SQL Server LOGEST Function

Updated 2023-11-01 20:14:15.017000

Description

Use the table-valued LOGEST to calculate the exponential curve for a series of x- and y-values. The LOGEST function returns the statistics that describe the calculated solution, including the coefficients (m), the standard error of the coefficients (se), the t statistic for each coefficient (tstat) and the associated p-values (pval), the coefficient of determination (rsq), the adjusted r-square value (rsqa) and the modified r-square value (rsqm), the standard error of the y estimate (sey), the F-observed value (F), the residual degrees of freedom (df), the regression sum of squares (ss_reg), and the residual some of squares (ss_resid).

In the case where we have one column of x-values and a column of y-values, the LOGEST solution is:

y=bm^x

For purposes of multi-linear regression, where there are multiple columns of x-values (and still a single column of y-values), the formula for the solution is described by the following equation:

y = m_n^{x_n} * m_{n-1}^{x_{n-1}} * m_{n-2}^{x_{n-2}}...m_1^{x_1} * m_0

Where n is the number of x-columns and m₀ is b. It's easy to see that by taking the natural logarithm, the exponential regression can be expressed as the linear regression of the natural logarithms:

\ln y = x_n * \ln m_n + x_{n-1} * \ln m_{n-1} + x_{n-2} * \ln m_{n-2} + ... + x_1 * \ln m_1 + \ln m_0

The function expects the input to be in row-column, or spreadsheet, format, rather than in third-normal form. Thus, the input into the function requires the specification of the column names for the x- and y-values, and the specification of a 1-based index identifying the column reference for the y-values. The column specifications and the table or view that contains the data are passed into the function as strings. The function dynamically creates SQL and the resultant table from the SQL is used as input into the OLS calculations.

LOGEST automatically detects collinearity and removes the right-most co-linear column resulting in a regression coefficient of 0 for that column.

Syntax

SELECT * FROM [westclintech].[wct].[LOGEST](
  <@TableName, nvarchar(max),>
 ,<@ColumnNames, nvarchar(4000),>
 ,<@GroupedColumnName, nvarchar(4000),>
 ,<@GroupedColumnValue, sql_variant,>
 ,<@Y_ColumnNumber, int,>
 ,<@Lconst, bit,>)

Arguments

@TableName

the name, as text, of the table or view that contains the values used in the calculation.

@ColumnNames

the name, as text, of the columns in the table or view specified by @TableName that contain the values used in the calculation. Data returned from the @ColumnNames must be of the type float or of a type that implicitly converts to float.

@GroupedColumnName

the name, as text, of the column in the table or view specified by @TableName which will be used for grouping the results.

@GroupedColumnValue

the column value to do the grouping on.

@Y_ColumnNumber

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

@Lconst

a logical value specifying whether to force the y-intercept value ( m 0) equal to 1. @LConst must be of the type bit or of a type that implicitly converts to bit.

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.

If a y-value is < 0 then NULL will be returned.

For more complicated queries, you can try the LOGEST_q function.

Examples

Example #1

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

SELECT *
INTO   #xy
  FROM (   VALUES (508.78, 0.34, 3.95, 1.75, 4.41, 49),
                  (276.59, 0.39, 3.72, 1.17, 3.48, 46.72),
                  (82.07, 0.41, 2.35, 1.64, 3.98, 13.03),
                  (225.66, 0.54, 5.08, 1.57, 3.21, 16.32),
                  (176.03, 0.35, 3.31, 1.19, 3.73, 34.16),
                  (148.2, 0.37, 4.46, 1.4, 4.36, 14.62),
                  (357.56, 0.38, 5.51, 1.63, 3.74, 27.97),
                  (470.35, 0.38, 3.74, 1.72, 3.94, 48.8),
                  (308.7, 0.48, 4.76, 1.55, 3.37, 25.84),
                  (219.6, 0.37, 5.03, 1.53, 4.37, 17.6)) n (y, x0, x1, x2, x3, x4);

This is what the data look like.

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

SELECT *
FROM wct.LOGEST(   '#xy',              --@TableName
                   'y,x0,x1,x2,x3,x4', --@ColumnNames
                   '',                 --@GroupedColumnName
                   NULL,               --@GroupedColumnValue
                   1,                  --@Y_ColumnNumber
                   'True'              --@Lconst
               );

Note that the table name and the column names are both bound by single-quotes, so that they are passed into the function as a string. Essentially the function is dynamically creating a SQL statement to SELECT the column names from the #xy table. This means that we can actually simplify the second parameter, since we are selecting all the columns in this particular table, by entering the following:

SELECT *
FROM wct.LOGEST(   '#xy', --@TableName
                   '*',   --@ColumnNames
                   '',    --@GroupedColumnName
                   NULL,  --@GroupedColumnValue
                   1,     --@Y_ColumnNumber
                   'True' --@Lconst
               );

The third parameter is entered as blank (2 single quotes), since there is no column on which we want to group the results for input into the calculation and the fourth parameter is NULL, since there is no grouped column name. The fifth parameter, the y-column value, specifies that the first column in the resultant table, which becomes the input into the calculation, contains the y-values. The last parameter specified that we want to calculate the y- intercept .

