root/applications/dual/SIDP/bakalarka/SIDP/text/prezentace/Prezentace.tex @ 1351

\documentclass{beamer}
\usepackage[czech]{babel}      % �ky psan�r�
\usepackage[T1]{fontenc}
\usepackage[cp1250]{inputenc}  % vstupn�nakov�ada: Windows 1250

\usepackage{amsmath} % bal�k pro pokro�ou matem. sazbu
\usepackage{algorithm}
\usepackage{algorithmic}
\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\cov}{Cov}
\DeclareMathOperator*{\tr}{tr}

\usetheme{Berlin}
\title{Stochastick�terativn�proximace dynamick� programov� (SIDP)}
\author{Miroslav Zima}
\institute{FJFI �UT}
\date{1. z� 2010}

\begin{document}

\begin{frame}
\titlepage
\end{frame}



\begin{frame}
\frametitle{Obsah}
\tableofcontents
\end{frame}


\section{�oha ��a neur�osti}
\begin{frame}{Syst�
\begin{itemize}[<+->]

\item Syst� jeho odezva na vstup $u_t$ p�alizaci �umu $v_{t+1}$
\begin{equation}
y_{t+1}=h_t(I_t,u_t,v_{t+1},\theta), \qquad t=0,\ldots,N-1,
\end{equation}
kde $I_t=(y_{t:0},u_{t-1:0})$ a $\theta$ je nezn� parametr. Zn� $I_0$, rozd�n�umu $v_t$ a apriorn�nformaci $f(\theta|T_0)$.

\item Bayesovsk��� z�� aposteriorn�ustoty $f(\theta|T_{t+1})$
\begin{equation}
f(\theta|T_{t+1}) = \frac{f(y_{t+1} | \theta, I_t,u_t) f(\theta| T_t)}
{\int f(y_{t+1} | \theta, I_t,u_t) f(\theta| T_t)\mathrm{d}\theta}.
\end{equation}

\item Hyperstav $H_t=(I_t,T_t)$ plat�\begin{equation}
H_{t+1}=f_t(H_t,u_t,y_{t+1}), \qquad  t=0,\ldots,N-1.
\end{equation} 

\end{itemize}
\end{frame}

\begin{frame}{Formulace �}
\begin{itemize}[<+->]


\item Ztr�v�unkce (ur�e kvalitu ��
\begin{equation}
g(y_{1:N},u_{0:N-1})=\sum_{t=0}^{N-1}g_t(y_{t+1},u_t).
\end{equation}

\item �d� strategie
\begin{equation}
\mu_t(H_t)=u_t, \qquad t=0,1,\ldots,N-1.
\end{equation}

\item Hled� $\pi=\mu_{0:N-1}$ minimalizuj� o��nou ztr�
\begin{equation}
J_\pi=\E_{\theta_0,v_{0:N-1}}\left\{\sum_{t=0}^{N-1}g_t(y_{t+1},\mu_t(H_t))\right\},
\end{equation}
\end{itemize}
\end{frame}

\begin{frame}{Pou�it�ynamick� programov� p��en�lohy �� aditivn�tr�u}

\begin{itemize}[<+->]

\item Princip optimality $\implies$ postupn�d konce �� horizontu minimalizujeme d����n�tr� 
\begin{gather}
J_N(H_N)=0,\\
J_t(H_t)=\min_{u_t \in U_t}\E_{y_{t+1}}\left\{g_t(y_{t+1},u_t)+J_{t+1}(H_{t+1})|H_t,u_t\right\},
\end{gather}
podm�y: $y_{t+1}=h_t(I_t,u_t,v_{t+1},\theta)$ a $H_{t+1}=f_t(H_t,u_t,y_{t+1})$  

\item Probl� v ��\begin{itemize}
\item v� st� hodnoty
\item minimalizace vzhledem k $u_t$
\item Analytick�e�en�bvykle nejde nal�  $\implies$ aproxima� metody.
\end{itemize}
\end{itemize}
\end{frame}

\section{Suboptim���p k ��omoc�IDP}
\begin{frame}{SIDP}

\begin{itemize}[<+->]
\item Iterativn�ynamick�rogramov�
\begin{itemize}
\item {Nam�o p�o nalezen�\pi$ konstruujeme $\pi_n \to \pi$}
\end{itemize}

\item Stochastick�proximace $=$ Metoda Monte Carlo
\begin{itemize}
\item O��n�tr� se aproximuje pr�m z $n$ realizac�tr�
\item Generov� realizace ztr� pro konkr��od $H_t$ 
\begin{enumerate}
\item \label{po} generujeme $v_t$ a $\theta$ (pou�ijeme $f(\theta,T_t)$)
\item aplikac��c� z�hu z�� $y_{t+1}=h_t(I_t,u_t,v_{t+1},\theta)$ 
\item k dosavadn�tr� p�me $g_t(y_{t+1},u_t)$
\item dopo�� $H_{t+1}=f_t(H_t,u_t,y_{t+1})$
\item interpolac�xtrapolac�\pi_n$ ur�e optim���h pro $H_{t+1}$
\item pokud $t<N-1 \implies$ \ref{po}.
\end{enumerate}
\end{itemize}
\end{itemize}
\end{frame}

\section{Srovn� SIDP s jin�todami} 
\begin{frame}{Integr�r s nezn�m ziskem}
\begin{itemize}[<+->]

\item V�syst�
\begin{equation}
y_{t+1}=y_t+\theta u_t+v_{t+1}, \qquad t=0,\ldots,N-1,
\end{equation}
kde $\theta\neq0$ nezn�,  �um $v_{t+1}\sim N(0,0.1)$ a $y_0=1$.

