aboutsummaryrefslogblamecommitdiff
path: root/docs/dev/reference/nlme.html
blob: e7844299221264949ef17755a0b079ea68791c49 (plain) (tree)
1
2
3
4
5
6
7
8
9
               
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
                                                                         
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                               
                                                              
 
                                                    
                                                                                    








                                                                                                                               
                                                                                                                                        


                                                      
                                     






                                                                                                 
                                            





















                                                                                                                             
            

                                       
                                                                  
                                             

      
                                       



                           
                                

                                                                          
                                                                                                                                                         





                                                                                
                                                                                                             
          
                                                                                                                                                                                 
 
                                                                                                                                    
 












                                                                                                                       
 











































































                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
                                                                 
                                                                                      

      
                                            

                     
                                                                                                             
      
                     


  
  
 
                
 
<!DOCTYPE html>
<!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1.0"><title>Helper functions to create nlme models from mmkin row objects — nlme_function • mkin</title><!-- jquery --><script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script><!-- Bootstrap --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/css/bootstrap.min.css" integrity="sha256-bZLfwXAP04zRMK2BjiO8iu9pf4FbLqX6zitd+tIvLhE=" crossorigin="anonymous"><script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/js/bootstrap.min.js" integrity="sha256-nuL8/2cJ5NDSSwnKD8VqreErSWHtnEP9E7AySL+1ev4=" crossorigin="anonymous"></script><!-- bootstrap-toc --><link rel="stylesheet" href="../bootstrap-toc.css"><script src="../bootstrap-toc.js"></script><!-- Font Awesome icons --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" integrity="sha256-mmgLkCYLUQbXn0B1SRqzHar6dCnv9oZFPEC1g1cwlkk=" crossorigin="anonymous"><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/v4-shims.min.css" integrity="sha256-wZjR52fzng1pJHwx4aV2AO3yyTOXrcDW7jBpJtTwVxw=" crossorigin="anonymous"><!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- pkgdown --><link href="../pkgdown.css" rel="stylesheet"><script src="../pkgdown.js"></script><meta property="og:title" content="Helper functions to create nlme models from mmkin row objects — nlme_function"><meta property="og:description" content="These functions facilitate setting up a nonlinear mixed effects model for
an mmkin row object. An mmkin row object is essentially a list of mkinfit
objects that have been obtained by fitting the same model to a list of
datasets. They are used internally by the nlme.mmkin() method."><meta name="robots" content="noindex"><!-- mathjax --><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script><!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script>
<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
<![endif]--></head><body data-spy="scroll" data-target="#toc">
    

    <div class="container template-reference-topic">
      <header><div class="navbar navbar-default navbar-fixed-top" role="navigation">
  <div class="container">
    <div class="navbar-header">
      <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false">
        <span class="sr-only">Toggle navigation</span>
        <span class="icon-bar"></span>
        <span class="icon-bar"></span>
        <span class="icon-bar"></span>
      </button>
      <span class="navbar-brand">
        <a class="navbar-link" href="../index.html">mkin</a>
        <span class="version label label-info" data-toggle="tooltip" data-placement="bottom" title="In-development version">1.1.0</span>
      </span>
    </div>

    <div id="navbar" class="navbar-collapse collapse">
      <ul class="nav navbar-nav"><li>
  <a href="../reference/index.html">Functions and data</a>
</li>
<li class="dropdown">
  <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">
    Articles
     
    <span class="caret"></span>
  </a>
  <ul class="dropdown-menu" role="menu"><li>
      <a href="../articles/mkin.html">Introduction to mkin</a>
    </li>
    <li>
      <a href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a>
    </li>
    <li>
      <a href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a>
    </li>
    <li>
      <a href="../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a>
    </li>
    <li>
      <a href="../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a>
    </li>
    <li>
      <a href="../articles/twa.html">Calculation of time weighted average concentrations with mkin</a>
    </li>
    <li>
      <a href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a>
    </li>
    <li>
      <a href="../articles/web_only/benchmarks.html">Some benchmark timings</a>
    </li>
  </ul></li>
<li>
  <a href="../news/index.html">News</a>
</li>
      </ul><ul class="nav navbar-nav navbar-right"><li>
  <a href="https://github.com/jranke/mkin/" class="external-link">
    <span class="fab fa-github fa-lg"></span>
     
