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<!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/html; charset=UTF-8" /><title>76.2. Multivariate Statistics Examples</title><link rel="stylesheet" type="text/css" href="stylesheet.css" /><link rev="made" href="pgsql-docs@lists.postgresql.org" /><meta name="generator" content="DocBook XSL Stylesheets Vsnapshot" /><link rel="prev" href="row-estimation-examples.html" title="76.1. Row Estimation Examples" /><link rel="next" href="planner-stats-security.html" title="76.3. Planner Statistics and Security" /></head><body id="docContent" class="container-fluid col-10"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="5" align="center">76.2. Multivariate Statistics Examples</th></tr><tr><td width="10%" align="left"><a accesskey="p" href="row-estimation-examples.html" title="76.1. Row Estimation Examples">Prev</a> </td><td width="10%" align="left"><a accesskey="u" href="planner-stats-details.html" title="Chapter 76. How the Planner Uses Statistics">Up</a></td><th width="60%" align="center">Chapter 76. How the Planner Uses Statistics</th><td width="10%" align="right"><a accesskey="h" href="index.html" title="PostgreSQL 16.3 Documentation">Home</a></td><td width="10%" align="right"> <a accesskey="n" href="planner-stats-security.html" title="76.3. Planner Statistics and Security">Next</a></td></tr></table><hr /></div><div class="sect1" id="MULTIVARIATE-STATISTICS-EXAMPLES"><div class="titlepage"><div><div><h2 class="title" style="clear: both">76.2. Multivariate Statistics Examples <a href="#MULTIVARIATE-STATISTICS-EXAMPLES" class="id_link">#</a></h2></div></div></div><div class="toc"><dl class="toc"><dt><span class="sect2"><a href="multivariate-statistics-examples.html#FUNCTIONAL-DEPENDENCIES">76.2.1. Functional Dependencies</a></span></dt><dt><span class="sect2"><a href="multivariate-statistics-examples.html#MULTIVARIATE-NDISTINCT-COUNTS">76.2.2. Multivariate N-Distinct Counts</a></span></dt><dt><span class="sect2"><a href="multivariate-statistics-examples.html#MCV-LISTS">76.2.3. MCV Lists</a></span></dt></dl></div><a id="id-1.10.27.5.2" class="indexterm"></a><div class="sect2" id="FUNCTIONAL-DEPENDENCIES"><div class="titlepage"><div><div><h3 class="title">76.2.1. Functional Dependencies <a href="#FUNCTIONAL-DEPENDENCIES" class="id_link">#</a></h3></div></div></div><p>
Multivariate correlation can be demonstrated with a very simple data set
— a table with two columns, both containing the same values:
</p><pre class="programlisting">
CREATE TABLE t (a INT, b INT);
INSERT INTO t SELECT i % 100, i % 100 FROM generate_series(1, 10000) s(i);
ANALYZE t;
</pre><p>
As explained in <a class="xref" href="planner-stats.html" title="14.2. Statistics Used by the Planner">Section 14.2</a>, the planner can determine
cardinality of <code class="structname">t</code> using the number of pages and
rows obtained from <code class="structname">pg_class</code>:
</p><pre class="programlisting">
SELECT relpages, reltuples FROM pg_class WHERE relname = 't';
relpages | reltuples
----------+-----------
45 | 10000
</pre><p>
The data distribution is very simple; there are only 100 distinct values
in each column, uniformly distributed.
</p><p>
The following example shows the result of estimating a <code class="literal">WHERE</code>
condition on the <code class="structfield">a</code> column:
</p><pre class="programlisting">
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1;
QUERY PLAN
-------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..170.00 rows=100 width=8) (actual rows=100 loops=1)
Filter: (a = 1)
Rows Removed by Filter: 9900
</pre><p>
The planner examines the condition and determines the selectivity
of this clause to be 1%. By comparing this estimate and the actual
number of rows, we see that the estimate is very accurate
(in fact exact, as the table is very small). Changing the
<code class="literal">WHERE</code> condition to use the <code class="structfield">b</code> column, an
identical plan is generated. But observe what happens if we apply the same
condition on both columns, combining them with <code class="literal">AND</code>:
</p><pre class="programlisting">
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-----------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=1 width=8) (actual rows=100 loops=1)
Filter: ((a = 1) AND (b = 1))
Rows Removed by Filter: 9900
</pre><p>
The planner estimates the selectivity for each condition individually,
arriving at the same 1% estimates as above. Then it assumes that the
conditions are independent, and so it multiplies their selectivities,
producing a final selectivity estimate of just 0.01%.
