You are viewing a plain text version of this content. The canonical link for it is here.
Posted to commits@lucene.apache.org by ct...@apache.org on 2021/01/28 16:33:32 UTC
svn commit: r1070647 [24/35] - in
/websites/production/lucene/content/solr/guide/8_8: ./
images/math-expressions/ meta-docs/
Modified: websites/production/lucene/content/solr/guide/8_8/segments-info.html
==============================================================================
--- websites/production/lucene/content/solr/guide/8_8/segments-info.html (original)
+++ websites/production/lucene/content/solr/guide/8_8/segments-info.html Thu Jan 28 16:33:25 2021
@@ -8,7 +8,7 @@
<meta name="description" content="">
<meta name="keywords" content=" ">
-<title>Segments Info | Apache Solr Reference Guide 8.8-DRAFT</title>
+<title>Segments Info | Apache Solr Reference Guide 8.8</title>
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/css/bootstrap.min.css" integrity="sha384-MCw98/SFnGE8fJT3GXwEOngsV7Zt27NXFoaoApmYm81iuXoPkFOJwJ8ERdknLPMO" crossorigin="anonymous">
<link rel="stylesheet" type="text/css" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.5.0/css/font-awesome.min.css">
@@ -68,7 +68,7 @@
</script>
</head>
-<body class="DRAFT" id="segments-info">
+<body class="" id="segments-info">
<div class="container-fluid">
<div class="row">
<nav id="sidebar" class="col-2 d-none d-md-block">
@@ -76,11 +76,6 @@
<div class="sidebar-header">
<div class="sidebarTitle text-center">Apache Solr Reference Guide</div>
- <p class="draft-notice">
- This is an unofficial DRAFT of the Guide for 8.8.
- <a href="https://lucene.apache.org/solr/guide/">Official releases are available from the Solr website</a>.
- </p>
-
<!--comment out this block if you want to hide search-->
<!--start search-->
@@ -872,11 +867,36 @@
</li>
<li class="sb-level2">
- <a href="math-expressions.html">Math Expressions</a>
+ <a href="math-expressions.html">Streaming Expressions and Math Expressions</a>
<ul>
<li class="sb-level3">
+ <a href="visualization.html">Visualization</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="math-start.html">Getting Started</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="loading.html">Loading Data</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="search-sample.html">Searching, Sampling and Aggregation</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="transform.html">Transforming Data</a>
+
+ </li>
+
+ <li class="sb-level3">
<a href="scalar-math.html">Scalar Math</a>
</li>
@@ -897,12 +917,12 @@
</li>
<li class="sb-level3">
- <a href="vectorization.html">Streams and Vectorization</a>
+ <a href="term-vectors.html">Text Analysis and Term Vectors</a>
</li>
<li class="sb-level3">
- <a href="term-vectors.html">Text Analysis and Term Vectors</a>
+ <a href="probability-distributions.html">Probability Distributions</a>
</li>
@@ -912,12 +932,12 @@
</li>
<li class="sb-level3">
- <a href="probability-distributions.html">Probability Distributions</a>
+ <a href="regression.html">Linear Regression</a>
</li>
<li class="sb-level3">
- <a href="simulations.html">Monte Carlo Simulations</a>
+ <a href="curve-fitting.html">Curve Fitting</a>
</li>
@@ -927,32 +947,32 @@
</li>
<li class="sb-level3">
- <a href="regression.html">Linear Regression</a>
+ <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
</li>
<li class="sb-level3">
- <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
+ <a href="dsp.html">Digital Signal Processing</a>
</li>
<li class="sb-level3">
- <a href="curve-fitting.html">Curve Fitting</a>
+ <a href="simulations.html">Monte Carlo Simulations</a>
</li>
<li class="sb-level3">
- <a href="dsp.html">Digital Signal Processing</a>
+ <a href="machine-learning.html">Machine Learning</a>
</li>
<li class="sb-level3">
- <a href="machine-learning.html">Machine Learning</a>
+ <a href="computational-geometry.html">Computational Geometry</a>
</li>
<li class="sb-level3">
- <a href="computational-geometry.html">Computational Geometry</a>
+ <a href="logs.html">Log Analytics</a>
</li>
@@ -1690,7 +1710,7 @@
<div class="row">
<div class="col-lg-12 footer">
©2021 Apache Software Foundation. All rights reserved. <br />
- Site Version: 8.8-DRAFT <br />Site last generated: 2021-01-19 <br />
+ Site Version: 8.8 <br />Site last generated: 2021-01-28 <br />
<p><img src="images/solr-sunOnly-small.png" alt="Apache Solr"/></p>
</div>
</div>
Modified: websites/production/lucene/content/solr/guide/8_8/setting-up-an-external-zookeeper-ensemble.html
==============================================================================
--- websites/production/lucene/content/solr/guide/8_8/setting-up-an-external-zookeeper-ensemble.html (original)
+++ websites/production/lucene/content/solr/guide/8_8/setting-up-an-external-zookeeper-ensemble.html Thu Jan 28 16:33:25 2021
@@ -8,7 +8,7 @@
<meta name="description" content="">
<meta name="keywords" content=" ">
-<title>Setting Up an External ZooKeeper Ensemble | Apache Solr Reference Guide 8.8-DRAFT</title>
+<title>Setting Up an External ZooKeeper Ensemble | Apache Solr Reference Guide 8.8</title>
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/css/bootstrap.min.css" integrity="sha384-MCw98/SFnGE8fJT3GXwEOngsV7Zt27NXFoaoApmYm81iuXoPkFOJwJ8ERdknLPMO" crossorigin="anonymous">
<link rel="stylesheet" type="text/css" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.5.0/css/font-awesome.min.css">
@@ -68,7 +68,7 @@
</script>
</head>
-<body class="DRAFT" id="setting-up-an-external-zookeeper-ensemble">
+<body class="" id="setting-up-an-external-zookeeper-ensemble">
<div class="container-fluid">
<div class="row">
<nav id="sidebar" class="col-2 d-none d-md-block">
@@ -76,11 +76,6 @@
<div class="sidebar-header">
<div class="sidebarTitle text-center">Apache Solr Reference Guide</div>
- <p class="draft-notice">
- This is an unofficial DRAFT of the Guide for 8.8.
