Google Git
Sign in
apache / madlib-site / da7bed6a2b56b966779ebd8366a01690f8b18cf3 / . / docs / v1.18.0 / group__grp__pca__project.html
blob: 24b15d426268750784dd8596b868a33ff813ba56 [file] [log] [blame]
<!-- HTML header for doxygen 1.8.4-->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
<meta http-equiv="X-UA-Compatible" content="IE=9"/>
<meta name="generator" content="Doxygen 1.8.13"/>
<meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/>
<title>MADlib: Principal Component Projection</title>
<link href="tabs.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="jquery.js"></script>
<script type="text/javascript" src="dynsections.js"></script>
<link href="navtree.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="resize.js"></script>
<script type="text/javascript" src="navtreedata.js"></script>
<script type="text/javascript" src="navtree.js"></script>
<script type="text/javascript">
$(document).ready(initResizable);
</script>
<link href="search/search.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="search/searchdata.js"></script>
<script type="text/javascript" src="search/search.js"></script>
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
jax: ["input/TeX","output/HTML-CSS"],
});
</script><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js"></script>
<!-- hack in the navigation tree -->
<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
<!-- google analytics -->
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-45382226-1', 'madlib.apache.org');
ga('send', 'pageview');
</script>
</head>
<body>
<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
<div id="titlearea">
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
<td id="projectlogo"><a href="http://madlib.apache.org"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
<span id="projectnumber">1.18.0</span>
</div>
<div id="projectbrief">User Documentation for Apache MADlib</div>
</td>
<td> <div id="MSearchBox" class="MSearchBoxInactive">
<span class="left">
<img id="MSearchSelect" src="search/mag_sel.png"
onmouseover="return searchBox.OnSearchSelectShow()"
onmouseout="return searchBox.OnSearchSelectHide()"
alt=""/>
<input type="text" id="MSearchField" value="Search" accesskey="S"
onfocus="searchBox.OnSearchFieldFocus(true)"
onblur="searchBox.OnSearchFieldFocus(false)"
onkeyup="searchBox.OnSearchFieldChange(event)"/>
</span><span class="right">
<a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a>
</span>
</div>
</td>
</tr>
</tbody>
</table>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.8.13 -->
<script type="text/javascript">
var searchBox = new SearchBox("searchBox", "search",false,'Search');
</script>
</div><!-- top -->
<div id="side-nav" class="ui-resizable side-nav-resizable">
<div id="nav-tree">
<div id="nav-tree-contents">
<div id="nav-sync" class="sync"></div>
</div>
</div>
<div id="splitbar" style="-moz-user-select:none;"
class="ui-resizable-handle">
</div>
</div>
<script type="text/javascript">
$(document).ready(function(){initNavTree('group__grp__pca__project.html','');});
</script>
<div id="doc-content">
<!-- window showing the filter options -->
<div id="MSearchSelectWindow"
onmouseover="return searchBox.OnSearchSelectShow()"
onmouseout="return searchBox.OnSearchSelectHide()"
onkeydown="return searchBox.OnSearchSelectKey(event)">
</div>
<!-- iframe showing the search results (closed by default) -->
<div id="MSearchResultsWindow">
<iframe src="javascript:void(0)" frameborder="0"
name="MSearchResults" id="MSearchResults">
</iframe>
</div>
<div class="header">
<div class="headertitle">
<div class="title">Principal Component Projection<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a> &raquo; <a class="el" href="group__grp__pca.html">Dimensionality Reduction</a></div></div> </div>
</div><!--header-->
