Another observation is also needed to be stressed. In spite of the application of the POLS some values of the |
|
fitting parameters are calculated also with relatively large error. This fact takes place in all cases when the fitting |
|
function contains large number of the fitting parameters. At first, we consider the fitting of the smoothed |
|
functions obtained by means of the POLS. These plots are shown on Fig.5a below. |
|
Figure 5a. These plots demonstrate the application of the BLR |
|
to the smoothed functions for calculation of the fitting |
|
parameters that initially are contained in the functions (8). |
|
The values of the calculated fitting parameters are shown in |
|
Table 2. |
|
Figure 5b. These plots demonstrate the application of BLR for initial |
|
data that are strongly distorted by initial error. For this case, the |
|
calculated values of the fitting parameters are deviated from the initial |
|
ones. It means that for successful application of the ECS method the |
|
accurate or smoothed data are needed. The values of the fitting |
|
parameters are collected in Table 2. |
11 |
After smoothing procedure the desired values of the fitting parameters can be obtained by means of BLR (41)
that are very close to the initial ones. See results presented in Table 2. If this smoothing procedure is not used then the values of the calculated fitting parameters are deviated from the initial ones (stressed by bold). The quality of the fitting procedure (that is described by the value of the relative error) is deteriorated in ten times. So, one can conclude that the smoothing procedure plays an important role in the data fitting procedure. Table 2. The set of the fitting parameters that are calculated for the function (1) which is subjected initially the POLS and without the smoothing procedure
1 |
2 |
|
A0 |
A1 |
A2 |
A3 |
RelErr(%) |
PCC |
0.9037 |
1.0198 |
3.0123 |
-4.4495 |
4.6132 |
5.5804 |
5.4677 |
4.4036 |
0.99882 |
(0.9) |
(1.0) |
(3) |
(-4) |
(4) |
(6) |
(6) |
|
|
0.4825 |
1.2827 |
2.3998 |
5.3297 |
-0.8596 |
4.1747 |
-3.6339 |
44.7565 |
0.8972 |
1.0801 |
0.9098 |
2.0723 |
-2.7777 |
5.5221 |
3.9546 |
3.8837 |
7.3585 |
0.99559 |
(1) |
(0.9) |
(2) |
(-2) |
(6) |
(4) |
(4) |
|
|
0.8718 |
0.9499 |
2.6112 |
-0.8617 |
6.5086 |
-1.6879 |
3.8017 |
51.9292 |
0.8645 |
12
Comments to the Table 2. The exact values of the fitting parameters are bolded and given in the line together with calculated values obtained for the smoothed curves. As one can notice from analysis of these values the POLS has undoubted advantages in comparison of the fitting procedure based on the ECs method applied to raw (initial) data. So, if the initial data are strongly deteriorated by initial error the POLS becomes an effective procedure. The amplitudes have more deviated values in comparison with the calculation of the values of the power-law exponents. This phenomenon is observed for the both cases.
1.7. Further generalizations and some recommendations to the usage of the ECs method.
We will show nontrivial examples of applications of the ECs method for consideration of many actual data proving its effectiveness in the lectures 2. The basic elements and concrete examples of application of the ECs methods were considered also in papers [10-19] and here we do not need to repeat some specific details. However, it is instructive to specify some
problems that can be investigated properly in the nearest future because they represent a general interest for the
mathematical statistics as a whole.
