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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
\institute{UZH}
\title[Introduction to \\Monte Carlo methods]{Introduction to \\Monte Carlo methods}
\date{\fixme}
\begin{document}
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\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Introduction to \\Monte Carlo methods}
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\quad
\vspace{3em}
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\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}
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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, \\ \fixme, 2015}
\end{center}
\end{frame}
}
\begin{frame}\frametitle{Literature}
\begin{enumerate}
\item J. M. Hammersley, D. C. Hamdscomb, ``Monte Carlo Methods'', London: Methuen \& Co. Ltd., New York: J. Wiley \& Sons Inc., 1964
\item I. M. Sobol, ``The Monte Carlo Method'', Mir Publishers, Moscow, 1975.
\item M. H. Kalos, P. A. Whitlock, ,,Monte Carlo Methods”, J. Wiley \& Sons Inc., New York, 1986
\item G. S. Fishman, ,,Monte Carlo: Concepts, Algorithms and Applications”, Springer, 1996.
\item R. Y. Rubinstein, D. P. Kroese, ,,Simulation and the Monte Carlo Method”, Second Edition, J. Wiley \& Sons Inc., 2008.
\item R. Korn, E. Korn, G. Kroisandt, ,,Monte Carlo methods and models in finance and insurance”, CRC Press,
Taylor \& Francis Group, 2010.
\item S. Jadach, ,,Practical Guide to Monte Carlo”, \href{http://arxiv.org/abs/physics/9906056}{arXiv:physics/9906056}, \href{http://cern.ch/jadach/MCguide/}{http://cern.ch/jadach/MCguide/}.
\end{enumerate}
\end{frame}
\begin{frame}\frametitle{Course Plan}
We will have 6 hourse of Monte Calro (MC) lectures. The lecutres will be devoted:\\
\hspace{1.5cm}
\begin{itemize}
\item 1 h: Mathematical introduction to MC methods.
\item 1 h: MC integration methods.
\item 2 h: Random numbers generators.
\item 0.5 h: Cool applications of MC methods.
\item 1.5h: Hands-on tutorial with MC methods.
\end{itemize}
\hspace{1cm} \\
The hands-on tutorial will consist of program templates in which you will have to implement couple of algoriths that were explained in the lecture. The solutions will be discussed on the last lecture.
\end{frame}
\begin{frame}\frametitle{Definitions}
\begin{footnotesize}
$\Rrightarrow$ Basic definition:\\
\begin{exampleblock}{}
Monte Carlo method is any technique that uses {\it{random numbers}} to solve a given mathematical problem.
\end{exampleblock}
% \vspace{0.5cm}
$\rightarrowtail$ Random number: For the purpose of this course we need to assume that we know what it is, although the formal definition is highly non-trivial.\\
\vspace{0.05cm}
$\Rrightarrow$ My favourite definition (Halton 1970): \begin{scriptsize}more complicated, but more accurate.\end{scriptsize}
\begin{exampleblock}{}
''Representing the solution of a problem as a parameter of a hypothetical population, and using a random sequence of numbers to construct a sample of the population, from which statistical estimates of the parameter can be obtained.''
\end{exampleblock}
To put this definition in mathematical language:\\
Let $F$ be a solution of a given mathematical problem. The estimate of the result $\hat{F}$:\\
\begin{equation*}
\hat{F}=f( \lbrace r_1, r_2, r_3,...,r_n \rbrace; ...),
\end{equation*}
where $\lbrace r_1, r_2, r_3,...,r_n \rbrace$ are random numbers.
\begin{center}
\color{red}{The problem we are solving doesn't need to be stochastic!}
\end{center}
\begin{scriptsize}
$\twoheadrightarrow$ One could wonder why are we trying to add all the stochastic properties to a deterministic problem. Those are the properties that allow to use all well known statistic theorems.
\end{scriptsize}
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{History of MC methods}
\begin{footnotesize}
\begin{itemize}
\item {\color{PineGreen}{G. Compte de Buffon (1777)}} - First documented usage of random numbers for integral computation (Buffon thrown niddle on the table with parrarel line; we will do a modern version of this exercise).
\item {\color{PineGreen}{Marquis de Laplace (1886)}} - Used the Buffon niddle to determine the value of $\pi$ number.
\item {\color{PineGreen}{Lord Kelvin (1901)}} - Thanks to drawing randomly numbered cards he managed he managed to calculate some integrals in kinematic gas theorem.
