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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, \\ 25 November, 2015} + \textcolor{normal text.fg!50!Comment}{Monte Carlo methods, \\ 10 March, 2016} + \end{center} \end{frame} } +\begin{frame}\frametitle{Classical methods of variance reduction} +\begin{footnotesize} + +\ARROW In Monte Carlo methods the statistical uncertainty is defined as: +\begin{align*} +\sigma = \dfrac{1}{\sqrt{N}}\sqrt{V(f)} +\end{align*} +\ARROW Obvious conclusion: +\begin{itemize} +\item To reduce the uncertainty one needs to increase $N$.\\ +$\rightrightarrows$ Slow convergence. In order to reduce the error by factor of 10 one needs to simulate factor of 100 more points! +\end{itemize} +\ARROW How ever the other handle ($V(f)$) can be changed! $\longrightarrow$ Lot's of theoretical effort goes\ + into reducing this factor.\\ +\ARROW We will discuss {\color{Mahogany}{four}} classical methods of variance reduction: +\begin{enumerate} +\item Stratified sampling. +\item Importance sampling. +\item Control variates. +\item Antithetic variates. +\end{enumerate} + + +\end{footnotesize} +\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 ā€” 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{frame}\frametitle{Disadvantages of classical variance reduction methods} +\begin{footnotesize} + + \ARROW All aforementioned methods(beside the Stratified sampling) require knowledge of the integration function!\\ + \ARROW If you use the method in the incorrect way, you can easily get the opposite effect than intendant. \\ + + \ARROW Successful application of then require non negligible effort before running the program.\\ + \ARROW A natural solution would be that our program is ''smart'' enough that on his won he will learn something about our function while he is trying to calculate the integral.\\ + \ARROW Similar techniques already were created for numerical integration!\\ +\ARROW Truly adaptive methods are nontrivial to code but are widely available in external packages as we will learn.\\ +\ARROW Naming conventions: \begin{itemize} -\item Mechanical -\item Physical +\item Integration \mc - software that is able to compute JUST! integrals. +\item Generator \mc - software that BESIDES beeing able to perform the integration is also capable of performing a generation of points accordingly to the integration function. \end{itemize} -$\color{PineGreen}\Rrightarrow$ Disadvantages of physical generators: + + + +\end{footnotesize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Schematic of running this kind of methods} +\begin{footnotesize} + \begin{enumerate} + \item Function probing (exploration): + \begin{itemize} +\item Recursive algorithm that searches for hipper-surfaces in which the function is approximately close. For evaluation of an integral in a given hipper-surface normally one uses numerical or \mc crude methods. In general it is not a easy task! +\item Often the function is approximated by a given set of elementary functions. + \end{itemize} +\item Calculation phase \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! +\item The integral is calculated using mostly using Stratified Sampling and Importance Sampling, depending on exploration phase. +\item If a \mc program has capability to generated distributions accordingly to the function of which we want to calculate the integral, it's in this place where it happens. +\end{itemize} + \end{enumerate} +\ARROW There are algorithms where the exploration phase is linked with calculation phase. For each of the optimisation phase the integral is calculated as well. The result will be weighted average of those integrals! +\begin{alertblock}{~} +This method might be bias! if in the extrapolation phase the function picks up a function peaks to late the whole method will lead to systematically bias results. +\end{alertblock} + + + +\end{footnotesize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{\texttt{RIWIAD} algorithm} +\begin{footnotesize} + \ARROW The first algorithm of this kind \texttt{RIWIAD} was proposed by Sheppeya \& Lautrupa in $1970$s. It was used to calculate integrals in cube $(0,1)^n$. \\ + \ARROW It worked as follows: + \begin{itemize} + \item At the begging the hipper-cube is divided in equal size sub cubes. In each of them the integral is calculated. + \item Based on the calculated integrals programs moves the boundaries to make the hipper-cubes smaller in the places where the function is greater and smaller where the function is smaller. + + \item The process starts over and continues over and over again. At each step the integral estimator and it's standard deviation is calculated. Form those a weighted average is constructed and it's standard deviation is constructed and its standard deviation. + + \item The process stops when the standard deviation reaches our desired sensitivity. + + + + \end{itemize} + +\ARROW Disadvantages: +\begin{itemize} +\item Hipper-cubes are always parallel to the coordinate axis. +\item Some are are divided even thought they didn't have to. +\item The weighted average might be a bias estimator. \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{footnotesize} +\end{frame} -\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{Friedmanns algorithm} +\begin{small} + \ARROW In the $1970$s J.Friedmann has also developed an adaptive \mc integration algorithm.