需要SQL Server查询来解决三阶多项式回归问题

Question

任何人都可以帮助一些SQL查询代码来提供三阶多项式回归的系数估计吗？

请假设我有一个X和Y数据值表,并想要估算a,b和c:

Y(X) = aX + bX^2 + cX^3 + E 

Answer 1

近似但快速的解决方案是从数据中采样 4 个有代表性的点并求解这些点的多项式方程。

1. 至于采样，您可以将数据拆分为相等的扇区并计算每个扇区的 X 和 Y 的平均值 -可以使用 X 值的四分位数、X 值的平均值min(x)+(max(x)-min(x))/4或您认为最合适的任何值来完成拆分.

   为了说明按四分位数（即按行号）抽样：

2. 至于求解，我使用numberempire.com来求解变量的这些*方程k,a,b,c：

   k + a*X1 + b*X1^2 + c*X1^3 - Y1 = 0,
   k + a*X2 + b*X2^2 + c*X2^3 - Y2 = 0,
   k + a*X3 + b*X3^2 + c*X3^3 - Y3 = 0,
   k + a*X4 + b*X4^2 + c*X4^3 - Y4 = 0
   

   *由于Y(X) = 0 + ax bx^2 + cx^3 + ?隐式包含 [0, 0] 点作为样本点之一，它会为不包含 [0, 0] 的数据集创建错误的近似值。我冒昧地解决了Y(X) = k + ax bx^2 + cx^3 + ?。

实际的 SQL 应该是这样的：

select
    -- returns 1 row with columns labeled K, A, B and C = coefficients in 3rd order polynomial equation for the 4 sample points
    -(X1*(X2p2*(X3p3*Y4-X4p3*Y3)+X2p3*(X4p2*Y3-X3p2*Y4)+(X3p2*X4p3-X3p3*X4p2)*Y2)+X1p2*(X2*(X4p3*Y3-X3p3*Y4)+X2p3*(X3*Y4-X4*Y3)+(X3p3*X4-X3*X4p3)*Y2)+X1p3*(X2*(X3p2*Y4-X4p2*Y3)+X2p2*(X4*Y3-X3*Y4)+(X3*X4p2-X3p2*X4)*Y2)+(X2*(X3p3*X4p2-X3p2*X4p3)+X2p2*(X3*X4p3-X3p3*X4)+X2p3*(X3p2*X4-X3*X4p2))*Y1)/(X1*(X2p2*(X4p3-X3p3)-X3p2*X4p3+X3p3*X4p2+X2p3*(X3p2-X4p2))+X2*(X3p2*X4p3-X3p3*X4p2)+X1p2*(X3*X4p3+X2*(X3p3-X4p3)+X2p3*(X4-X3)-X3p3*X4)+X2p2*(X3p3*X4-X3*X4p3)+X1p3*(X2*(X4p2-X3p2)-X3*X4p2+X3p2*X4+X2p2*(X3-X4))+X2p3*(X3*X4p2-X3p2*X4))  as k,
    (X1p2*(X2p3*(Y4-Y3)-X3p3*Y4+X4p3*Y3+(X3p3-X4p3)*Y2)+X2p2*(X3p3*Y4-X4p3*Y3)+X1p3*(X3p2*Y4+X2p2*(Y3-Y4)-X4p2*Y3+(X4p2-X3p2)*Y2)+X2p3*(X4p2*Y3-X3p2*Y4)+(X3p2*X4p3-X3p3*X4p2)*Y2+(X2p2*(X4p3-X3p3)-X3p2*X4p3+X3p3*X4p2+X2p3*(X3p2-X4p2))*Y1)/(X1*(X2p2*(X4p3-X3p3)-X3p2*X4p3+X3p3*X4p2+X2p3*(X3p2-X4p2))+X2*(X3p2*X4p3-X3p3*X4p2)+X1p2*(X3*X4p3+X2*(X3p3-X4p3)+X2p3*(X4-X3)-X3p3*X4)+X2p2*(X3p3*X4-X3*X4p3)+X1p3*(X2*(X4p2-X3p2)-X3*X4p2+X3p2*X4+X2p2*(X3-X4))+X2p3*(X3*X4p2-X3p2*X4))  as a,
    -(X1*(X2p3*(Y4-Y3)-X3p3*Y4+X4p3*Y3+(X3p3-X4p3)*Y2)+X2*(X3p3*Y4-X4p3*Y3)+X1p3*(X3*Y4+X2*(Y3-Y4)-X4*Y3+(X4-X3)*Y2)+X2p3*(X4*Y3-X3*Y4)+(X3*X4p3-X3p3*X4)*Y2+(X2*(X4p3-X3p3)-X3*X4p3+X3p3*X4+X2p3*(X3-X4))*Y1)/(X1*(X2p2*(X4p3-X3p3)-X3p2*X4p3+X3p3*X4p2+X2p3*(X3p2-X4p2))+X2*(X3p2*X4p3-X3p3*X4p2)+X1p2*(X3*X4p3+X2*(X3p3-X4p3)+X2p3*(X4-X3)-X3p3*X4)+X2p2*(X3p3*X4-X3*X4p3)+X1p3*(X2*(X4p2-X3p2)-X3*X4p2+X3p2*X4+X2p2*(X3-X4))+X2p3*(X3*X4p2-X3p2*X4))  as b,
    (X1*(X2p2*(Y4-Y3)-X3p2*Y4+X4p2*Y3+(X3p2-X4p2)*Y2)+X2*(X3p2*Y4-X4p2*Y3)+X1p2*(X3*Y4+X2*(Y3-Y4)-X4*Y3+(X4-X3)*Y2)+X2p2*(X4*Y3-X3*Y4)+(X3*X4p2-X3p2*X4)*Y2+(X2*(X4p2-X3p2)-X3*X4p2+X3p2*X4+X2p2*(X3-X4))*Y1)/(X1*(X2p2*(X4p3-X3p3)-X3p2*X4p3+X3p3*X4p2+X2p3*(X3p2-X4p2))+X2*(X3p2*X4p3-X3p3*X4p2)+X1p2*(X3*X4p3+X2*(X3p3-X4p3)+X2p3*(X4-X3)-X3p3*X4)+X2p2*(X3p3*X4-X3*X4p3)+X1p3*(X2*(X4p2-X3p2)-X3*X4p2+X3p2*X4+X2p2*(X3-X4))+X2p3*(X3*X4p2-X3p2*X4))  as c
  from (select
      samples.*,
      -- precomputing the powers should give better performance (at least i hope it would)
      power(X1,2) X1p2, power(X2,2) X2p2, power(X3,2) X3p2, power(X4,2) X4p2,
      power(Y1,3) Y1p3, power(Y2,3) Y2p3, power(Y3,3) Y3p3, power(Y4,3) Y4p3
    from (select
        avg(case when sector = 1 then x end) X1,
        avg(case when sector = 2 then x end) X2,
        avg(case when sector = 3 then x end) X3,
        avg(case when sector = 4 then x end) X4,
        avg(case when sector = 1 then y end) Y1,
        avg(case when sector = 2 then y end) Y2,
        avg(case when sector = 3 then y end) Y3,
        avg(case when sector = 4 then y end) Y4
      from (select x, y, 
          -- splitting to sectors 1 - 4 by row number (SQL Server version)
          ceiling(row_number() OVER (ORDER BY x asc) / count(*) * 4) sector
        from original_data
      )
    ) samples
  )

根据developer.mimer.com，这些可选功能需要在 SQL Server 中启用：

T611, "Elementary OLAP operations"
F591, "Derived tables"