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b/doc/src/Bayesian/src/spectral_lines.tex new file mode 100644 index 000000000..5658f90c3 --- /dev/null +++ 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}}} + 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PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}} +\expandafter\def\csname 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 000000000..9cf191f94 Binary files /dev/null and b/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_10_0.png differ diff --git a/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_22_0.png b/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_22_0.png new file mode 100644 index 000000000..bac073972 Binary files /dev/null and b/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_22_0.png differ diff --git a/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_37_0.png b/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_37_0.png new file mode 100644 index 000000000..13e0fff00 Binary files /dev/null and b/doc/src/Bayesian/src/spectral_lines_files/spectral_lines_37_0.png differ