  </a>
</li>
      </ul></div><!--/.nav-collapse -->
  </div><!--/.container -->
</div><!--/.navbar -->

      

      </header><div class="row">
  <div class="col-md-9 contents">
    <div class="page-header">
    <h1>Helper functions to create nlme models from mmkin row objects</h1>
    <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/R/nlme.R" class="external-link"><code>R/nlme.R</code></a></small>
    <div class="hidden name"><code>nlme.Rd</code></div>
    </div>

    <div class="ref-description">
    <p>These functions facilitate setting up a nonlinear mixed effects model for
an mmkin row object. An mmkin row object is essentially a list of mkinfit
objects that have been obtained by fitting the same model to a list of
datasets. They are used internally by the <code><a href="nlme.mmkin.html">nlme.mmkin()</a></code> method.</p>
    </div>

    <div id="ref-usage">
    <div class="sourceCode"><pre class="sourceCode r"><code><span class="fu">nlme_function</span><span class="op">(</span><span class="va">object</span><span class="op">)</span>

<span class="fu">nlme_data</span><span class="op">(</span><span class="va">object</span><span class="op">)</span></code></pre></div>
    </div>

    <div id="arguments">
    <h2>Arguments</h2>
    <dl><dt>object</dt>
<dd><p>An mmkin row object containing several fits of the same model to different datasets</p></dd>
</dl></div>
    <div id="value">
    <h2>Value</h2>
    <p>A function that can be used with nlme
A <code><a href="https://rdrr.io/pkg/nlme/man/groupedData.html" class="external-link">groupedData</a></code> object</p>
    </div>
    <div id="see-also">
    <h2>See also</h2>
    <div class="dont-index"><p><code><a href="nlme.mmkin.html">nlme.mmkin</a></code></p></div>
    </div>