This is a significant underestimate, as the actual number of rows
matching the conditions (100) is two orders of magnitude higher.
</p><p>
This problem can be fixed by creating a statistics object that
directs <code class="command">ANALYZE</code> to calculate functional-dependency
multivariate statistics on the two columns:
</p><pre class="programlisting">
CREATE STATISTICS stts (dependencies) ON a, b FROM t;
ANALYZE t;
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=100 width=8) (actual rows=100 loops=1)
Filter: ((a = 1) AND (b = 1))
Rows Removed by Filter: 9900
</pre><p>
</p></div><div class="sect2" id="MULTIVARIATE-NDISTINCT-COUNTS"><div class="titlepage"><div><div><h3 class="title">76.2.2. Multivariate N-Distinct Counts <a href="#MULTIVARIATE-NDISTINCT-COUNTS" class="id_link">#</a></h3></div></div></div><p>
A similar problem occurs with estimation of the cardinality of sets of
multiple columns, such as the number of groups that would be generated by
a <code class="command">GROUP BY</code> clause. When <code class="command">GROUP BY</code>
lists a single column, the n-distinct estimate (which is visible as the
estimated number of rows returned by the HashAggregate node) is very
accurate:
</p><pre class="programlisting">
EXPLAIN (ANALYZE, TIMING OFF) SELECT COUNT(*) FROM t GROUP BY a;
QUERY PLAN
-----------------------------------------------------------------------------------------
HashAggregate (cost=195.00..196.00 rows=100 width=12) (actual rows=100 loops=1)
Group Key: a
-> Seq Scan on t (cost=0.00..145.00 rows=10000 width=4) (actual rows=10000 loops=1)
</pre><p>
But without multivariate statistics, the estimate for the number of
groups in a query with two columns in <code class="command">GROUP BY</code>, as
in the following example, is off by an order of magnitude:
</p><pre class="programlisting">
EXPLAIN (ANALYZE, TIMING OFF) SELECT COUNT(*) FROM t GROUP BY a, b;
QUERY PLAN
--------------------------------------------------------------------------------------------
HashAggregate (cost=220.00..230.00 rows=1000 width=16) (actual rows=100 loops=1)
Group Key: a, b
-> Seq Scan on t (cost=0.00..145.00 rows=10000 width=8) (actual rows=10000 loops=1)
</pre><p>
By redefining the statistics object to include n-distinct counts for the
two columns, the estimate is much improved:
</p><pre class="programlisting">
DROP STATISTICS stts;
CREATE STATISTICS stts (dependencies, ndistinct) ON a, b FROM t;
ANALYZE t;
EXPLAIN (ANALYZE, TIMING OFF) SELECT COUNT(*) FROM t GROUP BY a, b;
QUERY PLAN
--------------------------------------------------------------------------------------------
HashAggregate (cost=220.00..221.00 rows=100 width=16) (actual rows=100 loops=1)
Group Key: a, b
-> Seq Scan on t (cost=0.00..145.00 rows=10000 width=8) (actual rows=10000 loops=1)
</pre><p>
</p></div><div class="sect2" id="MCV-LISTS"><div class="titlepage"><div><div><h3 class="title">76.2.3. MCV Lists <a href="#MCV-LISTS" class="id_link">#</a></h3></div></div></div><p>
As explained in <a class="xref" href="multivariate-statistics-examples.html#FUNCTIONAL-DEPENDENCIES" title="76.2.1. Functional Dependencies">Section 76.2.1</a>, functional
dependencies are very cheap and efficient type of statistics, but their
main limitation is their global nature (only tracking dependencies at
the column level, not between individual column values).
