Is there a way to calculate correlation in TSQL using OVER Clauses instead of CTE's?

Let's say you have a table with columns, Date, GroupID, X and Y.

CREATE TABLE #sample
  (
     [Date]  DATETIME,
     GroupID INT,
     X       FLOAT,
     Y       FLOAT
  )

DECLARE @date DATETIME = getdate()

INSERT INTO #sample VALUES(@date, 1, 1,3)
INSERT INTO #sample VALUES(DATEADD(d, 1, @date), 1, 1,1)
INSERT INTO #sample VALUES(DATEADD(d, 2, @date), 1, 4,2)
INSERT INTO #sample VALUES(DATEADD(d, 3, @date), 1, 3,3)
INSERT INTO #sample VALUES(DATEADD(d, 4, @date), 1, 6,4)
INSERT INTO #sample VALUES(DATEADD(d, 5, @date), 1, 7,5)
INSERT INTO #sample VALUES(DATEADD(d, 6, @date), 1, 1,6)

and you want to calculate the correlation of X and Y for each group. Currently I use CTEs which get a little messy:

;WITH DataAvgStd
     AS (SELECT GroupID,
                AVG(X)   AS XAvg,
                AVG(Y)   AS YAvg,
                STDEV(X) AS XStdev,
                STDEV(Y) AS YSTDev,
                COUNT(*) AS SampleSize
         FROM   #sample
         GROUP  BY GroupID),
     ExpectedVal
     AS (SELECT s.GroupID,
                SUM(( X - XAvg ) * ( Y - YAvg )) AS ExpectedValue
         FROM   #sample s
                JOIN DataAvgStd das
                  ON s.GroupID = das.GroupID
         GROUP  BY s.GroupID)
SELECT das.GroupID,
       ev.ExpectedValue / ( das.SampleSize - 1 ) / ( das.XStdev * das.YSTDev )
       AS
       Correlation
FROM   DataAvgStd das
       JOIN ExpectedVal ev
         ON das.GroupID = ev.GroupID

DROP TABLE #sample

It seems like there should be a way to use OVER and PARTITION to do this in one fell swoop without any subqueries. Ideally TSQL would have a function so you could write:

SELECT GroupID, CORR(X, Y) OVER(PARTITION BY GroupID)
FROM #sample
GROUP BY GroupID

Solution

Using this formula of corellation you cannot avoid all the nested queries even if you use over(). The thing is that you cannot use both group by and over in the same query and also you can not have nested aggregation functions e.g. sum(x - avg(x)). So you in best case scenario, according to your data, you will need to keep at least the with.

Your code will look like something like that

;WITH DataAvgStd
     AS (SELECT GroupID,
                STDEV(X) over(partition by GroupID) AS XStdev,
                STDEV(Y) over(partition by GroupID) AS YSTDev,
                COUNT(*) over(partition by GroupID) AS SampleSize,
                ( X - AVG(X) over(partition by GroupID)) * ( Y - AVG(Y) over(partition by GroupID)) AS ExpectedValue
         FROM   #sample s)         
SELECT distinct GroupID,
       SUM(ExpectedValue) over(partition by GroupID) / (SampleSize - 1 ) / ( XStdev * YSTDev ) AS Correlation
FROM DataAvgStd

An alternative is to use an equilevant formula for correlation as Wikipedia describes.

This can be written as

SELECT GroupID,
       Correlation=(COUNT(*) * SUM(X * Y) - SUM(X) * SUM(Y)) / 
                   (SQRT(COUNT(*) * SUM(X * X) - SUM(X) * SUM(x))
                    * SQRT(COUNT(*) * SUM(Y* Y) - SUM(Y) * SUM(Y)))
FROM #sample s
GROUP BY GroupID;