Wavelet Analysis Statistical Data
Mazurkin Peter Matveevich
Department of Environmental Engineering,Volga State University of Technology, YoshkarOla, Republic of Mari El, Russian Federation
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To cite this article:
Mazurkin Peter Matveevich. Wavelet Analysis Statistical Data. Advances in Sciences and Humanities. Vol. 1, No. 2, 2015, pp. 3044. doi: 10.11648/j.ash.20150102.11
Abstract: Examples of modeling, identification of the amount of asymmetric wavelet signal behavior of natural objects: the pulse of the electrocardiogram of a healthy person; natural drying samples meadow grass; mutual influence of forest cover and tilled territory; Crisis dynamics of the ruble and default 1998 y.; volume of patenting and forecast innovations in Russia until 2020 y.; dynamics of forest fires in the national park for the 19822011 y.; hour increments pulses alpha decay ^{239}Pu sample at the maximum of the solar eclipse; amplitude of gravitational waves from the orbital period of 10 pulsars in the model splashing Universe.
Keywords: Wavelet Signal, Generalized Model, Identification, Examples, Patterns
1. Introduction
Unlike deductive approach to wavelet analysis [1,2,17] proceeding from the equations of classical mathematics inductive approach when statistical selection is primary is offered and concerning it the structure and values of parameters of the generalized wave function [9,10,11,15,21,23] is identified.
Any phenomenon (time cut) or process (change in time) according to sound tabular statistical quantitative data (a numerical field) inductively can be identified the sum of asymmetric wavelet signals of a look
(1)
, ,
where  indicator (dependent variable),  number of the making statistical model (1),  the number of members of model depending on achievement of the remains from (1) error of measurements,  explanatory variable,  amplitude (half) of fluctuation (ordinate),  halfcycle of fluctuation (abscissa),  the parameters of model (1) determined in the program environment CurveExpert (URL: http://www.curveexpert.net/).
On a formula (1) with two fundamental physical constants (Napier's number or number of time) and (Archimedes's number or number of space) the quantized wavelet signal is formed from within the studied phenomenon and/or process.
Further we will show some different examples of the wavelet analysis.
2. Asymmetric Signals of the Electrocardiogram
The purpose  receiving mathematical model of a signal of the electrocardiogram [13]. One period of an electrocardiographic signal without pathology in a reference system is set , where:  time measured in milliseconds (ms);  heart tension in millivolts (mV). The signal is given in figure 1.
In process of complication of a formula from the previous session of search more and more exact records of coefficients (model parameters) are accepted. Process comes to the end with the difficult nonlinear polynomial equation. The received equation will be mathematical model of this one signal of the electrocardiogram.
The schedule of binomial model corresponding to the QRS complex of heart of the healthy person is given in figure 2.
On the remains from the model in figure 2 corresponding to a tooth of T of a signal of the electrocardiogram of heart of the healthy person it is shown in figure 3.
From the fifth component on the thirty third new asymmetric signals are received. But their accessory to fluctuations of cardiac muscles the implicit. Apparently, schedules of tension of electric signals can belong and to other human organs.
Parameters of model (1) and coefficient of correlation are specified in table 1.
 Amplitude of fluctuation, mV  Halfcycle and shift of fluctuation, ms  Correl. coef.  







