From 6348b521028b1aba042dd53fbaac805fb971b096 Mon Sep 17 00:00:00 2001 From: Christian Forssen Date: Tue, 15 May 2018 08:16:51 +0200 Subject: [PATCH] Spectral lines problem added to Bayesian chapter --- doc/src/Bayesian/src/data_spectral_lines.txt | 100 ++++ doc/src/Bayesian/src/spectral_lines.tex | 497 ++++++++++++++++++ .../spectral_lines_10_0.png | Bin 0 -> 10600 bytes .../spectral_lines_22_0.png | Bin 0 -> 10728 bytes .../spectral_lines_37_0.png | Bin 0 -> 196160 bytes 5 files changed, 597 insertions(+) create mode 100644 doc/src/Bayesian/src/data_spectral_lines.txt create mode 100644 doc/src/Bayesian/src/spectral_lines.tex create mode 100644 doc/src/Bayesian/src/spectral_lines_files/spectral_lines_10_0.png create mode 100644 doc/src/Bayesian/src/spectral_lines_files/spectral_lines_22_0.png create mode 100644 doc/src/Bayesian/src/spectral_lines_files/spectral_lines_37_0.png diff --git a/doc/src/Bayesian/src/data_spectral_lines.txt b/doc/src/Bayesian/src/data_spectral_lines.txt new 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b/doc/src/Bayesian/src/spectral_lines.tex @@ -0,0 +1,497 @@ + +% Default to the notebook output style + + + + +% Inherit from the specified cell style. + + + + + +\documentclass{article} + + + + \usepackage{graphicx} % Used to insert images + \usepackage{adjustbox} % Used to constrain images to a maximum size + \usepackage{color} % Allow colors to be defined + \usepackage{enumerate} % Needed for markdown enumerations to work + \usepackage{geometry} % Used to adjust the document margins + \usepackage{amsmath} % Equations + \usepackage{amssymb} % Equations + \usepackage{eurosym} % defines \euro + \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support + \usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document + \usepackage{fancyvrb} % verbatim replacement that allows latex + \usepackage{grffile} % extends the file name processing of package graphics + % to support a larger range + % The hyperref package gives us a pdf with properly built + % internal navigation ('pdf bookmarks' for the table of contents, + % internal cross-reference links, web links for URLs, etc.) + \usepackage{hyperref} + \usepackage{longtable} % longtable support required by pandoc >1.10 + \usepackage{booktabs} % table support for pandoc > 1.12.2 + \usepackage{ulem} % ulem is needed to support strikethroughs (\sout) + + + + + \definecolor{orange}{cmyk}{0,0.4,0.8,0.2} + \definecolor{darkorange}{rgb}{.71,0.21,0.01} + \definecolor{darkgreen}{rgb}{.12,.54,.11} + \definecolor{myteal}{rgb}{.26, .44, .56} + \definecolor{gray}{gray}{0.45} + \definecolor{lightgray}{gray}{.95} + \definecolor{mediumgray}{gray}{.8} + \definecolor{inputbackground}{rgb}{.95, .95, .85} + \definecolor{outputbackground}{rgb}{.95, .95, .95} + \definecolor{traceback}{rgb}{1, .95, .95} + % ansi colors + \definecolor{red}{rgb}{.6,0,0} + \definecolor{green}{rgb}{0,.65,0} + \definecolor{brown}{rgb}{0.6,0.6,0} + \definecolor{blue}{rgb}{0,.145,.698} + \definecolor{purple}{rgb}{.698,.145,.698} + \definecolor{cyan}{rgb}{0,.698,.698} + \definecolor{lightgray}{gray}{0.5} + + % bright ansi colors + \definecolor{darkgray}{gray}{0.25} + \definecolor{lightred}{rgb}{1.0,0.39,0.28} + \definecolor{lightgreen}{rgb}{0.48,0.99,0.0} + \definecolor{lightblue}{rgb}{0.53,0.81,0.92} + \definecolor{lightpurple}{rgb}{0.87,0.63,0.87} + \definecolor{lightcyan}{rgb}{0.5,1.0,0.83} + + % commands and environments needed by pandoc snippets + % extracted from the output of `pandoc -s` + \providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} + \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}} + % Add ',fontsize=\small' for more characters per line + \newenvironment{Shaded}{}{} + \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}} + \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}} + \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}} + \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}} + \newcommand{\RegionMarkerTok}[1]{{#1}} + \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\NormalTok}[1]{{#1}} + + % Additional commands for more recent versions of Pandoc + \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}} + \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}} + 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exist. + \def\br{\hspace*{\fill} \\* } + % Math Jax compatability definitions + \def\gt{>} + \def\lt{<} + % Document parameters + \title{A spectral line problem} + + \author{Christian Forss\'en, Department of Physics, Chalmers} + + + % Pygments definitions + +\makeatletter +\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax% + \let\PY@ul=\relax \let\PY@tc=\relax% + \let\PY@bc=\relax \let\PY@ff=\relax} +\def\PY@tok#1{\csname PY@tok@#1\endcsname} +\def\PY@toks#1+{\ifx\relax#1\empty\else% + \PY@tok{#1}\expandafter\PY@toks\fi} +\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{% + \PY@it{\PY@bf{\PY@ff{#1}}}}}}} +\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}} + +\expandafter\def\csname PY@tok@s\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@kd\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} 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PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@kn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}} +\expandafter\def\csname PY@tok@nl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.63,0.00}{##1}}} +\expandafter\def\csname PY@tok@nb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@nn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@gi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.63,0.00}{##1}}} +\expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}} +\expandafter\def\csname