from above of the samples.](images/7-percep-plane.pdf){height=5.7cm}
 ![Gaussian of the samples on the projection
-  line.](images/percep-proj.pdf){height=5.7cm}
+  line.](images/7-percep-proj.pdf){height=5.7cm}
 
 Aerial and lateral views of the projection direction, in blue, and the cut, in
 red.