<div class="contents">
<div class="toc"><b>Contents</b> <ul>
<li class="level1">
<a href="#project">Projection Function</a> </li>
<li class="level1">
<a href="#examples">Examples</a> </li>
<li class="level1">
<a href="#notes">Notes</a> </li>
<li class="level1">
<a href="#background_project">Technical Background</a> </li>
<li class="level1">
<a href="#related">Related Topics</a> </li>
</ul>
</div><p>Principal component projection is a mathematical procedure that projects high dimensional data onto a lower dimensional space. This lower dimensional space is defined by the \( k \) principal components with the highest variance in the training data.</p>
<p>More details on the mathematics of PCA can be found in <a class="el" href="group__grp__pca__train.html">Principal Component Analysis</a> and some details about principal component projection calculations can be found in the <a class="el" href="group__grp__pca__project.html#background_project">Technical Background</a>.</p>
<p><a class="anchor" id="project"></a></p><dl class="section user"><dt>Projection Function</dt><dd>The projection functions are slightly different for dense and sparse matrices. For dense matrices: <pre class="syntax">
madlib.pca_project( source_table,
pc_table,
out_table,
row_id,
residual_table,
result_summary_table
)
</pre> For sparse matrices: <pre class="syntax">
madlib.pca_sparse_project( source_table,
pc_table,
out_table,
row_id,
col_id, -- Sparse matrices only
val_id, -- Sparse matrices only
row_dim, -- Sparse matrices only
col_dim, -- Sparse matrices only
residual_table,
result_summary_table
)
</pre></dd></dl>
<p><b>Arguments</b> </p><dl class="arglist">
<dt>source_table </dt>
<dd><p class="startdd">TEXT. Source table name. Identical to <a class="el" href="pca_8sql__in.html#a31abf88e67a446a4f789764aa2c61e85">pca_train</a>, the input data matrix should have \( N \) rows and \( M \) columns, where \( N \) is the number of data points, and \( M \) is the number of features for each data point.</p>
<p>The input table for <em> pca_project </em> is expected to be in the one of the two standard MADlib dense matrix formats, and the sparse input table for <em> pca_sparse_project </em> should be in the standard MADlib sparse matrix format. These formats are described in the documentation for <a class="el" href="group__grp__pca__train.html">Principal Component Analysis</a>.</p>
<p class="enddd"></p>
</dd>
<dt>pc_table </dt>
<dd><p class="startdd">TEXT. Table name for the table containing principal components. </p>
<p class="enddd"></p>
</dd>
<dt>out_table </dt>
<dd><p class="startdd">TEXT. Name of the table that will contain the low-dimensional representation of the input data.</p>
<p>The <em>out_table</em> encodes a dense matrix with the projection onto the principal components. The table has the following columns:</p>
<table class="output">
<tr>
<th>row_id </th><td>Row id of the output matrix. </td></tr>
<tr>
<th>row_vec </th><td>A vector containing elements in the row of the matrix. </td></tr>
</table>
<p class="enddd"></p>
</dd>
<dt>row_id </dt>
<dd><p class="startdd">TEXT. Column name containing the row IDs in the input source table. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and <em>N</em>. For dense matrix format, it should contain all continguous integers from 1 to <em>N</em> describing the full matrix.</p>
<p class="enddd"></p>
</dd>
<dt>col_id </dt>
<dd><p class="startdd">TEXT. Column name containing the column IDs in sparse matrix representation. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and <em>M</em>. <em>This parameter applies to sparse matrices only.</em></p>
<p class="enddd"></p>
</dd>
<dt>val_id </dt>
<dd><p class="startdd">TEXT. Name of 'val_id' column in sparse matrix representation defining the values of the nonzero entries. <em>This parameter applies to sparse matrices only.</em></p>
