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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
\institute{UZH}
\title[Random number generators and application]{Random number generators and application}
\date{\fixme}


\begin{document}
\tikzstyle{every picture}+=[remember picture]

{
\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
\begin{frame}[c]%{\phantom{title page}} 
\begin{center}
\begin{center}
	\begin{columns}
		\begin{column}{0.9\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Random number generators and application}
		\end{column}
		\begin{column}{0.2\textwidth}
		  %\includegraphics[width=\textwidth]{SHiP-2}
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\end{center}
	\quad
	\vspace{3em}
\begin{columns}
\begin{column}{0.44\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}

\end{column}
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\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{Experimental Methods in Particle Physics, \\ 19 November, 2015}
\end{center}
\end{frame}
}


\begin{frame}\frametitle{Random and pseudorandom numbers}
  \begin{exampleblock}{John von Neumann:}                                                                                                                                                                                                                                                                                                
''Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.''
        \end{exampleblock} 

$\color{PineGreen}\Rrightarrow$ Random number: a given value that is taken by a random variable $ \twoheadrightarrow$ by definition cannot be predicted.\\
%$\color{PineGreen}\Rrightarrow$ Sequence of random numbers $\twoheadrightarrow$ 
$\color{PineGreen}\Rrightarrow$ Sources of truly random numbers:
\begin{itemize}
\item Mechanical
\item Physical
\end{itemize}
$\color{PineGreen}\Rrightarrow$ Disadvantages of physical generators:
\begin{itemize}
\item To slow for typical applications, especially the mechanical ones!
\item Not stable; small changes in boundary conditions might lead to completely different results!
\end{itemize}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Random numbers - history remark}
$\color{PineGreen}\Rrightarrow$ In the past there were books with random numbers:
\begin{center}
\includegraphics[width=0.55\textwidth]{images/million-random-digits-open.jpg}
  \end{center}
$\color{PineGreen}\Rrightarrow$ It's obvious that they didn't become very popular ;)\\
$\color{PineGreen}\Rrightarrow$ This methods are comming back!\\
$\color{PineGreen}\twoheadrightarrow$ Storage device are getting more cheap and bigger (CD, DVD).\\
$\color{PineGreen}\twoheadrightarrow$ 1995: G. Marsaglia, $650\rm MB$ of random numbers, ''White and Black Noise''.


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Pseudorandom numbers}
$\color{PineGreen}\Rrightarrow$ Pseudorandom numbers are numbers that are generated accordingly to strict mathematical formula. \\
$\color{PineGreen}\looparrowright$ Strictly speaking they are non random numbers, how ever they have all the statistical properties of random numbers.\\
$\color{PineGreen}\looparrowright$  Discussing those properties is a wide topic so let's just say that without knowing the formula they are generated by one cannot say if those numbers are random or not.\\
$\color{PineGreen}\Rrightarrow$  Mathematical methods of producing pseudorandom numbers:
\begin{itemize}
\item Good statistical properties of generated numbers.
\item Easy to use and fast!
\item Reproducible!
\end{itemize}
$\color{PineGreen}\Rrightarrow$ Since mathematical pseudorandom genrators are dominantly:  pseudorandom $\rightarrowtail$ random.


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Middle square generator; von Neumann}
$\color{PineGreen}\Rrightarrow$  The first mathematical generator (middle square) was proposed by von Neumann (1964).\\{~}\\
$\color{PineGreen}\looparrowright$ Formula: $
\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
  boxrule=0pt,frame hidden]{ X_n= \lfloor X_{n-1}^2\cdot 10 ^{-m} \rfloor - \lfloor X_{n-1}^2\cdot 10^{-3m} \rfloor}
$

$\color{PineGreen}\looparrowright$ where $X_0$ is a constant (seed), $\lfloor\cdot\rfloor$ is the cut-off of a number to integer.\\
$\color{PineGreen}\Rrightarrow$  Example:\\
{~}{~}Let's put $m=2$ and $X_0=2045$:\\
\begin{columns}
\column{0.1\textwidth}
{~}
\column{0.4\textwidth}
$\color{PineGreen}\looparrowright$ $X_0^2=\underbrace{04}_{\rm rej}1820\underbrace{25}_{\rm rej}$
\column{0.3\textwidth}
 $\Rightarrow X_1=1820$
\end{columns}
\begin{columns}
\column{0.1\textwidth}
{~}
\column{0.4\textwidth}
$\color{PineGreen}\looparrowright$ $X_1^2=\underbrace{03}_{\rm rej}3124\underbrace{00}_{\rm rej}$
\column{0.3\textwidth}
 $\Rightarrow X_1=3124$
\end{columns}
$\color{PineGreen}\looparrowright$ Simple generator but unfortunately quite bad generator. Firstly the sequences are very short and strongly dependent on the $X_0$ number.