 
1  2.59574e028  16.74766  0.078955  1.04811  0  0  0  0  0.9780 
2  1.42992e149  63.76683  0.013904  1.30529  10.59449  0.0022405  1.34410  3.18773  
3  3.40070e033  14.31631  0.000120  1.77400  21.15976  0.21683  0.87313  0.52152  
4  1.67937e064  25.57819  0.021603  1.01816  244.7042  0.0030608  0.98941  5.23804  0.6740 
The first three components give coefficient of correlation 0.9780. Therefore the equation (1) quite sufficient to facilitate to the cardiologist diagnostics of heart according to the digital electrocardiogram.
The signal of the electrocardiogram is measured in 0.001 seconds to 2200 ms. Processing of signals in seismic exploration demands a time scale also in 0.001 seconds to 2500 ms [1, page 29, fig. 1.1]. In neurodynamics the time scale in 1 second till 1000 seconds is applied [2, page 90, fig. 2.2a]. In different examples of measurements and the wavelet analysis other units of time are applied.
3. Dynamics of Mass of Moisture During Natural Drying of Tests of a Grass
In the course of natural drying of the cutoff elevated part of a grass there are, at least, two processes: first, dehydration yet not of dead cellular structures from action of heat of air; secondly, moisture deduction by the deadened cages before evaporation of free moisture. In this example a unit of time is 1 hour to 2000 h [16].
Dynamics of mass of test No. 1 during natural drying of the cutoff test of a grass to ready hay on a trial platform is characterized by the equation (1) in a look
(2)
, ,
,
,
,
,
,
.
In a formula (2) there are following symbols:  the mass of test of a grass in dynamics of natural drying, g/m2,  mass of dry hay or grass of an air and dry state, g/m2,  the mass of water in test of a grass in dynamics of its drying, g/m2,  oscillatory indignation of cages of the cutoff grass, g/m2,  oscillatory return of free water cages of a grass, g/m2,  time of natural drying from the moment of cutting of test of a grass, h, = 656.388  the mass of initial moisture in grass test, g/m2,  amplitude (half) of indignation of cages of test of a grass, g/m2,  amplitude (half) of indignation of the cages trying to reanimate circulation of moisture and to receive mineral substances, g/m2,  the period (half) of indignation on moisture loss, h,  the period (half) of indignation on recession of a metabolism, h.
The schedules constructed on a formula (2) are given in figure 4, and it has accurate physical interpretation of behavior of a grass after its cut.
The maximum relative error is equal . The increased waves are shown in figure 5. During the first 500 h natural drying in test of a grass there is a sharp fluctuation of dynamic weight .
From schedules it is visible that the dynamic component quickly dies away in limits till 200 o'clock drying, and less dynamic component proceeds.
From four components on all 18 models type (2) criteria for an ecological and landscape assessment of a meadow are received:
1) rational time of natural drying of the cutoff grass to ready hay, h;
2) optimum time of drying, h;
3) maximum time of drying of a grass, h;
4) air and dry mass of test of a grass or hay, g/m2;
5) minimum mass of test , g/m2;
6) the maximum mass of a grass after a cut, g/m2;
7) the actual mass of test of the cutoff grass, g/m2;
8) coefficient of dynamism of mass of the dried grass from humidity of air, as the relation of the third member (2) to the first;
9) coefficient of dynamism of oscillatory return of moisture grass cages when drying;
10) the general coefficient of dynamism as the relation of the third with the fourth to the sum of the first and second members;
11) wave coefficient of dynamism of drying and moistening (the sum of the third and fourth components to the first);
12) humidity of test, %: a) in a statics (the second part to the first); b) in dynamics (the sum of the second, third and fourth parts to the first).
Further we will show the analysis of the land registry of the Kirov region in time cut (year) on mutual influence of the wood (a biosphere kernel) and arable lands (an agriculture kernel).
4. Influence of Plowing on Woodiness of the Territory
The model of influence of plowing on woodiness was executed in the program environment CurveExpert1.38 [14].
On all 39 points (fig. 6a) the trend as the law of exponential death is received
(3)
After identification (1) dependence with two waves was received (fig. 6b):
(4)
,
,
,
.

a) regularity trend 

b) trend and accounting of two waves of a tremor 
Both fluctuations have a negative sign, therefore indignations crisis. In figure 6b three clusters of municipalities having different style of behavior are visible.
5. Influence of Woodiness of on Plowed of the Territory
And as woodiness of the territory influences on percentage of plowing? [14].
The trend received (fig. 7a) a formula
(5)
Connectivity is better in comparison with structure . Thus, at the heart of relationship with the earth nevertheless the ancient tradition of fire agriculture which nowadays and developments is in the heat of growth in the territory of the Amazon River remained. In the Kirov region this technology to the wood isn't applied.
After identification of a trend and two wavelets of crisis oscillatory indignation of rural regions of the Kirov region the look formula was received (fig. 7b)

