Lecture 1 (part 2)
In what cases the non-linear fitting problem is reduced to the well-known linear least squares (LLS) method?
The ECs method
1.6 Is it possible to select the most suitable hypothesis from two alternative ones?
1.7. Further generalizations and some recommendations to the usage of the ECs method.
1.8. Concluding remarks
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1.6. Is it possible to select the most suitable hypothesis from two alternative ones? |
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Here we want to show that besides the calculation of the desired set of constants Ck (k=1,2,…,s) the ECs methods |
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helps to select the proper hypothesis verifying two Basic Linear Relationships, which are tuned for selection and |
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simultaneous verification of the competitive hypothesis based on a finite set of measured points. The general |
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idea is the following. Each verified function has own differential equation and if the value of the initial error is |
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rather small then there is one-to-one correspondence between the function and its differential equation which |
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should be satisfied by this partial solution. – (Теорема Пикара, задача Коши)). So, a "strange" function ystr(x) |
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being presented in the eigen-coordinates belonging to a "native" function ynat(x) should receive an additional |
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dependence of the constants Ck (k=1,2,…,s) against the variable x. In other words, the set of straight lines |
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(35) |
BLR |
identifying the "native" function (Ck = const (k=1,2,…,s)) is distorted when the "strange" function (which does not |
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satisfy the set of equalities (27)) is verified. This simple procedure (using additional information based on the |
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properties of the corresponding differential equations) makes this method more preferable and adequate in |
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comparison with traditional methods when two and alternative hypothesis [2] are compared. |
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Let us consider a simple example that illustrates this observation. We choose two simple functions that are |
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similar to each other |
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(36)
After simple manipulations, it is easy to find the BLRs for both functions. For y1(x) the BLR can be written as
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Try to reproduce these expressions as a |
(37a) |
self-tested exercise! |
After calculation of the fitting parameters b1 and the last parameter A1 is found easily from the relations given below
(37b)
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For the function y2(x) one can obtain another BLR
(38a) |
(38b) |
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The plots of these functions (36) visually closed to each other
and they are given on Fig.4(a).
Figure 4a. Here we show two different functions (expressions (36)) marked by red points (y1(x)) and cyan stars (y2(x)) respectively. The
values of the parameters are the following: (b1=0.5, A1=10, =0.7;
b2=0.37, A2=10, = 0.84). The value of the error does not exceed 2% .
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Is it possible to recognize the proper hypothesis and notice the distortions that can appear in the corresponding BLRs? The calculations show that for more reliable recognition of the "native" hypothesis it is convenient to present the straight lines in the form
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What is happened if we pass two |
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(39) |
hypotheses (y1,2(x)) through the |
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ECs tuned for recognition of the |
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first hypothesis? The result is |
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presented by Fig.4(b). |
Figure 4b. On the central figure we show the fitting of the "native"
function to expressions (36). On the figure shown on the right-hand we
show the results of the fitting of the "strange" function y2(x) to the same
expressions (36). Visually one can see that there are no principal
differences between these two results. But one quantitative result
remains important at any level of the external error. The value of the
relative error remains less if the chosen hypothesis corresponds to the rule: "native" function" – "native" BLR. For this case the value of the relative error equals 2.495% in comparison with the situation when the "strange" function corresponds to the BLR tuned for recognition of another hypothesis. For the second situation: "strange" function
-"native" BLR the value of the relative error is equaled to 3.352%.
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We remind here that the value of the relative error is defined by expression (4). Is it possible to differentiate the small differences between two hypothesis in terms of distortions that can appear in the behavior of the constants
C1,2?
Figure 4c. This figure demonstrates the desired distortions that are appeared in the value of the constants, when the "native" function corresponds to the "native" BLR (red points) and when the "strange"
function y2(x) is passing through the same BLR (white points). The
distortions are noticeable. One can notice these distortions in distribution of the absolute error. For native function this distribution looks as uniform, in the opposite case the remnant function is clearly noticeable.
Figure 4d. The increasing the value of the initial error (up to 3.5%)
makes the recognition procedure more uncertain. However, an "experienced eye" can notice possible distortions.
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The verification of the second hypothesis y2(x), which becomes "native for the BRL given by expressions (38), leads to the same results. The value of the relative error is decreased when the rule "native" hypothesis –
"native" BLR is satisfied. In opposite cases the distortions in behavior of the constants (that are expressed by
Eqns. (31)) are noticeable. In the presence of the external error the recognition procedure becomes more
uncertain but one basic feature is conserved. The distortions evoked by external error keeps the straight line in the middle of the given segment while the distortions evoked by the verification of the "strange" function can go out from the middle of the calculated segment.
Figure 4e. This figure demonstrates the behavior of the distortions in the case of large external error (3.5%). One can notice that in the case of verification of the "native function" to its "native" BLR the distortions are concentrated in the vicinity of the middle of the distorted
segment. When the correspondence of the "strange"
function to the same BLR is verified this observation is
violated. Distortions evoked by the presence of the "strange" functions are leaving out the region of distortions, generated by external error. On this figure a specific "hockey stick" located on the left-hand side is clearly noticeable.
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In the end of this paragraph it is instructive to formulate some general observations that can be useful in selection of the proper hypothesis. Usually, the researcher does not know the true hypothesis and judges about the "truth" of the hypothesis chosen based on preliminary information (some theoretical results justifying the selection of the proper hypothesis) and the results of the fitting procedure. The ECs method gives additional
information about the adequacy of the hypothesis selected. Let us formulate these observations in the form of
some recommendations.
Recommendation 1. Try to work with "clean" data containing large number of measured points. If the values of the external error are rather high (as it is presented on Fig2a, for example, then it is useful to use the POLS for smoothing of initial data.
Recommendation 2. If possible, for each selected hypothesis it is necessary to calculate its BLR that was
preliminary verified on mimic data.
Recommendation 3. If the calculation of the BLR is not possible the smaller values of the relative error (Eqn. (11)) can serve as an initial quantitative criterion in selection of the proper hypothesis.
Recommendation 4. The specific behavior of the constants C1,2 and their distortions calculated from
expressions (39) will give additional information for selection of true hypothesis.
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In conclusion, we want to go back to the example presented by the functions (8) and (9). After smoothing these functions, it is necessary to have the BLR for the recognition of the function having at least 6 fitting parameters. Following to general recommendations it is necessary to obtain the differential equation that is satisfied by functions expressed in the form of expressions (8). It is easy to notice that the corresponding differential equation is the conventional Euler equation of the third order that can be written in the form
(40)
Any function of the type
(41)
at certain initial conditions satisfies to the differential equation (40). Integrating (40) three times one can obtain
the desired BLR of the type
(42)
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From this BRL the values of the constants C1,2,3
can be found. After that the desired power-law constants are found as the roots of the cubic
equation
(43)
(44)
The values of the constants C4,5,6 entering into (42) contain the unknown values of the derivatives at the initial point x0 and are not interesting for their calculation. So, they can be omitted. The values of the constants A0,
A1,2,3 are found from (41) by the conventional LLSM, when the values of the power-law parameters from (44) are known. In realization of the fitting procedure we want to mark the importance of the POLS, because the large
values of the initial errors distorted essentially the values of the initial fitting parameters.
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