\item {\color{PineGreen}{W. S. Gosse (better knows as Student) (1908)}} - Used similar way as Lord Kelvin to get random numbers to prove \textit{t}-Student distribution.
\item {\color{PineGreen}{Enrico Fermi (1930) }} - First mechanical device (\texttt{FERMIAC}) for random number generations. Solved neutron transport equations in the nuclear plants.
\item {\color{PineGreen}{S. Ulam, R. Feynman, J. von Neumann et. al.}} - First massive usage of random numbers. Most applications were in Manhattan project to calculate neutron scattering and absorption. \\
In {\color{NavyBlue}{Los Alamos}} the name {\color{Mahogany}{Monte Carlo}} was created as kryptonim of this kind of calculations.
\end{itemize}
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{Euler number determination}
\begin{footnotesize}
$\Rrightarrow$ As mentioned before \mc~methods can be used to solve problems that \textbf{do not} have stochastic nature! All the integrals calculated in Los Alamos during the Manhattan project are nowadays solvable without any \mc~methods.\\
$\rightarrowtail$ Let's give a trivial example of solving a non stochastic problem: calculating Euler number $e$. We know that $e=2.7182818...$.
$\Rrightarrow$ To calculate the $\hat{e}$ we will use the following algorithm:
\begin{itemize}
\item We generate a random number in range $(0,1)$ (in stat. $\mathcal{U}(0,1)$) until the number we generate is smaller then the previous one, aka we get the following sequence:
\begin{align*}
x_1<x_2<...<x_{n-1}>x_{n}
\end{align*}
\item We store the number $n$. We repeat this experiment $N$ times and calculate the arithmetic average of $n$. The obtained value is an statistical estimator of $e$:
\begin{align*}
\hat{e}= \dfrac{1}{N}\sum_{i=1}^N n_i \xrightarrow{N\to \infty} e .
\end{align*}
\end{itemize}
$\Rrightarrow$ Numerical example:
\begin{tabular}{r c c}
$N$ & $\hat{e}$ & $\hat{e} - e$\\
100 & $2.760000$ & $0.041718$ \\
10000 & $2.725000$ & $0.006718$ \\
1000000 & $2.718891$ & $0.000609$ \\
100000000 & $2.718328$ & $0.000046$\\
\end{tabular}
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{Monte Carlo and integration}
\begin{footnotesize}
$\hookrightarrow$ {\color{BrickRed}{\textbf{All MC calculations are equivalent to preforming an integration.}}}\\
$\rightrightarrows$ Assumptions: $r_i$ random numbers from $\mathcal{U}(0,1)$. The MC result:
\begin{align*}
F=F(r_1,r_2,...r_n)
\end{align*}
is unbias estimator of an integral:
\begin{align*}
I=\int_0^1...\int_0^1 F(x_1,x_2,...,x_n)dx_1,dx_2...,dx_n
\end{align*}
aka the expected value of the $I$ integral is:
\begin{align*}
E(F)=I.
\end{align*}
\begin{exampleblock}{}
$\Rrightarrow$ This mathematical identity is the most useful property of the MC methods. It is a link between mathematical analysis and statistic world. Now we can use the best of the both world!
\end{exampleblock}
If we want to calculate the integral in different range then $(0,1)$ we just scale the the previous result:
\begin{align*}
\dfrac{1}{N}\sum_{i=1}^N f(x_i) \xrightarrow{N\to \infty} E(f)=\dfrac{1}{b-a}\int_a^b f(x)dx
\end{align*}
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{Uncertainty from Monte Carlo methods}
\begin{footnotesize}
$\Rrightarrow$ In practice we do not have $N\to \infty$ so we will never know the exact result of an integral :(\\
$\longmapsto$ Let's use the {\color{BrickRed}{statistical}} world and estimate the uncertainty of an integral in this case :)\\
$\rightarrowtail$ A variance of a MC integral:
\begin{align*}
V(\hat{I}) = \dfrac{1}{n} = \dfrac{1}{n} \Big\lbrace E(f^2) - E^2(f) \Big\rbrace = \dfrac{1}{n} \Big\lbrace \dfrac{1}{b-a} \int_a^b f^2(x)dx - I^2 \Big\rbrace
\end{align*}
\begin{alertblock}{}
$\looparrowright$ To calculate $V(\hat{I})$ one needs to know the value of $I$!