a) regularity trend 

b) trend and accounting of two waves of a tremor 
(6)
,
,
,
.
In total six more wavelet functions from which four are adaptation indignations (a positive sign ahead of a component), and two – crisis fluctuations were found. Thus, the total of crisis indignations is also equal to four. In total in statistical model nine members are formed.
The first adaptation happened (fig. 8a) on a formula
(7)
,
.
The second adaptation (fig. 8b) happened on dependence
(8)
,
.
The adaptation of waves goes from left to right, reducing values of the remains of an indicator of plowing of the territory. Thus it is noticeable that rural areas with the high level of woodiness don't react to an indicator of plowing as in their territories the forest industry is developed.


a) first adaptation of plowing  b) second adaptation of plowing 


c) third adaptation of plowing  d) fourth wave of adaptation 


e) crisis on the seventh wave  f) crisis on the eighth wave 
The third adaptation (fig. 8с) happened on expression
(9)
,
.
The fourth adaptation took place (fig. 8d) on the wave equation
(10)
,
.
The third crisis wave taking into account the first, earlier found two wave working out the equations (6) in figure 8e the crisis equation is shown
(11)
,
.
The eighth component receives the constant frequency of fluctuation.
It appeared that Verhoshizhemsky and Sloboda rural areas have equal woodiness of 55.4%. However at them, respectively, plowing of the territory is equal 28.4 and 17.1%. This difference in 28.4 – 17.1 = 11.3% is also shown by the program CurveExpert1.38.
The fourth crisis wave turned out after an exception of these two rural areas (fig. 8f) in the form of model of crisis indignation of a look
(12)
,
.
On character of an arrangement of the remains it was visible that also the subsequent wave components of model are possible. However the maximum relative error already came nearer to an error of measurements of the area of the territory. Therefore there is no need for further building of a design of the general statistical regularity. It is clear, that in the Kirov region woodiness parameter still influences change of coefficient of a plowing of the territory.
6. Crisis Dynamics of Ruble Exchange Rate
The fragment of tabular model for an identification of function of type is given in table 2 (on days from 06.01.1993 till 31.12.2004 only 2522 supervision) [7]. For 06.01.1993 value of the explaining variable .
Date  , day  , rub/$ 
22.01.1993  22  493.0000 
20.01.1993  20  474.5000 
15.01.1993  15  442.0000 
…  …  … 
08.01.2004  4066  29.2450 
06.01.2004  4064  29.4545 
05.01.2004  4063  29.4545 
The main model of crisis of ruble is received by computing opportunities of the program environment CurveExpert1.40 (fig. 9, fig. 10) in the form of a formula
(13)
,
,
,
,
, , ,
, .
Each component (13) makes physical (economic) sense.
The first component under the law of exponential death (Pareto's law in economy, Zipf in biology, Mandelbrot in physics and Laplace in mathematics) shows natural aspiration of the Russian economic system to decrease by 06.01.1993 in the gathered course of domestic currency = 417 rub/$. The second component under the biotechnical law shows stressful excitement of the Russian financial system.
Together two components give a trend in the form of not the wave (determined) binomial formula (fig. 9).
Other three members are wave functions which define computing approaches of quantum informatics in some sense. But, despite a rupture of the actual ruble exchange rate because of denomination by 1000 times, us it was proved that is possible to describe a default smooth functions. Thereby once again is confirmed that social and economic crisis of any power can be described statistical models on good quality selections.
The third component in the form of mega fluctuation is the main, showing chaotic while clearly unwitting economic management. This oscillatory indignation nearly brought to collapse and is similar to behavior of obviously upset natural biological systems.
The fourth and fifth components show a ruble default. This process of a default us is described by a type hyperbole
,
(14)
where  the parameters of model (14) identified in the program environment.


