Standard deviation in

It is shown in ../FitData.doc htm and ../ExtremePow.doc htm that the value of a set of M constants c = {c₁,c₂,..c_M}can be determined by minimizing c² with respect to these constants.

(1)

In this case the function F is this process of arriving at this set of constants.

              (2)

The standard deviation in F with data points f_i with known standard deviations δ_i is shown in the first section of  ../../definitions/Random/Errinfun.doc htm to be

    (3)

 

The value of c_I found by the minimization the c² in (1) is an example of the function F of {f_i}.

In the process of minimization the matrix A and vector B are defined by

        (4)

    (5)

The changes in the constants c on the last minimization step are given by.

             (6)

This can be used to find the derivative of c_I with respect to a single point in the data set. 

  (7)

The coefficient of the partial of the second derivative array is exactly zero at the minimum of c². The partial of the second term with respect to f_i is given by taking the partial of (5) with respect to f_i. 

 

(8)

So that

       (9)

Insert (9) into (3) to find

          (10)

Expand the sums

               (11)

 

Evaluate the sum on i first

                (12)

The term inside the { } in (12) is very similar to the original A_{K J} defined in (4)

The Error Matrix

                The usual assumption is ɛ_i = d_i.  Then the term in { } in (12) is A_K,J. Since A^-1_J,L  is the inverse of A_K,L, the sum on L yields

            (13)

And the sum on K yields

       (14)

Thus 2N_powA_I,J^-1 is the “error matrix”.  This differs from the ususal notation by the factors 2 and N_pow.  The factor of 2 is due the fact that in the definitions of A and B in equations (4) and (5) the twos were not cancelled out.  This form above makes A and B the actual second and first derivatives of c² but introduces a 2 into the final equation for the standard deviation.  The N_pow is the result of minimizing a power other than 2 in (1).  Nowhere in the derivation of (14) is it assumed that (1) is reduced to any particular value.

                The result in (14) means that if we have a very accurate set of values, small d_i’s resulting from 128 hours of data, and we repeat the minimizations using shorter runs with larger values of d_i’s, for example a whole series I_S = 1 to M_s of d_I’s resulting from 8 hours of data.  That the standard deviation defined by (14) is also that given by

(15)

                If the c² defined in (1) with N_pw = 1 is significantly greater than N, then either the function f_A(x_i) is not capable of representing the data or maybe d_I is too small.  A common “fix” is to assume that

  (16)

If (14)is multiplied by (16), equations (14) and (15) will not give the same results.  The error has been increased to compensate for an inadequate fit.  .

 

The Standard Deviation in     

                If F(x,{c{f}} is simply c_K, then the partials in (14) become d_JK and d_KI leaving

   (17)

 

The Standard Deviation in

                                (18)

                           (19)

So that  (14) becomes

   (20)

A shortcut to the error equation is to first write

     (21)

Expand so that all terms are of the form d_Id_J, then substitute A_I,J^-1 for the d_Id_J products.  Thus for F = c₁c₇

(22)

The Standard Deviation in        

                            (23)

                               (24)

After a bit of algebra equivalent to that in (20)

                (25)

Note that for J=K, that the error becomes zero as it should.

The Standard Deviation in     

(26)

The Standard Deviation in a Linear Function

    (27)

Using (14)

  (28)

The Standard Deviation in a Non-Linear Function

The function F(x,c) differs from that defined in (2) in that it is a function also of x.  When this is carried through all of the derivations, equation (14) is reached, but the partials become functions of x.

               (29)