Career, Family And Living For The Lord
-
A Twenty-Five Year History

by

James Thomas Lee, Jr.
12/25/97

Copyrighted 1995 by James Thomas Lee, Jr.
Copyright Number:  XXx xxx-xxx

Appendices

Appendix C. Polynomial Least Squares Regression {320 words}

Again, like with the linear least squares technique from above, polynomial least squares begins with the definition of a deviation-squared function.

1. Therefore, let f = SUM (y(i) - a1x(i)**2 - a2x(i) - a3)**2, where a1, a2, and a3 are coefficients, 


                      y(i)  => the observed value for y,

                      x(i)  => the observed value for x.

2. Find Partial(f) / Partial(a1) , Partial(f) / Partial(a2) , Partial(f) / Partial(a3), and set to zero to minimize the variance.

3. This will yield the following equations: 


             Partial(f)
        a.   -----------    =  2 SUM (y(i) - a1x(i)**2 - a2x(i) - a3 ) (-x(i)**2) = 0
             Partial(a1)


             Partial(f)
        b.   -----------    =  2 SUM (y(i) - a1x(i)**2 - a2x(i) - a3 ) (-x(i)) =  0
             Partial(a2)


             Partial(f)
        c.   -----------    =  SUM (y(i) - a1x(i)**2 - a2x(i) - a3 ) (-1)  =    0
             Partial(a3)

4. The above equations from Step 3 can be set up in the following matrix format: 


  |                  |       |                                           |  |    |
  | SUM(x(i)**2y(i)) |       | SUM(x(i)**4)    SUM(x(i)**3   SUM(x(i)**2 |  | a1 |
  |                  |       |                                           |  |    |
  | SUM(x(i)y(i))    |   =   | SUM(x(i)**3)    SUM(x(i)**2    SUM(x(i))  |  | a2 |
  |                  |       |                                           |  |    |
  | SUM(y(i))        |       | SUM(x(i)**2)      SUM(x(i)          n     |  | a3 |
  |                  |       |                                           |  |    |

5. Determinant of Matrix A. 


Let   a1 = nSUM(x(i)**2) - (SUM(x(i))**2

      a2 = nSUM(x(i)**3) - SUM(x(i))SUM(x(i)**2)

      a3 = SUM(x(i))SUM(x(i)**3) - (SUM(x(i)**2)**2

      a4 = nSUM(x(i)**4) - (SUM(x(i)**2))**2

      a5 = SUM(x(i))SUM(x(i)**4) - SUM(x(i)**2)SUM(x(i)**3)

      a6 = SUM(x(i)**2)SUM(x(i)**3) - (SUM(x(i)**3))**2


det(A) = SUM(x(i)**4)(a1) - SUM(x(i)**3)(a2) + SUM(x(i)**2)(a3)



6.  Using co-factors,

                        |                                |
                        |    a1         -a2        a3    |
                        |  -----       -----     -----   |
                        |  det(A)      det(A)    det(A)  |
                        |                                |
                        |                                |
                        |   -a2          a4       -a5    |
Inverse of A (A**-1) =  |  -----       -----     -----   |
                        |  det(A)      det(A)    det(A)  |
                        |                                |
                        |                                |
                        |    a3         -a5        a6    |
                        |  -----       -----     -----   |
                        |  det(A)      det(A)    det(A)  |
                        |                                |

7. Using A**-1, the coefficients, a1, a2, and a3, can be found and are as follows: 



          (SUM(x(i)**2y(i))(a1)  -   (SUM(x(i)y(i))(a2)  +   (SUM(y(i))(a3)
 a1 = ------------------------------------------------------------------------
                                    det(A)


          (SUM(x(i)**2y(i))(-a2)  +  (SUM(x(i)y(i))(a4)  +   (SUM(y(i))(-a5)
 a2 = ------------------------------------------------------------------------
                                    det(A)


          (SUM(x(i)**2y(i))(a3)  +  (SUM(x(i)y(i))(-a5)  +   (SUM(y(i))(a6)
 a3 = ------------------------------------------------------------------------
                                    det(A)

Appendix D. The Implementation of a Special Language Interpreter

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