SQL Server LLS_q Function

Updated 2024-03-28 21:24:59.197000

Description

Use the table-valued function LLS_q when you want to perform a Linear Least Squares calculation and you to adjust the tolerance for collinearity. For more information on Linear Least Squares see the LINEST_q function.

Syntax

SELECT * FROM [westclintech].[wct].[LLS_q] (
   <@Matrix_RangeQuery, nvarchar(max),>
  ,<@y_ColumnNumber, int,>
  ,<@LConst, bit,>
  ,<@Tol, float,>)

Arguments

@Matrix_RangeQuery

The SELECT statement, as a string, which, when executed, creates the resultant table of x- and y-values which will be used in the calculation. Data returned from the @Matrix_RangeQuery must be of a type float or of a type that intrinsically converts to float.

@y_ColumnNumber

the index into the array identifying the column containing the y-values. The index value must be between 1 and n, where n is the number of columns in the resultant table created by the @Matrix_RangeQuery. @y_ColumnNumber must be of the type int or of a type that implicitly converts to int.

@Tol

The tolerance values. The tolerance is used internally in QR decomposition to determine when to stop the pivoting. When the tolerance ins sufficiently small, the regression will continue for columns that might otherwise be considered collinear.

Return Type

table

Remarks

If @Lconst is NULL then @LConst is set to 'True'.

If @LConst is true than the number of rows must be greater than the number of columns.

If @Lconst is false than the number of rows must be greater than or equal to the number of columns.

For simpler queries, you can try the LINEST function.

If @Tol is NULL then @Tol = 0.000000015036712.

Examples

Example #1

We put the x- and y-data into a temp table, #xy.

SELECT *
INTO   #xy
  FROM (   VALUES (6.1, 1.21, 4.35, 5.42, 6.45, 138.08),
                  (7.95, 0.97, 2.79, 4.73, 8.14, 80.3),
                  (8.53, 9.73, 9.7, 1.16, 9.05, 284.45),
                  (7.4, 1.61, 9.9, 8.8, 4.38, 226.66),
                  (7.42, 4.58, 0.06, 8.97, 8.75, 112.37),
                  (6.19, 3.56, 9.69, 8.7, 5.67, 168.73),
                  (8.44, 3.85, 8.23, 1.05, 0.92, 160.38),
                  (0.84, 3.86, 7.85, 2.14, 3.03, 129.26),
                  (0.37, 7.33, 5.07, 8.06, 3.25, 170.39),
                  (7.48, 0.68, 8.34, 2.98, 2.81, 188.53),
                  (5.33, 3.51, 7.03, 6.49, 7.54, 131.1)) n (x1, x2, x3, x4, x5, y);

This is what the data look like.

To invoke the table-valued function LLS_q , we enter the following SQL.

SELECT *
FROM wct.LLS_q(   'SELECT * FROM #xy', --@Matrix_RangeQuery
                  6,                   --@y_columnNumber
                  'True',              --@Lconst
                  NULL                 --@Tol
              );

The following results are produced.

The results are returned in 3^rd normal form. You can use standard SQL commands to re-format the results. For example, if you wanted to produce the the coefficients in a format similar to output of the Excel Data Analysis Regression Tool, you could use the following SQL.

;WITH mycte
as (SELECT *
    FROM wct.LLS_q(   'SELECT * FROM #xy', --@Matrix_RangeQuery
                      6,                   --@y_columnNumber
                      'True',              --@Lconst
                      NULL                 --@Tol
                  ) p
   )
SELECT d.col_name,
       d.m,
       d.se,
       d.tstat,
       d.pval,
       m - wct.T_INV_2T(.05, m.stat_val) * se as [Lower Confidence Level],
       m + wct.T_INV_2T(.05, m.stat_val) * se as [Upper Confidence Level]
FROM mycte p
    PIVOT
    (
        MAX(stat_val)
        FOR stat_name in (m, se, tstat, pval)
    ) d
    CROSS JOIN mycte m
WHERE stat_name = 'df'
      AND d.col_name IS NOT NULL;

This produces the following result.