<p class="enddd"></p>
</dd>
<dt>row_dim </dt>
<dd><p class="startdd">INTEGER. The actual number of rows in the matrix. That is, if the matrix was transformed into dense format, this is the number of rows it would have. <em>This parameter applies to sparse matrices only.</em></p>
<p class="enddd"></p>
</dd>
<dt>col_dim </dt>
<dd><p class="startdd">INTEGER. The actual number of columns in the matrix. That is, if the matrix was transformed into dense format, this is the number of columns it would have. <em>This parameter applies to sparse matrices only.</em></p>
<dl class="section note"><dt>Note</dt><dd>The parameters 'row_dim' and 'col_dim' could actually be inferred from the sparse matrix representation, so they will be removed in the future. For now they are maintained for backward compatability so you must enter them. Making 'row_dim' or 'col_dim' larger than the actual matrix has the effect of padding it with zeros, which is probably not useful.</dd></dl>
</dd>
<dt>residual_table (optional) </dt>
<dd><p class="startdd">TEXT, default: NULL. Name of the optional residual table.</p>
<p>The <em>residual_table</em> encodes a dense residual matrix. The table has the following columns:</p>
<table class="output">
<tr>
<th>row_id </th><td>Row id of the output matrix. </td></tr>
<tr>
<th>row_vec </th><td>A vector containing elements in the row of the residual matrix. </td></tr>
</table>
<p class="enddd"></p>
</dd>
<dt>result_summary_table (optional) </dt>
<dd><p class="startdd">TEXT, default: NULL. Name of the optional summary table.</p>
<p class="enddd">The <em>result_summary_table</em> contains information about the performance time of the PCA projection. The table has the following columns: </p><table class="output">
<tr>
<th>exec_time </th><td>Elapsed time (ms) for execution of the function. </td></tr>
<tr>
<th>residual_norm </th><td>Absolute error of the residuals. </td></tr>
<tr>
<th>relative_residual_norm </th><td>Relative error of the residuals. </td></tr>
</table>
</dd>
</dl>
<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1">
<li>View online help for the PCA projection function: <pre class="example">
SELECT madlib.pca_project();
</pre></li>
<li>Create sample data in dense matrix form: <pre class="example">
DROP TABLE IF EXISTS mat;
CREATE TABLE mat (id integer,
row_vec double precision[]
);
INSERT INTO mat VALUES
(1, '{1,2,3}'),
(2, '{2,1,2}'),
(3, '{3,2,1}');
</pre></li>
<li>Run the PCA function for a specified number of principal components and view the results: <pre class="example">
DROP TABLE IF EXISTS result_table, result_table_mean;
SELECT madlib.pca_train('mat', -- Source table
'result_table', -- Output table
'id', -- Row id of source table
2); -- Number of principal components
SELECT * FROM result_table ORDER BY row_id;
</pre> <pre class="result">
row_id | principal_components | std_dev | proportion
--------+--------------------------------------------------------------+-------------------+-------------------
1 | {0.707106781186547,-6.93889390390723e-18,-0.707106781186548} | 1.41421356237309 | 0.857142857142244
2 | {0,1,0} | 0.577350269189626 | 0.142857142857041
(2 rows)
</pre></li>
<li>Project the original data to a lower dimensional representation and view the result of the projection: <pre class="example">
DROP TABLE IF EXISTS residual_table, result_summary_table, out_table;
SELECT madlib.pca_project( 'mat',
'result_table',
'out_table',
'id',
'residual_table',
'result_summary_table'
);
SELECT * FROM out_table ORDER BY row_id;
</pre> <pre class="result">
row_id | row_vec
--------+--------------------------------------
1 | {-1.41421356237309,-0.33333333333}
2 | {2.77555756157677e-17,0.66666666667}
3 | {1.41421356237309,-0.33333333333}
(3 rows)
