Morten Hjorth-Jensen
515 lines
34 KiB
HTML
515 lines
34 KiB
HTML
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<!-- navigation toc: --> <li><a href="._Statistics-bs001.html#___sec0" style="font-size: 80%;"> Domains and probabilities</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs002.html#___sec1" style="font-size: 80%;"> Tossing a dice</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs003.html#___sec2" style="font-size: 80%;"> Stochastic variables</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs004.html#___sec3" style="font-size: 80%;"> Stochastic variables and the main concepts, the discrete case</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs005.html#___sec4" style="font-size: 80%;"> Stochastic variables and the main concepts, the continuous case</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs006.html#___sec5" style="font-size: 80%;"> The cumulative probability</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs007.html#___sec6" style="font-size: 80%;"> Properties of PDFs</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs008.html#___sec7" style="font-size: 80%;"> Important distributions, the uniform distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs009.html#___sec8" style="font-size: 80%;"> Gaussian distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs010.html#___sec9" style="font-size: 80%;"> Exponential distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs011.html#___sec10" style="font-size: 80%;"> Expectation values</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs012.html#___sec11" style="font-size: 80%;"> Stochastic variables and the main concepts, mean values</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs013.html#___sec12" style="font-size: 80%;"> Stochastic variables and the main concepts, central moments, the variance</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs014.html#___sec13" style="font-size: 80%;"> Probability Distribution Functions</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs015.html#___sec14" style="font-size: 80%;"> Probability Distribution Functions</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs016.html#___sec15" style="font-size: 80%;"> The three famous Probability Distribution Functions</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs017.html#___sec16" style="font-size: 80%;"> Probability Distribution Functions, the normal distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs018.html#___sec17" style="font-size: 80%;"> Probability Distribution Functions, the normal distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs019.html#___sec18" style="font-size: 80%;"> Probability Distribution Functions, the cumulative distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs020.html#___sec19" style="font-size: 80%;"> Probability Distribution Functions, other important distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs021.html#___sec20" style="font-size: 80%;"> Probability Distribution Functions, the binomial distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs022.html#___sec21" style="font-size: 80%;"> Probability Distribution Functions, Poisson's distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs023.html#___sec22" style="font-size: 80%;"> Probability Distribution Functions, Poisson's distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs024.html#___sec23" style="font-size: 80%;"> Meet the covariance!</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs025.html#___sec24" style="font-size: 80%;"> Meet the covariance in matrix disguise</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs026.html#___sec25" style="font-size: 80%;"> Meet the covariance, uncorrelated events</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs027.html#___sec26" style="font-size: 80%;"> Numerical experiments and the covariance</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs028.html#___sec27" style="font-size: 80%;"> Numerical experiments and the covariance</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs029.html#___sec28" style="font-size: 80%;"> Numerical experiments and the covariance, actual situations</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs030.html#___sec29" style="font-size: 80%;"> Numerical experiments and the covariance, our observables</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs031.html#___sec30" style="font-size: 80%;"> Numerical experiments and the covariance, the sample variance</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs032.html#___sec31" style="font-size: 80%;"> Numerical experiments and the covariance, central limit theorem</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs033.html#___sec32" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs034.html#___sec33" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs035.html#___sec34" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs036.html#___sec35" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs037.html#___sec36" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation, sample variance</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs038.html#___sec37" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs039.html#___sec38" style="font-size: 80%;"><b>Random Numbers</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs040.html#___sec39" style="font-size: 80%;"><b>Random Numbers, better name: pseudo random numbers</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs041.html#___sec40" style="font-size: 80%;"><b>Random number generator RNG</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs042.html#___sec41" style="font-size: 80%;"><b>Random number generator RNG and periodic outputs</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs043.html#___sec42" style="font-size: 80%;"><b>Random number generator RNG and its period</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs044.html#___sec43" style="font-size: 80%;"><b>Random number generator RNG, other examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs045.html#___sec44" style="font-size: 80%;"><b>Random number generator RNG, other examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs046.html#___sec45" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs047.html#___sec46" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._Statistics-bs048.html#___sec47" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs049.html#___sec48" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs050.html#___sec49" style="font-size: 80%;"><b>Random number generator RNG, RAN0 code</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs051.html#___sec50" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs052.html#___sec51" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs053.html#___sec52" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs054.html#___sec53" style="font-size: 80%;"> Simple demonstration of RNGs using python</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs055.html#___sec54" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec55" style="font-size: 80%;"> Correlation function and which random number generators should I use</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs057.html#___sec56" style="font-size: 80%;"> Correlation function and which random number generators should I use</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs058.html#___sec57" style="font-size: 80%;"><b>Which RNG should I use?