\\ + \ARROW The algorithm was as follows: + \begin{itemize} \item A probe function is constructed using a combination of Cauchy functions (Briet-Wigner), in which the peaks correspond to the local maxima of the integration function. In order to do so one needs to study the eigen functions in the neighbourhood of each peak (nasty thing...). + + \item The Briet-Wigner is it falls down to $0$ slower then a Gauss distribution. + + \item The integral and the standard deviation is calculated based on the weighted averaged based on the probe function. + + \end{itemize} + \begin{alertblock}{Disadvantage:} +Cannot be applied to functions that cannot be approximated with small number of Briet-Wigner functions. +\end{alertblock} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\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: +\end{small} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{\texttt{DIVIONNE2} algorithm} +\begin{footnotesize} +\ARROW J.Friedmann (1977): adaptive algorithm for \mc~integration based on recursive division of the integration area(available in the CERBLIB package).\\ +\ARROW The algorithm: \begin{itemize} -\item Good statistical properties of generated numbers. -\item Easy to use and fast! -\item Reproducible! +\item Multidimensional division of the hipper-cube. We divide each of the initial sub cubes to minimalise the spread of the function. +\item After this the integral is calculated using Stratified Sampling. +\item We can generate a events accordingly to this function with this method. \end{itemize} -$\color{PineGreen}\Rrightarrow$ Since mathematical pseudorandom genrators are dominantly: pseudorandom $\rightarrowtail$ random. + +\ARROW Disadvantages: +\begin{itemize} +\item Hipper-cubes are always parallel to the coordinate axis. +\end{itemize} +\ARROW Advantages: +\begin{itemize} +\item Because we divide only one hipper-cube at the time, the procedure doesn't get bias as easily the \text{RIWID} does. +\end{itemize} -\end{frame} +\end{footnotesize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}\frametitle{\texttt{VEGAS} algorithm} +\begin{footnotesize} +\ARROW J. G. P. Lepage (1978): adaptive algorithm for \mc~integration based on iterative division of the integration area(similar to \texttt{RIWID}).\\ +\ARROW Let's calculate: $\int_0^1 f(x)dx$. +\begin{itemize} +\item We generate M random points from $\mathcal{U}(0,1)$. We calculate from them the integral and standard deviation. +\item Now we divide the integration region in N equal subdivisions: +\begin{align*} +0=x_0q$. Then: $b_n=b_{n-p}~{\rm xor}~b_{n-q}$ - -$\color{PineGreen}\Rrightarrow$ How to get numbers from bits (for example):\\ -$U_i = \sum_{j=1}^L 2^{-j} b_{is+j},~s{ -\includegraphics[width=0.95\textwidth]{images/gen1.png} -} -\only<2>{ -\includegraphics[width=0.95\textwidth]{images/gen2.png} -} -\only<3>{ -\includegraphics[width=0.95\textwidth]{images/gen3.png} -} -\only<4>{ -\includegraphics[width=0.95\textwidth]{images/gen4.png} -} -\only<5>{ -\includegraphics[width=0.95\textwidth]{images/gen5.png} -} -\only<6>{ -\includegraphics[width=0.95\textwidth]{images/gen6.png} -} -\end{small} -\end{frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Detector simulation} - -\begin{small} -$\color{PineGreen}\Rrightarrow$ Things do not get simpler on the detector side simulation.\\ -$\color{PineGreen}\Rrightarrow$ Lots of effects need to be taken into account: -\begin{columns} -\column{0.2in} -{~} -\column{2in} -$\color{PineGreen}\rightarrowtail$ Bremsstrahlung\\ -$\color{PineGreen}\rightarrowtail$ Interactions with different detector materials\\ -$\color{PineGreen}\rightarrowtail$ Particle identification\\ -$\color{PineGreen}\rightarrowtail$ Showers\\ -\column{3in} -\includegraphics[width=0.95\textwidth]{{images/lhcb2_h-640x408}.jpg} -\end{columns} -$\color{PineGreen}\Rrightarrow$ Example of generators:\\ -$\color{PineGreen}\rightarrowtail$ FLUKA\\ -$\color{PineGreen}\rightarrowtail$ Geant - -\end{small} -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Method of Moments} - -\begin{small} -$\color{PineGreen}\Rrightarrow$ Now real cool things!