population & sample covariance, standard deviation Aggregate in PostgreSQL
背景
PostgreSQL自带了一些常用的统计学聚合函数, 非常好用.
本文介绍一下方差和标准差的一些聚合函数.
总体方差 : population covariance
总体标准差 : population standard deviation
样本方差 : sample covariance
样本标准差 : sample standard deviation
均值 : mean
样本均值和样本方差的介绍 :
http://en.wikipedia.org/wiki/Sample_mean_and_sample_covariance
均值介绍 :
http://en.wikipedia.org/wiki/Mean
对方差, 标准差, 均值不了解的话, 建议参考网易公开课, 统计学.
浅显易懂.
http://v.163.com/special/Khan/khstatistics.html
http://v.ku6.com/playlist/index_6598382.html
PostgreSQL计算方差, 标准差的聚合函数如下 :
http://www.postgresql.org/docs/devel/static/functions-aggregate.html
| 函数 | 参数类型 | 返回值类型 | 描述 |
|---|---|---|---|
| stddev(expression) | smallint, int, bigint, real, double precision, or numeric | double precision for floating-point arguments, otherwise numeric | historical alias for stddev_samp |
| stddev_pop(expression) | smallint, int, bigint, real, double precision, or numeric | double precision for floating-point arguments, otherwise numeric | population standard deviation of the input values |
| stddev_samp(expression) | smallint, int, bigint, real, double precision, or numeric | double precision for floating-point arguments, otherwise numeric | sample standard deviation of the input values |
| variance(expression) | smallint, int, bigint, real, double precision, or numeric | double precision for floating-point arguments, otherwise numeric | historical alias for var_samp |
| var_pop(expression) | smallint, int, bigint, real, double precision, or numeric | double precision for floating-point arguments, otherwise numeric | population variance of the input values (square of the population standard deviation) |
| var_samp(expression) | smallint, int, bigint, real, double precision, or numeric | double precision for floating-point arguments, otherwise numeric | sample variance of the input values (square of the sample standard deviation) |
其中stddev和variance是stddev_samp和var_samp的别名.
这些函数用于计算数据集的总体/样本 方差,总体/样本 标准差.
例如 :
1,2,3,100 这组数据共4个值, 总体均值和样本均值分别为 :
(1+2+3+100)/4 = 26.5
总体方差 : ((1-26.5)^2 + (2-26.5)^2 + (3-26.5)^2 + (100-26.5)^2)/4 = 1801.25
样本方差 : ((1-26.5)^2 + (2-26.5)^2 + (3-26.5)^2 + (100-26.5)^2)/(4-1) = 2401.6666….
总体标准差 : 平方根(总体方差) = 42.4411357058220109
样本标准差 : 平方根(样本方差) = 49.0068022489395513
使用PostgreSQL计算如下 :
postgres=# select variance(id) from (values(1),(2),(3),(100)) as t(id);
variance
-----------------------
2401.6666666666666667
(1 row)
postgres=# select var_pop(id) from (values(1),(2),(3),(100)) as t(id);
var_pop
-----------------------
1801.2500000000000000
(1 row)
postgres=# select var_samp(id) from (values(1),(2),(3),(100)) as t(id);
var_samp
-----------------------
2401.6666666666666667
(1 row)
postgres=# select stddev(id) from (values(1),(2),(3),(100)) as t(id);
stddev
---------------------
49.0068022489395513
(1 row)
postgres=# select stddev_pop(id) from (values(1),(2),(3),(100)) as t(id);
stddev_pop
---------------------
42.4411357058220109
(1 row)
postgres=# select stddev_samp(id) from (values(1),(2),(3),(100)) as t(id);
stddev_samp
---------------------
49.0068022489395513
(1 row)
参考
1. src/backend/utils/adt/float.c
/*
* =========================
* FLOAT AGGREGATE OPERATORS
* =========================
*
* float8_accum - accumulate for AVG(), variance aggregates, etc.
