SQL Server COVM Function

Updated 2024-03-04 21:11:59.947000

Description

Use the table-valued function COVM to calculate a sample covariance matrix. The sample covariance of a matrix (A) is calculated by subtracting the column mean (µ) from each element in a column, calculating X^TX and then dividing the result by n-1, where n is the number of rows. For example:

A=\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&_{33}&a_{34}\end{bmatrix}

X=\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&_{33}&a_{34}\end{bmatrix}-\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\\1&1&1\\\end{bmatrix}\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&_{33}&a_{34}\end{bmatrix}*\frac{1}{n}

\mathrm{Cov(A)=(X^TX)/(n-1)}

Syntax

SELECT * FROM [westclintech].[wct].[COVM](
  <@Matrix_RangeQuery, nvarchar(max),>
 ,<@Is3N, bit,>)

Arguments

@Matrix_RangeQuery

the SELECT statement, as text, used to return the input matrix for this function. The SELECT statement specifies the column names from the table or view or can be used to enter the matrix values directly. Data returned from the @Matrix_RangeQuery select must be of the type float or of a type that implicitly converts to float.

@Is3N

a bit value identifying the form for the resultant table returned by @Matrix_RangeQuery. Enter 'True' for a resultant table in 3rd normal form. Enter 'False' for a de-normalized table in 'spreadsheet' form.

Return Type

table

Remarks

If @Is3N is NULL then @Is3N = 'False'.

Use MCOV for a matrix stored as a string.

If @Is3N is 'True' then the resultant table should be returned as row, column, and value.

If the array contains NULL, then NULL will be returned.

The function returns an error if the array contains a non-numeric value.

If the supplied input is a vector, then result will be the (sample) variance.

Examples

In this example @Matrix_RangeQuery returns the matrix from a derived table embodied in the statement. The matrix is in spreadsheet form.

SELECT *
FROM wct.COVM(
                 'SELECT
          x1,x2,x3,x4,x5,x6,x7,x8
        FROM (VALUES
           (1,1,1,1,1,1,1,1)
          ,(2,3,4,5,6,7,8,9)
          ,(4,9,16,25,36,49,64,81)
        )n(x1,x2,x3,x4,x5,x6,x7,x8)',
                 'False'
             );

This produces the following result.

Note that the results are returned in third-normal form. If we wanted to a more traditional (de-normalized) presentation of the results, we can us the PIVOT function. Also note that we can use * to select all the columns. We have rounded the results to 2 decimal place for ease of viewing.

SELECT ROUND([0], 2) as [0],
       ROUND([1], 2) as [1],
       ROUND([2], 2) as [2],
       ROUND([3], 2) as [3],
       ROUND([4], 2) as [4],
       ROUND([5], 2) as [5],
       ROUND([6], 2) as [6],
       ROUND([7], 2) as [7]
FROM
(
    SELECT *
    FROM wct.COVM(
                     'SELECT
          *
        FROM (VALUES
           (1,1,1,1,1,1,1,1)
          ,(2,3,4,5,6,7,8,9)
          ,(4,9,16,25,36,49,64,81)
        )n(x1,x2,x3,x4,x5,x6,x7,x8)',
                     'False'
                 )
) d
PIVOT
(
    SUM(ItemValue)
    for ColNum in ([0], [1], [2], [3], [4], [5], [6], [7])
) as P;

This produces the following result.

Let's put several matrices into a table and calculate the covariance matrix for each. We will use CROSS APPLY to calculate to the covariance matrix for each matrix. Note that we have to convert the matrix identifier (which is defined as int) to a varchar to include it in the WHERE clause of @Matrix_RangeQuery.

CREATE TABLE #c
(
    Matrix int,
    rn int,
    x1 float,
    x2 float,
    x3 float,
    PRIMARY KEY (
                    Matrix,
                    rn
                )
);
INSERT INTO #c
VALUES
(100, 1, -11, -41, 36);
INSERT INTO #c
VALUES
(100, 2, -31, 41, -47);
INSERT INTO #c
VALUES
(100, 3, 48, -38, 33);
INSERT INTO #c
VALUES
(100, 4, 8, 44, -10);
INSERT INTO #c
VALUES
(101, 1, 39, 6, -7);
INSERT INTO #c
VALUES
(101, 2, 33, -49, 16);
INSERT INTO #c
VALUES
(101, 3, 14, 29, 13);
INSERT INTO #c
VALUES
(101, 4, 35, -38, -50);
INSERT INTO #c
VALUES
(101, 5, 9, -32, -25);
INSERT INTO #c
VALUES
(102, 1, 29, 49, -17);
INSERT INTO #c
VALUES
(102, 2, 35, 28, 28);
INSERT INTO #c
VALUES
(102, 3, -34, -29, -49);
INSERT INTO #c
VALUES
(102, 4, 0, -5, 0);
INSERT INTO #c
VALUES
(102, 5, -17, 14, 24);
INSERT INTO #c
VALUES
(102, 6, 44, 3, -23);
SELECT n.MATRIX,
       k.*
FROM
(SELECT DISTINCT MATRIX FROM #c) n
    CROSS APPLY wct.COVM('SELECT
          x1,x2,x3
        FROM
          #c
       WHERE MATRIX = ' + cast(n.Matrix as varchar(max)) + ' ORDER by rn', 'False')
                 k;

This produces the following result.

In this example we calculate the covariance matrix from a derived table in 3^rd normal form.

SELECT *
FROM wct.COVM(
                 'SELECT
          rownum,colnum,itemvalue
        FROM (VALUES
               (0,0,1),(0,1,1),(0,2,1),(0,3,1),(0,4,1),(0,5,1),(0,6,1),(0,7,1)
              ,(1,0,2),(1,1,3),(1,2,4),(1,3,5),(1,4,6),(1,5,7),(1,6,8),(1,7,9)
              ,(2,0,4),(2,1,9),(2,2,16),(2,3,25),(2,4,36),(2,5,49),(2,6,64),(2,7,
                        81)
        )n(rownum,colnum,itemvalue)',
                 'True'
             );

This produces the following result.

See Also

MCOV - Calculate a sample covariance matrix.

CORRM - Table-valued function to calculate the correlation matrix

COVARIANCE_S - Aggregate function to calculate the sample covariance

COVARIANCE_P - Aggregate function to calculate the population covariance