</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs059.html#___sec58" style="font-size: 80%;"> How to use the Mersenne generator</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs060.html#___sec59" style="font-size: 80%;"> Why blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs061.html#___sec60" style="font-size: 80%;"> Why blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs062.html#___sec61" style="font-size: 80%;"> Code to demonstrate the calculation of the autocorrelation function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs063.html#___sec62" style="font-size: 80%;"> What is blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs064.html#___sec63" style="font-size: 80%;"> What is blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs065.html#___sec64" style="font-size: 80%;"> What is blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs066.html#___sec65" style="font-size: 80%;"> Implementation</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs067.html#___sec66" style="font-size: 80%;"> Actual implementation with code, main function</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs068.html#___sec67" style="font-size: 80%;"> The Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs069.html#___sec68" style="font-size: 80%;"> Bootstrapping</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs070.html#___sec69" style="font-size: 80%;"> Bootstrapping, recipe</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs071.html#___sec70" style="font-size: 80%;"> Bootstrapping, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs072.html#___sec71" style="font-size: 80%;"> Jackknife, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"</a></li>
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<a name="part0056"></a>
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<!-- !split -->
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<h2 id="___sec55" class="anchor">Correlation function and which random number generators should I use </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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The program here computes the correlation function for one of the standard functions included with the c++ compiler.
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<p>
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<!-- code=c++ (!bc cppcod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic">// This function computes the autocorrelation function for </span>
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<span style="color: #408080; font-style: italic">// the standard c++ random number generator</span>
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<span style="color: #BC7A00">#include</span> <span style="color: #408080; font-style: italic"><fstream></span><span style="color: #BC7A00"></span>
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<span style="color: #BC7A00">#include</span> <span style="color: #408080; font-style: italic"><iomanip></span><span style="color: #BC7A00"></span>
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<span style="color: #BC7A00">#include</span> <span style="color: #408080; font-style: italic"><iostream></span><span style="color: #BC7A00"></span>
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<span style="color: #BC7A00">#include</span> <span style="color: #408080; font-style: italic"><cmath></span><span style="color: #BC7A00"></span>
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<span style="color: #008000; font-weight: bold">using</span> <span style="color: #008000; font-weight: bold">namespace</span> std;
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<span style="color: #408080; font-style: italic">// output file as global variable</span>
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ofstream ofile;
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<span style="color: #408080; font-style: italic">// Main function begins here </span>
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<span style="color: #B00040">int</span> <span style="color: #0000FF">main</span>(<span style="color: #B00040">int</span> argc, <span style="color: #B00040">char</span><span style="color: #666666">*</span> argv[])
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{
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<span style="color: #B00040">int</span> n;
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<span style="color: #B00040">char</span> <span style="color: #666666">*</span>outfilename;
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cin <span style="color: #666666">>></span> n;
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<span style="color: #B00040">double</span> MCint <span style="color: #666666">=</span> <span style="color: #666666">0.</span>; <span style="color: #B00040">double</span> MCintsqr2<span style="color: #666666">=0.</span>;
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<span style="color: #B00040">double</span> invers_period <span style="color: #666666">=</span> <span style="color: #666666">1./</span>RAND_MAX; <span style="color: #408080; font-style: italic">// initialise the random number generator</span>
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srand(time(<span style="color: #008000">NULL</span>)); <span style="color: #408080; font-style: italic">// This produces the so-called seed in MC jargon</span>
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<span style="color: #408080; font-style: italic">// Compute the variance and the mean value of the uniform distribution</span>
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<span style="color: #408080; font-style: italic">// Compute also the specific values x for each cycle in order to be able to</span>
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<span style="color: #408080; font-style: italic">// the covariance and the correlation function </span>
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<span style="color: #408080; font-style: italic">// Read in output file, abort if there are too few command-line arguments</span>
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<span style="color: #008000; font-weight: bold">if</span>( argc <span style="color: #666666"><=</span> <span style="color: #666666">2</span> ){
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cout <span style="color: #666666"><<</span> <span style="color: #BA2121">"Bad Usage: "</span> <span style="color: #666666"><<</span> argv[<span style="color: #666666">0</span>] <span style="color: #666666"><<</span>