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^[1].

;WITH mycte
   as (SELECT *
         FROM wct.LOGEST('#xy', '*', '', NULL, 1, 1) p )
SELECT d.col_name,
       d.m,
       d.se,
       d.tstat,
       d.pval,
       LOG(m) - wct.T_INV_2T(.05, m.stat_val) * se as [Lower Confidence Level],
       LOG(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 [Residual df],
              Obs - df - 1 as [Regression df],
              ss_reg as [Regression SS],
              ss_resid as [Residual SS],
              F
         FROM (   SELECT stat_name,
                         stat_val
                    FROM wct.LOGEST('#xy', '*', '', NULL, 1, 1)
                   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.

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.LOGEST(   '#xy',  --@TableName
                   '*',    --@ColumnNames
                   '',     --@GroupedColumnName
                   NULL,   --@GroupedColumnValue
                   1,      --@Y_ColumnNumber
                   'False' --@Lconst
               )

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,
       LOG(m) - wct.T_INV_2T(.05, m.stat_val) * se as [Lower Confidence Level],
       LOG(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 1 (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,
       LOG(m) - wct.T_INV_2T(.05, m.stat_val) * se as [Lower Confidence Level],
       LOG(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

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 LOGEST calculation. This allows us to compute the ordinary least squares values for multiple sets of data in a single SELECT statement.

SELECT *
INTO   #xy
  FROM (   VALUES ('Test1', 3.93, 57.91),
                  ('Test1', 4.4, 76.2),
                  ('Test1', 3.33, 20.78),
                  ('Test1', 1.88, 5.06),
                  ('Test1', 3.68, 32.53),
                  ('Test1', 4.81, 55.32),
                  ('Test1', 4.27, 57.83),
                  ('Test1', 3.11, 15.62),
                  ('Test1', 3.81, 22.93),
                  ('Test1', 3.36, 20.58),
                  ('Test2', 12.56, 5.19),
                  ('Test2', 544.52, 66.68),
                  ('Test2', 955.35, 96.94),
                  ('Test2', 478.54, 61.1),
                  ('Test2', 565.07, 68.28),
                  ('Test2', 879.44, 91.75),
                  ('Test2', 988.58, 99.19),
                  ('Test2', 218.15, 36.2),
                  ('Test2', 523.75, 64.97),
                  ('Test2', 653.03, 75.2),
                  ('Test3', 10011.32, 73.6),
                  ('Test3', 14224.21, 86.71),
                  ('Test3', 12616.39, 81.99),
                  ('Test3', 731.51, 21.7),
                  ('Test3', 83.07, 7.84),
                  ('Test3', 7420.4, 64),
                  ('Test3', 3470.08, 44.89),
                  ('Test3', 4320.23, 49.72),
                  ('Test3', 257.08, 13.31),
                  ('Test3', 4083.32, 48.43),
                  ('Test4', 6.75, 33.34),
                  ('Test4', 10.66, 95.33),
                  ('Test4', 8.78, 64.92),
                  ('Test4', 8.46, 57.19),
                  ('Test4', 8.58, 58.07),
                  ('Test4', 8.13, 64.62),
                  ('Test4', 8.83, 74.75),
                  ('Test4', 8.56, 59.43),
                  ('Test4', 9.35, 73.64),
                  ('Test4', 7.4, 46.11),
                  ('Test5', 67.86, 26.69),
                  ('Test5', 198.1, 66.08),
                  ('Test5', 46.65, 18.53),
                  ('Test5', 246.06, 79.92),
                  ('Test5', 82.08, 32.66),
                  ('Test5', 306.95, 97.34),
                  ('Test5', 315.58, 98.06),
                  ('Test5', 231.01, 76.41),
                  ('Test5', 214.47, 71.35),
                  ('Test5', 155.25, 54.05)) n (testid, y, x0);

Let's say wanted to run LOGEST for the all the data where the testid is equal to Test3. We could simply enter the following statement.

SELECT *
  FROM wct.LOGEST('#xy', 'y,x0', 'testid', 'Test3', 1, 'True');

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.LOGEST('#xy', 'y,x0', 'testid', n.testid, 1, 'True') 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.LOGEST('#xy', 'y,x0', 'testid', n.testid, 1, 'True') 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.

^[1]The Excel Regression Tool does not directly support exponential regression. You would need to convert the y-values to ln(y). This means that the coefficients returned are the natural log of the coefficients.

See Also

LINEST_Q - 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