我有以下代码块，使用线性回归（最小二乘法）计算趋势线的公式。它只找到X轴和Y轴的R-Squared和核心系数值。
如果X和Y轴是int和float，这将计算精确的值。
CREATE FUNCTION [dbo].[LinearReqression] (@Data AS XML)
RETURNS TABLE AS RETURN (
WITH Array AS (
SELECT x = n.value('@x', 'float'),
y = n.value('@y', 'float')
FROM @Data.nodes('/r/n') v(n)
),
Medians AS (
SELECT xbar = AVG(x), ybar = AVG(y)
FROM Array ),
BetaCalc AS (
SELECT Beta = SUM(xdelta * (y - ybar)) / NULLIF(SUM(xdelta * xdelta), 0)
FROM Array
CROSS JOIN Medians
CROSS APPLY ( SELECT xdelta = (x - xbar) ) xd ),
AlphaCalc AS (
SELECT Alpha = ybar - xbar * beta
FROM Medians
CROSS JOIN BetaCalc),
SSCalc AS (
SELECT SS_tot = SUM((y - ybar) * (y - ybar)),
SS_err = SUM((y - (Alpha + Beta * x)) * (y - (Alpha + Beta * x)))
FROM Array
CROSS JOIN Medians
CROSS JOIN AlphaCalc
CROSS JOIN BetaCalc )
SELECT r_squared = CASE WHEN SS_tot = 0 THEN 1.0
ELSE 1.0 - ( SS_err / SS_tot ) END,
Alpha, Beta
FROM AlphaCalc
CROSS JOIN BetaCalc
CROSS JOIN SSCalc
)
用法：
DECLARE @DataTable TABLE (
SourceID INT,
x Date,
y FLOAT
) ;
INSERT INTO @DataTable ( SourceID, x, y )
SELECT ID = 0, x = 1.2, y = 1.0
UNION ALL SELECT 1, 1.6, 1
UNION ALL SELECT 2, 2.0, 1.5
UNION ALL SELECT 3, 2.0, 1.75
UNION ALL SELECT 4, 2.1, 1.85
UNION ALL SELECT 5, 2.1, 2
UNION ALL SELECT 6, 2.2, 3
UNION ALL SELECT 7, 2.2, 3
UNION ALL SELECT 8, 2.3, 3.5
UNION ALL SELECT 9, 2.4, 4
UNION ALL SELECT 10, 2.5, 4
UNION ALL SELECT 11, 3, 4.5 ;
-- Create and view XML data array
DECLARE @DataXML XML ;
SET @DataXML = (
SELECT -- FLOAT values are formatted in XML like "1.000000000000000e+000", increasing the character count
-- Converting them to VARCHAR first keeps the XML small without sacrificing precision
-- They are unpacked as FLOAT in the function either way
[@x] = CAST(x AS VARCHAR(20)),
[@y] = CAST(y AS VARCHAR(20))
FROM @DataTable
FOR XML PATH('n'), ROOT('r') ) ;
SELECT @DataXML ;
-- Get the results
SELECT * FROM dbo.LinearReqression (@DataXML) ;
我需要什么
在我的情况下，X轴也可能是日期列？那么如何用日期列计算相同的回归分析呢？
请帮我实现日期栏的回归分析。
答案 0 :(得分：2)
简短回答是：计算日期的趋势线与计算浮动的趋势线几乎相同。
对于日期，您可以选择一些开始日期，并将开始日期和日期之间的天数用作X。
我没有检查你的功能本身，我认为那里的公式是正确的。
另外，我不明白为什么你从表中生成XML并将其解析回函数内的表。效率很低。你可以简单地通过表格。
我使用你的函数来制作两个变体：用于处理浮点数和处理日期。 我在这个例子中使用SQL Server 2008。
首先创建一个用户定义的表类型，这样我们就可以将表传递给函数：
CREATE TYPE [dbo].[FloatRegressionDataTableType] AS TABLE(
[x] [float] NOT NULL,
[y] [float] NOT NULL
)
GO
然后创建接受这样的表的函数：
CREATE FUNCTION [dbo].[LinearRegressionFloat] (@ParamData dbo.FloatRegressionDataTableType READONLY)
RETURNS TABLE AS RETURN (
WITH Array AS (
SELECT x,
y
FROM @ParamData
),
Medians AS (
SELECT xbar = AVG(x), ybar = AVG(y)
FROM Array ),
BetaCalc AS (
SELECT Beta = SUM(xdelta * (y - ybar)) / NULLIF(SUM(xdelta * xdelta), 0)
FROM Array