</pre> Check the error in the projection: <pre class="example">
SELECT * FROM result_summary_table;
</pre> <pre class="result">
exec_time | residual_norm | relative_residual_norm
---------------+-------------------+------------------------
331.792116165 | 5.89383520611e-16 | 9.68940539229e-17
(1 row)
</pre> Check the residuals: <pre class="example">
SELECT * FROM residual_table ORDER BY row_id;
</pre> <pre class="result">
row_id | row_vec
--------+--------------------------------------------------------------------
1 | {-2.22044604925031e-16,-1.11022302462516e-16,3.33066907387547e-16}
2 | {-1.12243865646685e-18,0,4.7381731349413e-17}
3 | {2.22044604925031e-16,1.11022302462516e-16,-3.33066907387547e-16}
(3 rows)
</pre></li>
<li>Now we use grouping in dense form to learn different models for different groups. First, we create sample data in dense matrix form with a grouping column. Note we actually have different matrix sizes for the different groups, which is allowed for dense: <pre class="example">
DROP TABLE IF EXISTS mat_group;
CREATE TABLE mat_group (
id integer,
row_vec double precision[],
matrix_id integer
);
INSERT INTO mat_group VALUES
(1, '{1,2,3}', 1),
(2, '{2,1,2}', 1),
(3, '{3,2,1}', 1),
(4, '{1,2,3,4,5}', 2),
(5, '{2,5,2,4,1}', 2),
(6, '{5,4,3,2,1}', 2);
</pre></li>
<li>Run the PCA function with grouping for a specified proportion of variance and view the results: <pre class="example">
DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
SELECT madlib.pca_train('mat_group', -- Source table
'result_table_group', -- Output table
'id', -- Row id of source table
0.8, -- Proportion of variance
'matrix_id'); -- Grouping column
SELECT * FROM result_table_group ORDER BY matrix_id, row_id;
</pre> <pre class="result">
row_id | principal_components | std_dev | proportion | matrix_id
--------+------------------------------------------------------------------------------------------------+-----------------+-------------------+-----------
1 | {0.707106781186548,0,-0.707106781186547} | 1.4142135623731 | 0.857142857142245 | 1
1 | {-0.555378486712784,-0.388303582074091,0.0442457354870796,0.255566375612852,0.688115693174023} | 3.2315220311722 | 0.764102534485173 | 2
2 | {0.587384101786277,-0.485138064894743,0.311532046315153,-0.449458074050715,0.347212037159181} | 1.795531127192 | 0.235897465516047 | 2
(3 rows)
</pre></li>
<li>Run the PCA projection on subsets of an input table based on grouping columns. Note that the parameter 'pc_table' used for projection must be generated in training using the same grouping columns. <pre class="example">
DROP TABLE IF EXISTS mat_group_projected;
SELECT madlib.pca_project('mat_group',
'result_table_group',
'mat_group_projected',
'id');
SELECT * FROM mat_group_projected ORDER BY matrix_id, row_id;
</pre> <pre class="result">
row_id | row_vec | matrix_id
--------+---------------------------------------+-----------
1 | {1.4142135623731} | 1
2 | {7.40148683087139e-17} | 1
3 | {-1.4142135623731} | 1
4 | {-3.59290479201926,0.559694003674779} | 2
5 | {0.924092949098971,-2.00871628417505} | 2
6 | {2.66881184290186,1.44902228049511} | 2
(6 rows)
</pre></li>
<li>Now let's look at sparse matrices. Create sample data in sparse matrix form: <pre class="example">
DROP TABLE IF EXISTS mat_sparse;
CREATE TABLE mat_sparse (
row_id integer,
col_id integer,
value double precision
);
INSERT INTO mat_sparse VALUES
(1, 1, 1.0),
(2, 2, 2.0),
(3, 3, 3.0),
(4, 4, 4.0),
(1, 5, 5.0),
(2, 4, 6.0),
(3, 2, 7.0),
(4, 3, 8.0);
</pre> As an aside, this is what the sparse matrix above looks like when put in dense form: <pre class="example">
DROP TABLE IF EXISTS mat_dense;
SELECT madlib.matrix_densify('mat_sparse',
'row=row_id, col=col_id, val=value',
'mat_dense');
SELECT * FROM mat_dense ORDER BY row_id;
</pre> <pre class="result">
