Morten Hjorth-Jensen
457 lines
26 KiB
HTML
457 lines
26 KiB
HTML
<!--
|
|
Automatically generated HTML file from DocOnce source
|
|
(https://github.com/hplgit/doconce/)
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
|
<meta name="description" content="Data Analysis and Machine Learning: Elements of Probability Theory">
|
|
|
|
<title>Data Analysis and Machine Learning: Elements of Probability Theory</title>
|
|
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
|
<!-- not necessary
|
|
<link href="http://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
|
-->
|
|
|
|
<style type="text/css">
|
|
|
|
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
|
.dropdown-menu {
|
|
height: auto;
|
|
max-height: 400px;
|
|
overflow-x: hidden;
|
|
}
|
|
|
|
/* Adds an invisible element before each target to offset for the navigation
|
|
bar */
|
|
.anchor::before {
|
|
content:"";
|
|
display:block;
|
|
height:50px; /* fixed header height for style bootstrap */
|
|
margin:-50px 0 0; /* negative fixed header height */
|
|
}
|
|
</style>
|
|
|
|
|
|
<script src="http://sagecell.sagemath.org/static/jquery.min.js"></script>
|
|
<script src="http://sagecell.sagemath.org/embedded_sagecell.js"></script>
|
|
<link rel="stylesheet" type="text/css" href="https://sagecell.sagemath.org/static/sagecell_embed.css">
|
|
<script>
|
|
$(function () {
|
|
// Make the div with id 'mycell' a Sage cell
|
|
sagecell.makeSagecell({inputLocation: '#mycell',
|
|
template: sagecell.templates.minimal,
|
|
evalButtonText: 'Activate'});
|
|
// Make *any* div with class 'compute' a Sage cell
|
|
sagecell.makeSagecell({inputLocation: 'div.compute',
|
|
evalButtonText: 'Evaluate'});
|
|
});
|
|
</script>
|
|
|
|
</head>
|
|
|
|
<!-- tocinfo
|
|
{'highest level': 1,
|
|
'sections': [('Domains and probabilities', 2, None, '___sec0'),
|
|
('Tossing a dice', 2, None, '___sec1'),
|
|
('Stochastic variables', 2, None, '___sec2'),
|
|
('Stochastic variables and the main concepts, the discrete case',
|
|
2,
|
|
None,
|
|
'___sec3'),
|
|
('Stochastic variables and the main concepts, the continuous case',
|
|
2,
|
|
None,
|
|
'___sec4'),
|
|
('The cumulative probability', 2, None, '___sec5'),
|
|
('Properties of PDFs', 2, None, '___sec6'),
|
|
('Important distributions, the uniform distribution',
|
|
2,
|
|
None,
|
|
'___sec7'),
|
|
('Gaussian distribution', 2, None, '___sec8'),
|
|
('Exponential distribution', 2, None, '___sec9'),
|
|
('Expectation values', 2, None, '___sec10'),
|
|
('Stochastic variables and the main concepts, mean values',
|
|
2,
|
|
None,
|
|
'___sec11'),
|
|
('Stochastic variables and the main concepts, central moments, the variance',
|
|
2,
|
|
None,
|
|
'___sec12'),
|
|
('Probability Distribution Functions', 2, None, '___sec13'),
|
|
('Probability Distribution Functions', 2, None, '___sec14'),
|
|
('The three famous Probability Distribution Functions',
|
|
2,
|
|
None,
|
|
'___sec15'),
|
|
('Probability Distribution Functions, the normal distribution',
|
|
2,
|
|
None,
|
|
'___sec16'),
|
|
('Probability Distribution Functions, the normal distribution',
|
|
2,
|
|
None,
|
|
'___sec17'),
|
|
('Probability Distribution Functions, the cumulative distribution',
|
|
2,
|
|
None,
|
|
'___sec18'),
|
|
('Probability Distribution Functions, other important distribution',
|
|
2,
|
|
None,
|
|
'___sec19'),
|
|
('Probability Distribution Functions, the binomial distribution',
|
|
2,
|
|
None,
|
|
'___sec20'),
|
|
("Probability Distribution Functions, Poisson's distribution",
|
|
2,
|
|
None,
|
|
'___sec21'),
|
|
("Probability Distribution Functions, Poisson's distribution",
|
|
2,
|
|
None,
|
|
'___sec22'),
|
|
('Meet the covariance!', 2, None, '___sec23'),
|
|
('Meet the covariance in matrix disguise',
|
|
2,
|
|
None,
|
|
'___sec24'),
|
|
('Meet the covariance, uncorrelated events',
|
|
2,
|
|
None,
|
|
'___sec25'),
|
|
('Numerical experiments and the covariance',
|
|
2,
|
|
None,
|
|
'___sec26'),
|
|
('Numerical experiments and the covariance',
|
|
2,
|
|
None,
|
|
'___sec27'),
|
|
('Numerical experiments and the covariance, actual situations',
|
|
2,
|
|
None,
|
|
'___sec28'),
|
|
('Numerical experiments and the covariance, our observables',
|
|
2,
|
|
None,
|
|
'___sec29'),
|
|
('Numerical experiments and the covariance, the sample variance',
|
|
2,
|
|
None,
|
|
'___sec30'),
|
|
('Numerical experiments and the covariance, central limit theorem',
|
|
2,
|
|
None,
|
|
'___sec31'),
|
|
('Definition of Correlation Functions and Standard Deviation',