</p><p>
This section introduces multivariate variant of <acronym class="acronym">MCV</acronym>
(most-common values) lists, a straightforward extension of the per-column
statistics described in <a class="xref" href="row-estimation-examples.html" title="76.1. Row Estimation Examples">Section 76.1</a>. These
statistics address the limitation by storing individual values, but it is
naturally more expensive, both in terms of building the statistics in
<code class="command">ANALYZE</code>, storage and planning time.
</p><p>
Let's look at the query from <a class="xref" href="multivariate-statistics-examples.html#FUNCTIONAL-DEPENDENCIES" title="76.2.1. Functional Dependencies">Section 76.2.1</a>
again, but this time with a <acronym class="acronym">MCV</acronym> list created on the
same set of columns (be sure to drop the functional dependencies, to
make sure the planner uses the newly created statistics).
</p><pre class="programlisting">
DROP STATISTICS stts;
CREATE STATISTICS stts2 (mcv) ON a, b FROM t;
ANALYZE t;
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 1;
QUERY PLAN
-------------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=100 width=8) (actual rows=100 loops=1)
Filter: ((a = 1) AND (b = 1))
Rows Removed by Filter: 9900
</pre><p>
The estimate is as accurate as with the functional dependencies, mostly
thanks to the table being fairly small and having a simple distribution
with a low number of distinct values. Before looking at the second query,
which was not handled by functional dependencies particularly well,
let's inspect the <acronym class="acronym">MCV</acronym> list a bit.
</p><p>
Inspecting the <acronym class="acronym">MCV</acronym> list is possible using
<code class="function">pg_mcv_list_items</code> set-returning function.
</p><pre class="programlisting">
SELECT m.* FROM pg_statistic_ext join pg_statistic_ext_data on (oid = stxoid),
pg_mcv_list_items(stxdmcv) m WHERE stxname = 'stts2';
index | values | nulls | frequency | base_frequency
-------+----------+-------+-----------+----------------
0 | {0, 0} | {f,f} | 0.01 | 0.0001
1 | {1, 1} | {f,f} | 0.01 | 0.0001
...
49 | {49, 49} | {f,f} | 0.01 | 0.0001
50 | {50, 50} | {f,f} | 0.01 | 0.0001
...
97 | {97, 97} | {f,f} | 0.01 | 0.0001
98 | {98, 98} | {f,f} | 0.01 | 0.0001
99 | {99, 99} | {f,f} | 0.01 | 0.0001
(100 rows)
</pre><p>
This confirms there are 100 distinct combinations in the two columns, and
all of them are about equally likely (1% frequency for each one). The
base frequency is the frequency computed from per-column statistics, as if
there were no multi-column statistics. Had there been any null values in
either of the columns, this would be identified in the
<code class="structfield">nulls</code> column.
</p><p>
When estimating the selectivity, the planner applies all the conditions
on items in the <acronym class="acronym">MCV</acronym> list, and then sums the frequencies
of the matching ones. See <code class="function">mcv_clauselist_selectivity</code>
in <code class="filename">src/backend/statistics/mcv.c</code> for details.
</p><p>
Compared to functional dependencies, <acronym class="acronym">MCV</acronym> lists have two
major advantages. Firstly, the list stores actual values, making it possible
to decide which combinations are compatible.
</p><pre class="programlisting">
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a = 1 AND b = 10;
QUERY PLAN
---------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=1 width=8) (actual rows=0 loops=1)
Filter: ((a = 1) AND (b = 10))
Rows Removed by Filter: 10000
</pre><p>
Secondly, <acronym class="acronym">MCV</acronym> lists handle a wider range of clause types,
not just equality clauses like functional dependencies. For example,
consider the following range query for the same table:
</p><pre class="programlisting">
EXPLAIN (ANALYZE, TIMING OFF) SELECT * FROM t WHERE a <= 49 AND b > 49;
QUERY PLAN
---------------------------------------------------------------------------
Seq Scan on t (cost=0.00..195.00 rows=1 width=8) (actual rows=0 loops=1)
Filter: ((a <= 49) AND (b > 49))
Rows Removed by Filter: 10000
</pre><p>
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