Similarly, if you wanted to reformat the results to produce the equivalent of the ANOVA table from the Excel Data Analysis Regression tool, you could use the following SQL.

;WITH mycte
as (SELECT df as [Regression df],
           Obs - df - 1 as [Residual df],
           ss_reg as [Regression SS],
           ss_resid as [Residual SS],
           F
    FROM
    (
        SELECT stat_name,
               stat_val
        FROM wct.LLS_q(   'SELECT * FROM #xy', --@Matrix_RangeQuery
                          6,                   --@y_columnNumber
                          'True',              --@Lconst
                          NULL                 --@Tol
                      ) p
        WHERE stat_name in ( 'ss_reg', 'ss_resid', 'F', 'df' )
        UNION
        SELECT 'Obs',
               COUNT(*)
        FROM #xy
    ) d
    PIVOT
    (
        MAX(stat_val)
        FOR stat_name in (df, F, ss_reg, ss_resid, Obs)
    ) p
   )
SELECT 'Regression',
       [Regression df] as DF,
       [Regression SS] as SS,
       [Regression SS] / [Regression df] as MS,
       F,
       wct.F_DIST_RT(F, [Regression df], [Residual df]) as [Significance F]
FROM mycte
UNION ALL
SELECT 'Residual',
       [Residual df],
       [Residual SS],
       [Residual SS] / [Residual df] as [Residual MS],
       NULL,
       NULL
FROM mycte
UNION ALL
SELECT 'Total',
       [Regression df] + [Residual df],
       [Regression SS] + [Residual SS],
       NULL,
       NULL,
       NULL
FROM mycte;

This produces the following result.

Since the function executes the @Matrix_RangeQuery values, we did not need to put the x- and y-values into a temporary table and could have simply included them in @Matrix_RangeQuery , like this.

SELECT *
  FROM wct.LINEST_q(
           'SELECT
          *
       FROM (VALUES
           (6.1,1.21,4.35,5.42,6.45,138.08)
          ,(7.95,0.97,2.79,4.73,8.14,80.3)
          ,(8.53,9.73,9.7,1.16,9.05,284.45)
          ,(7.4,1.61,9.9,8.8,4.38,226.66)
          ,(7.42,4.58,0.06,8.97,8.75,112.37)
          ,(6.19,3.56,9.69,8.7,5.67,168.73)
          ,(8.44,3.85,8.23,1.05,0.92,160.38)
          ,(0.84,3.86,7.85,2.14,3.03,129.26)
          ,(0.37,7.33,5.07,8.06,3.25,170.39)
          ,(7.48,0.68,8.34,2.98,2.81,188.53)
          ,(5.33,3.51,7.03,6.49,7.54,131.1)
          )n(x1,x2,x3,x4,x5,y)',            --@Matrix_RangeQuery
           6,                               --@y_columnNumber
           'True',                          --@Lconst
          NULL                              --@Tol
    );

Example #2

Using the same data as Example #1, if we wanted to calculate the coefficients without a y-intercept, we would modify function call to make the last parameter FALSE. This time, however, we will put the results into a temporary table, #L0, and reformat the results using the #L0 table.

SELECT *
INTO #L0
FROM wct.LLS_q(   'SELECT * FROM #xy', --@Matrix_RangeQuery
                  6,                   --@y_columnNumber
                  'False',             --@Lconst
                  NULL                 --@Tol
              );

The #L0 table should contain the following data.

We can use the same technique as in Example #1 to reformat the coefficient statistics.