row_id | value
--------+-------------
1 | {1,0,0,0,5}
2 | {0,2,0,6,0}
3 | {0,7,3,0,0}
4 | {0,0,8,4,0}
(4 rows)
</pre></li>
<li>Run the PCA sparse function for a specified number of principal components and view the results: <pre class="example">DROP TABLE IF EXISTS result_table, result_table_mean;
SELECT madlib.pca_sparse_train( 'mat_sparse', -- Source table
'result_table', -- Output table
'row_id', -- Row id of source table
'col_id', -- Column id of source table
'value', -- Value of matrix at row_id, col_id
4, -- Actual number of rows in the matrix
5, -- Actual number of columns in the matrix
3); -- Number of principal components
SELECT * FROM result_table ORDER BY row_id;
</pre> Result (with principal components truncated for readability): <pre class="result">
row_id | principal_components | std_dev | proportion
--------+----------------------------------------------+------------------+-------------------
1 | {-0.0876046030186158,-0.0968983772909994,... | 4.21362803829554 | 0.436590030617467
2 | {-0.0647272661608605,0.877639526308692,... | 3.68408023747461 | 0.333748701544697
3 | {-0.0780380267884855,0.177956517174911,... | 3.05606908060098 | 0.229661267837836
(3 rows)
</pre></li>
<li>Project the original sparse data to low-dimensional representation: <pre class="example">
DROP TABLE IF EXISTS mat_sparse_out;
SELECT madlib.pca_sparse_project(
'mat_sparse',
'result_table',
'mat_sparse_out',
'row_id',
'col_id',
'value',
4,
5
);
SELECT * FROM mat_sparse_out ORDER BY row_id;
</pre> <pre class="result">
row_id | row_vec
--------+---------------------------------------------------------
1 | {4.66617015032369,-2.63552220635847,2.1865220849604}
2 | {0.228360685652383,-1.21616275892926,-4.46864627611561}
3 | {0.672067460100428,5.45249627172823,0.56445525585642}
4 | {-5.5665982960765,-1.6008113064405,1.71766893529879}
(4 rows)
</pre></li>
<li>Now we use grouping in sparse form to learn different models for different groups. First, we create sample data in sparse matrix form with a grouping column: <pre class="example">
DROP TABLE IF EXISTS mat_sparse_group;
CREATE TABLE mat_sparse_group (
row_id integer,
col_id integer,
value double precision,
matrix_id integer);
INSERT INTO mat_sparse_group VALUES
(1, 1, 1.0, 1),
(2, 2, 2.0, 1),
(3, 3, 3.0, 1),
(4, 4, 4.0, 1),
(1, 5, 5.0, 1),
(2, 4, 6.0, 2),
(3, 2, 7.0, 2),
(4, 3, 8.0, 2);
</pre></li>
<li>Run the PCA function with grouping for a specified proportion of variance and view the results: <pre class="example">
DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
SELECT madlib.pca_sparse_train( 'mat_sparse_group', -- Source table
'result_table_group', -- Output table
'row_id', -- Row id of source table
'col_id', -- Column id of source table
'value', -- Value of matrix at row_id, col_id
4, -- Actual number of rows in the matrix
5, -- Actual number of columns in the matrix
0.8, -- Proportion of variance
'matrix_id');
SELECT * FROM result_table_group ORDER BY matrix_id, row_id;
</pre> Result (with principal components truncated for readability): <pre class="result">
row_id | principal_components | std_dev | proportion | matrix_id
--------+--------------------------------------------+------------------+-------------------+-----------
1 | {-0.17805696611353,0.0681313257646983,... | 2.73659933165925 | 0.544652792875481 | 1
2 | {-0.0492086814863993,0.149371585357526,... | 2.06058314533194 | 0.308800210823714 | 1
1 | {0,-0.479486114660443,... | 4.40325305087975 | 0.520500333693473 | 2
2 | {0,0.689230898585949,... | 3.7435566458567 | 0.376220573442628 | 2
(4 rows)
</pre></li>
<li>Projection in sparse format with grouping: <pre class="example">
DROP TABLE IF EXISTS mat_sparse_group_projected;
SELECT madlib.pca_sparse_project(
'mat_sparse_group',
'result_table_group',
'mat_sparse_group_projected',
'row_id',