\end{alertblock}
$\Rrightarrow$ In practice $V(\hat{I})$ is calculated via estimator:
\begin{columns}
\column{2in}
\begin{align*}
\hat{V}(\hat{I})=\dfrac{1}{n}\hat{V}(f),
\end{align*}
\column{3in}
\begin{align*}
\hat{V}(f) = \dfrac{1}{n-1}\sum_{i=1}^n \Big[ f(x_i)-\dfrac{1}{n} \sum_{i=1}^nf(x_i)\Big]^2.
\end{align*}
\end{columns}
$\Rrightarrow$ MC estimator of standard deviation: $\hat{\sigma}=\sqrt{\hat{V}(\hat{I})}$
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{Buffon needle - $\pi$ number calculus}
\begin{footnotesize}
$\Rrightarrow$ Buffon needle (Buffon 1777, Laplace 1886):
We are throwing a needle (of length $l$) on to a surface covered with parallel lines width distance $L$. If a thrown needle touches a line we count a hit, else miss. Knowing the number of hits and misses one can calculate the $\pi$ number.
\vspace{0.3cm}
\begin{columns}
\column{0.1in}
{~}
\column{2in}
{\color{ForestGreen}{Experiment:}}
\column{2.8in}
{\color{Cerulean}{Theory:}}
\end{columns}
\begin{columns}
\column{0.1in}
{~}
\column{2in}
\includegraphics[width=0.9\textwidth]{images/buffon.png}\\
$n$ - number of hits\\
$N$ number of hits and misses,\\
aka number of tries.
\column{2.8in}
$\Rightarrow$ x - angle between needle and horizontal line, $x \in \mathcal{U}(0,\pi)$.
$\Rightarrow$ the probability density function (\pdf) for x:
\begin{align*}
\rho(x)=\dfrac{1}{\pi}
\end{align*}
$\Rightarrow$ $p(x)$ probability to hit a line for a given x value:
\begin{align*}
p(x)=\dfrac{l}{L}\vert \cos x \vert
\end{align*}
$\Rightarrow$ Total hit probability:
\begin{align*}
P = E[p(x)]=\int_0^{\pi}p(x)\rho(x)dx=\dfrac{2l}{\pi L}
\end{align*}
\end{columns}
Now one can calculate $\hat{P}$ from MC : $\hat{P}=\dfrac{n}{N} \xrightarrow{N\to \infty} P= \dfrac{2l}{\pi L} \Rightarrow \hat{\pi}=\dfrac{2Nl}{nL}$
\end{footnotesize}
\end{frame}
\begin{frame}\frametitle{Buffon needle - Simplest Carlo method}
\begin{footnotesize}
{\color{MidnightBlue}{Monte Carlo type ''heads or tails''}}\\
Let's use the summery of $p(x)$ function nad take $0<x<\frac{\pi}{2}$.\\
$\Rightarrow$ Algorithm:\\
\begin{columns}
\column{0.1in}
{~}
\column{3.2in}
Generate 2 dim. distribution:
\begin{align*}
(x,y): \mathcal{U}(0,\dfrac{\pi}{2})\times \mathcal{U}(0,1) {\rm{~and~}}
\end{align*}
\begin{align*}
y
\begin{cases}
\leq p(x): & \text{hit},\\
> p(x): & \text{miss}.
\end{cases}
\end{align*}
\column{2.5in}
\includegraphics[width=0.75\textwidth]{images/result.png}
\end{columns}
Let's define weight function: $w(x,y)=\Theta(p(x)-y)$, \\
where $\Theta(x)$ is the step function.\\
$\rightarrowtail$ \pdf : $\varrho(x,y)=\rho(x)g(y)=\frac{2}{\pi} \cdot 1$\\
$\Rightarrow$ Integrated probability:
\begin{align*}
P=E(w)=\int w(x,y) \varrho(x,y)dx dy = \dfrac{2l}{\pi L} \xleftarrow{N\to \infty}\hat{P}=\frac{1}{N} \sum_{i=1}^N w(x_i,y_i)= \dfrac{n}{N}
\end{align*}
Standard deviation for $\hat{P}$: $\hat{\sigma}=\dfrac{1}{\sqrt{N-1}}\sqrt{\dfrac{n}{N}\Big(1-\dfrac{n}{N}\Big)} $
\end{footnotesize}
\end{frame}
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mchrzasz