- <a href="https://lucene.apache.org/solr/guide/">Official releases are available from the Solr website</a>.
- </p>
-
<!--comment out this block if you want to hide search-->
<!--start search-->
@@ -872,11 +867,36 @@
</li>
<li class="sb-level2">
- <a href="math-expressions.html">Math Expressions</a>
+ <a href="math-expressions.html">Streaming Expressions and Math Expressions</a>
<ul>
<li class="sb-level3">
+ <a href="visualization.html">Visualization</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="math-start.html">Getting Started</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="loading.html">Loading Data</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="search-sample.html">Searching, Sampling and Aggregation</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="transform.html">Transforming Data</a>
+
+ </li>
+
+ <li class="sb-level3">
<a href="scalar-math.html">Scalar Math</a>
</li>
@@ -897,12 +917,12 @@
</li>
<li class="sb-level3">
- <a href="vectorization.html">Streams and Vectorization</a>
+ <a href="term-vectors.html">Text Analysis and Term Vectors</a>
</li>
<li class="sb-level3">
- <a href="term-vectors.html">Text Analysis and Term Vectors</a>
+ <a href="probability-distributions.html">Probability Distributions</a>
</li>
@@ -912,12 +932,12 @@
</li>
<li class="sb-level3">
- <a href="probability-distributions.html">Probability Distributions</a>
+ <a href="regression.html">Linear Regression</a>
</li>
<li class="sb-level3">
- <a href="simulations.html">Monte Carlo Simulations</a>
+ <a href="curve-fitting.html">Curve Fitting</a>
</li>
@@ -927,32 +947,32 @@
</li>
<li class="sb-level3">
- <a href="regression.html">Linear Regression</a>
+ <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
</li>
<li class="sb-level3">
- <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
+ <a href="dsp.html">Digital Signal Processing</a>
</li>
<li class="sb-level3">
- <a href="curve-fitting.html">Curve Fitting</a>
+ <a href="simulations.html">Monte Carlo Simulations</a>
</li>
<li class="sb-level3">
- <a href="dsp.html">Digital Signal Processing</a>
+ <a href="machine-learning.html">Machine Learning</a>
</li>
<li class="sb-level3">
- <a href="machine-learning.html">Machine Learning</a>
+ <a href="computational-geometry.html">Computational Geometry</a>
</li>
<li class="sb-level3">
- <a href="computational-geometry.html">Computational Geometry</a>
+ <a href="logs.html">Log Analytics</a>
</li>
@@ -1918,7 +1938,7 @@ REM -a option on start script, those opt
<div class="row">
<div class="col-lg-12 footer">
©2021 Apache Software Foundation. All rights reserved. <br />
- Site Version: 8.8-DRAFT <br />Site last generated: 2021-01-19 <br />
+ Site Version: 8.8 <br />Site last generated: 2021-01-28 <br />
<p><img src="images/solr-sunOnly-small.png" alt="Apache Solr"/></p>
</div>
</div>
Modified: websites/production/lucene/content/solr/guide/8_8/shard-management.html
==============================================================================
--- websites/production/lucene/content/solr/guide/8_8/shard-management.html (original)
+++ websites/production/lucene/content/solr/guide/8_8/shard-management.html Thu Jan 28 16:33:25 2021
@@ -8,7 +8,7 @@
<meta name="description" content="">
<meta name="keywords" content=" ">
-<title>Shard Management Commands | Apache Solr Reference Guide 8.8-DRAFT</title>
+<title>Shard Management Commands | Apache Solr Reference Guide 8.8</title>
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/css/bootstrap.min.css" integrity="sha384-MCw98/SFnGE8fJT3GXwEOngsV7Zt27NXFoaoApmYm81iuXoPkFOJwJ8ERdknLPMO" crossorigin="anonymous">
<link rel="stylesheet" type="text/css" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.5.0/css/font-awesome.min.css">
@@ -68,7 +68,7 @@
</script>
</head>
-<body class="DRAFT" id="shard-management">
+<body class="" id="shard-management">
<div class="container-fluid">
<div class="row">
<nav id="sidebar" class="col-2 d-none d-md-block">
@@ -76,11 +76,6 @@
<div class="sidebar-header">
<div class="sidebarTitle text-center">Apache Solr Reference Guide</div>
- <p class="draft-notice">
- This is an unofficial DRAFT of the Guide for 8.8.
- <a href="https://lucene.apache.org/solr/guide/">Official releases are available from the Solr website</a>.