\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Linear generators}

$\color{PineGreen}\Rrightarrow$ General equation:
\begin{equation}
\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
  boxrule=0pt,frame hidden]{X_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~m    ,} \nonumber
\end{equation}
$\color{PineGreen}\looparrowright$ where $a_i,c,m$ are parameters of a generator(integer numbers).\\
$\color{PineGreen}\looparrowright$  Generator initialization $\color{PineGreen} \rightleftarrows$ setting those parameters.\\
$\color{PineGreen}\Rrightarrow$ Very old generators. (often used in Pascal, or first C versions):
\begin{columns}
\column{0.1\textwidth}
{~}
\column{0.8\textwidth}
$k=1:~X_n=(aX_{n-1} +c )~{\rm mod} m,$
$c=
\begin{cases}
=0, {\rm multiplicative geneator}\\
\neq 0, {\rm mix geneator}
\end{cases}
$
\column{0.1\textwidth}
{~}


\end{columns}
$\color{PineGreen}\Rrightarrow$ The period can be achieved by tuning the seed parameters:\\
$P_{{\rm max}}=
\begin{cases}
2^{L-2};~{\rm for~} m=2^L\\
m-1;~{\rm for~} m= {\rm prime~number}
\end{cases}
$
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Linear generators; examples}

$\color{PineGreen}\Rrightarrow$ General equation:
\begin{equation}
\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
  boxrule=0pt,frame hidden]{X_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~m    ,} \nonumber
\end{equation}
$\color{PineGreen}\looparrowright$ where $a_i,c,m$ are parameters of a generator(integer numbers).\\
$\color{PineGreen}\looparrowright$  Generator initialization $\color{PineGreen} \rightleftarrows$ setting those parameters.\\
$\color{PineGreen}\Rrightarrow$ Very old generators. (often used in Pascal, or first C versions):
\begin{columns}
\column{0.1\textwidth}
{~}
\column{0.8\textwidth}
$k=1:~X_n=(aX_{n-1} +c )~{\rm mod} m,$
$c=
\begin{cases}
=0, {\rm multiplicative~geneator}\\
\neq 0, {\rm mix~geneator}
\end{cases}
$
\column{0.1\textwidth}
{~}


\end{columns}
$\color{PineGreen}\Rrightarrow$ The period can be achieved by tuning the seed parameters:\\
$P_{{\rm max}}=
\begin{cases}
2^{L-2};~{\rm for~} m=2^L\\
m-1;~{\rm for~} m= {\rm prime~number}
\end{cases}
$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Shift register generator}
\begin{small}
$\color{PineGreen}\Rrightarrow$ General equation:
\begin{equation}
\tcbhighmath[fuzzy halo=0.5mm with PineGreen!50!white,arc=0.1pt,
  boxrule=0pt,frame hidden]{b_n=(a_1X_{n-1} + a_2X_{n-2}+...+a_kX_{n-k}+c)~{\rm mod}~2    ,} \nonumber
\end{equation}
where $a_i \subset(\lbrace0,1\rbrace )$\\
$\color{PineGreen}\Rrightarrow$ Super fast and easy to implement due to: $(a+b) ~{\rm mod}~2 = a~{\rm xor}~b$
\begin{center}
\begin{tabular}{||c|c|c||}
\hline \hline a & b & a xor b\\ \hline
0 & 0 & 0\\
1 & 0 & 1\\
0 & 1 & 1\\
1 & 1 & 0\\ \hline \hline
\end{tabular}
\end{center}
$\color{PineGreen}\Rrightarrow$ Maximal period is $2^k-1$.\\
$\color{PineGreen}\Rrightarrow$ Example (Tausworths generator):\\
$a_p=a_q=1$, other $a_i=0$ and $p>q$. Then: $b_n=b_{n-p}~{\rm xor}~b_{n-q}$

$\color{PineGreen}\Rrightarrow$ How to get numbers from bits (for example):\\
$U_i = \sum_{j=1}^L 2^{-j} b_{is+j},~s<L$.

\end{small}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Fibonacci generator}
\begin{small}
$\color{PineGreen}\Rrightarrow$ In 1202 Fibonacci with Leonardo in Piza:
\begin{equation}
f_n=f_{n-2}+f_{n-1},~n\geqslant 2 \nonumber
\end{equation}
$\color{PineGreen}\Rrightarrow$ Based on this first generator was created (Taussky and Todd, 1956):
\begin{equation}
X_n=(X_{n-2}+X_{n-1})~{\rm mod}~m,~n\geqslant 2 \nonumber
\end{equation}
This generator isn't so good in terms of statistics tests.\\
$\color{PineGreen}\Rrightarrow$ Generalization:
\begin{equation}
X_n=(X_{n-r} \odot X_{n-s})~{\rm mod}~m,~n \geqslant r ,~s \geqslant 1 \nonumber
\end{equation}\begin{center}

\begin{tabular}{||c|c|c||}
\hline \hline $\bigodot$ & $P_{max}$ & Stat. properties \\ \hline
$+,-$ & $(2^r-1)2^{L-1}$ & good \\
$x$ & $(2^r-1)2^{L-13}$ & very good \\
$xor$ & $(2^r-1)$ & poor \\ \hline \hline 
\end{tabular}


\end{center}

\end{small}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Multiply with carry, generator}
\begin{small}


\end{small}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Subtract with borrow, generator}
\begin{small}


\end{small}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Non linear generators}
\begin{small}


\end{small}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Non linear generators}
\begin{small}


\end{small}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%% Stat tests$





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