diff --git a/notes/docs/bibliography.bib b/notes/docs/bibliography.bib new file mode 100644 index 0000000..64d8f15 --- /dev/null +++ b/notes/docs/bibliography.bib @@ -0,0 +1,7 @@ +@article{cock41, + title={The distribution of a product from several sources to numerous localities}, + author={F. L. Hitchcock}, + year={2942}, + journal={Journal of Mathematical Physics}, + pages={224 - 230} +} diff --git a/notes/docs/bibliography.csl b/notes/docs/bibliography.csl new file mode 100644 index 0000000..86c3511 --- /dev/null +++ b/notes/docs/bibliography.csl @@ -0,0 +1,14 @@ + + diff --git a/notes/sections/0.md b/notes/sections/0.md index 49a4cb9..40efbf9 100644 --- a/notes/sections/0.md +++ b/notes/sections/0.md @@ -80,4 +80,7 @@ header-includes: | \captionsetup{width=11cm} \usepackage{stmaryrd} ``` + +bibliography: docs/bibliography.bib +csl: docs/bibliography.csl --- diff --git a/notes/sections/1.md b/notes/sections/1.md index da0f2f9..f052eda 100644 --- a/notes/sections/1.md +++ b/notes/sections/1.md @@ -1,4 +1,4 @@ -# Exercise 1 +# Exercise 1 {#sec:Landau} ## Random numbers following the Landau distribution @@ -6,7 +6,7 @@ The Landau distribution is a probability density function which can be defined as follows: $$ - f(x) = \int \limits_{0}^{+ \infty} dt \, e^{-t log(t) -xt} \sin (\pi t) + f(x) = \int \limits_{0}^{+ \infty} dt \, e^{-t \log(t) -xt} \sin (\pi t) $$ ![Landau distribution.](images/landau-small.pdf){width=50%} @@ -18,7 +18,7 @@ was used. For the purpose of visualizing the resulting sample, the data was put into an histogram and plotted with matplotlib. The result is shown in @fig:landau. -![Example of N points generated with the `gsl_ran_landau()` +![Example of N = 10'000 points generated with the `gsl_ran_landau()` function and plotted in a 100-bins histogram ranging from -10 to 80.](images/landau-hist.png){#fig:landau} @@ -41,7 +41,7 @@ $$ where: - $x$ runs over the sample, - - $F(x)$ is the Landau cumulative distribution and function + - $F(x)$ is the Landau cumulative distribution function, - $F_N(x)$ is the empirical cumulative distribution function of the sample. If $N$ numbers have been generated, for every point $x$, diff --git a/notes/sections/4.md b/notes/sections/4.md index 718d18a..f86c4c7 100644 --- a/notes/sections/4.md +++ b/notes/sections/4.md @@ -119,7 +119,7 @@ $$ $$ from which, the integral $I$ can now be computed. The edges of the integral -are fixed bt the fact that the total momentum can not exceed $P_{\text{max}}$: +are fixed by the fact that the total momentum can not exceed $P_{\text{max}}$: $$ I = \int @@ -218,7 +218,7 @@ $$ p_h = j \cdot w + \frac{w}{2} = w \left( 1 + \frac{1}{2} \right) $$ -The following result was obtained: +For $p_{\text{max}} = 10$, the following result was obtained: ![Histogram of the obtained distribution.](images/dip.pdf) diff --git a/notes/sections/6.md b/notes/sections/6.md index e1a148f..a8fb3ad 100644 --- a/notes/sections/6.md +++ b/notes/sections/6.md @@ -91,9 +91,9 @@ of bins default set $n = 150$. In @fig:original an example is shown. ![Example of an intensity histogram.](images/fraun-original.pdf){#fig:original} -## Gaussian noise convolution {#sec:convolution} +## Gaussian convolution {#sec:convolution} -The sample must then be smeared with a Gaussian noise with the aim to recover +The sample must then be 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 a odd number $m$ of bins and the same bin width of the previous one, but a smaller number of them ($m < n$), was @@ -370,7 +370,7 @@ $P^{\star}$ is the flipped point spread function. When implemented, this method results in an easy step-wise routine: - create a flipped copy of the kernel; - - elect a zero-order estimate for {$c_i$}; + - choose a zero-order estimate for {$c_i$}; - compute the convolutions with the method described in @sec:convolution, the product and the division at each step; - proceed until a given number of reiterations is achieved. @@ -393,27 +393,27 @@ deconvolved with RL is located below. As can be seen, increasig 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, remarkably similar to the -original signal. 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 deconvoluted 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. +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 deconvoluted 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. It was also implemented the possibility to add a Poisson noise to the -convoluted histogram to check weather the deconvolution is affected or not by -this kind of noise. 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 Poisson noise seems to affect partially the -deconvolution. When the FFT method was applied, it adds little spikes nearly -everywhere on the curve but it is particularly evident on the edges of the -curve, where the expected data are very small. This is because the technique is -very accurate and hence returns nearly the exact original data which, in this -case, is the expected one to which the Poisson noise is added. -On the other hand, the Richardson-Lucy routine is less affected by this further -complication being already inaccurate in itself. +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.
![Convolved signal.](images/fraun-conv-0.05.pdf){width=12cm} @@ -454,3 +454,35 @@ Results for $\sigma = \Delta \theta$, where $\Delta \theta$ is the bin width. Results for $\sigma = \Delta \theta$, with Poisson noise.
+ +In order to quantify the similarity of the 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 +are interpreted as two different ways of piling up a certain amount of dirt over +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 of transportation problem. + +\textcolor{red}{earth mover's distance} + +In this case, where the EMD must be applied to two histograms, the procedure +simplifies a lot boiling down to the difference of the comulative functions of +the two histograms. + +These distances were used to build their empirical cumulative distribution. + +\textcolor{red}{empirical distribution} + +At 95% confidence level, the compatibility of the deconvolved signal with +the original one cannot be disporoved if its distance from the original signal +is grater than \textcolor{red}{value}. + +\textcolor{red}{counts} diff --git a/notes/todo b/notes/todo index 415d5e3..ba64fcb 100644 --- a/notes/todo +++ b/notes/todo @@ -1,2 +1,3 @@ -- rifare tutti i grafici con le scritte enormi -- aggiungere 4 e 5 nel readme +- cambiare simbolo convoluzione +- aggiungere citazioni e referenze +- rifare grafici senza bordino