- </p>
-
<!--comment out this block if you want to hide search-->
<!--start search-->
@@ -872,11 +867,36 @@
</li>
<li class="sb-level2">
- <a href="math-expressions.html">Math Expressions</a>
+ <a href="math-expressions.html">Streaming Expressions and Math Expressions</a>
<ul>
<li class="sb-level3">
+ <a href="visualization.html">Visualization</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="math-start.html">Getting Started</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="loading.html">Loading Data</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="search-sample.html">Searching, Sampling and Aggregation</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="transform.html">Transforming Data</a>
+
+ </li>
+
+ <li class="sb-level3">
<a href="scalar-math.html">Scalar Math</a>
</li>
@@ -897,12 +917,12 @@
</li>
<li class="sb-level3">
- <a href="vectorization.html">Streams and Vectorization</a>
+ <a href="term-vectors.html">Text Analysis and Term Vectors</a>
</li>
<li class="sb-level3">
- <a href="term-vectors.html">Text Analysis and Term Vectors</a>
+ <a href="probability-distributions.html">Probability Distributions</a>
</li>
@@ -912,12 +932,12 @@
</li>
<li class="sb-level3">
- <a href="probability-distributions.html">Probability Distributions</a>
+ <a href="regression.html">Linear Regression</a>
</li>
<li class="sb-level3">
- <a href="simulations.html">Monte Carlo Simulations</a>
+ <a href="curve-fitting.html">Curve Fitting</a>
</li>
@@ -927,32 +947,32 @@
</li>
<li class="sb-level3">
- <a href="regression.html">Linear Regression</a>
+ <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
</li>
<li class="sb-level3">
- <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
+ <a href="dsp.html">Digital Signal Processing</a>
</li>
<li class="sb-level3">
- <a href="curve-fitting.html">Curve Fitting</a>
+ <a href="simulations.html">Monte Carlo Simulations</a>
</li>
<li class="sb-level3">
- <a href="dsp.html">Digital Signal Processing</a>
+ <a href="machine-learning.html">Machine Learning</a>
</li>
<li class="sb-level3">
- <a href="machine-learning.html">Machine Learning</a>
+ <a href="computational-geometry.html">Computational Geometry</a>
</li>
<li class="sb-level3">
- <a href="computational-geometry.html">Computational Geometry</a>
+ <a href="logs.html">Log Analytics</a>
</li>
@@ -1842,7 +1862,7 @@ This is an expert level command, and sho
<div class="row">
<div class="col-lg-12 footer">
©2021 Apache Software Foundation. All rights reserved. <br />
- Site Version: 8.8-DRAFT <br />Site last generated: 2021-01-19 <br />
+ Site Version: 8.8 <br />Site last generated: 2021-01-28 <br />
<p><img src="images/solr-sunOnly-small.png" alt="Apache Solr"/></p>
</div>
</div>
Modified: websites/production/lucene/content/solr/guide/8_8/shards-and-indexing-data-in-solrcloud.html
==============================================================================
--- websites/production/lucene/content/solr/guide/8_8/shards-and-indexing-data-in-solrcloud.html (original)
+++ websites/production/lucene/content/solr/guide/8_8/shards-and-indexing-data-in-solrcloud.html Thu Jan 28 16:33:25 2021
@@ -8,7 +8,7 @@
<meta name="description" content="">
<meta name="keywords" content=" ">
-<title>Shards and Indexing Data in SolrCloud | Apache Solr Reference Guide 8.8-DRAFT</title>
+<title>Shards and Indexing Data in SolrCloud | Apache Solr Reference Guide 8.8</title>
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/css/bootstrap.min.css" integrity="sha384-MCw98/SFnGE8fJT3GXwEOngsV7Zt27NXFoaoApmYm81iuXoPkFOJwJ8ERdknLPMO" crossorigin="anonymous">
<link rel="stylesheet" type="text/css" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.5.0/css/font-awesome.min.css">
@@ -68,7 +68,7 @@
</script>
</head>
-<body class="DRAFT" id="shards-and-indexing-data-in-solrcloud">
+<body class="" id="shards-and-indexing-data-in-solrcloud">
<div class="container-fluid">
<div class="row">
<nav id="sidebar" class="col-2 d-none d-md-block">
@@ -76,11 +76,6 @@
<div class="sidebar-header">
<div class="sidebarTitle text-center">Apache Solr Reference Guide</div>
- <p class="draft-notice">
- This is an unofficial DRAFT of the Guide for 8.8.
- <a href="https://lucene.apache.org/solr/guide/">Official releases are available from the Solr website</a>.
- </p>
-
<!--comment out this block if you want to hide search-->
<!--start search-->
@@ -872,11 +867,36 @@
</li>
<li class="sb-level2">
- <a href="math-expressions.html">Math Expressions</a>
+ <a href="math-expressions.html">Streaming Expressions and Math Expressions</a>
<ul>
<li class="sb-level3">
+ <a href="visualization.html">Visualization</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="math-start.html">Getting Started</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="loading.html">Loading Data</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="search-sample.html">Searching, Sampling and Aggregation</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="transform.html">Transforming Data</a>
+
+ </li>
+
+ <li class="sb-level3">
<a href="scalar-math.html">Scalar Math</a>
</li>
@@ -897,12 +917,12 @@
</li>
<li class="sb-level3">
- <a href="vectorization.html">Streams and Vectorization</a>
+ <a href="term-vectors.html">Text Analysis and Term Vectors</a>
</li>
<li class="sb-level3">
- <a href="term-vectors.html">Text Analysis and Term Vectors</a>
+ <a href="probability-distributions.html">Probability Distributions</a>
</li>
@@ -912,12 +932,12 @@
</li>
<li class="sb-level3">
- <a href="probability-distributions.html">Probability Distributions</a>
+ <a href="regression.html">Linear Regression</a>
</li>
<li class="sb-level3">
- <a href="simulations.html">Monte Carlo Simulations</a>
+ <a href="curve-fitting.html">Curve Fitting</a>
</li>
@@ -927,32 +947,32 @@
</li>
<li class="sb-level3">
- <a href="regression.html">Linear Regression</a>
+ <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
</li>
<li class="sb-level3">
- <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
+ <a href="dsp.html">Digital Signal Processing</a>
</li>
<li class="sb-level3">
- <a href="curve-fitting.html">Curve Fitting</a>
+ <a href="simulations.html">Monte Carlo Simulations</a>
</li>
<li class="sb-level3">
- <a href="dsp.html">Digital Signal Processing</a>
+ <a href="machine-learning.html">Machine Learning</a>
</li>
<li class="sb-level3">
- <a href="machine-learning.html">Machine Learning</a>
+ <a href="computational-geometry.html">Computational Geometry</a>
</li>
<li class="sb-level3">
- <a href="computational-geometry.html">Computational Geometry</a>
+ <a href="logs.html">Log Analytics</a>
</li>
@@ -1778,7 +1798,7 @@ Using <code>autoSoftCommit</code> or <co
<div class="row">
<div class="col-lg-12 footer">
©2021 Apache Software Foundation. All rights reserved. <br />
- Site Version: 8.8-DRAFT <br />Site last generated: 2021-01-19 <br />
+ Site Version: 8.8 <br />Site last generated: 2021-01-28 <br />
<p><img src="images/solr-sunOnly-small.png" alt="Apache Solr"/></p>
</div>
</div>
Modified: websites/production/lucene/content/solr/guide/8_8/simulations.html
==============================================================================
--- websites/production/lucene/content/solr/guide/8_8/simulations.html (original)
+++ websites/production/lucene/content/solr/guide/8_8/simulations.html Thu Jan 28 16:33:25 2021
@@ -8,7 +8,7 @@
<meta name="description" content="">
<meta name="keywords" content=" ">
-<title>Monte Carlo Simulations | Apache Solr Reference Guide 8.8-DRAFT</title>
+<title>Monte Carlo Simulations | Apache Solr Reference Guide 8.8</title>
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/css/bootstrap.min.css" integrity="sha384-MCw98/SFnGE8fJT3GXwEOngsV7Zt27NXFoaoApmYm81iuXoPkFOJwJ8ERdknLPMO" crossorigin="anonymous">
<link rel="stylesheet" type="text/css" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.5.0/css/font-awesome.min.css">
@@ -68,7 +68,7 @@
</script>
</head>
-<body class="DRAFT" id="simulations">
+<body class="" id="simulations">
<div class="container-fluid">
<div class="row">
<nav id="sidebar" class="col-2 d-none d-md-block">
@@ -76,11 +76,6 @@
<div class="sidebar-header">
<div class="sidebarTitle text-center">Apache Solr Reference Guide</div>
- <p class="draft-notice">
- This is an unofficial DRAFT of the Guide for 8.8.