PY@tok@err\endcsname{\def\PY@bc##1{\setlength{\fboxsep}{0pt}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}} +\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit} + +\def\PYZbs{\char`\\} +\def\PYZus{\char`\_} +\def\PYZob{\char`\{} +\def\PYZcb{\char`\}} +\def\PYZca{\char`\^} +\def\PYZam{\char`\&} +\def\PYZlt{\char`\<} +\def\PYZgt{\char`\>} +\def\PYZsh{\char`\#} +\def\PYZpc{\char`\%} +\def\PYZdl{\char`\$} +\def\PYZhy{\char`\-} +\def\PYZsq{\char`\'} +\def\PYZdq{\char`\"} +\def\PYZti{\char`\~} +% for compatibility with earlier versions +\def\PYZat{@} +\def\PYZlb{[} +\def\PYZrb{]} +\makeatother + + + % Exact colors from NB + \definecolor{incolor}{rgb}{0.0, 0.0, 0.5} + \definecolor{outcolor}{rgb}{0.545, 0.0, 0.0} + + + + + % Prevent overflowing lines due to hard-to-break entities + \sloppy + % Setup hyperref package + \hypersetup{ + breaklinks=true, % so long urls are correctly broken across lines + colorlinks=true, + urlcolor=blue, + linkcolor=darkorange, + citecolor=darkgreen, + } + % Slightly bigger margins than the latex defaults + + \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} + + + + \begin{document} + + + \maketitle + + + + + See e.g.~Section 4.2 in Sivia for a similar problem formulation. In +short, we have data from a spectroscopy experiment that supposedly shows +a number of spectral lines. The ideal spectrum ca be expressed as + +\[ G(x) = \sum_{j=1}^M A_j f(x,x_j),\] + +where \(A_j\) is the amplitude of the \(j\)th line, and \(x_j\) +represents its position. If all the spectral lines were Gaussians of +width \(W\), for example, then + +\[ f(x,x_j) = \exp \left[ - \frac{(x-x_j)^2}{2 W^2} \right]\] + + includes a background signal \(\{ B_k\}\). We use the label `\(k\)' to +enumerate the positions \(\{x_k\}\). + +The ideal spectrum according to our model is therefore + +\[ F_k \equiv F(x_k) = G(x_k) + B(x_k).\] + + The experimental data is denoted \(\{ D_k\}\). This data also includes +measurement errors \(\{ \varepsilon_k\}\) that are assumed to be +independent and identically distributed (IID) normal with some variance +\(\sigma_k\). The measured data is then related to the ideal spectrum by + +\[ D_k \equiv D(x_k) = G(x_k) + B(x_k) + \varepsilon(x_k).\] + +A simulated data set is shown in the figure. + \begin{center} + \adjustimage{max size={0.8\linewidth}{0.8\paperheight}}{spectral_lines_files/spectral_lines_22_0.png} + \end{center} + { \hspace*{\fill} \\} + + +\paragraph{Problem task:} +The task is to infer the positions (\(x_j\)) and amplitudes (\(A_j\)) of +the spectral lines from the experimental data. + +\subsection*{Known information} +% +Let us use a model that assumes two spectral lines. In addition we +assert the following known information: +% +\begin{enumerate} +\item a known, constant background ($B$). +\item a known, natural width +(W) of the spectral lines (the same for both). +\item a known variance ($\sigma_k$) for the IID +normal experimental errors. +\item a known and relevant interval $[x_\mathrm{min}, +x_\mathrm{max}]$ in position space. +\end{enumerate} +% +The following numerical data is used in this