|
|
2,
|
|
None,
|
|
'___sec32'),
|
|
('Definition of Correlation Functions and Standard Deviation',
|
|
2,
|
|
None,
|
|
'___sec33'),
|
|
('Definition of Correlation Functions and Standard Deviation',
|
|
2,
|
|
None,
|
|
'___sec34'),
|
|
('Definition of Correlation Functions and Standard Deviation',
|
|
2,
|
|
None,
|
|
'___sec35'),
|
|
('Definition of Correlation Functions and Standard Deviation, sample variance',
|
|
2,
|
|
None,
|
|
'___sec36'),
|
|
('Definition of Correlation Functions and Standard Deviation',
|
|
2,
|
|
None,
|
|
'___sec37'),
|
|
('Random Numbers', 1, None, '___sec38'),
|
|
('Random Numbers, better name: pseudo random numbers',
|
|
1,
|
|
None,
|
|
'___sec39'),
|
|
('Random number generator RNG', 1, None, '___sec40'),
|
|
('Random number generator RNG and periodic outputs',
|
|
1,
|
|
None,
|
|
'___sec41'),
|
|
('Random number generator RNG and its period',
|
|
1,
|
|
None,
|
|
'___sec42'),
|
|
('Random number generator RNG, other examples',
|
|
1,
|
|
None,
|
|
'___sec43'),
|
|
('Random number generator RNG, other examples',
|
|
1,
|
|
None,
|
|
'___sec44'),
|
|
('Random number generator RNG, RAN0', 1, None, '___sec45'),
|
|
('Random number generator RNG, RAN0', 1, None, '___sec46'),
|
|
('Random number generator RNG, RAN0', 1, None, '___sec47'),
|
|
('Random number generator RNG, RAN0', 1, None, '___sec48'),
|
|
('Random number generator RNG, RAN0 code', 1, None, '___sec49'),
|
|
('Properties of Selected Random Number Generators',
|
|
2,
|
|
None,
|
|
'___sec50'),
|
|
('Properties of Selected Random Number Generators',
|
|
2,
|
|
None,
|
|
'___sec51'),
|
|
('Properties of Selected Random Number Generators',
|
|
2,
|
|
None,
|
|
'___sec52'),
|
|
('Simple demonstration of RNGs using python',
|
|
2,
|
|
None,
|
|
'___sec53'),
|
|
('Properties of Selected Random Number Generators',
|
|
2,
|
|
None,
|
|
'___sec54'),
|
|
('Correlation function and which random number generators should I use',
|
|
2,
|
|
None,
|
|
'___sec55'),
|
|
('Correlation function and which random number generators should I use',
|
|
2,
|
|
None,
|
|
'___sec56'),
|
|
('Which RNG should I use?', 1, None, '___sec57'),
|
|
('How to use the Mersenne generator', 2, None, '___sec58'),
|
|
('Why blocking?', 2, None, '___sec59'),
|
|
('Why blocking?', 2, None, '___sec60'),
|
|
('Code to demonstrate the calculation of the autocorrelation function',
|
|
2,
|
|
None,
|
|
'___sec61'),
|
|
('What is blocking?', 2, None, '___sec62'),
|
|
('What is blocking?', 2, None, '___sec63'),
|
|
('What is blocking?', 2, None, '___sec64'),
|
|
('Implementation', 2, None, '___sec65'),
|
|
('Actual implementation with code, main function',
|
|
2,
|
|
None,
|
|
'___sec66'),
|
|
('The Bootstrap method', 2, None, '___sec67'),
|
|
('Bootstrapping', 2, None, '___sec68'),
|
|
('Bootstrapping, recipe', 2, None, '___sec69'),
|
|
('Bootstrapping, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"',
|
|
2,
|
|
None,
|
|
'___sec70'),
|
|
('Jackknife, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"',
|
|
2,
|
|
None,
|
|
'___sec71')]}
|
|
end of tocinfo -->
|
|
|
|
<body>
|
|
|
|
|
|
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
TeX: {
|
|
equationNumbers: { autoNumber: "none" },
|
|
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript"
|
|
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
|
</script>
|
|
|
|
|
|
|
|
|
|
<!-- Bootstrap navigation bar -->
|
|
<div class="navbar navbar-default navbar-fixed-top">
|
|
<div class="navbar-header">
|
|
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
</button>
|
|
<a class="navbar-brand" href="Statistics-bs.html">Data Analysis and Machine Learning: Elements of Probability Theory</a>
|
|
</div>
|
|
|
|
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
|
<ul class="nav navbar-nav navbar-right">
|
|
<li class="dropdown">
|
|
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
|
<ul class="dropdown-menu">