- <a href="https://lucene.apache.org/solr/guide/">Official releases are available from the Solr website</a>.
- </p>
-
<!--comment out this block if you want to hide search-->
<!--start search-->
@@ -872,11 +867,36 @@
</li>
<li class="sb-level2">
- <a href="math-expressions.html">Math Expressions</a>
+ <a href="math-expressions.html">Streaming Expressions and Math Expressions</a>
<ul>
<li class="sb-level3">
+ <a href="visualization.html">Visualization</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="math-start.html">Getting Started</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="loading.html">Loading Data</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="search-sample.html">Searching, Sampling and Aggregation</a>
+
+ </li>
+
+ <li class="sb-level3">
+ <a href="transform.html">Transforming Data</a>
+
+ </li>
+
+ <li class="sb-level3">
<a href="scalar-math.html">Scalar Math</a>
</li>
@@ -897,12 +917,12 @@
</li>
<li class="sb-level3">
- <a href="vectorization.html">Streams and Vectorization</a>
+ <a href="term-vectors.html">Text Analysis and Term Vectors</a>
</li>
<li class="sb-level3">
- <a href="term-vectors.html">Text Analysis and Term Vectors</a>
+ <a href="probability-distributions.html">Probability Distributions</a>
</li>
@@ -912,12 +932,12 @@
</li>
<li class="sb-level3">
- <a href="probability-distributions.html">Probability Distributions</a>
+ <a href="regression.html">Linear Regression</a>
</li>
<li class="sb-level3">
- <a href="simulations.html">Monte Carlo Simulations</a>
+ <a href="curve-fitting.html">Curve Fitting</a>
</li>
@@ -927,32 +947,32 @@
</li>
<li class="sb-level3">
- <a href="regression.html">Linear Regression</a>
+ <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
</li>
<li class="sb-level3">
- <a href="numerical-analysis.html">Interpolation, Derivatives and Integrals</a>
+ <a href="dsp.html">Digital Signal Processing</a>
</li>
<li class="sb-level3">
- <a href="curve-fitting.html">Curve Fitting</a>
+ <a href="simulations.html">Monte Carlo Simulations</a>
</li>
<li class="sb-level3">
- <a href="dsp.html">Digital Signal Processing</a>
+ <a href="machine-learning.html">Machine Learning</a>
</li>
<li class="sb-level3">
- <a href="machine-learning.html">Machine Learning</a>
+ <a href="computational-geometry.html">Computational Geometry</a>
</li>
<li class="sb-level3">
- <a href="computational-geometry.html">Computational Geometry</a>
+ <a href="logs.html">Log Analytics</a>
</li>
@@ -1654,10 +1674,21 @@
<nav class="toc float-right justify-content-end">
<ul class="sectlevel1">
-<li><a href="#uncorrelated-simulations">Uncorrelated Simulations</a></li>
-<li><a href="#correlated-simulations">Correlated Simulations</a>
+<li><a href="#random-time-series">Random Time Series</a>
+<ul class="sectlevel2">
+<li><a href="#autocorrelation">Autocorrelation</a></li>
+<li><a href="#visualizing-the-distribution">Visualizing the Distribution</a></li>
+<li><a href="#fitting-the-distribution">Fitting the Distribution</a></li>
+<li><a href="#monte-carlo">Monte Carlo</a></li>
+<li><a href="#random-walk">Random Walk</a></li>
+</ul>
+</li>
+<li><a href="#multivariate-normal-distribution">Multivariate Normal Distribution</a>
<ul class="sectlevel2">
-<li><a href="#correlation-effects-on-the-probability-model">Correlation Effects on the Probability Model</a></li>
+<li><a href="#correlation-and-covariance">Correlation and Covariance</a></li>
+<li><a href="#covariance-matrix">Covariance Matrix</a></li>
+<li><a href="#multivariate-simulation">Multivariate Simulation</a></li>
+<li><a href="#the-effect-of-correlation">The Effect of Correlation</a></li>
</ul>
</li>
</ul>
@@ -1666,137 +1697,154 @@
<section class="content">
<section id="preamble" aria-label="Preamble"><p>Monte Carlo simulations are commonly used to model the behavior of
-stochastic systems. This section describes
-how to perform both uncorrelated and correlated Monte Carlo simulations
-using the sampling capabilities of the probability distribution framework.</p></section>
-<section class="sect1"><h2 id="uncorrelated-simulations">Uncorrelated Simulations</h2><p>Uncorrelated Monte Carlo simulations model stochastic systems with the assumption
-that the underlying random variables move independently of each other.