trend from two members  first indignation 


trend and first wave  first hyperbole of a default 


trend and two waves  second hyperbole of a default 
In figure 10 the third wave of hyperbolic fluctuation is shown for the points which are near date of a default and it is characterized by a formula (14) with model parameters:
rub/$, , , day, days.
From a formula (13) it is visible that there were two defaults. If not urgent measures of financial stabilization, on elements of the third member ruble exchange rate would reach the level of 1.50797 106 rub/$ in the beginning, and then 23035.35 rub/$. On 06.01.1993 the periods are equal: 2288.21462 576.4; 743.5; 44.6 days.
7. Patenting Volume in Russia
The number of patents for intellectual property in our country changes [12] according to schedules in figure 11 on a formula
(15)
,
,
,
,
,
,
,
.


binomial trend  first wave indignation 


second wave indignation  dynamics of patents 
The first component (15) is the law of exponential growth in a full form. It is natural part of a trend. And the second component shows crisis (a negative sign before the member) change of number of the Russian patents. As a result of binomial trend in figure 1 receives a tendency of some increase from 1998 to 2007. But after 2007 the second component starts exceeding sharply the natural first member of the equation (15). Understanding of the strongest crisis in innovative development of the country in the second member of the equation (15) would allow to take effective measures and to reduce rates of recession of volume of patenting. Both fluctuations positive (a positive sign before components), amplitude of the first fluctuation has recession, and the second ‒ nature of stressful excitement. Thus the first wave indignation sharply becomes frequent on the frequency that shows controllability loss by innovative processes.
At a halfcycle of 7.24 years increase will reach till one year in 2019 and further will become negative on value (a half of amplitude will become equal only 1108 pieces). Influence of the first fluctuation since 2020 can be neglected. The second fluctuation will end in 2015.
If genotypic distance from a prototype to the invention isn't raised, according to the forecast a formula (15) patenting volume will sharply increase, but thus patents won't raise a world technological level of the letout domestic production.
As a result of, as before, more than 70% of patents will be sent on obsolete to the technician and technology. Since 2017 the volume of patenting will reach till crisis 2007, but each patent will have very small increments on novelty, complexity of a technical solution. For radical reform of the industry and the valid transition to an innovative way of development of the country and growth of technological base of economy remained only 2017  2014 = 3 years.
8. Dynamic number of Forest Fires as Series of Signals
As an example in figure 12 schedules of the physical and mathematical analysis of dynamics of forest fires for 19822011 in «Mary Chodra» on several wavelets are shown (models aren't given). And on the revealed regularities it would be possible to execute the indicative inertial forecast for the forecast horizon equal to the forecast basis (to a period from zero before the last measurement of forest fire), and also the working inertial forecast for one third of the period of last measurements of forest fires. As a result efficiency of suppression of the fires increases. Application of the offered method [15,18,20] expands possibilities of territorial environmental monitoring, increases functionality of system like «Forest Patrol».
Consider one of the affected forest block 27 Alchick. For mathematical analysis left only the strong biotechnical regularities and the correlation coefficient is not less than 0.7. The total period of measurements with 27.07.1992 on 22.05.1994 is only 4282 – 3648 = 634 days. In such a short period of less than two years on the forest block 27 only 9 forest fires.


fourth fluctuation  fifth fluctuation 


sixth fluctuation  seventh fluctuation 


eighth fluctuation  ninth fluctuation 
The period of the fire extinguishing efficiency (fig. 13) is determined by the formula
(16)
The graph shows that with the growth period of the fire increases the efficiency of quenching, i.e. the personnel of the fire service forestry starting to become more active with a greater intensity. The first component of the trend is the exponential growth. The forestry staff is focused on stress arousal (the second component of the trend in biotechnological law), when the period of the fire changes from one to seven hours.
9. Alpha Decay at a Solar Eclipse
Influence of three celestial bodies of the Sun, Earth and the moon on intensity of hour increments of alpha activity of a sample ^{239}Pu are shown in articles [36]. For example, after identification the formula was received (fig. 14) fivemembered (on a limit of opportunities of the program CurveExpert environment)
(17)
, ,
,
,
,
,
,
, ,
,
,
where – time in hours, beginning of 0405 o'clock on April 07, 2005, – sentries of excess of number of impulses, , , , and – components of the model (17), , and – amplitudes (half) of the first, second and third oscillatory indignation of a gravitational wave, , и – halfcycles of fluctuations of alpha decay of a sample 239Pu.