* float4_accum - same, but input data is float4
* float8_avg - produce final result for float AVG()
* float8_var_samp - produce final result for float VAR_SAMP()
* float8_var_pop - produce final result for float VAR_POP()
* float8_stddev_samp - produce final result for float STDDEV_SAMP()
* float8_stddev_pop - produce final result for float STDDEV_POP()
*
* The transition datatype for all these aggregates is a 3-element array
* of float8, holding the values N, sum(X), sum(X*X) in that order.
*
* Note that we represent N as a float to avoid having to build a special
* datatype. Given a reasonable floating-point implementation, there should
* be no accuracy loss unless N exceeds 2 ^ 52 or so (by which time the
* user will have doubtless lost interest anyway...)
*/
..................
Datum
float8_var_pop(PG_FUNCTION_ARGS)
{
ArrayType *transarray = PG_GETARG_ARRAYTYPE_P(0);
float8 *transvalues;
float8 N,
sumX,
sumX2,
numerator;
transvalues = check_float8_array(transarray, "float8_var_pop", 3);
N = transvalues[0];
sumX = transvalues[1];
sumX2 = transvalues[2];
/* Population variance is undefined when N is 0, so return NULL */
if (N == 0.0)
PG_RETURN_NULL();
numerator = N * sumX2 - sumX * sumX;
CHECKFLOATVAL(numerator, isinf(sumX2) || isinf(sumX), true);
/* Watch out for roundoff error producing a negative numerator */
if (numerator <= 0.0)
PG_RETURN_FLOAT8(0.0);
PG_RETURN_FLOAT8(numerator / (N * N));
}
Datum
float8_var_samp(PG_FUNCTION_ARGS)
{
ArrayType *transarray = PG_GETARG_ARRAYTYPE_P(0);
float8 *transvalues;
float8 N,
sumX,
sumX2,
numerator;
transvalues = check_float8_array(transarray, "float8_var_samp", 3);
N = transvalues[0];
sumX = transvalues[1];
sumX2 = transvalues[2];
/* Sample variance is undefined when N is 0 or 1, so return NULL */
if (N <= 1.0)
PG_RETURN_NULL();
numerator = N * sumX2 - sumX * sumX;
CHECKFLOATVAL(numerator, isinf(sumX2) || isinf(sumX), true);
/* Watch out for roundoff error producing a negative numerator */
if (numerator <= 0.0)
PG_RETURN_FLOAT8(0.0);
PG_RETURN_FLOAT8(numerator / (N * (N - 1.0)));
}
Datum
float8_stddev_pop(PG_FUNCTION_ARGS)
{
ArrayType *transarray = PG_GETARG_ARRAYTYPE_P(0);
float8 *transvalues;
float8 N,
sumX,
sumX2,
numerator;
transvalues = check_float8_array(transarray, "float8_stddev_pop", 3);
N = transvalues[0];
sumX = transvalues[1];
sumX2 = transvalues[2];
/* Population stddev is undefined when N is 0, so return NULL */
if (N == 0.0)
PG_RETURN_NULL();
numerator = N * sumX2 - sumX * sumX;
CHECKFLOATVAL(numerator, isinf(sumX2) || isinf(sumX), true);
/* Watch out for roundoff error producing a negative numerator */
if (numerator <= 0.0)
PG_RETURN_FLOAT8(0.0);
PG_RETURN_FLOAT8(sqrt(numerator / (N * N)));
}
Datum
float8_stddev_samp(PG_FUNCTION_ARGS)
{
ArrayType *transarray = PG_GETARG_ARRAYTYPE_P(0);
float8 *transvalues;
float8 N,
sumX,
sumX2,
numerator;
transvalues = check_float8_array(transarray, "float8_stddev_samp", 3);
N = transvalues[0];
sumX = transvalues[1];
sumX2 = transvalues[2];
/* Sample stddev is undefined when N is 0 or 1, so return NULL */
if (N <= 1.0)
PG_RETURN_NULL();
numerator = N * sumX2 - sumX * sumX;
CHECKFLOATVAL(numerator, isinf(sumX2) || isinf(sumX), true);
/* Watch out for roundoff error producing a negative numerator */
if (numerator <= 0.0)
PG_RETURN_FLOAT8(0.0);
PG_RETURN_FLOAT8(sqrt(numerator / (N * (N - 1.0))));
}