-A simple example of a Monte Carlo simulation using two independently changing random variables
-is described below.</p>
-<p>In this example a Monte Carlo simulation is used to determine the probability that a simple hinge assembly will
-fall within a required length specification.</p>
-<p>The hinge has two components A and B. The combined length of the two components must be less then 5 centimeters
-to fall within specification.</p>
-<p>A random sampling of lengths for component A has shown that its length conforms to a
-normal distribution with a mean of 2.2 centimeters and a standard deviation of .0195
-centimeters.</p>
-<p>A random sampling of lengths for component B has shown that its length conforms
-to a normal distribution with a mean of 2.71 centimeters and a standard deviation of .0198 centimeters.</p>
-<div class="listingblock"><pre class="rouge highlight"><code class="language-text" data-lang="text"><pre class="highlight"><code>let(componentA=normalDistribution(2.2, .0195), <i class="conum" data-value="1"></i>
- componentB=normalDistribution(2.71, .0198), <i class="conum" data-value="2"></i>
- simresults=monteCarlo(sampleA=sample(componentA), <i class="conum" data-value="3"></i>
- sampleB=sample(componentB),
- add(sampleA, sampleB), <i class="conum" data-value="4"></i>
- 100000), <i class="conum" data-value="5"></i>
- simmodel=empiricalDistribution(simresults), <i class="conum" data-value="6"></i>
- prob=cumulativeProbability(simmodel, 5)) <i class="conum" data-value="7"></i>
-</code></pre></code></pre></div>
-<p>The Monte Carlo simulation below performs the following steps:</p>
-<div class="colist arabic"><table><tr><td><i class="conum" data-value="1"></i><b>1</b></td><td>A normal distribution with a mean of 2.2 and a standard deviation of .0195 is created to model the length of <code>componentA</code>.</td></tr><tr><td><i class="conum" data-value="2"></i><b>2</b></td><td>A normal distribution with a mean of 2.71 and a standard deviation of .0198 is created to model the length of <code>componentB</code>.</td></tr><tr><td><i class="conum" data-value="3"></i><b>3</b></td><td>The <code>monteCarlo</code> function samples from the <code>componentA</code> and <code>componentB</code> distributions and sets the values to variables <code>sampleA</code> and <code>sampleB</code>.</td></tr><tr><td><i class="conum" data-value="4"></i><b>4</b></td><td>It then calls the <code>add(sampleA, sampleB)</code>* function to find the combined lengths of the samples.</td></tr><tr><td><i class="conum" data-value="5"></i><b>5</b></td><td>The <code>monteCarlo</code> function runs a s
et number of times, 100000, and collects the results in an array. Each
-time the function is called new samples are drawn from the <code>componentA</code>
-and <code>componentB</code> distributions. On each run, the <code>add</code> function adds the two samples to calculate the combined length.
-The result of each run is collected in an array and assigned to the <code>simresults</code> variable.</td></tr><tr><td><i class="conum" data-value="6"></i><b>6</b></td><td>An <code>empiricalDistribution</code> function is then created from the <code>simresults</code> array to model the distribution of the
-simulation results.</td></tr><tr><td><i class="conum" data-value="7"></i><b>7</b></td><td>Finally, the <code>cumulativeProbability</code> function is called on the <code>simmodel</code> to determine the cumulative probability
-that the combined length of the components is 5 or less.</td></tr></table></div>
-<p>Based on the simulation there is .9994371944629039 probability that the combined length of a component pair will
-be 5 or less:</p>
-<div class="listingblock"><pre class="rouge highlight"><code class="language-json" data-lang="json"><pre class="highlight"><code><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"result-set"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"docs"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="background-color: #f8f8f8">[</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"prob"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #009999">0.9994371944629039</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">},</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"EOF"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #000000;font-weight: bold">true</span><span style="background-color: #f8f8f8">,</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"RESPONSE_TIME"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #009999">660</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">}</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">]</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">}</span><span style="color: #bbbbbb">
-</span><span style="background-color: #f8f8f8">}</span></code></pre></code></pre></div></section>
-<section class="sect1"><h2 id="correlated-simulations">Correlated Simulations</h2><p>The simulation above assumes that the lengths of <code>componentA</code> and <code>componentB</code> vary independently.
-What would happen to the probability model if there was a correlation between the lengths of
-<code>componentA</code> and <code>componentB</code>?</p>
-<p>In the example below a database containing assembled pairs of components is used to determine
-if there is a correlation between the lengths of the components, and how the correlation effects the model.</p>
-<p>Before performing a simulation of the effects of correlation on the probability model its
-useful to understand what the correlation is between the lengths of <code>componentA</code> and <code>componentB</code>.</p>
-<div class="listingblock"><pre class="rouge highlight"><code class="language-text" data-lang="text"><pre class="highlight"><code>let(a=random(collection5, q="*:*", rows="5000", fl="componentA_d, componentB_d"), <i class="conum" data-value="1"></i>
- b=col(a, componentA_d)), <i class="conum" data-value="2"></i>
- c=col(a, componentB_d)),
- d=corr(b, c)) <i class="conum" data-value="3"></i>
-</code></pre></code></pre></div>
-<div class="colist arabic"><table><tr><td><i class="conum" data-value="1"></i><b>1</b></td><td>In the example, 5000 random samples are selected from a collection of assembled hinges.
-Each sample contains lengths of the components in the fields <code>componentA_d</code> and <code>componentB_d</code>.</td></tr><tr><td><i class="conum" data-value="2"></i><b>2</b></td><td>Both fields are then vectorized. The <strong>componentA_d</strong> vector is stored in
-variable <strong><code>b</code></strong> and the <strong>componentB_d</strong> variable is stored in variable <strong><code>c</code></strong>.</td></tr><tr><td><i class="conum" data-value="3"></i><b>3</b></td><td>Then the correlation of the two vectors is calculated using the <code>corr</code> function.</td></tr></table></div>
-<p>Note from the result that the outcome from <code>corr</code> is 0.9996931313216989.
