SQL Server POLYFIT Function

Updated 2024-03-06 21:30:48.917000

Description

Use the table-valued function POLYFIT for calculating the coefficients of a polynomial p(x) of degree n that fits the x- and y-values supplied to the function. The result is a two-column table having n+1 rows, containing the polynomial coefficients in descending powers:

y=p_1x^n+p_2x^{n-1}+\dots+p_nx^1+p_{n+1}x^0

Syntax

SELECT * FROM [westclintech].[wct].[POLYFIT](
  <@TableName, nvarchar(max),>
 ,<@x_ColumnName, nvarchar(4000),>
 ,<@y_ColumnName, nvarchar(4000),>
 ,<@GroupedColumnName, nvarchar(4000),>
 ,<@Matrix_GroupedColumnValue, sql_variant,>
 ,<@n, int,>)

Arguments

@x_ColumnName

the name, as text, of the column in the table or view specified by @TableName that contains the x-values to be used in the POLYFIT calculation. Data returned from the @x_ColumnName must be of the type float or of a type that implicitly converts to float.

@y_ColumnName

the name, as text, of the column in the table or view specified by @TableName that contains the y-values to be used in the POLYFIT calculation. Data returned from the @y_ColumnName 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.

@n

an integer specifying the degree of the polynomial.

Return Type

table

Remarks

The x- and y-values are passed to the function as pairs

If x is NULL or y is NULL, the pair is not used in the calculation.

@n must be less than the number of rows in @TableName for the @GroupedColumnValue.

Use the POLYFIT_q function for more complicated queries.

Use the POLYVAL function to evaluate the polynomial for an x-value.

Examples

In this example, we will use the SeriesFloat function to generate a series of x-values equally spaced in the interval [0, 2.5] and then evaluate the error function, ERF, at those points. We will specify an approximating polynomial of 6 degrees.

SET NOCOUNT ON;
SELECT SeriesValue as x,
       wct.ERF(SeriesValue) as y
INTO #erf
FROM wct.SeriesFloat(0, 2.5, 0.1, NULL, NULL);
SELECT *
FROM wct.POLYFIT(   '#erf', --@TableName
                    'x',    --@x_ColumnName
                    'y',    --@y_ColumnName
                    '',     --@GroupedColumnName
                    NULL,   --@GroupedColumnValue
                    6       --@n
                );
DROP TABLE #erf;

This produces the following result.

This means that there are 7 coefficients. We could run the following SQL to see the structure of the approximating polynomial.

SET NOCOUNT ON;
SELECT SeriesValue as x,
       wct.ERF(SeriesValue) as y
INTO #erf
FROM wct.SeriesFloat(0, 2.5, 0.1, NULL, NULL);
SELECT *,
       ROW_NUMBER() OVER (ORDER BY coe_num DESC) - 1 as pow
FROM wct.POLYFIT(   '#erf', --@TableName
                    'x',    --@x_ColumnName
                    'y',    --@y_ColumnName
                    '',     --@GroupedColumnName
                    NULL,   --@GroupedColumnValue
                    6       --@n
                );
DROP TABLE #erf;

This produces the following result.

Thus, the approximating polynomial would be:

0.00841937x ⁶ - 0.0982996x⁵ + 0.421736x⁴ - 0.743463x³ + 0.147104x² + 1.10645x + 0.000441174

We can inspect the fit of the approximating polynomial by creating a table of the x- and y-values and evaluating the approximating polynomial for each x and comparing it to the actual y-values. While we could do this using the output of the POLYFIT_q table-valued function, it is much simpler to just use the POLYVAL function.

SET NOCOUNT ON;
SELECT SeriesValue as x,
       wct.ERF(SeriesValue) as y
INTO #erf
FROM wct.SeriesFloat(0, 2.5, 0.1, NULL, NULL);
SELECT a.x,
       ROUND(a.y, 8) as y,
       ROUND(wct.POLYVAL(b.x, b.y, 6, a.x), 8) as f,
       ROUND(wct.POLYVAL(b.x, b.y, 6, a.x) - a.y, 8) as [f - y]
FROM #erf a,
     #erf b
GROUP BY a.x,
         a.y
ORDER BY 1;
DROP TABLE #erf;

This produces the following result.