example: + + \begin{Verbatim}[commandchars=\\\{\}] +Constant background: B\_k = B = 0.2 +Natural width of spectral lines: W = 0.1 +Variance for IID normal exp errors: s = 0.05 +Relevant range in position space: [xmin, xmax] = [0.0, 2.0] + \end{Verbatim} + + +\subsection*{Suggested solution strategy---known number of spectral lines}\label{solution-strategy} + +Assuming that our model \(M\) describes a spectrum with two spectral +lines, it will have five model parameters. We denote them by the vector +\(\vec{\alpha}\). These are the amplitudes and positions of the two +lines (the width is assumed to be known), and the +constant background. We order them as follows: + +\[ \vec{\alpha} = (A_0, x_0, A_1, x_1, B).\] + +The background strength ($B$) is a \emph{nuisance parameter} in the sense that +we're not really interested in its value, we just need to marginalize +over it. + + We start with wirting down Bayes' theorem + +\[ p(\alpha | \{D_k\}, I) = \frac{p(\{D_k\} | \vec{\alpha}, I) p(\vec{\alpha}|I)}{p(\{D_k\} | I)},\] + +where the prior information includes expectations of the number of peaks +(two in this case, their natural width, the experimental errors, etc). +In the following, we will ignore the denominator (the \emph{evidence} or +\emph{marginal likelihood}) as it just constitutes a normalization +factor to the posterior pdf (the left hand side), which is the quantity +that we are interested in. + + It is your task, however, to make reasonable assumptions for the prior +(\emph{hint:} uniform ones are easy to work with). We can assume that +the available information implies that we expect the amplitude to be a +positive quantity, and that it is expeted to be of order 1 (i.e.~not +10). The peak positions can be assumed to be in the {[}xmin, xmax{]} +range of the data, and the background signal is supposedly at least an +order of magnitude smaller than the peak amplitudes. + + Concerning the \emph{likelihood} (the first factor in the nominator), +our information on the IID errors of the experiment leads to the +least-squares likelihood (see Sivia, Sec. 3.5): + +\[ p(\{D_k\} | \vec{\alpha}, I) \propto \exp(-\chi^2/2),\] + +where the chi-squared function is defined by the sum of squared +\emph{residuals} + +\[ \chi^2 = \sum_{k=1}^N \left( \frac{F_k - D_k}{\sigma_k} \right)^2.\] + +Note that the normalization factor of the likelihood (involving the +product of terms \(\sqrt{2\pi}\sigma_k\)) has been omitted as it does +not depend on the parameters of our model. + +\subsection*{Simulated data} + +It can be +loaded from the file \texttt{data\_spectral\_lines.txt}. The columns +of this file correspond to the data triples: $(x_k, D_k, \sigma_k)$, as +also listed below. + + \begin{Verbatim}[commandchars=\\\{\}] +\# x\_k D\_k s\_k +\# ---- ---- --- + 0.0000 0.2046 0.1 + 0.0202 0.2546 0.1 + 0.0404 0.1027 0.1 + 0.0606 0.1307 0.1 + 0.0808 0.0852 0.1 + 0.1010 0.3205 0.1 + 0.1212 0.2864 0.1 + 