-This means that <code>componentA_d</code> and *<code>componentB_d</code> are almost perfectly correlated.</p>
-<div class="listingblock"><pre class="rouge highlight"><code class="language-json" data-lang="json"><pre class="highlight"><code><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"result-set"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"docs"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="background-color: #f8f8f8">[</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"d"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #009999">0.9996931313216989</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">},</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"EOF"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #000000;font-weight: bold">true</span><span style="background-color: #f8f8f8">,</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"RESPONSE_TIME"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #009999">309</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">}</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">]</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">}</span><span style="color: #bbbbbb">
-</span><span style="background-color: #f8f8f8">}</span></code></pre></code></pre></div>
-<section class="sect2"><h3 id="correlation-effects-on-the-probability-model">Correlation Effects on the Probability Model</h3><p>The example below explores how to use a multivariate normal distribution function
-to model how correlation effects the probability of hinge defects.</p>
-<p>In this example 5000 random samples are selected from a collection
-containing length data for assembled hinges. Each sample contains
-the fields <code>componentA_d</code> and <code>componentB_d</code>.</p>
-<p>Both fields are then vectorized. The <code>componentA_d</code> vector is stored in
-variable <strong><code>b</code></strong> and the <code>componentB_d</code> variable is stored in variable <strong><code>c</code></strong>.</p>
-<p>An array is created that contains the means of the two vectorized fields.</p>
-<p>Then both vectors are added to a matrix which is transposed. This creates
-an observation matrix where each row contains one observation of
-<code>componentA_d</code> and <code>componentB_d</code>. A covariance matrix is then created from the columns of
-the observation matrix with the
-<code>cov</code> function. The covariance matrix describes the covariance between <code>componentA_d</code> and <code>componentB_d</code>.</p>
-<p>The <code>multivariateNormalDistribution</code> function is then called with the
-array of means for the two fields and the covariance matrix. The model
-for the multivariate normal distribution is stored in variable <strong><code>g</code></strong>.</p>
-<p>The <code>monteCarlo</code> function then calls the function <code>add(sample(g))</code> 50000 times
-and collections the results in a vector. Each time the function is called a single sample
-is drawn from the multivariate normal distribution. Each sample is a vector containing
-one <code>componentA</code> and <code>componentB</code> pair. The <code>add</code> function adds the values in the vector to
-calculate the length of the pair. Over the long term the samples drawn from the
-multivariate normal distribution will conform to the covariance matrix used to construct it.</p>
-<p>Just as in the non-correlated example an empirical distribution is used to model probabilities
-of the simulation vector and the <code>cumulativeProbability</code> function is used to compute the cumulative
-probability that the combined component length will be 5 centimeters or less.</p>
-<p>Notice that the probability of a hinge meeting specification has dropped to 0.9889517439980468.
-This is because the strong correlation
-between the lengths of components means that their lengths rise together causing more hinges to
-fall out of the 5 centimeter specification.</p>
-<div class="listingblock"><pre class="rouge highlight"><code class="language-text" data-lang="text"><pre class="highlight"><code>let(a=random(hinges, q="*:*", rows="5000", fl="componentA_d, componentB_d"),
- b=col(a, componentA_d),
- c=col(a, componentB_d),
- cor=corr(b,c),
- d=array(mean(b), mean(c)),
- e=transpose(matrix(b, c)),
- f=cov(e),
- g=multiVariateNormalDistribution(d, f),
- h=monteCarlo(add(sample(g)), 50000),
- i=empiricalDistribution(h),
- j=cumulativeProbability(i, 5))</code></pre></code></pre></div>
-<p>When this expression is sent to the <code>/stream</code> handler it responds with:</p>
-<div class="listingblock"><pre class="rouge highlight"><code class="language-json" data-lang="json"><pre class="highlight"><code><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"result-set"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"docs"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="background-color: #f8f8f8">[</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"j"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #009999">0.9889517439980468</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">},</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">{</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"EOF"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #000000;font-weight: bold">true</span><span style="background-color: #f8f8f8">,</span><span style="color: #bbbbbb">
- </span><span style="color: #000080">"RESPONSE_TIME"</span><span style="background-color: #f8f8f8">:</span><span style="color: #bbbbbb"> </span><span style="color: #009999">599</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">}</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">]</span><span style="color: #bbbbbb">
- </span><span style="background-color: #f8f8f8">}</span><span style="color: #bbbbbb">
-</span><span style="background-color: #f8f8f8">}</span></code></pre></code></pre></div></section></section>
+stochastic (random) systems. This section of the user guide covers
+the basics of performing Monte Carlo simulations with Math Expressions.</p></section>
+<section class="sect1"><h2 id="random-time-series">Random Time Series</h2><p>The daily movement of stock prices is often described as a "random walk".
+But what does that really mean, and how is this different than a random time series?
+The examples below will use Monte Carlo simulations to explore both "random walks"
+and random time series.</p>
+<p>A useful first step in understanding the difference is to visualize
+daily stock returns, calculated as closing price minus opening price, as a time series.</p>
+<p>The example below uses the <code>search</code> function to return 1000 days of daily stock
+returns for the ticker <strong>CVX</strong> (Chevron). The <code>change_d</code> field, which is the
+change in price for the day, is then plotted as a time series.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk1.png" alt="randomwalk1"></div>
+<p>Notice that the time series of daily price changes moves randomly above and
+below zero. Some days the stock is up, some days its down, but there
+does not seem to be a noticeable pattern or any dependency between steps. This is a hint
+that this is a <strong>random time series</strong>.</p>
+<section class="sect2"><h3 id="autocorrelation">Autocorrelation</h3><p>Autocorrelation measures the degree to which a signal is correlated with itself.
+ Autocorrelation can be used to determine
+if a vector contains a signal or if there is dependency between values in a time series. If there is no
+signal and no dependency between values in the time series then the time series is random.</p>
+<p>It’s useful to plot the autocorrelation of the <code>change_d</code> vector to confirm that it is indeed random.</p>
+<p>In the example below the search results are set to a variable and then the <code>change_d</code> field is vectorized and stored in variable <code>b</code>.
+Then the <code>conv</code> (convolution) function is used to autocorrelate
+the <code>change_d</code> vector.
+Notice that the <code>conv</code> function is simply "convolving" the <code>change_d</code> vector
+with a reversed copy of itself.
+This is the technique for performing autocorrelation using convolution.