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs001.html#___sec0" style="font-size: 80%;"> Domains and probabilities</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs002.html#___sec1" style="font-size: 80%;"> Tossing a dice</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs003.html#___sec2" style="font-size: 80%;"> Stochastic variables</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs004.html#___sec3" style="font-size: 80%;"> Stochastic variables and the main concepts, the discrete case</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs005.html#___sec4" style="font-size: 80%;"> Stochastic variables and the main concepts, the continuous case</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs006.html#___sec5" style="font-size: 80%;"> The cumulative probability</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs007.html#___sec6" style="font-size: 80%;"> Properties of PDFs</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs008.html#___sec7" style="font-size: 80%;"> Important distributions, the uniform distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs009.html#___sec8" style="font-size: 80%;"> Gaussian distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs010.html#___sec9" style="font-size: 80%;"> Exponential distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs011.html#___sec10" style="font-size: 80%;"> Expectation values</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs012.html#___sec11" style="font-size: 80%;"> Stochastic variables and the main concepts, mean values</a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs014.html#___sec13" style="font-size: 80%;"> Probability Distribution Functions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs015.html#___sec14" style="font-size: 80%;"> Probability Distribution Functions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs016.html#___sec15" style="font-size: 80%;"> The three famous Probability Distribution Functions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs017.html#___sec16" style="font-size: 80%;"> Probability Distribution Functions, the normal distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs018.html#___sec17" style="font-size: 80%;"> Probability Distribution Functions, the normal distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs019.html#___sec18" style="font-size: 80%;"> Probability Distribution Functions, the cumulative distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs020.html#___sec19" style="font-size: 80%;"> Probability Distribution Functions, other important distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs021.html#___sec20" style="font-size: 80%;"> Probability Distribution Functions, the binomial distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs022.html#___sec21" style="font-size: 80%;"> Probability Distribution Functions, Poisson's distribution</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs023.html#___sec22" style="font-size: 80%;"> Probability Distribution Functions, Poisson's distribution</a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs026.html#___sec25" style="font-size: 80%;"> Meet the covariance, uncorrelated events</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs027.html#___sec26" style="font-size: 80%;"> Numerical experiments and the covariance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs028.html#___sec27" style="font-size: 80%;"> Numerical experiments and the covariance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs029.html#___sec28" style="font-size: 80%;"> Numerical experiments and the covariance, actual situations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs030.html#___sec29" style="font-size: 80%;"> Numerical experiments and the covariance, our observables</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs031.html#___sec30" style="font-size: 80%;"> Numerical experiments and the covariance, the sample variance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs032.html#___sec31" style="font-size: 80%;"> Numerical experiments and the covariance, central limit theorem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs033.html#___sec32" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec33" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs035.html#___sec34" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs036.html#___sec35" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs037.html#___sec36" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation, sample variance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs038.html#___sec37" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs039.html#___sec38" style="font-size: 80%;"><b>Random Numbers</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs040.html#___sec39" style="font-size: 80%;"><b>Random Numbers, better name: pseudo random numbers</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs041.html#___sec40" style="font-size: 80%;"><b>Random number generator RNG</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs042.html#___sec41" style="font-size: 