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{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-11-06T10:22:26","documenter_version":"1.1.2"}}
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>BLUE Example · EFTfitter.jl</title><meta name="title" content="BLUE Example · EFTfitter.jl"/><meta property="og:title" content="BLUE Example · EFTfitter.jl"/><meta property="twitter:title" content="BLUE Example · EFTfitter.jl"/><meta name="description" content="Documentation for EFTfitter.jl."/><meta property="og:description" content="Documentation for EFTfitter.jl."/><meta property="twitter:description" content="Documentation for EFTfitter.jl."/><meta property="og:url" content="https://github.com/tudo-physik-e4/EFTfitter.jl/BLUE/"/><meta property="twitter:url" content="https://github.com/tudo-physik-e4/EFTfitter.jl/BLUE/"/><link rel="canonical" href="https://github.com/tudo-physik-e4/EFTfitter.jl/BLUE/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">EFTfitter.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../installation/">Installation</a></li><li><a class="tocitem" href="../tutorial/">Tutorial</a></li><li><a class="tocitem" href="../advanced_tutorial/">Advanced Tutorial</a></li><li class="is-active"><a class="tocitem" href>BLUE Example</a></li><li><a class="tocitem" href="../plotting/">Plotting</a></li><li><a class="tocitem" href="../api/">API</a></li><li><a class="tocitem" href="../LICENSE/">LICENSE</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>BLUE Example</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>BLUE Example</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/tudo-physik-e4/EFTfitter.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/tudo-physik-e4/EFTfitter.jl/blob/main/docs/src/BLUE.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="EFTfitter.jl-BLUE-Example"><a class="docs-heading-anchor" href="#EFTfitter.jl-BLUE-Example">EFTfitter.jl - BLUE Example</a><a id="EFTfitter.jl-BLUE-Example-1"></a><a class="docs-heading-anchor-permalink" href="#EFTfitter.jl-BLUE-Example" title="Permalink"></a></h1><p>When using multiple measurements of a single observable and a uniform prior for the parameter representing the combined value, the combination of measurements performed with EFTfitter.jl yields the same results as the Best Linear Unbiased Estimator (BLUE) method.</p><p>Here, we demonstrate this by using the examples of the paper &quot;How to combine correlated estimates of a single physical quantity&quot; by L. Lyons, D. Gibaut and P. Clifford (https://www.sciencedirect.com/science/article/pii/0168900288900186). All numbers are taken from the example on charm particle lifetime experiments in section 5. A factor of 10^13 is applied for convenience.</p><pre><code class="language-julia hljs">using EFTfitter
using BAT
using IntervalSets
using Statistics
using StatsBase
using LinearAlgebra
using Plots</code></pre><p>We need one parameter for the best estimator and choose a uniform distribution in the range 8 to 14 as prior:</p><pre><code class="language-julia hljs">parameters = BAT.NamedTupleDist(
τ = 8..14,
)</code></pre><p>When combining multiple measurements of the same observable, only a function returning the combination parameter is needed:</p><pre><code class="language-julia hljs">estimator(params) = params.τ</code></pre><p>In Eq. (17&#39;) of the reference paper the following covariance matrix is given:</p><pre><code class="language-julia hljs">covariance = [2.74 1.15 0.86 1.31;
1.15 1.67 0.82 1.32;
0.86 0.82 2.12 1.05;
1.31 1.32 1.05 2.93]</code></pre><p>For using this in EFTfitter.jl, we first need to convert the covariance matrix into a correlation matrix and the corresponding uncertainty values:</p><pre><code class="language-julia hljs">corr, unc = EFTfitter.cov_to_cor(covariance)

measurements = (
τ1 = Measurement(estimator, 9.5, uncertainties = (stat=unc[1],) ),
τ2 = Measurement(estimator, 11.9, uncertainties = (stat=unc[2],) ),
τ3 = Measurement(estimator, 11.1, uncertainties = (stat=unc[3],) ),
τ4 = Measurement(estimator, 8.9, uncertainties = (stat=unc[4],) ),
)

correlations = (
stat = Correlation(corr),
)</code></pre><p>construct an <code>EFTfitterModel</code>:</p><pre><code class="language-julia hljs">model = EFTfitterModel(parameters, measurements, correlations)
posterior = PosteriorMeasure(model);</code></pre><p>sample the posterior with BAT.jl:</p><pre><code class="language-julia hljs">algorithm = MCMCSampling(mcalg =MetropolisHastings(), nsteps = 10^6, nchains = 4)
samples = bat_sample(posterior, algorithm).result</code></pre><p>plot the posterior distribution for the combination parameter τ:</p><pre><code class="language-julia hljs">plot(samples, :τ, mean=true)</code></pre><p><img src="../plots/plot_blue.png" alt="blue plots"/></p><p>print numerical results of combination:</p><pre><code class="language-julia hljs">println(&quot;Mode: $(mode(samples).τ)&quot;)
println(&quot;Mean: $(mean(samples).τ) ± $(std(samples).τ)&quot;)
```
Mode: 11.15985
Mean: 11.15471 ± 0.80180
```</code></pre><h3 id="Comparison-with-BLUE-method"><a class="docs-heading-anchor" href="#Comparison-with-BLUE-method">Comparison with BLUE method</a><a id="Comparison-with-BLUE-method-1"></a><a class="docs-heading-anchor-permalink" href="#Comparison-with-BLUE-method" title="Permalink"></a></h3><pre><code class="language-julia hljs">blue = BLUE(model)
println(&quot;BLUE: $(blue.value) ± $(blue.unc)&quot;)
println(&quot;BLUE weights: $(blue.weights)&quot;)
```
BLUE: 11.15983 ± 1.28604
BLUE weights: [0.145, 0.470, 0.347, 0.038]
```</code></pre><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../advanced_tutorial/">« Advanced Tutorial</a><a class="docs-footer-nextpage" href="../plotting/">Plotting »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 6 November 2023 10:22">Monday 6 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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License:</p><p>Copyright (c) 2020 Cornelius Grunwald</p><p>Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the &quot;Software&quot;), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:</p><p>The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.</p><p>THE SOFTWARE IS PROVIDED &quot;AS IS&quot;, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../api/">« API</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Monday 6 November 2023 10:22">Monday 6 November 2023</span>. 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