+The <a href="dsp.html#dsp">Signal Processing</a> section
+of the user guide covers both convolution and autocorrelation in detail.
+In this section we’ll just discuss the plot.</p>
+<p>The plot shows the intensity of correlation that is calculated as the <code>change_d</code> vector is slid across itself by the <code>conv</code> function.
+Notice in the plot there is long period of low intensity correlation that appears to be random.
+Then in the center a peak of high intensity correlation where the vectors
+are directly lined up.
+This is followed by another long period of low intensity correlation.</p>
+<p>This is the autocorrelation plot of pure noise.
+The daily stock changes appear to be a random time series.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk2.png" alt="randomwalk2"></div></section>
+<section class="sect2"><h3 id="visualizing-the-distribution">Visualizing the Distribution</h3><p>The random daily changes in stock prices cannot be predicted, but they can be modeled with a probability distribution.
+To model the time series we’ll start by visualizing the distribution of the <code>change_d</code> vector.
+In the example below the <code>change_d</code> vector is plotted using the <code>empiricalDistribution</code> function to create an 11 bin
+histogram of the data.
+Notice that the distribution appears to be normally distributed.
+Daily stock price changes do tend to be normally distributed although <strong>CVX</strong> was chosen specifically for this example because of this characteristic.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk3.png" alt="randomwalk3"></div></section>
+<section class="sect2"><h3 id="fitting-the-distribution">Fitting the Distribution</h3><p>The <code>ks</code> test can be used to determine if the distribution of a vector of data fits a reference distribution.
+In the example below the <code>ks</code> test is performed with a <strong>normal distribution</strong> with the <strong>mean</strong> (<code>mean</code>) and <strong>standard deviation</strong> (<code>stddev</code>) of the <code>change_d</code> vector as the reference distribution.
+The <code>ks</code> test is checking the reference distribution against the <code>change_d</code> vector itself to see if it fits a normal distribution.</p>
+<p>Notice in the example below the <code>ks</code> test reports a p-value of .16278.
+A p-value of .05 or less is typically used to invalidate the null hypothesis of the test which is that the vector could have been drawn from the reference distribution.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk4.png" alt="randomwalk4"></div>
+<p>The <code>ks</code> test, which tends to be fairly sensitive, has confirmed the visualization which appeared to be normal.
+Because of this the normal distribution with the <strong>mean</strong> and <strong>standard deviation</strong> of the <code>change_d</code> vector will be used to represent the daily stock returns for Chevron in the Monte Carlo simulations below.</p></section>
+<section class="sect2"><h3 id="monte-carlo">Monte Carlo</h3><p>Now that we have fit a distribution to the daily stock return data we can use the <code>monteCarlo</code> function to run a simulation using the distribution.</p>
+<p>The <code>monteCarlo</code> function runs a specified number of times.
+On each run it sets a series of variables and runs one final function which returns a single numeric value.
+The <code>monteCarlo</code> function collects the results of each run in a vector and returns it.
+The final function typically has one or more variables that are drawn from probability distributions on each run.
+The <code>sample</code> function is used to draw the samples.</p>
+<p>The simulation’s result array can then be treated as an empirical distribution to understand the probabilities of the simulation results.</p>
+<p>The example below uses the <code>monteCarlo</code> function to simulate a distribution for the total return of 100 days of stock returns.</p>
+<p>In the example a <code>normalDistribution</code> is created from the <strong>mean</strong> and <strong>standard deviation</strong> of the <code>change_d</code> vector.
+The <code>monteCarlo</code> function then draws 100 samples from the normal distribution to represent 100 days of stock returns and sets the vector of samples to the variable <code>d</code>.</p>
+<p>The <code>add</code> function then calculates the total return
+from the 100 day sample.
+The output of the <code>add</code> function is collected by the <code>monteCarlo</code> function.
+This is repeated 50000 times, with each run drawing a different set of samples from the normal distribution.</p>
+<p>The result of the simulation is set to variable <code>s</code>, which contains
+the total returns from the 50000 runs.</p>
+<p>The <code>empiricalDistribution</code> function is then used to visualize the output of the simulation as a 50 bin histogram.
+The distribution visualizes the probability of the different total
+returns from 100 days of stock returns for ticker <strong>CVX</strong>.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk5.png" alt="randomwalk5"></div>
+<p>The <code>probability</code> and <code>cumulativeProbability</code> functions can then used to
+learn more about the <code>empiricalDistribution</code>.
+For example the <code>probability</code> function can be used to calculate the probability of a non-negative return from 100 days of stock returns.</p>
+<p>The example below uses the <code>probability</code> function to return the probability of a
+return between the range of 0 and 40 from the <code>empiricalDistribution</code>
+of the simulation.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk5.1.png" alt="randomwalk5.1"></div></section>
+<section class="sect2"><h3 id="random-walk">Random Walk</h3><p>The <code>monteCarlo</code> function can also be used to model a random walk of
+daily stock prices from the <code>normalDistribution</code> of daily stock returns.
+A random walk is a time series where each step is calculated by adding a random sample to the previous step.
+This creates a time series where each value is dependent on the previous value, which simulates the autocorrelation of stock prices.</p>
+<p>In the example below the random walk is achieved by adding a random sample to the variable <code>v</code> on each Monte Carlo iteration.
+The variable <code>v</code> is maintained between iterations so each iteration uses the previous value of <code>v</code>.
+The <code>double</code> function is the final function run each iteration, which simply returns the value of <code>v</code> as a double.
+The example iterates 1000 times to create a random walk with 1000 steps.</p>
+<div class="imageblock"><img src="images/math-expressions/randomwalk6.png" alt="randomwalk6"></div>
+<p>Notice the autocorrelation in the daily stock prices caused by the dependency
+between steps produces a very different plot then the
+random daily change in stock price.</p></section></section>
+<section class="sect1"><h2 id="multivariate-normal-distribution">Multivariate Normal Distribution</h2><p>The <code>multiVariateNormalDistribution</code> function can be used to model and simulate
+two or more normally distributed variables.