linear trend  first fluctuation 


second fluctuation  third fluctuation 


schedule of fivemembered (17)  the remains after model (17) 
In a formula (17) the first component received constant value, and the second member has linear change. The initial stage for three fluctuations is respectively equal 213.83259 27.7, 22.18104 5.1 = const and 27.67903 15.4 h.
The general model of dynamics of hour increments of impulses includes 43 members.
10. Fractal Analysis of Wavelets
The quantity of impulses of radioactive decay always exceeds с1. Therefore for fractal group we will enter levels [36].
Zero level is equal to the maximum rest of =14034 hour increments of impulses divided on 300, i.e. .
After identification the formula of rank decrease in the maximum remains in the form of the binomial equation was received
(18)
,
.
The first member is Mandelbrot's law for fractal decomposition. The remains show that within ranks 011 one more wave indignation of parameters of models of wavelets is possible. Further models settle down more quietly.
11. Fractals of a Maximum of a Solar Eclipse
The model includes three fractal levels [5]. The first level is equal . Then decrease with turns out 46.8 to 7.4 by 6.3 times. The second level of fractal group at corresponds to three waves at ranks .
All members of model have correlation coefficient more than 0.3 and therefore each wave regularity is significant.
In figure 15 schedules of members with ranks 611 are given.


sixth component  seventh component 


eighth component  ninth component 


tenth component  11th component 
Thus the 11th component as if closes group, receiving amplitude under the law of exponential death. This fact means that the stage of a maximum of solar activity is influenced by the previous stage of a laminar phase. Therefore all wavelets are interconnected on a time scale by infinite wavelets. And finitedimensional wavelets show a stage or part of the general process.
The last wavelet signal received coefficient of correlation 0.8041 and it refers regularity №43 in group with strong factorial communications. Thus, we proved possibility of statistical modeling of indignations of gravitational waves from action of a maximum of a solar eclipse.
12. Model of the Lapping Universe
From schedules in figure 16 it is visible that gravitational oscillatory indignation of star systems on amplitude happens in the range of a cycle time of pulsars 0.11.0 days.

binomial trend 

wavelet signal from 10 pulsars 

second asymmetric wavele 

schedule of model (19) 
All four members give model of the lapping Universe (fig. 16) of a look
(19)
,
,
,
,
,
,
,
,
From a formula (19) it is visible that the periods of fluctuations of amplitude of gravitational waves under a condition are equal at the first wavelet 20.024095462 = 0.048190924 and second fluctuation 20.13093392 = 0.26186784 (it is necessary to consider still a multiplier 1022).
In this interval there are seven star systems, except three pulsars of PSR J0700+6418, PSR J1829+2456 and PSR J1439−5501. They are located to the right of a pointer point 1,0 [8, 19]. The interval of a cycle time of pulsars in the second fluctuation changes from 0.4 to 2.2 – 2.3 days. Two fluctuations are in the range of 0.4 – 1.0 days.
The attitude of the second fluctuation towards the first will be equal 5.43398. Frequency of two fluctuations of a gravitational wave under a condition at the possible pulsar which is infinitely quickly rotating theoretically will be equal 20.750805е22 and 3.818721е22.
To two aperiodic fluctuations of a trend which were expressed by the modified Laplace's laws, two asymmetric wavelet are added.
13. Conclusions
Applicability of statistical model (1) to any natural objects, first of all to biological [2225] and biotechnical is proved, beginning from an impulse of heart of the person to model of the lapping Universe. For example, in gravitational waves between the Sun, Earth and the Moon in the period of a maximum of a solar eclipse there are multilevel processes which can precisely be caught on hour increments of impulses of alpha decay from a sample ^{239}Pu.
And, at last, there is a group of regularities, on the remains comparable with a margin error measurements. This fact proves that the method of measurement of indignations of gravitational waves of the device with alpha decay impulses ^{239}Pu is very exact, allowing to carry out registration on separate impulses.
References