\item Ztr�v�unkce je kvadratick�\begin{equation}
g(y_{1:N},u_{0:N-1})=\sum_{t=0}^{N-1}y_{t+1}^2.
\end{equation}

\item P�klad $\cov(v_{t+1},\theta_t)=0$ a $T_0=(\hat{\theta}_0,P_0)$ $\implies$ lze z�at konkr��var rce $H_{t+1}=f_t(H_t,u_t,y_{t+1})$. 
\item Hyperstav syst� $H_t$ tvo�ktor $(y_t,\hat{\theta}_t,P_t)$ (lze ho vhodnou transformaci redukovat na dim=2).
\end{itemize}
\end{frame}

\begin{frame}{Konkuren� metody}
\begin{itemize}[<+->]

\item Certainty equivalence (CEC)
\begin{itemize}

\item pro n�h �� z�hu je br�bodov�d $\hat{\theta}$
\item je mo�n��at analytick�yj�en�ro optim���c��h
\end{itemize}

\item Cautious control (CC)
\begin{itemize}

\item hled�e optim���n�a horizontu d�y $N=1$
\item je mo�n��at analytick�yj�en�ro optim���c��h
\end{itemize}

\item Klasick�rick�tup k  dynamick� programov� (DP)
\begin{itemize}
\item Pro v� st� hodnoty je pou�ita Simpsonova metoda
\item K minimalizaci ztr� slou��ednoduch�nterpola� metoda.
\end{itemize}

\end{itemize}
\end{frame}

\begin{frame}{Kvantitativn�rovn�}
\begin{itemize}

\item Dosa�en�tr� - pr�p�000 simulac�\item $\theta$ = prvn�enulov�ealizace veli�y s rozd�n�$N(\hat{\theta}_0,P_0)$
\end{itemize}

\begin{figure}
\centering
\begin{minipage}[c]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{N=10,P=10}
\end{minipage}
\begin{minipage}[c]{0.49\textwidth}
\begin{itemize}
\item Metoda CEC neposkytuje pou�iteln��n�
\item Metoda CC funguje pouze p�hat{\theta}_0 \neq 0$. Tehdy dosahuje nejni���tr�.
\item SIDP a DP dob�d��dy. P�p��o�e� identifikaci m�IDP hor���y (diskretizace).
\end{itemize}
\end{minipage}
\end{figure}
\end{frame}


\begin{frame}{Kvalitativn�rovn� SIDP a DP}
\begin{itemize}
\item �tnost konkr�� realizac�tr� - 1000 simulac�
\end{itemize}

\begin{figure}
\centering
\begin{minipage}[c]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{SIDP4}
\end{minipage}
\begin{minipage}[c]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{DP4}
\end{minipage}
\end{figure}

\begin{itemize}

\item SIDP se vyh�razn�patn� ��$>10$)
\item DP �t� nab����elkov�tr� ($<1$)
\item SIDP navrhuje opatrn���hy $\implies$ pomalej��dentifikace $\theta$
\end{itemize}
\end{frame}

\begin{frame}{Test robustnosti}
\begin{itemize}

\item Dosa�en�tr� vhledem k apriorn�nformaci - 1000 simulac�
\item $\theta$ = prvn�enulov�ealizace n�dn�eli�y s rozd�n�$N(1,10)$, r�apriorn�nformace na $\hat{\theta}_0$, rozptyl $P_0=10$.
\end{itemize}

\begin{figure}
\centering
\begin{minipage}[c]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{rob}
\end{minipage}
\begin{minipage}[c]{0.49\textwidth}
\begin{itemize}
\item Vy���dolnost metody SIDP v�patn�priorn�nformaci
\item SIDP poskytuje dobr��n� pro $\hat{\theta}_0=-10$, p� skute� $\theta=1$
\end{itemize}
\end{minipage}
\end{figure}
\end{frame}


\begin{frame}{Z�r}
\begin{itemize}[<+->]

\item P�ty BP
\begin{itemize}
\item �oha ��a neur�osti - teoretick�e�en�omoc�ynamick� programov�
\item Algoritmus SIDP jako mo�n�proxima� metoda
\item Implementace SIDP pro ��ednoduch� syst�, porovn� s dal�� metodami
\end{itemize}
\item Mo�n�al��r�
\begin{itemize}
\item Navrhnout algoritmus, kter�yl m� v�n��� a st� poskytoval dobr��n�\item nap�_t=u^{(1)}_t+u^{(2)}_t$, kde $u^{(1)}_t$ minimalizuje ztr� (nap�d CC) a $u^{(2)}_t$ bud�yst�za �m jeho lep��dentifikace (pou�it�IDP).
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\huge{D�ji za pozornost}
\end{frame}

\end{document}