# 70.2.1. 欄位相依性

Multivariate correlation can be demonstrated with a very simple data set — a table with two columns, both containing the same values:

`CREATE TABLE t (a INT, b INT);INSERT INTO t SELECT i % 100, i % 100 FROM generate_series(1, 10000) s(i);ANALYZE t;`

`SELECT relpages, reltuples FROM pg_class WHERE relname = 't';​ relpages | reltuples----------+-----------       45 |     10000`

The data distribution is very simple; there are only 100 distinct values in each column, uniformly distributed.

The following example shows the result of estimating a `WHERE` condition on the `a` column:

`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`

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 `WHERE` condition to use the `b` column, an identical plan is generated. But observe what happens if we apply the same condition on both columns, combining them with `AND`:

`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`

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.

This problem can be fixed by creating a statistics object that directs `ANALYZE` to calculate functional-dependency multivariate statistics on the two columns:

`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`

# 70.2.2. 多元 N-Distinct 組合數量

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 `GROUP BY` clause. When `GROUP BY` 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:

`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)`

But without multivariate statistics, the estimate for the number of groups in a query with two columns in `GROUP BY`, as in the following example, is off by an order of magnitude:

`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)`

By redefining the statistics object to include n-distinct counts for the two columns, the estimate is much improved:

`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)`

# 70.2.3. MCV Lists (最常見值列表)

`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`

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 MCV list a bit.

Inspecting the MCV list is possible using `pg_mcv_list_items` set-returning function.

`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)`

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 `nulls` column.

When estimating the selectivity, the planner applies all the conditions on items in the MCV list, and then sums the frequencies of the matching ones. See `mcv_clauselist_selectivity` in `src/backend/statistics/mcv.c` for details.

Compared to functional dependencies, MCV lists have two major advantages. Firstly, the list stores actual values, making it possible to decide which combinations are compatible.

`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`

Secondly, MCV 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:

`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`
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