Theil-Sen estimator function in T-SQL

Question

Does anyone have a Theil-Sen regression function written in T-SQL?

I found one written in Perl, but I'm not able to recode it into SQL.

Answer 1

I was lying when I said, I'm not able to recode it into SQL. I was just too lazy. Here is the code with an example of usage.

The Code is based on a TheiSen perl library, using QuickMedian. Let's define a new table type to easily pass our data to the procedure.

CREATE TYPE dbo.TheilSenInputDataTableType AS TABLE 
(
    ID INT IDENTITY(1,1),
    x REAL, 
    y REAL
)

Please note the ID column, which is important here as our solution uses CROSS APPLY statement in order to achieve correct interpretation of the inner loop found in TheilSen.pm.

my ($x1,$x2,$y1,$y2);
foreach my $i(0 .. $n-2){
    $y1 = $y->[$i];
    $x1 = $x->[$i];
    foreach my $j($i+1 .. $n-1){
        $y2 = $y->[$j];
        $x2 = $x->[$j];

We will also need a new data type for storing an array of real type values.

CREATE TYPE [dbo].[RealArray] AS TABLE(
    [val] [real] NULL
)

Here is the f_QuickMedian function, returning median for given array. Credit for this one goes to Itzik Ben-Gan.

CREATE FUNCTION [dbo].[f_QuickMedian](@RealArray RealArray READONLY)
RETURNS REAL
AS
BEGIN
    DECLARE @Median REAL;
    DECLARE @QMedian REAL;

    SELECT @Median = AVG(1.0 * val)
    FROM
    (
        SELECT o.val, rn = ROW_NUMBER() OVER (ORDER BY o.val), c.c
        FROM @RealArray AS o
        CROSS JOIN (SELECT c = COUNT(*) FROM @RealArray) AS c
    ) AS x
    WHERE rn IN ((c + 1)/2, (c + 2)/2);

    SELECT TOP 1 @QMedian = val FROM @RealArray
    ORDER BY ABS(val - @Median) ASC, val DESC

    RETURN @QMedian
END

And the p_TheilSen Estimator:

CREATE PROCEDURE [dbo].[p_TheilSen](
      @TheilSenInput TheilSenInputDataTableType READONLY
    , @m Real OUTPUT
    , @c Real OUTPUT
)
AS
BEGIN
    DECLARE 
        @m_arr RealArray
      , @c_arr RealArray;       

    INSERT INTO @m_arr
        SELECT m
        FROM 
        (
            SELECT  
                t1.x as x1
                , t1.y as y1
                , t2o.x as x2
                , t2o.y as y2
                , t2o.y-t1.y as [y2 - y1]
                , t2o.x-t1.x as [x2 - x1]
                , CASE WHEN (t2o.x <> t1.x) THEN  CAST((t2o.y-t1.y) AS Real)/(t2o.x-t1.x) ELSE NULL END AS [($y2-$y1)/($x2-$x1)]
                , CASE WHEN t1.y = t2o.y THEN 0
                  ELSE
                    CASE WHEN t1.x = t2o.x THEN NULL
                        ELSE 
                        -- push @M, ($y2-$y1)/($x2-$x1);
                        CAST((t2o.y-t1.y) AS Real)/(t2o.x-t1.x)
                    END
                  END as m
            FROM @TheilSenInput t1
            CROSS APPLY
                    (
                    SELECT  t2.x, t2.y
                    FROM    @TheilSenInput t2
                    WHERE   t2.ID > t1.ID
                     ) t2o
        ) t
        WHERE m IS NOT NULL 

    SELECT @m = dbo.f_QuickMedian(@m_arr)

    INSERT INTO @c_arr
        SELECT y - (@m * x)
            FROM @TheilSenInput

    SELECT @c = dbo.f_QuickMedian(@c_arr)

END

Example:

DECLARE 
      @in TheilSenInputDataTableType
    , @m Real
    , @c Real

INSERT INTO @in(x,y) VALUES (10.79,118.99)
INSERT INTO @in(x,y) VALUES (10.8,120.76)
INSERT INTO @in(x,y) VALUES (10.86,122.71)
INSERT INTO @in(x,y) VALUES (10.93,125.48)
INSERT INTO @in(x,y) VALUES (10.99,127.31)
INSERT INTO @in(x,y) VALUES (10.96,130.06)
INSERT INTO @in(x,y) VALUES (10.98,132.41)
INSERT INTO @in(x,y) VALUES (11.03,135.89)
INSERT INTO @in(x,y) VALUES (11.08,139.02)
INSERT INTO @in(x,y) VALUES (11.1,140.25)
INSERT INTO @in(x,y) VALUES (11.19,145.61)
INSERT INTO @in(x,y) VALUES (11.25,153.45)
INSERT INTO @in(x,y) VALUES (11.4,158.03)
INSERT INTO @in(x,y) VALUES (11.61,162.72)
INSERT INTO @in(x,y) VALUES (11.69,167.67)
INSERT INTO @in(x,y) VALUES (11.91,172.86)
INSERT INTO @in(x,y) VALUES (12.07,177.52)
INSERT INTO @in(x,y) VALUES (12.32,182.09)


EXEC p_TheilSen @in, @m = @m OUTPUT, @c = @c OUTPUT

SELECT @m
SELECT @c

Returns:

m = 52.7079
c = -448.4853

Just for a comparision, perl version returns following values for the same data set:

m = 52.7078651685394
c = -448.484943820225

I use TheilSen estimator to calculate DaysToFill metric for filesystems. Enjoy!

Answer 2

I checked also, for T-SQL, Oracle and servers in general (too complex to be written in pure SQL).

However, you might be interested in this (a scientific/statistical package for Python). The algorithm is implemented there also and in Python. Python is a language which humans have at least some chance of being able to understand, unlike Perl .

Your question intrigued me and I dug around. There are C and C++ libraries which contain this algorithm - and it's also available in a few R packages. And @srutzky 's post also looks interesting.

+1 for an interesting question BTW - and welcome to the forum :-)

Answer 3

This most likely would be a good fit for doing something in SQLCLR, similar to the following Question / Answer (also here on DBA.SE):

Is there a SQL Server implementation of the Longest Common Substring problem?

When I have time later, I will see how feasible this would be.