SELECT d.col_name,
       d.m,
       d.se,
       d.tstat,
       d.pval,
       m - wct.T_INV_2T(.05, m.stat_val) * se as [Lower Confidence Level],
       m + wct.T_INV_2T(.05, m.stat_val) * se as [Upper Confidence Level]
  FROM #L0 p
    PIVOT (   MAX(stat_val)
              FOR stat_name in (m, se, tstat, pval)) d
 CROSS JOIN #L0 m
 WHERE m.stat_name = 'df'
   AND d.col_name IS NOT NULL;

This produces the following result.

Note that even though we specified no intercept, an intercept row is still created with a regression coefficient of 0 (keeping the results consistent with the Excel Data Analysis Regression). It is simple enough to exclude this from the output (like R does) simply by adding another condition to the WHERE clause.

SELECT d.col_name,
       d.m,
       d.se,
       d.tstat,
       d.pval,
       m - wct.T_INV_2T(.05, m.stat_val) * se as [Lower Confidence Level],
       m + wct.T_INV_2T(.05, m.stat_val) * se as [Upper Confidence Level]
  FROM #L0 p
    PIVOT (   MAX(stat_val)
              FOR stat_name in (m, se, tstat, pval)) d
 CROSS JOIN #L0 m
 WHERE m.stat_name = 'df'
   AND d.col_name IS NOT NULL
   AND d.col_name  <> 'Intercept';

The following SQL can be used to reproduce the ANOVA table.

;WITH mycte
   as (SELECT df as [Residual df],
              --,Obs - df - 1 as [Regression df]
              Obs - df as [Regression df],
              ss_reg as [Regression SS],
              ss_resid as [Residual SS],
              F
         FROM (   SELECT stat_name,
                         stat_Val
                    FROM #L0
                   WHERE stat_name in ( 'ss_reg', 'ss_resid', 'F', 'df' )
                  UNION
                  SELECT 'Obs',
                         COUNT(*)
                    FROM #xy) d
         PIVOT (   MAX(stat_val)
                   FOR stat_name in (df, F, ss_reg, ss_resid, Obs)) p)
SELECT 'Regression',
       [Regression df] as DF,
       [Regression SS] as SS,
       [Regression SS] / [Regression df] as MS,
       F,
       wct.F_DIST_RT(F, [Regression df], [Residual df]) as [Significance F]
  FROM mycte
UNION ALL
SELECT 'Residual',
       [Residual df],
       [Residual SS],
       [Residual SS] / [Residual df] as [Residual MS],
       NULL,
       NULL
  FROM mycte
UNION ALL
SELECT 'Total',
       [Regression df] + [Residual df],
       [Regression SS] + [Residual SS],
       NULL,
       NULL,
       NULL
  FROM mycte;

This produces the following result.

Note that the calculation of the regression degrees of freedom needs to be adjusted to reflect the absence on intercept. It also worthwhile noting that the calculation in this SQL produces a Significance F that agrees with R while Excel produces a different value (which seems to be caused be subtracting 1 from the residual degrees of freedom).

Example #3

Here's another example. Instead of just having a matrix of x- and y-values in our table, we are going to add a column, testid, which is a way of grouping x- and y-values together for purposes of doing the LINEST_q calculation. This allows us to compute the ordinary least squares values for multiple sets of data in a single SELECT statement.

Second, we take the x-values and raise them to the powers of 2, 3, 4, 5 thus solving for the equation:

