The principal difference between the previous result (57) and expression (65) is that the nonlinear dependence between initial coefficients {Ak} and coefficients {Ck} figuring in the corresponding BLR generates

the polynomial dependence for the variable t in the expression for dispersion (t) and the local minimal points in this expression are possible.

From a set of three possible minimal points the global minimal point from 0 < tmin <1

(66)

should be chosen. By analogy with this example one can consider example (47b). Therefore, based on this concrete example we show how to solve the problem of calculation the desired fitting constants if some variables are located in confidence intervals.

The problem of elimination of depending constants. Another problem that can be a serious drawback in

application of the conventional LLSM and mentioned above is the dependence of the constants Ck (k=1,2,…,s)

entering into the BLR from each other. In this case the direct application of the LLSM becomes impossible because the value of the determinant of the main correlation matrix

(67)

22

reduces to zero. In these cases the Gramm-Schmidt orthogonal procedure is becoming useless because the basic relationship (21) is applicable only for linear independent set of initial vectors. So, this case requires a special treatment. The simplest procedure that can be applicable for this case is the elimination of dependent variable. In order to understand better this case let us consider simple example. Let us suppose that the BLR has the

following structure

(68)

The constant C2 =F(C1,C3) is dependent constant. We can exclude the constant C2 using the following linear procedure

(69)

Subtracting expression (75) from (74) we receive finally the BLR for independent variables

(70)

Here new variables are defined as

23

(71)

This procedure can be continued

(72)

Definitely, we apply this procedure in the general case obtaining finally the structure of the type

(73)

24

Relationships (73) it is natural to define as the final eigen-coordinates (ECs).

The simplest form of the BLR (78) was obtained by another way based on the following considerations. Let us suppose that the corresponding BLR after exclusion of the dependent variables can be presented in the form

(74)

If in the last equation the parameter Ck is considered as an independent variable then any Cp (from the right-side

of equation (73)) in accordance with ideas discussed above (see Eqns.(51),(54)) can be written as

(75a)

The unknown constants (ap, bp) are found by the LLSM from equations

(75b)

After calculation of these constants the desired functions figuring in (78) are found from the following relationships

25

(76)

1.8. Concluding remarks and questions for self-testing

In the second part of the first lecture we demonstrate how to increase the limits of applicability of the linear least squares method. If the differential equation for the fitting function contains linear combination of the

constants like in Eqn. (47) then after integration one can try to apply the ECs method.

In subsequent lectures a reader will see how to generalize the LLSM for the functional least square method. This method is appeared in the result of the searching the intermediate model for quasi-reproducible experiments.

26

Questions for self-testing

1.How the conventional fitting problem is formulated?

2.What is the regression model, curve?

3.Try to formulate the basic drawback of the nonlinear regression problem?

4.How to formulate the linear least square method? In what cases it can be applicable?

5.What is the BLR and how to obtain it?

6.Why it is necessary to eliminate the mean value of the error < (x)> from the BLR ?

7.If you have two competitive hypotheses for the fitting of available data – what of two is more preferable?

6. Compare two competitive hypotheses

The center of these distributions x0 is supposed to be known.

8. Find the BLR for the three-exponential function

3

y3 (x) A0 Ak exp k x

k1

9.Find the BLR for the combination of exponential and power-law functions

y(x) A0 A1x p A2 exp x

The power-law parameter p (p > 0) is supposed to be known. How this result is changed if the power-law parameter p is located in the interval

[pmin, pmax]?

10.Find the BLR for the combination of exponential and power-law functions

y(x) A1x A2 x2 A3 exp x

29