+It also incorporates the <strong>correlation</strong> between variables into the model which allows for the study of how correlation effects the possible outcomes.</p>
+<p>In the examples below a simulation of the total daily returns of two
+stocks is explored.
+The <strong>ALL</strong> ticker (<strong>Allstate</strong>) is used along with the <strong>CVX</strong> ticker (<strong>Chevron</strong>) from the previous examples.</p>
+<section class="sect2"><h3 id="correlation-and-covariance">Correlation and Covariance</h3><p>The multivariate simulations show the effect of correlation on possible
+outcomes.
+Before getting started with actual simulations it’s useful to first understand the correlation and covariance between the Allstate and Chevron stock returns.</p>
+<p>The example below runs two searches to retrieve the daily stock returns
+for all Allstate and Chevron.
+The <code>change_d</code> vectors from both returns are read into variables (<code>all</code> and <code>cvx</code>) and Pearson’s correlation is calculated for the two vectors with the <code>corr</code> function.</p>
+<div class="imageblock"><img src="images/math-expressions/corrsim1.png" alt="corrsim1"></div>
+<p>Covariance is an unscaled measure of correlation.
+Covariance is the measure used by the multivariate simulations so it’s useful to also compute the covariance for the two stock returns.
+The example below computes the covariance.</p>
+<div class="imageblock"><img src="images/math-expressions/corrsim2.png" alt="corrsim2"></div></section>
+<section class="sect2"><h3 id="covariance-matrix">Covariance Matrix</h3><p>A covariance matrix is actually whats needed by the
+<code>multiVariateNormalDistribution</code> as it contains both the variance of the
+two stock return vectors and the covariance between the two
+vectors.
+The <code>cov</code> function will compute the covariance matrix for the
+the columns of a matrix.</p>
+<p>The example below demonstrates how to compute the covariance matrix by adding the <code>all</code> and <code>cvx</code> vectors as rows to a matrix.
+The matrix is then transposed with the <code>transpose</code> function so that the <code>all</code> vector is the first column and the <code>cvx</code> vector is the second column.</p>
+<p>The <code>cov</code> function then computes the covariance matrix for the columns of the matrix and returns the result.</p>
+<div class="imageblock"><img src="images/math-expressions/corrsim3.png" alt="corrsim3"></div>
+<p>The covariance matrix is a square matrix which contains the
+variance of each vector and the covariance between the
+vectors as follows:</p>
+<div class="listingblock"><pre class="rouge highlight"><code class="language-text" data-lang="text"><pre class="highlight"><code> all cvx
+all [0.12294442137237226, 0.13106056985285258],
+cvx [0.13106056985285258, 0.7409729840230235]</code></pre></code></pre></div></section>
+<section class="sect2"><h3 id="multivariate-simulation">Multivariate Simulation</h3><p>The example below demonstrates a Monte Carlo simulation with two stock tickers using the
+<code>multiVariateNormalDistribution</code>.</p>
+<p>In the example, result sets with the <code>change_d</code> field for both stock tickers, <code>all</code> (Allstate) and <code>cvx</code> (Chevron),
+are retrieved and read into vectors.</p>
+<p>A matrix is then created from the two vectors and is transposed so
+the matrix contains two columns, one with the <code>all</code> vector and one with the <code>cvx</code> vector.</p>
+<p>Then the <code>multiVariateNormalDistribution</code> is created with two parameters. The first parameter is an array of <code>mean</code> values.
+In this case the means for the <code>all</code> vector and the <code>cvx</code> vector.
+The second parameter is the covariance matrix which was created from the 2-column matrix of the two vectors.</p>
+<p>The <code>monteCarlo</code> function then performs the simulation by drawing 100 samples from the <code>multiVariateNormalDistribution</code> on each iteration.
+Each sample set is a matrix with 100 rows and 2 columns containing stock return samples from the <code>all</code> and <code>cvx</code> distributions.
+The distributions of the columns will match the normal distributions used to create the <code>multiVariateNormalDistribution</code>.
+The covariance of the sample columns will match the covariance matrix.</p>
+<p>On each iteration the <code>grandSum</code> function is used to sum all the values of the sample matrix to get the total stock returns for both stocks.</p>
+<p>The output of the simulation is a vector which can be treated as an empirical distribution in exactly the same manner as the single stock ticker simulation.
+In this example it is plotted as a 50 bin histogram which visualizes the probability of the different total returns from 100 days of stock returns for the tickers <code>all</code> and <code>cvx</code>.</p>
+<div class="imageblock"><img src="images/math-expressions/mnorm.png" alt="mnorm"></div></section>
+<section class="sect2"><h3 id="the-effect-of-correlation">The Effect of Correlation</h3><p>The covariance matrix can be changed to study the effect on the simulation.
+The example below demonstrates this by providing a hard coded covariance matrix with a higher covariance value for the two vectors.
+This results is a simulated outcome distribution with a higher standard deviation or larger spread from the mean.
+This measures the degree that higher correlation produces higher volatility
+in the random walk.</p>
+<div class="imageblock"><img src="images/math-expressions/mnorm2.png" alt="mnorm2"></div></section></section>
</section>
@@ -1811,10 +1859,10 @@ fall out of the 5 centimeter specificati
<nav class="scrollnav row">
<div class="col-lg-12">
- <a class="btn btn-primary prev float-left" href="probability-distributions.html">Probability Distributions</a>
+ <a class="btn btn-primary prev float-left" href="dsp.html">Digital Signal Processing</a>
- <a class="btn btn-primary next float-right" href="time-series.html">Time Series</a>
+ <a class="btn btn-primary next float-right" href="machine-learning.html">Machine Learning</a>
</div>
</nav>
@@ -1826,7 +1874,7 @@ fall out of the 5 centimeter specificati
<div class="row">
<div class="col-lg-12 footer">
©2021 Apache Software Foundation. All rights reserved. <br />
- Site Version: 8.8-DRAFT <br />Site last generated: 2021-01-19 <br />
+ Site Version: 8.8 <br />Site last generated: 2021-01-28 <br />
<p><img src="images/solr-sunOnly-small.png" alt="Apache Solr"/></p>
</div>
</div>