'col_id',
'value',
4,
5
);
SELECT * FROM mat_sparse_group_projected ORDER BY matrix_id, row_id;
</pre> <pre class="result">
row_id | row_vec | matrix_id
--------+-----------------------------------------+-----------
1 | {-4.00039298524261,-0.626820612715982} | 1
2 | {0.765350785238575,0.951348276645455} | 1
3 | {1.04951017256904,2.22388180170356} | 1
4 | {2.185532027435,-2.54840946563303} | 1
1 | {-0.627846810195469,-0.685031603549092} | 2
2 | {-1.64754249747757,-4.7662114622896} | 2
3 | {-3.98424961281857,4.13958468655255} | 2
4 | {6.25963892049161,1.31165837928614} | 2
(8 rows)
</pre></li>
</ol>
</dd></dl>
<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd><ul>
<li>This function is intended to operate on the principal component tables generated by <em> pca_train </em> or <em> pca_sparse_train</em>. The MADlib PCA functions generate a table containing the column-means in addition to a table containing the principal components. If this table is not found by the MADlib projection function, it will trigger an error. As long the principal component tables are created with MADlib functions, then the column-means table will be automatically found by the MADlib projection functions.</li>
<li>Because of the centering step in PCA projection (see "Technical Background"), sparse matrices almost always become dense during the projection process. Thus, this implementation automatically densifies sparse matrix input, and there should be no expected performance improvement in using sparse matrix input over dense matrix input.</li>
<li>Table names can be optionally schema qualified (current_schemas() is searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').</li>
<li>If the input table for pca_project (pca_sparse_project) contains grouping columns, the same grouping columns must be used in the training function used to generate the principal components too.</li>
</ul>
</dd></dl>
<p><a class="anchor" id="background_project"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
<p>Given a table containing some principal components \( \boldsymbol P \) and some input data \( \boldsymbol X \), the low-dimensional representation \( {\boldsymbol X}&#39; \) is computed as </p><p class="formulaDsp">
\begin{align*} {\boldsymbol {\hat{X}}} &amp; = {\boldsymbol X} - \vec{e} \hat{x}^T \\ {\boldsymbol X}&#39; &amp; = {\boldsymbol {\hat {X}}} {\boldsymbol P}. \end{align*}
</p>
<p> where \(\hat{x} \) is the column means of \( \boldsymbol X \) and \( \vec{e} \) is the vector of all ones. This step is equivalent to centering the data around the origin.</p>
<p>The residual table \( \boldsymbol R \) is a measure of how well the low-dimensional representation approximates the true input data, and is computed as </p><p class="formulaDsp">
\[ {\boldsymbol R} = {\boldsymbol {\hat{X}}} - {\boldsymbol X}&#39; {\boldsymbol P}^T. \]
</p>
<p> A residual matrix with entries mostly close to zero indicates a good representation.</p>
<p>The residual norm \( r \) is simply </p><p class="formulaDsp">
\[ r = \|{\boldsymbol R}\|_F \]
</p>
<p> where \( \|\cdot\|_F \) is the Frobenius norm. The relative residual norm \( r&#39; \) is </p><p class="formulaDsp">
\[ r&#39; = \frac{ \|{\boldsymbol R}\|_F }{\|{\boldsymbol X}\|_F } \]
</p>
<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="pca__project_8sql__in.html" title="Principal Component Analysis Projection. ">pca_project.sql_in</a> documenting the SQL functions</dd></dl>
<p><a class="el" href="group__grp__pca__train.html">Principal Component Analysis</a> </p>
</div><!-- contents -->
</div><!-- doc-content -->
<!-- start footer part -->
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="footer">Generated on Wed Mar 31 2021 20:45:50 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
</ul>
</div>
</body>
</html>