y = B₀ + B₁x + B₂x² + B₃x³ + B₄x⁴ + B₅x⁵

SELECT *
INTO   #xy
  FROM (   VALUES ('Wampler5', 7590001, 0),
                  ('Wampler5', -20479994, 1),
                  ('Wampler5', 20480063, 2),
                  ('Wampler5', -20479636, 3),
                  ('Wampler5', 25231365, 4),
                  ('Wampler5', -20476094, 5),
                  ('Wampler5', 20489331, 6),
                  ('Wampler5', -20460392, 7),
                  ('Wampler5', 18417449, 8),
                  ('Wampler5', -20413570, 9),
                  ('Wampler5', 20591111, 10),
                  ('Wampler5', -20302844, 11),
                  ('Wampler5', 18651453, 12),
                  ('Wampler5', -20077766, 13),
                  ('Wampler5', 21059195, 14),
                  ('Wampler5', -19666384, 15),
                  ('Wampler5', 26348481, 16),
                  ('Wampler5', -18971402, 17),
                  ('Wampler5', 22480719, 18),
                  ('Wampler5', -17866340, 19),
                  ('Wampler5', 10958421, 20),
                  ('Wampler4', 75901, 0),
                  ('Wampler4', -204794, 1),
                  ('Wampler4', 204863, 2),
                  ('Wampler4', -204436, 3),
                  ('Wampler4', 253665, 4),
                  ('Wampler4', -200894, 5),
                  ('Wampler4', 214131, 6),
                  ('Wampler4', -185192, 7),
                  ('Wampler4', 221249, 8),
                  ('Wampler4', -138370, 9),
                  ('Wampler4', 315911, 10),
                  ('Wampler4', -27644, 11),
                  ('Wampler4', 455253, 12),
                  ('Wampler4', 197434, 13),
                  ('Wampler4', 783995, 14),
                  ('Wampler4', 608816, 15),
                  ('Wampler4', 1370781, 16),
                  ('Wampler4', 1303798, 17),
                  ('Wampler4', 2205519, 18),
                  ('Wampler4', 2408860, 19),
                  ('Wampler4', 3444321, 20),
                  ('Wampler3', 760, 0),
                  ('Wampler3', -2042, 1),
                  ('Wampler3', 2111, 2),
                  ('Wampler3', -1684, 3),
                  ('Wampler3', 3888, 4),
                  ('Wampler3', 1858, 5),
                  ('Wampler3', 11379, 6),
                  ('Wampler3', 17560, 7),
                  ('Wampler3', 39287, 8),
                  ('Wampler3', 64382, 9),
                  ('Wampler3', 113159, 10),
                  ('Wampler3', 175108, 11),
                  ('Wampler3', 273291, 12),
                  ('Wampler3', 400186, 13),
                  ('Wampler3', 581243, 14),
                  ('Wampler3', 811568, 15),
                  ('Wampler3', 1121004, 16),
                  ('Wampler3', 1506550, 17),
                  ('Wampler3', 2002767, 18),
                  ('Wampler3', 2611612, 19),
                  ('Wampler3', 3369180, 20),
                  ('Wampler2', 1, 0),
                  ('Wampler2', 1.11111, 1),
                  ('Wampler2', 1.24992, 2),
                  ('Wampler2', 1.42753, 3),
                  ('Wampler2', 1.65984, 4),
                  ('Wampler2', 1.96875, 5),
                  ('Wampler2', 2.38336, 6),
                  ('Wampler2', 2.94117, 7),
                  ('Wampler2', 3.68928, 8),
                  ('Wampler2', 4.68559, 9),
                  ('Wampler2', 6, 10),
                  ('Wampler2', 7.71561, 11),
                  ('Wampler2', 9.92992, 12),
                  ('Wampler2', 12.75603, 13),
                  ('Wampler2', 16.32384, 14),
                  ('Wampler2', 20.78125, 15),
                  ('Wampler2', 26.29536, 16),
                  ('Wampler2', 33.05367, 17),
                  ('Wampler2', 41.26528, 18),
                  ('Wampler2', 51.16209, 19),
                  ('Wampler2', 63, 20),
                  ('Wampler1', 1, 0),
                  ('Wampler1', 6, 1),
                  ('Wampler1', 63, 2),
                  ('Wampler1', 364, 3),
                  ('Wampler1', 1365, 4),
                  ('Wampler1', 3906, 5),
                  ('Wampler1', 9331, 6),
                  ('Wampler1', 19608, 7),
                  ('Wampler1', 37449, 8),
                  ('Wampler1', 66430, 9),
                  ('Wampler1', 111111, 10),
                  ('Wampler1', 177156, 11),
                  ('Wampler1', 271453, 12),
                  ('Wampler1', 402234, 13),
                  ('Wampler1', 579195, 14),
                  ('Wampler1', 813616, 15),
                  ('Wampler1', 1118481, 16),
                  ('Wampler1', 1508598, 17),
                  ('Wampler1', 2000719, 18),
                  ('Wampler1', 2613660, 19),
                  ('Wampler1', 3368421, 20)) n (testid, y, x);