CROSS JOIN Medians
CROSS APPLY ( SELECT xdelta = (x - xbar) ) xd ),
AlphaCalc AS (
SELECT Alpha = ybar - xbar * beta
FROM Medians
CROSS JOIN BetaCalc),
SSCalc AS (
SELECT SS_tot = SUM((y - ybar) * (y - ybar)),
SS_err = SUM((y - (Alpha + Beta * x)) * (y - (Alpha + Beta * x)))
FROM Array
CROSS JOIN Medians
CROSS JOIN AlphaCalc
CROSS JOIN BetaCalc )
SELECT r_squared = CASE WHEN SS_tot = 0 THEN 1.0
ELSE 1.0 - ( SS_err / SS_tot ) END,
Alpha, Beta
FROM AlphaCalc
CROSS JOIN BetaCalc
CROSS JOIN SSCalc
)
GO
非常相似，为包含日期的表创建一个类型：
CREATE TYPE [dbo].[DateRegressionDataTableType] AS TABLE(
[x] [date] NOT NULL,
[y] [float] NOT NULL
)
GO
并创建一个接受此类表的函数。对于每个给定日期，它使用2001-01-01计算x与给定日期DATEDIFF之间的天数，然后将结果转换为浮动以确保其余计算正确。您可以尝试删除转换为浮动，您将看到不同的结果。您可以选择任何其他开始日期，但不必是2001-01-01。
CREATE FUNCTION [dbo].[LinearRegressionDate] (@ParamData dbo.DateRegressionDataTableType READONLY)
RETURNS TABLE AS RETURN (
WITH Array AS (
SELECT CAST(DATEDIFF(day, '2001-01-01', x) AS float) AS x,
y
FROM @ParamData
),
Medians AS (
SELECT xbar = AVG(x), ybar = AVG(y)
FROM Array ),
BetaCalc AS (
SELECT Beta = SUM(xdelta * (y - ybar)) / NULLIF(SUM(xdelta * xdelta), 0)
FROM Array
CROSS JOIN Medians
CROSS APPLY ( SELECT xdelta = (x - xbar) ) xd ),
AlphaCalc AS (
SELECT Alpha = ybar - xbar * beta
FROM Medians
CROSS JOIN BetaCalc),
SSCalc AS (
SELECT SS_tot = SUM((y - ybar) * (y - ybar)),
SS_err = SUM((y - (Alpha + Beta * x)) * (y - (Alpha + Beta * x)))
FROM Array
CROSS JOIN Medians
CROSS JOIN AlphaCalc
CROSS JOIN BetaCalc )
SELECT r_squared = CASE WHEN SS_tot = 0 THEN 1.0
ELSE 1.0 - ( SS_err / SS_tot ) END,
Alpha, Beta
FROM AlphaCalc
CROSS JOIN BetaCalc
CROSS JOIN SSCalc
)
GO
这是测试功能的方法：
-- test float data
DECLARE @FloatDataTable [dbo].[FloatRegressionDataTableType];
INSERT INTO @FloatDataTable (x, y)
VALUES
(1.2, 1.0)
,(1.6, 1)
,(2.0, 1.5)
,(2.0, 1.75)
,(2.1, 1.85)
,(2.1, 2)
,(2.2, 3)
,(2.2, 3)
,(2.3, 3.5)
,(2.4, 4)
,(2.5, 4)
,(3, 4.5);
SELECT * FROM dbo.LinearRegressionFloat(@FloatDataTable);
-- test date data
DECLARE @DateDataTable [dbo].[DateRegressionDataTableType];
INSERT INTO @DateDataTable (x, y)
VALUES
('2001-01-13', 1.0)
,('2001-01-17', 1)
,('2001-01-21', 1.5)
,('2001-01-21', 1.75)
,('2001-01-22', 1.85)
,('2001-01-22', 2)
,('2001-01-23', 3)
,('2001-01-23', 3)
,('2001-01-24', 3.5)
,('2001-01-25', 4)
,('2001-01-26', 4)
,('2001-01-31', 4.5);
SELECT * FROM dbo.LinearRegressionDate(@DateDataTable);
以下是两个结果集：
r_squared Alpha Beta
----------------------------------------------------------
0.798224907472009 -2.66524390243902 2.46417682926829
r_squared Alpha Beta
----------------------------------------------------------
0.79822490747201 -2.66524390243902 0.246417682926829