At first, we should explain why this method received the definition as the "eigen-coordinates" (ECs) method. The origin of this definition is the following. As one can see above besides the fitting procedure corresponding to the global fitting minimum (the seed values of the fitting parameters are absent) it has a possibility to select the most suitable hypothesis. Imagine that we want to verify numerically the eigen-values for the stationary Schrödinger quantum equation
(45)
13
Here the set k forms the eigen-functions, H-is the Hamiltonian of a system. If the set k forms a set of eigen- |
|
functions then the set of the functions |
|
|
(46) |
in coordinates (Yk, k) should give a set of straight segments with slopes equaled Ek. If the set of Yk does not |
|
coincide with eigen-functions of the chosen Hamiltonian then this set of segments is distorted and the eigen- |
|
values Ek start to depend on some current variable. In the definition of this method, we want to stress this |
|
important property. |
|
As one can see above the eigen-coordinates (ECs) method allows in solving of some basic problems of the theory |
|
of hypothesis admission. This method is based on the presentation of some analytical function F(x, A) initially |
|
containing a set of non-linear fitting parameters to a new set of the fitting parameters С(A), which are becoming |
|
linear with respect to the chosen function yi F({x}, A). |
|
Such presentation becomes possible if the chosen function satisfies to linear/nonlinear differential equation with |
|
a new set of parameters С(A) forming a linear combination with respect to independent variable x, dependent |
|
variable y and the corresponding derivatives. In other words, the applicability of the ECs method is based on the |
|
following structure of the corresponding linear/nonlinear differential equation |
|
|
(47) |
The set of functions |
k = 1, 2,… is determined totally by the chosen function |
|
yi=F({ x(j)}, A). |
|
14 |
For example, the functions defined by relationships admit the application of EC=method
(48) |
(49) |
The dimension of the vector С=(С1, С2, ... Сk) connected with the initial vector А=(a1, a2, ... ak) by linear/nonlinear |
|
relationships coincides with dimension (number of the fitting components) of the initial vector A or can be less. |
|
Any other function presented in these ECs because of the peculiarities of their construction will be deformed and |
|
accepts the form of a curve. These invertible deformations are provided by Picard theorem [9]. According to this |
|
theorem the function which satisfies to the corresponding differential equation at the given initial conditions is |
|
unique. So, the problem of the analytical relationship identification is visually simplified and hypothesis about the |
|
acceptance or non-acceptance of the chosen function F(x, A) for relationship of Y with X is based on the high level |
15 |
of its significance (at least in the limits of the external error dispersion). |
For example, the most of special functions used in mathematical physics satisfy to the structure of differential equation (DE) [8] presented by equation (47) and the most usual functions also, when the initial set of the fitting parameters presented by the vector А = (a1,a2, ... ak) that enters into DE by a linear way (see Eqns. (49)). So, in
comparison with the previous structure (5) the basic linear relationship of the type (47) can be nonlinear and
includes in itself the fitting function y and its derivatives y', y" and etc. In these cases the integration procedure of
the initial BLR (46) is necessary. As we saw above the integration procedure alongside with the POLS decreases
the value of the initial error.
The usage of a priori information. Let us suppose that a priori some constant Ck is known. It means that it is located in some interval: ak Ck bk. How to take into account this information and makes the calculation in the
frame of the LLSM more definite and statistically stable?
In this case we introduce a new variable t located in the interval
(50)
The limiting values of ak, bk are supposed to be known. For this case the minimized value of the total error accepts
the form
(51)
16
In this case it is necessary to minimize the value of the dispersion 0 corresponding to the error j with respect to the values of the remaining constants Cp (p k)
(52)
Minimization of expression (51) with respect to the constants Cp (p k) leads to the system of linear equations of
the type
(53)
From the solution of linear system of equations (53) it follows that the supposition (50) leads to the conclusion that any constant Cp starts to depend on variable t, forming a "fork" located between the unknown constants
ap and bp
(54)
17
(55)
The variable t is independent variable and so the unknown limits entering in (54) are calculated at t = 0 and t = 1.
Therefore, for these limiting cases we have the relationships
(56)
These two BLRs help to find the unknown limits [ap, bp] for other variables if the values of ak and bk are known.
After calculation of the desired limits of the constants Cp (p k) the values of the errors from (48) 0,j 1,j become
known. Then the total dispersion is minimized with respect to variable t
(57)
The desired values are the following. It is easy to calculate the extreme values tmin and min: the value tmin is located in the interval [0,1]
18
(58)
So, the desired values of the unknown constants are found as
(59)
One can conclude that in the presence of a priory information it is necessary to find only the value of tmin in order
to calculate the true values of the desired constants (59) at t = tmin.
If an initial fitting constant Ak entering into initial fitting function F(x, A) satisfies to relationship (50) then it is
difficult to show the general procedure in derivation of expressions for the constants {Ak} because the relationship
between constants A(C) and C(A) are nonlinear. The finding of the relationship of type (50) for the desired constant Ak should be considered separately. Let us consider some examples from (48) showing that solution of
this problem is not so difficult. For example (48a) let as suppose that one exponent lies in the interval [a1,b1]. So,
in this case the constant can be presented as
(60)
19
The basic linear relationship for the function (48a) has the form
(61)
Inserting the variable (60) into the BLR (61) one can obtain the following relationship
(62)
20
The relationship (62) should be minimized at t = 0 and t = 1. These calculations realized by means of the conventional LLSM will give us the confidence limits for two other variables 2 and C3.
(63)
These limiting values (a2,3 and b2,3) by nonlinear way related with the values a1 and b1 are known.
Then we define the errors
(64)
Expression for dispersion that is minimized with respect to the variable t accepts the form
(65)
21