Let's say wanted to run LLS_q for the all the data where the testid is equal to Wampler3. We could simply enter the following statement.

SELECT *
FROM wct.LLS_q(
                  'SELECT y,x,POWER(x,2),POWER(x,3),POWER(x,4),POWER(x,5) FROM #xy WHERE testid = ''Wampler3''',
                  1,
                  'True',
                  NULL
              );

This produces the following result.

You will notice that the column names for the calculated columns are Column1, Column2, Column3, and Column4. This is because we have not assigned them a name and these are the default column names. Even though Column1 does not refer to the first column in the resultant table from our dynamic SQL, it is the first column with no name. To get more descriptive column names, you can simply assign the names in the @ColumnNames variable as in this example.

SELECT *
FROM wct.LLS_q(
                  'SELECT y,x,POWER(x,2) as [x^2],POWER(x,3) as [x^3],POWER(x,4) as [x^4],POWER(x,5) as [x^5] FROM #xy WHERE testid = ''Wampler3''',
                  1,
                  'True',
                  NULL
              );

This produces the following result.

In the following SQL we return selected values for each test.

SELECT p.testid,
       p.rsq,
       p.rsqa,
       p.rsqm,
       p.F,
       p.df,
       p.ss_reg,
       p.ss_resid
FROM
(
    SELECT n.testid,
           k.stat_name,
           k.stat_val
    FROM
    (SELECT DISTINCT testid FROM #xy) n
        CROSS APPLY wct.LLS_q(
                                 'SELECT y,x,POWER(x,2) as [x^2],POWER(x,3) as [x^3],POWER(x,4) as [x^4],POWER(x,5) as [x^5] FROM #xy WHERE testid = '''
                                 + n.testid + '''',
                                 1,
                                 'True',
                                 NULL
                             ) k
) d
PIVOT
(
    SUM(stat_val)
    FOR stat_name IN (F, df, ss_reg, ss_resid, rsq, rsqm, rsqa)
) p;

This produces the following result.

In this SQL we select the coefficient statistics for each of the tests.

SELECT p.testid,
       p.col_name,
       p.m,
       p.se,
       p.tstat,
       p.pval
FROM
(
    SELECT n.testid,
           k.stat_name,
           k.idx,
           k.col_name,
           k.stat_val
    FROM
    (SELECT DISTINCT testid FROM #xy) n
        CROSS APPLY wct.LLS_q(
                                 'SELECT y,x,POWER(x,2) as [x^2],POWER(x,3) as [x^3],POWER(x,4) as [x^4],POWER(x,5) as [x^5] FROM #xy WHERE testid = '''
                                 + n.testid + '''',
                                 1,
                                 'True',
                                 NULL
                             ) k
    WHERE k.idx IS NOT NULL
) d
PIVOT
(
    SUM(stat_val)
    FOR stat_name IN (m, se, tstat, pval)
) p
ORDER BY testid,
         idx;

This produces the following result.

See Also

LINEST - the Ordinary Least Squares (OLS) solution for a series of x-values and y-values

LOGEST_Q - Exponential curve that best fits a series of x- and y-values

TREND - for simpler queries

GROWTH - calculate predicted exponential growth using existing values

TRENDMX - Calculate the values along a linear trend for multiple x values

GROWTHMX - Values along an exponential trend

FORECAST - The predicted y-value for a given x-value