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Identification of Trends and Wave Regularities according to Statistical Data of Fluctuations of Mass of Glaciers | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
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Identification of Trends and Wave Regularities according to Statistical Data of Fluctuations of Mass of Glaciers

Mazurkin PM*

Doctor of Engineering, Academician of Russian Academy of Natural History and Russian Academy of Natural Sciences, Volga State University of Technology, Russia

*Corresponding Author:

Mazurkin PM
Doctor of Engineering, professor
Academician of Russian Academy of Natural
H istory and Russian Academy of
Natural Sciences
Volga State University of Technology
Tel: 2156561646
E-mail: [email protected]

Received Date: November 14, 2016 Accepted Date: November 28, 2016 Published Date: November 30, 2016

Citation: Mazurkin PM (2016) Identification of Trends and Wave Regularities according to Statistical Data of Fluctuations of Mass of Glaciers. J Phys Math 7: 205. doi: 10.4172/2090-0902.1000205

Copyright: © 2016 Mazurkin PM. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The method of identification is shown on the example of tabular data of measurements of six parameters of subgroup of EEE on fluctuations in 2013 at balance of mass of 2528 glaciers of Earth. The equations of a trend and oscillatory indignations on the basis of steady laws on the generalized wave function in the form of an asymmetric wavelet signal with variables of amplitude and the period of fluctuation are received. Schedules of components of the generalized model of a wavelet signal allow to see visually a picture of mutual influence of all six parameters of subgroup of EEE at fluctuations of mass of glaciers. On the revealed equations it is possible to carry out the amplitude-frequency analysis.


Glaciers; Fluctuation; Balance of weight; Factors; Regularities


Unlike deductive approach to wavelet analysis 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 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

Equation (1)

where y - indicator (dependent variable), i - number of the making statistical model (1), m - the number of members of model depending on achievement of the remains from (1) error of measurements, x - explanatory variable, Ai-amplitude (half) of fluctuation (ordinate), pi - half-cycle of fluctuation (abscissa), a1...a8.

-The parameters of model (1) determined in the program environment CurveExpert.

On a formula (1) with two fundamental physical constants e (Napier's number or number of time) and π (Archimedes's number or number of space) the quantized wavelet signal is formed from within the studied phenomenon and/or process.

Basic Data for Statistical Modeling

In geoecology different types of natural objects [1], and among them, undoubtedly are considered by us, the important place is taken by glaciers as climate regulators. In article of regularity of distribution of glaciers for 2013 are given for the first time. On basic data [1] the fragment of the matrix including six factors is shown in Table 1 (identifiers are accepted according to the original). Data are located asymmetrically in Table 2.

1     100     100
2     398     500
71     1073     -63
72 -54.78136 -68.40169 1038 1015 -644 371
73 -54.78182 -68.40249 1036 947 -540 406
101 -54.78048 -68.40171 1073 743 -984 -241
102 -16.3032 -68.108 5053     -958
103 -16.3025 -68.1083 5056     -1385
620     4058     120
621     3796 20 -3719 -3699
622     3828 99 -2565 -2466
2526 60.40417 -148.9067 1053     -2090
2527 60.41974 -148.9207 1283     0
2528 60.42495 -148.9371 1367     960

Table 1: Matrix of basic data for statistical modelling.

Influencing factors x Dependent factors (indicatorsy)
EEE6-POINT LATITUDE [decimaldegree] 1623 1623 1623 388 388 1591
EEE7-POINT LONGITUDE [decimaldegree] 1623 1623 1623 388 388 1591
EEE8-POINT ELEVATION [ma.s.l.] 1623 1623 2467 531 531 2467
EEE9-POINT WINTER BALANCE [mmw.e.] 388 388 531 531 531 531
EEE10-POINT SUMMER BALANCE [mmw.e.] 388 388 531 531 531 531
EEE11-POINT ANNUAL BALANCE [mmw.e.] 1591 1591 2467 531 531 2467

Table 2: Quantity of the measured points of balance of mass of glaciers, piece.

Each of factors can have two states: first, a factor as the influencing variable x; secondly, the same factor is accepted as a dependent indicator of y. On them it is possible to assume that amplitude and the period of fluctuations on the general model. (1) submit to the biotechnical law [2-5]. Because of uncertainty of the direction of a vector "better → worse" at values of each factor we don't reveal regularity the rank distributions. Therefore the rating of glaciers is also not carried out.