    <div id="ref-examples">
    <h2>Examples</h2>
    <div class="sourceCode"><pre class="sourceCode r"><code><span class="r-in"><span class="va">sampling_times</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">28</span>, <span class="fl">60</span>, <span class="fl">90</span>, <span class="fl">120</span><span class="op">)</span></span>
<span class="r-in"><span class="va">m_SFO</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d_SFO_1</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkinpredict.html">mkinpredict</a></span><span class="op">(</span><span class="va">m_SFO</span>,</span>
<span class="r-in">  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>k_parent <span class="op">=</span> <span class="fl">0.1</span><span class="op">)</span>,</span>
<span class="r-in">  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fl">98</span><span class="op">)</span>, <span class="va">sampling_times</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d_SFO_1_long</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">d_SFO_1</span>, time <span class="op">=</span> <span class="st">"time"</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d_SFO_2</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkinpredict.html">mkinpredict</a></span><span class="op">(</span><span class="va">m_SFO</span>,</span>
<span class="r-in">  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>k_parent <span class="op">=</span> <span class="fl">0.05</span><span class="op">)</span>,</span>
<span class="r-in">  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fl">102</span><span class="op">)</span>, <span class="va">sampling_times</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d_SFO_2_long</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">d_SFO_2</span>, time <span class="op">=</span> <span class="st">"time"</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d_SFO_3</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkinpredict.html">mkinpredict</a></span><span class="op">(</span><span class="va">m_SFO</span>,</span>
<span class="r-in">  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>k_parent <span class="op">=</span> <span class="fl">0.02</span><span class="op">)</span>,</span>
<span class="r-in">  <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fl">103</span><span class="op">)</span>, <span class="va">sampling_times</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d_SFO_3_long</span> <span class="op">&lt;-</span> <span class="fu"><a href="mkin_wide_to_long.html">mkin_wide_to_long</a></span><span class="op">(</span><span class="va">d_SFO_3</span>, time <span class="op">=</span> <span class="st">"time"</span><span class="op">)</span></span>
<span class="r-in"></span>
<span class="r-in"><span class="va">d1</span> <span class="op">&lt;-</span> <span class="fu"><a href="add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_SFO_1</span>, <span class="kw">function</span><span class="op">(</span><span class="va">value</span><span class="op">)</span> <span class="fl">3</span>, n <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d2</span> <span class="op">&lt;-</span> <span class="fu"><a href="add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_SFO_2</span>, <span class="kw">function</span><span class="op">(</span><span class="va">value</span><span class="op">)</span> <span class="fl">2</span>, n <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span>
<span class="r-in"><span class="va">d3</span> <span class="op">&lt;-</span> <span class="fu"><a href="add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_SFO_3</span>, <span class="kw">function</span><span class="op">(</span><span class="va">value</span><span class="op">)</span> <span class="fl">4</span>, n <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span>
<span class="r-in"><span class="va">ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>d1 <span class="op">=</span> <span class="va">d1</span>, d2 <span class="op">=</span> <span class="va">d2</span>, d3 <span class="op">=</span> <span class="va">d3</span><span class="op">)</span></span>
<span class="r-in"></span>
<span class="r-in"><span class="va">f</span> <span class="op">&lt;-</span> <span class="fu"><a href="mmkin.html">mmkin</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="va">ds</span>, cores <span class="op">=</span> <span class="fl">1</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span class="r-in"><span class="va">mean_dp</span> <span class="op">&lt;-</span> <span class="fu"><a href="mean_degparms.html">mean_degparms</a></span><span class="op">(</span><span class="va">f</span><span class="op">)</span></span>
<span class="r-in"><span class="va">grouped_data</span> <span class="op">&lt;-</span> <span class="fu">nlme_data</span><span class="op">(</span><span class="va">f</span><span class="op">)</span></span>
<span class="r-in"><span class="va">nlme_f</span> <span class="op">&lt;-</span> <span class="fu">nlme_function</span><span class="op">(</span><span class="va">f</span><span class="op">)</span></span>
<span class="r-in"><span class="co"># These assignments are necessary for these objects to be</span></span>
<span class="r-in"><span class="co"># visible to nlme and augPred when evaluation is done by</span></span>
<span class="r-in"><span class="co"># pkgdown to generate the html docs.</span></span>
<span class="r-in"><span class="fu"><a href="https://rdrr.io/r/base/assign.html" class="external-link">assign</a></span><span class="op">(</span><span class="st">"nlme_f"</span>, <span class="va">nlme_f</span>, <span class="fu"><a href="https://rdrr.io/r/base/environment.html" class="external-link">globalenv</a></span><span class="op">(</span><span class="op">)</span><span class="op">)</span></span>