0.1414 0.3102 0.1 + 0.1616 0.2397 0.1 + 0.1818 0.2488 0.1 + 0.2020 0.1408 0.1 + 0.2222 0.2958 0.1 + 0.2424 0.1438 0.1 + 0.2626 0.1668 0.1 + 0.2828 0.1811 0.1 + 0.3030 0.1604 0.1 + 0.3232 0.2430 0.1 + 0.3434 0.1885 0.1 + 0.3636 0.1967 0.1 + 0.3838 0.1896 0.1 + 0.4040 0.2673 0.1 + 0.4242 0.1697 0.1 + 0.4444 0.1913 0.1 + 0.4646 0.2212 0.1 + 0.4848 0.1178 0.1 + 0.5051 0.1760 0.1 + 0.5253 0.2272 0.1 + 0.5455 0.2592 0.1 + 0.5657 0.2100 0.1 + 0.5859 0.1700 0.1 + 0.6061 0.2788 0.1 + 0.6263 0.2007 0.1 + 0.6465 0.2217 0.1 + 0.6667 0.2420 0.1 + 0.6869 0.2485 0.1 + 0.7071 0.2507 0.1 + 0.7273 0.3364 0.1 + 0.7475 0.2984 0.1 + 0.7677 0.3841 0.1 + 0.7879 0.4480 0.1 + 0.8081 0.4845 0.1 + 0.8283 0.5976 0.1 + 0.8485 0.6578 0.1 + 0.8687 0.7028 0.1 + 0.8889 0.7288 0.1 + 0.9091 0.7914 0.1 + 0.9293 0.8692 0.1 + 0.9495 0.8534 0.1 + 0.9697 0.6716 0.1 + 0.9899 0.9239 0.1 + 1.0101 0.8303 0.1 + 1.0303 0.9112 0.1 + 1.0505 1.0990 0.1 + 1.0707 1.1252 0.1 + 1.0909 1.3663 0.1 + 1.1111 1.4013 0.1 + 1.1313 1.3731 0.1 + 1.1515 1.3388 0.1 + 1.1717 1.4010 0.1 + 1.1919 1.3674 0.1 + 1.2121 1.1981 0.1 + 1.2323 1.1509 0.1 + 1.2525 0.9372 0.1 + 1.2727 0.8055 0.1 + 1.2929 0.6170 0.1 + 1.3131 0.5362 0.1 + 1.3333 0.4122 0.1 + 1.3535 0.2570 0.1 + 1.3737 0.4218 0.1 + 1.3939 0.2539 0.1 + 1.4141 0.2424 0.1 + 1.4343 0.2876 0.1 + 1.4545 0.2171 0.1 + 1.4747 0.2169 0.1 + 1.4949 0.2246 0.1 + 1.5152 0.1440 0.1 + 1.5354 0.2770 0.1 + 1.5556 0.2183 0.1 + 1.5758 0.1696 0.1 + 1.5960 0.2466 0.1 + 1.6162 0.2178 0.1 + 1.6364 0.1524 0.1 + 1.6566 0.2076 0.1 + 1.6768 0.2000 0.1 + 1.6970 0.2168 0.1 + 1.7172 0.2187 0.1 + 1.7374 0.2357 0.1 + 1.7576 0.1400 0.1 + 1.7778 0.2001 0.1 + 1.7980 0.1539 0.1 + 1.8182 0.2248 0.1 + 1.8384 0.1334 0.1 + 1.8586 0.0905 0.1 + 1.8788 0.1782 0.1 + 1.8990 0.3273 0.1 + 1.9192 0.2575 0.1 + 1.9394 0.2042 0.1 + 1.9596 0.3243 0.1 + 1.9798 0.1903 0.1 + 2.0000 0.1907 0.1 + \end{Verbatim} + + \subsection*{Possible extension 1:}\label{extension-1} + +Assume that the detectors are characterized by a finite resolution that +is described by a resolution function so that the ideal data is related +to the ideal spectrum +\[ \tilde{D}_k = \int G(x) R(x_k - x) dx + B(x_k) + \sigma(x_k),\] where +we have implicitly assumed that the resolution function \(R(x)\) does +not vary with position in writing the blurring process as a convolution +integral. + +\subsection*{Possible extension 2:}\label{extension-2} + +A possible extension of this problem is to compare the model evidences +for two competing hypothesis of the number of spectral lines in the +data. E.g. compute the evidence for a model \(M_1\) that corresponds to +the hypothesis that the experimental data can be explained with \emph{a +single} spectral line, and model \(M_2\) that assumes \emph{two} +spectral lines. + + % Add a bibliography block to the postdoc + + + + \end{document} diff --git a/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_10_0.png