80%;"><b>Random number generator RNG and periodic outputs</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs043.html#___sec42" style="font-size: 80%;"><b>Random number generator RNG and its period</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs044.html#___sec43" style="font-size: 80%;"><b>Random number generator RNG, other examples</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs045.html#___sec44" style="font-size: 80%;"><b>Random number generator RNG, other examples</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs046.html#___sec45" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs047.html#___sec46" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- 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="._Statistics-bs056.html#___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>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs067.html#___sec66" style="font-size: 80%;"> Actual implementation with code, main function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs068.html#___sec67" style="font-size: 80%;"> The Bootstrap method</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs069.html#___sec68" style="font-size: 80%;"> Bootstrapping</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs070.html#___sec69" style="font-size: 80%;"> Bootstrapping, recipe</a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div> <!-- end of navigation bar -->
|
|
|
|
<div class="container">
|
|
|
|
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
|
|
|
<a name="part0034"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec33" class="anchor">Definition of Correlation Functions and Standard Deviation </h2>
|
|
<div class="panel panel-default">
|
|
<div class="panel-body">
|
|
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
|
Our estimate of the true average \( \mu_{X} \) is the sample mean \( \langle X_m \rangle \)
|
|
|
|
<p>
|
|
If the
|
|
observables are uncorrelated, then the covariance is zero and we obtain a total variance
|
|
which agrees with the central limit theorem. Correlations may often be present in our data set, resulting in a non-zero covariance. The first term is normally called the uncorrelated
|
|
contribution.
|
|
Computationally the uncorrelated first term is much easier to treat
|
|
efficiently than the second.
|
|
We just accumulate separately the values \( x^2 \) and \( x \) for every
|
|
measurement \( x \) we receive. The correlation term, though, has to be
|
|
calculated at the end of the experiment since we need all the
|
|
measurements to calculate the cross terms. Therefore, all measurements
|
|
have to be stored throughout the experiment.
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._Statistics-bs033.html">«</a></li>
|
|
<li><a href="._Statistics-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._Statistics-bs026.html">27</a></li>
|
|
<li><a href="._Statistics-bs027.html">28</a></li>
|
|
<li><a href="._Statistics-bs028.html">29</a></li>
|
|
<li><a href="._Statistics-bs029.html">30</a></li>
|
|
<li><a href="._Statistics-bs030.html">31</a></li>
|
|
<li><a href="._Statistics-bs031.html">32</a></li>
|
|
<li><a href="._Statistics-bs032.html">33</a></li>
|
|
<li><a href="._Statistics-bs033.html">34</a></li>
|
|
<li class="active"><a href="._Statistics-bs034.html">35</a></li>
|
|
<li><a href="._Statistics-bs035.html">36</a></li>
|
|
<li><a href="._Statistics-bs036.html">37</a></li>
|
|
<li><a href="._Statistics-bs037.html">38</a></li>
|
|
<li><a href="._Statistics-bs038.html">39</a></li>
|
|
<li><a href="._Statistics-bs039.html">40</a></li>
|
|
<li><a href="._Statistics-bs040.html">41</a></li>
|
|
<li><a href="._Statistics-bs041.html">42</a></li>
|
|
<li><a href="._Statistics-bs042.html">43</a></li>
|
|
<li><a href="._Statistics-bs043.html">44</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._Statistics-bs072.html">73</a></li>
|
|
<li><a href="._Statistics-bs035.html">»</a></li>
|
|
</ul>
|
|
<!-- ------------------- end of main content --------------- -->
|
|
|
|
</div> <!-- end container -->
|
|
<!-- include javascript, jQuery *first* -->
|
|
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
|
|
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
|
|
|
|
<!-- Bootstrap footer
|
|
<footer>
|
|
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
|
</footer>
|
|
-->
|
|
|
|
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
|
|
|
|
</body>
|
|
</html>
|
|
|
|
|