<span class="r-in"><span class="fu"><a href="https://rdrr.io/r/base/assign.html" class="external-link">assign</a></span><span class="op">(</span><span class="st">"grouped_data"</span>, <span class="va">grouped_data</span>, <span class="fu"><a href="https://rdrr.io/r/base/environment.html" class="external-link">globalenv</a></span><span class="op">(</span><span class="op">)</span><span class="op">)</span></span>
<span class="r-in"></span>
<span class="r-in"><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://svn.r-project.org/R-packages/trunk/nlme/" class="external-link">nlme</a></span><span class="op">)</span></span>
<span class="r-in"><span class="va">m_nlme</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">value</span> <span class="op">~</span> <span class="fu">nlme_f</span><span class="op">(</span><span class="va">name</span>, <span class="va">time</span>, <span class="va">parent_0</span>, <span class="va">log_k_parent_sink</span><span class="op">)</span>,</span>
<span class="r-in">  data <span class="op">=</span> <span class="va">grouped_data</span>,</span>
<span class="r-in">  fixed <span class="op">=</span> <span class="va">parent_0</span> <span class="op">+</span> <span class="va">log_k_parent_sink</span> <span class="op">~</span> <span class="fl">1</span>,</span>
<span class="r-in">  random <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/pdDiag.html" class="external-link">pdDiag</a></span><span class="op">(</span><span class="va">parent_0</span> <span class="op">+</span> <span class="va">log_k_parent_sink</span> <span class="op">~</span> <span class="fl">1</span><span class="op">)</span>,</span>
<span class="r-in">  start <span class="op">=</span> <span class="va">mean_dp</span><span class="op">)</span></span>
<span class="r-in"><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">m_nlme</span><span class="op">)</span></span>
<span class="r-out co"><span class="r-pr">#&gt;</span> Nonlinear mixed-effects model fit by maximum likelihood</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>   Model: value ~ nlme_f(name, time, parent_0, log_k_parent_sink) </span>
<span class="r-out co"><span class="r-pr">#&gt;</span>   Data: grouped_data </span>
<span class="r-out co"><span class="r-pr">#&gt;</span>        AIC      BIC    logLik</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>   300.6824 310.2426 -145.3412</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> Random effects:</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>  Formula: list(parent_0 ~ 1, log_k_parent_sink ~ 1)</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>  Level: ds</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>  Structure: Diagonal</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>         parent_0 log_k_parent_sink Residual</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> StdDev: 1.697361         0.6801209 3.666073</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> Fixed effects:  parent_0 + log_k_parent_sink ~ 1 </span>
<span class="r-out co"><span class="r-pr">#&gt;</span>                       Value Std.Error DF  t-value p-value</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> parent_0          100.99378 1.3890416 46 72.70753       0</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> log_k_parent_sink  -3.07521 0.4018589 46 -7.65246       0</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>  Correlation: </span>
<span class="r-out co"><span class="r-pr">#&gt;</span>                   prnt_0</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> log_k_parent_sink 0.027 </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> Standardized Within-Group Residuals:</span>
<span class="r-out co"><span class="r-pr">#&gt;</span>        Min         Q1        Med         Q3        Max </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> -1.9942823 -0.5622565  0.1791579  0.7165038  2.0704781 </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> </span>
<span class="r-out co"><span class="r-pr">#&gt;</span> Number of Observations: 50</span>
<span class="r-out co"><span class="r-pr">#&gt;</span> Number of Groups: 3 </span>
<span class="r-in"><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/pkg/nlme/man/augPred.html" class="external-link">augPred</a></span><span class="op">(</span><span class="va">m_nlme</span>, level <span class="op">=</span> <span class="fl">0</span><span class="op">:</span><span class="fl">1</span><span class="op">)</span>, layout <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">3</span>, <span class="fl">1</span><span class="op">)</span><span class="op">)</span></span>
<span class="r-plt img"><img src="nlme-1.png" alt="" width="700" height="433"></span>
<span class="r-in"><span class="co"># augPred does not work on fits with more than one state</span></span>
<span class="r-in"><span class="co"># variable</span></span>
<span class="r-in"><span class="co">#</span></span>
<span class="r-in"><span class="co"># The procedure is greatly simplified by the nlme.mmkin function</span></span>
<span class="r-in"><span class="va">f_nlme</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/pkg/nlme/man/nlme.html" class="external-link">nlme</a></span><span class="op">(</span><span class="va">f</span><span class="op">)</span></span>
<span class="r-in"><span class="fu"><a href="https://rdrr.io/r/graphics/plot.default.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_nlme</span><span class="op">)</span></span>
<span class="r-plt img"><img src="nlme-2.png" alt="" width="700" height="433"></span>
</code></pre></div>
    </div>
  </div>
  <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
    <nav id="toc" data-toggle="toc" class="sticky-top"><h2 data-toc-skip>Contents</h2>
    </nav></div>
</div>


      <footer><div class="copyright">
  <p></p><p>Developed by Johannes Ranke.</p>
</div>

<div class="pkgdown">
  <p></p><p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.0.2.</p>
</div>

      </footer></div>

  


  

  </body></html>

Contact - Imprint