Here and below the symbol (A B) defines the scalar product in the discrete space of the given number of the measured points

(20)

Therefore, one can conclude that many functions containing initially the unknown set of the fitting parameters satisfy to differential equations where the initial set of nonlinear parameters forms the desired linear combination. In this case the ECs method reduces the problem of the nonlinear fitting to the well known LLSM. It opens a quite new possibility of the calculation of the fitting parameters of many statistical distributions by means of simple and well-developed method. Another possibility is opened in the fitting of many special

functions, which normally is expressed by means of infinite series, and their fitting to actual data presents a

specific and difficult problem. For example, let us consider the differential equation of the second order

(21)

At a = + +1, b = - , c = , this differential equation describes the solution presented in the form of

linear combination of two degenerated hypergeometric functions [8] ( accepts non-integer values)

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(22)

(23)

As it is known, the hypergeometric function is widely used in the mathematical physics for description of

different natural phenomena. If the fitting parameters , , are not known then the fitting by means of the function y(x) (22) of some actual data presents itself a nontrivial problem. The problem can be solved easily by means of differential equation (21). If one integrates this differential equation two times then we obtain the BLR of the type

(24)

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(25)

Here the constant C4 including the unknown value of the first derivative in the initial point x0 is not essential for

the calculations and so it is omitted. The function y(u) determines the actual data that pretends on description by

the hypothesis (22). From equations (24) and (25) the unknown constants C1,2,3 can be found by means of the conventional LLSM. From the calculated values of the constants Ck (k=1,2,3) one can find the desired values of

the unknown parameters , , from equations

(26)

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The unknown constants A1,2 are found from (22) by the LLSM, because other fitting constants entering in it are known. Other examples of application of the ECs method will be considered below.

2. The usage of the orthogonal variables

As it is known that in practical applications of the LLSM the determinant might have the value close to zero.

When it can be happened? Any researcher should know that all fitting constants Ck entering into BLR (5) should

be independent from each other. Possible errors can distort also the initial BLR and create a situation when determinant is becoming close to zero. In these uncomfortable cases the values of the fitting parameters Ck found

from (5) contains large errors and sometimes cannot be calculated. For these cases one suggests the transformation of initial variables Xk(x) to another set of variables that orthogonal each other. It makes the

functions statistically independent to each other and helps to avoid the zeros in the corresponding determinants, which are appeared in calculation of the fitting coefficients Cp. We present some BLR in the form

(27)

(28)

(29)

k=1,2,…, s. As before, the parentheses in (29) determine the value of scalar product in the discrete space of

dimension N, where N is determined by the number of measured points.

(30)

It is instructive to write down the first four functions

(31)

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It is easy to notice from (31) that new set of the functions { (xj )}s is orthogonal, i.e.

(32)

The initial set of constants Ck is found from the linear system of equations

(33)

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Questions for self-testing:

1.What is regression as a procedure?

2.Why this procedure is important? (Reduction and prediction)

3.Smoothing of data. In what cases this procedure is important?

4.Why POLS is more preferable in comparison with other approaches?

5.The conditions of applicability of the ECs method?

6.In what cases the orthogonalizaiton procedure is important?

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Questions, Comments or Remarks?

Thank for your patience!

29

LLSM

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