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<span style="color: #BA2121">" read also output file and number of cycles on same line"</span> <span style="color: #666666"><<</span> endl;
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exit(<span style="color: #666666">1</span>);
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}
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<span style="color: #008000; font-weight: bold">else</span>{
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outfilename<span style="color: #666666">=</span>argv[<span style="color: #666666">1</span>];
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}
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ofile.open(outfilename);
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<span style="color: #408080; font-style: italic">// Get the number of Monte-Carlo samples</span>
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n <span style="color: #666666">=</span> atoi(argv[<span style="color: #666666">2</span>]);
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<span style="color: #B00040">double</span> <span style="color: #666666">*</span>X;
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X <span style="color: #666666">=</span> <span style="color: #008000; font-weight: bold">new</span> <span style="color: #B00040">double</span>[n];
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<span style="color: #008000; font-weight: bold">for</span> (<span style="color: #B00040">int</span> i <span style="color: #666666">=</span> <span style="color: #666666">0</span>; i <span style="color: #666666"><</span> n; i<span style="color: #666666">++</span>){
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<span style="color: #B00040">double</span> x <span style="color: #666666">=</span> <span style="color: #B00040">double</span>(rand())<span style="color: #666666">*</span>invers_period;
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X[i] <span style="color: #666666">=</span> x;
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MCint <span style="color: #666666">+=</span> x;
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MCintsqr2 <span style="color: #666666">+=</span> x<span style="color: #666666">*</span>x;
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}
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<span style="color: #B00040">double</span> Mean <span style="color: #666666">=</span> MCint<span style="color: #666666">/</span>((<span style="color: #B00040">double</span>) n );
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MCintsqr2 <span style="color: #666666">=</span> MCintsqr2<span style="color: #666666">/</span>((<span style="color: #B00040">double</span>) n );
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<span style="color: #B00040">double</span> STDev <span style="color: #666666">=</span> sqrt(MCintsqr2<span style="color: #666666">-</span>Mean<span style="color: #666666">*</span>Mean);
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<span style="color: #B00040">double</span> Variance <span style="color: #666666">=</span> MCintsqr2<span style="color: #666666">-</span>Mean<span style="color: #666666">*</span>Mean;
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<span style="color: #408080; font-style: italic">// Write mean value and standard deviation </span>
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cout <span style="color: #666666"><<</span> <span style="color: #BA2121">" Standard deviation= "</span> <span style="color: #666666"><<</span> STDev <span style="color: #666666"><<</span> <span style="color: #BA2121">" Integral = "</span> <span style="color: #666666"><<</span> Mean <span style="color: #666666"><<</span> endl;
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<span style="color: #408080; font-style: italic">// Now we compute the autocorrelation function</span>
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<span style="color: #B00040">double</span> <span style="color: #666666">*</span>autocor; autocor <span style="color: #666666">=</span> <span style="color: #008000; font-weight: bold">new</span> <span style="color: #B00040">double</span>[n];
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<span style="color: #008000; font-weight: bold">for</span> (<span style="color: #B00040">int</span> j <span style="color: #666666">=</span> <span style="color: #666666">0</span>; j <span style="color: #666666"><</span> n; j<span style="color: #666666">++</span>){
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<span style="color: #B00040">double</span> sum <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>;
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<span style="color: #008000; font-weight: bold">for</span> (<span style="color: #B00040">int</span> k <span style="color: #666666">=</span> <span style="color: #666666">0</span>; k <span style="color: #666666"><</span> (n<span style="color: #666666">-</span>j); k<span style="color: #666666">++</span>){
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sum <span style="color: #666666">+=</span> (X[k]<span style="color: #666666">-</span>Mean)<span style="color: #666666">*</span>(X[k<span style="color: #666666">+</span>j]<span style="color: #666666">-</span>Mean);
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}
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autocor[j] <span style="color: #666666">=</span> sum<span style="color: #666666">/</span>Variance<span style="color: #666666">/</span>((<span style="color: #B00040">double</span>) n );
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ofile <span style="color: #666666"><<</span> setiosflags(ios<span style="color: #666666">::</span>showpoint <span style="color: #666666">|</span> ios<span style="color: #666666">::</span>uppercase);
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ofile <span style="color: #666666"><<</span> setw(<span style="color: #666666">15</span>) <span style="color: #666666"><<</span> setprecision(<span style="color: #666666">8</span>) <span style="color: #666666"><<</span> j;
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ofile <span style="color: #666666"><<</span> setw(<span style="color: #666666">15</span>) <span style="color: #666666"><<</span> setprecision(<span style="color: #666666">8</span>) <span style="color: #666666"><<</span> autocor[j] <span style="color: #666666"><<</span> endl;
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}
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ofile.close(); <span style="color: #408080; font-style: italic">// close output file</span>
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<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">0</span>;
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} <span style="color: #408080; font-style: italic">// end of main program </span>
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</pre></div>
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