\\ -$\color{PineGreen}\Rrightarrow$ Let's consider we want to study a rare decay: $\PB^{\pm} \to \PK^{\pm} \Pmu \Pmu$. The decay is described by the following PDF: -\begin{equation} -\dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =\dfrac{3}{4}(1-F_H)(1-\cos^2 \theta_l)+F_H/2 + A_{FB}\cos \theta_l \nonumber -\end{equation} -$\color{PineGreen}\Rrightarrow$ PDF by construction is normalized: $\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} =1$ -\begin{columns} -\column{0.1in} -{~} -\column{2.2in} -\begin{itemize} -\item Normally we do a likelihood fit and we are done. -\item There is a second way! -\end{itemize} -\column{2.8in} -\includegraphics[width=0.95\textwidth]{images/Kmumu_LL.png} -\end{columns} - -\end{small} -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Method of Moments} -\begin{small} -$\color{PineGreen}\Rrightarrow$ Let's calculate the integrals: -\begin{equation} -\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} \nonumber -\end{equation} -\begin{equation} -\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15} \nonumber -\end{equation} -$\color{PineGreen}\Rrightarrow$ So we can get our parameters that we searched for by doing a integration. So now what?\pause \\ -$\color{PineGreen}\Rrightarrow$ Well nature is the best random number generator so let's take the data and treat and calculate the integral estimates: -\begin{equation} -\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos \theta_l = \dfrac{2}{3}A_{FB} = \dfrac{1}{N} \sum_{i=1}^N \cos \theta_{l,i} \nonumber -\end{equation} -\begin{equation} -\int_{-1}^{1} \dfrac{1}{\Gamma}\dfrac{d^2\Gamma}{dq^2 d\cos \theta_l} \cdot \cos^2 \theta_l = \dfrac{1}{5} + \dfrac{2 F_H}{15}= \dfrac{1}{N} \sum_{i=1}^N \cos^2 \theta_{l,i} \nonumber -\end{equation} - -\end{small} -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Method of Moments} -\begin{small} -$\color{PineGreen}\Rrightarrow$ So what did we do? -\begin{itemize} -\item We have just estimated a parameters of interests without using any fit!! -\end{itemize} -$\color{PineGreen}\Rrightarrow$ Pros and cones of method of moments: -\begin{itemize} -\item {\color{PineGreen}{Are very immune to bias.}} -\item {\color{PineGreen}{Do not suffer from boundary problems.}} -\item {\color{PineGreen}{Require less statistic to work then likelihood fit.}} -\item {\color{PineGreen}{They always have a Gaussian error}}. -\item {\color{Red}{Estimator has a larger uncertainty.}} - - -\end{itemize} - -\end{small} -\end{frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Method of Moments, uncertainty estimator} -\begin{small} -\begin{columns} -\column{0.1in} -{~} -\column{2.5in} -$\color{PineGreen}\Rrightarrow$ It can be proven that Method of Moments estimator converges slower then the maximum likelihood fit. -\column{2.5in} -\includegraphics[width=0.95\textwidth]{images/S7.png} -\end{columns} - -\begin{columns} -\column{0.1in} -{~} -\column{2.5in} -\includegraphics[width=0.95\textwidth]{images/S8F_950.png} -\column{2.5in} -\includegraphics[width=0.95\textwidth]{images/S8.png} -\end{columns} - -\end{small} -\end{frame} - - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}\frametitle{Other application of MC - testing your analysis} -\begin{small} -$\color{PineGreen}\Rrightarrow$ Probably the biggest application of MC methods in HEP are validations of your experimental methodology. The procedure is as follows: -\begin{itemize} -\item Define your analysis methodology: selection, efficiency corrections, parameters you want to measure. -\item Simulate an assembly of simulation events for different values of parameters you want to measure. -\item Do the analysis on this pseudo data. -\item See if you are getting back what you have simulated. -\end{itemize} - -\end{small} -\end{frame} - -\begin{frame}\frametitle{Testing your analysis, $\rm \color{RubineRed}{Lecture2/Test\_met}$} -\begin{small} -$\color{PineGreen}\Rrightarrow$ Probably the biggest application of MC methods in HEP are validations of your experimental methodology. The procedure is as follows: -\begin{itemize} -\item Define your analysis methodology: selection, efficiency corrections, parameters you want to measure. -\item Simulate an assembly of simulation events for different values of parameters you want to measure. -\item Do the analysis on this pseudo data. -\item See if you are getting back what you have simulated. -\end{itemize} -\includegraphics[width=0.4\textwidth]{{images/mean_estimator}.png} -\end{small} -\end{frame} - - -\begin{frame}\frametitle{Wrap up} -\begin{small} -$\color{PineGreen}\Rrightarrow$ Things to remember: -\begin{itemize} -\item Computer cannot produce random numbers, only pseudorandom numbers. -\item We use pseudorandon numbers as random numbers if they are statistically acting the same as random numbers. -\item Linear generators are not commonly used nowadays. -\item State of the art generators are the ones based on Kolomogorows theorem. -\item MC methods used to simulate physics process, detector response and validating the estimators. -\end{itemize} -\end{small} -\end{frame} - - - \backupbegin \begin{frame}\frametitle{Backup}