The approximate analytical expression provides a 'universal' quantitative reduction of any random sequence to a set of parameters (An, n), including also the AUC and ymax values.
These fitting parameters allow in separation of the values (amplitudes) of a random sequence yj onto the optimal statistical
groups (clusters) n with parameters (An, n) that correspond to the reduced description of the random sequence considered.
In cases, when the volume of the sampling is large (N > 100) this reduced presentation can be more informative with respect to an external factor then the numerical evaluation of the k roots from the system of equations. For this case one can fit each function and entering into expression separately and then compare their proximity in terms of the fitting parameters (An, n). The comparison
of higher moments forming (in general) two different samplings (kN) is more precise and adequate. At k = 1, 2 we obtain the conventional reduction expressed in terms of mean value and standard deviation. The "universal" description expressed in terms of the initial integer moments as 1, 2 can be unsatisfactory in
most cases.
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Correlation of two different random sequences.
If the parameters and b do not depend on p then two random sequences compared can be considered as statistically close to each other. If (p) anf b(p) then the sequences compared are statistically different.
This statement is confirmed by numerical experiments.
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3.4 Different generalizations of the GMV- function
Definition of complex moments
Definition of complex moments
Multi-dimens.
GMV-function
The SFM based on usage of higher (fractional) moments can be used |
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for construction of calibration curves, which can show the variations |
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of distinct quantitative parameters characterizing any random |
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sequence (containing a trend) to respect of the desired external factor |
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(concentration of an additive, value of the external field, temperature, |
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pressure, pH-factor and etc.). |
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It is necessary to mark here that possible application of the SFM method depends on the ratio between identified number of additional points (k) and the volume of the sampling (N). If number of distinctive points is limited (k/N << 1) then the reduction based on calculation of the lower discrete moments is preferable. If it is necessary to compare large samplings (k/N 1, N >> 1) when many initial moments are close to each other then approach based on approximate analytical function with subsequent calculation of the fitting parameters by the ECs method is more preferable.
3.4 Relationship of the fractional moments with the nonextensive Tsallis entropy
Another possible step for generalization of the GMV
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Generalization
of the FMs
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q=1
Any random sequence has own Tsallis entropy, which can be expressed “quantitatively” in terms of this set
of the fitting parameters.
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3.4 Basic inequalities
Minkovsky inequality
Cauchy inequality
3.5 The Generalized Pearson’s Correlation Function (GPCF) in the space of the moments
PCC!
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On this figure we show three types of correlations: strong correlations Lm=1, average correlations (Lm > M) and weak correlations M Lm). This universal behavior allows to determine the complete correlation factor.
Matrix of pair correlations cfmin CC 1,
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Here we show the segment of the random sequence corresponding to beta- distribution. We choose it as the pattern distribution.
Here we show the test sequence wich represents a mixture of two distributions: beta- distribution and exponential one. What kind of behavior we might expect if one compare the pattern sequence with test one shifting the short sequence with the step equaled 50 points. So we obtain (1000-100)/50 =18 intervals, which are needed to be tested for recognition of the possible statistical proximity.
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