b/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_10_0.png new file mode 100644 index 0000000000000000000000000000000000000000..9cf191f94a372d119bc152e336555a0d8559cd32 GIT binary patch literal 10600 zcmaia1yt1C*Y40rr*udNLk}$=;Y*_+AR;xOLnuRcm+}(Q5=sgZ(&f-4;?N+749(C2 z(h^F22jB0#>;CV$_q(jcVwiJ&C-!ge=REt_C+fbg1~~~c2?PQm*V0rqfIx7j5$N2_UwP1pi-i(KK;~KuBq>esM-;gU%ojPKcJO$^)Owty$mK zX5N82d-am(ftHdPX)#$7?L1zKLKIQ75%?sI&YGO~Q~s|;!;@-IAHHm@9yJnVE6$vn z`@A*BrqW;w*By1Bba$@d)bidqU3AcGIs4%%&h9MnuG;R{A!+FRub1=^>^c(f+~e|x zH+_A?M@JGBX*4c2>NDrB$;Lg8j3ie6vI*hhTI3KQ;o?dv%;V(ZdiKA4yitiKq`@Oz zy2~bs9viH zjg+TQ@?909YrILZ!%kOEJ3bc;-rGwp6 z{>F4{jF*EuXI z-B6Zzrmo|>wgWXjHMXI9!wQXhz@_N<)GDkk^jcl!yt0H;8`Cxyxk5y<6c4MOckne z)Wb~ZemtfpFku2-twm=e04AfMg z57wAp^`iD>rC?<$D6fuA{kz0{C*iS?nea+3XGFx!gJfOpa4?S^YwTT^6z!m&fPJBr zn6fdFyjw4miydQCE0lHBoZl2AT`|$cTjH_(r(htng?X)_yfI#1KduUJ1mAyN1L%Rqj{N>W%bYYFGJufBuDk?U0#w#L<9_uG<+pA|D3Clp*+a3ZmXuDYT2M+LX+ticyD9uooZM&rxJ zd~lU@5cDZ4Hef+XV2?gO?(g4SUAe1q?;0VlNBuW3mqogfM*)+=dC@p>ZePlM-J6klgp8**t5oeH<3U;nCbUFi zTa-wV@=U-5Z#M{WHzH$FwFMa6T{4N$nj)oq08?$Y1gzy$Pe*}UFYI!urd)~g>N1rl zQXp&@`DV?30d?EGSrR-bu)gg?IXJ?PwY?GQ4mWnOULH%b6l1O6qj~da|Kiu;A3P7wM0z{phm23{)B97#E!{VPqjM2U zq>6Sog5hv}wE0VV+IG6|)(%3%T~aZvyv^^mbLup`?H9RPePN;WEY^g?__!Zrj3s2< zNV;8#@l)$IN|JBt73DJ{w#Cpb^QQ8YXV6^)Rtw9H+|n1VjaoKS*mkjgnzX%qm||?; z?=Izj;EFC>efZP2;G}WRplbXG2g7rDp^iR3#yf7#(&!Szttm;3B=BT*7uJZ2k6NAgjK&_6pM8NF2J+_Cyp0Z9 z2FF_$9ye|6UESeYY07eZO`#YDr;;Zw*38`_1-E$pub%zRF);6QM_a?4{A;a#p>3Y8 zKRHzKp9bwXzU9ZO`r6HCdh2UfQzw9_yQJubZf0humN021-3HD|31H*Cv4yUoh1wka zFNp+eDClYKWpA6kP|Y;7`M!U%Nws+UoeDicZzza`ehNN4)BZK!-vH!nK0*@pK|}VR zhDNai;-%ABn-`=Z6= z%#iyRrp2xVo;S4Wu4b0J&0>S?g8IpATx)XK%lO&p!|aeM1VX6ceXt8I(Id<^Y~N-< z>KFF0hA?X_U?8w&7e-S>6-p3OG{cJ|0JTkC+-r$XCPNbQBI5w%J- zpJwSSU4t&51z53NCo!d!g5g)w^9B~Htm=BBX$_{*Ju3d0i5DyhMQ1KT(!2L^K_5l8 zJae_PQ5i86grgcftskzr(L|C6`?o<%`c{8Wm(Zp;yEP$-jz{K(jO73$Eq)uB1CDK| zd^xI>X*!tnM1qrW22pgX7-oy7MdV`a`A$To?fbpS@N)6UaXW`-7KGSCM z2GWxmrfDV87J>$f6A8rK+Wudv`*}MJ&3xrHB!sx&sHWDxdca->XM*Afq-tEzCT7wP zePp4`O+|nkQakZVN(Mtkk|KQTuI!JC$~uP=DTWi7jCpsqxVt!2p6(2>o+%Y;kG{c& z54<9$ewScY_OED}qFbW{9dk%^%rp#fx5qTY^v5|p_LddNNd?NwqneM!1kGi+;Bq^$ zrz`1rn4x_Y_tN&jzv8LTP&P()boOEY%um+D^;?))$+2X-svWGDvBSpx;o6GK=hw*U zCyQRkBeDr^ZoU>EM!(hxt;g96VShWzc)%jEMr-04L3s0ZR{Y~}N#E2Pnd}bZ4Pin< zTY0;(F0>)ARc7o;j3c5$Qp~YD7U!=JG+jP+Ym)<~qwKJPKVQ#^X{MQ zoG{y6IAVZJISX}EihkW3+8qOP?@+1VDuYJ8 zPs>;aKZ$Lu($Gh;(gIGCo+E|uVZ2L~#TT`;jRg1gsc;HyS0_C=JE!90(l=W<^dL`> zckAeM{|shE7i3+2e*T%=D(2T3eGj(*9ehIRuH>rAx9}~vhE{1A|GV76OdN8_ey$p3 z!>aK%q7KG`EQxw1L{RoGM(WTkFXE7LfMf};L+O~_++@czlGIc=bkl9)iQqNG!YbXG zSfKbn*Pt1iD(UHO?2eIqChaAOajVRgOD(#99TUaJy1PBOIh)0k1fAY~H&0+1A`0PH z@X(LM#q;gqHK@8;j?i{hpiVtTKS-+q3Y%bMGSW{7w=ttrM3x7(!wN&gjYR?v`qXHk z4#V3z9&8r%1uOw|`Kl#UWbwM6d@o+p^Ijx5>Nm*$n%7%}Mri841`E-f*BbX48<~B$ z)-321{uN{NrPHUTsw_GTb=0_=6OY%fo+tRMdcBy7cm3Bn^%K|NLAk!<4Mk@5Z!}Xi zRpymNyX05}E71gd2Kn!2_|%Z>NeN=?blw*OOEnWQWmSgh#M&Mdzd4Dyb@`#OM)4o! 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