It looks like the approximating polynomial is accurate to about 3 or 4 decimal places. In fact if we graphed the results, they would look like this:

We can see that in this range, [0, 2.5] the fit between the y-values and the f-values is quite good. However, what if we extend the range of the interval from 2.5 to 6? We will simply change SeriesFloat to stop at 6.0 rather than 2.5 and then graph the output.

SELECT SeriesValue as x,
       wct.ERF(SeriesValue) as y
INTO #erf
FROM wct.SeriesFloat(0, 6.0, 0.1, NULL, NULL);
SELECT a.x,
       a.y,
       wct.POLYVAL(b.x, b.y, 6, a.x) as f,
       wct.POLYVAL(b.x, b.y, 6, a.x) - a.y as [f - y]
FROM #erf a,
     #erf b
GROUP BY a.x,
         a.y
ORDER BY 1;

This produces the following graph.

It's important to note that coefficients in the interval [0, 6] are different than the coefficients in the interval [0, 2.5] because there were more x- and y-values passed to the function. What if we wanted to use the coefficients from the [0, 2.5] interval to predict the values in the [0, 6] interval?

SELECT SeriesValue as x,
       wct.ERF(SeriesValue) as y
INTO #erf
FROM wct.SeriesFloat(0, 2.5, 0.1, NULL, NULL);
SELECT SeriesValue as x,
       wct.ERF(SeriesValue) as y,
       wct.POLYVAL(b.x, b.y, 6, a.SeriesValue) as f,
       wct.POLYVAL(b.x, b.y, 6, a.SeriesValue) - wct.ERF(SeriesValue) as [f - y]
FROM wct.SeriesFloat(0, 6.0, 0.1, NULL, NULL) a ,
     #erf b
GROUP BY a.SeriesValue,
         wct.ERF(SeriesValue)
ORDER BY 1;
DROP TABLE #erf;

This produces the following graph.

As you can see, the polynomial approximation quickly diverges from the function in the region above 2.5.

Let's look at a different example. In this example we will look at some interest rate values over a time-horizon and calculate the 3^rd degree polynomial approximation. The x-values represent the number of years and the y-values are the decimal values of the interest rate (.01 = 1%).

SET NOCOUNT ON;
SELECT *
INTO #y
from
(
    VALUES
        (1, 0.0028),
        (2, 0.0056),
        (3, 0.0085),
        (5, 0.0164),
        (7, 0.0235),
        (10, 0.0299),
        (20, 0.0382),
        (30, 0.0406)
) n (x, y);
SELECT *
FROM wct.POLYFIT('#y', 'x', 'y', '', NULL, 3);
DROP TABLE #y;

This produces the following result.

While we would always recommend using the POLYVAL function to calculate the polynomial approximation for any x-value, in this example we will use the results of the POLYFIT_q TVF to calculate the rates for each year from 1 through 30 out to 4 decimal places.

SET NOCOUNT ON;
SELECT *
INTO #y
from
(
    VALUES
        (1, 0.0028),
        (2, 0.0056),
        (3, 0.0085),
        (5, 0.0164),
        (7, 0.0235),
        (10, 0.0299),
        (20, 0.0382),
        (30, 0.0406)
) n (x, y);
SELECT *,
       ROW_NUMBER() OVER (order by coe_num DESC) - 1 as pow
into #coe
FROM wct.POLYFIT('#y', 'x', 'y', '', NULL, 3);
SELECT SeriesValue as y,
       ROUND(SUM(POWER(SeriesValue, pow) * coe_val), 4) as r
FROM wct.SeriesInt(1, 30, NULL, NULL, NULL) ,
     #coe
GROUP BY SeriesValue
ORDER BY 1;
DROP TABLE #coe;
DROP TABLE #y;

This produces the following result.