In this regard the correlation coefficient the rank distributions is equal 1, and the factorial analysis we will carry out at all 62 – 6=30 binary relations.

A rating of Factors as the Influencing Variables and Indicators

The method of the factorial analysis offered by us allows not to think a priori of ratios between separate parameters of the studied system (system contains 2528 glaciers in our example from the Table 1). As a result the psychological barrier at researchers gets off: many binary relations for them will be unexpected. Therefore as our practice [5] showed, the factorial analysis the unique equation of type (1) allows to find unexpected scientific solutions in the field of research [6-11]. If some factorial communications are unusual and thus are highly adequate, new technical solutions are theoretically shown here. And often at the level of inventions of world novelty [11]. Thus the repeating process identification of the law (1) on one binary relation we designated the wavelet analysis [8,10].

Without waves, that is changes only on amplitude at very long wave, incommensurably bigger on the fluctuation period to an interval of measurements, are formed the so-called determined binary relations. All of them are special cases of a formula (1).

The square correlation matrix received after the analysis of the binary relations between all six variables accepted on basic data from Table 1 is given in Table 3. Here the rating of factors is given.

Influencing factors x Dependent factors Indicators y Sum Σr Place Ix
EEE6   1 0.9732 0.9491 0.4425 0.3985 0.4551 4.2184 2
EEE7 0.9688 1 0.4305 0.4154 0.4629 0.4156 3.6932 3
EEE8 0.8178 0.8611 1 0.7201 0.6835 0.4421 4.5246 1
EEE9 0.2237 0.4492 0.5431 1 0.2265 0.4721 2.9146 6
EEE10 0.6408 0.5571 0.1557 0.2301 1 0.7309 3.3146 5
EEE11 0.3703 0.4978 0.1719 0.5363 0.7592 1 3.3355 4
Sum Σ r 4.0214 4.3384 3.2503 3.3444 3.5306 3.5158 22.0009 -
Place Iy 2 1 6 5 3 4 - 0.6111

Table 3: Correlation matrix and rating of factors on the binary relations.

The coefficient of a correlative variation for 2528 glaciers of Earth is equal 22.0009/62=0.6111. This criterion is applied when comparing various systems, for example, of different groups of glaciers with each other. Perhaps also comparison of all glaciers with other objects sew planets, for example, with deserts or with agroecological classes of a soil cover.

From six influencing variables on the first place there was EEE8 factor. On the second place EEE6 factor, and was located on the third place – EEE7. Among dependent indicators on the first place there is EEE7 factor. In second place is the factor EEE6, and the third -EEE10.

Strong Binary Relations

At correlation coefficient more than 0.7 binary relations between factors become strong (tab. 4As a rule, the accounting of wave indignation, additional to a trend, gives significant increase in adequacy to the revealed regularity on a formula (1). But glaciers do not yet have good data, because many of the values of the six factors have empty cells: the best matrix is illustrated in Table 4 completely filled with cells.

Influencing factors x Dependent factors indicators y
EEE6-POINT LATITUDE   0.9732 0.9491      
EEE7-POINT LONGITUDE 0.9688          
EEE8-POINT ELEVATION 0.8178 0.8611   0.7201    
EEE10-POINT SUMMER BALANCE           0.7309
EEE11-POINT ANNUAL BALANCE         0.7592  

Table 4: Correlation matrix of the strong binary relations at r>0.7.

Because of low completeness of a matrix from 30 formulas received only eight strong communications that makes 22.2%. Thus as the influencing variable EEE9 factor -POINT WINTER BALANCE was excluded. But as dependent indicators remained all six factors. From eight strong communications three (37.50%) treat trends.

Hierarchy of strong communications following (Table 5): 1) 0.9732 – ???7=f(EEE6); 2) 0.9688 – EEE6=f(EEE7); 3) 0.9491 – EEE8=f(EEE6); 4) 0.8611 – EEE7=f(EEE8); 5) 0.8178 – EEE6=f(EEE8); 6) 0.7592 – EEE10=f(EEE11); 7) 0.7309 – EEE11=f(EEE10); 8) 0.7201 – EEE9=f(EEE8).

In Table 5 parameters of statistical models on the general formula (1) which values are written down in a compact matrix form with significant Figures 1-7 are given


Figure 1: Schedules of models of influence of a factor of EEE6 on change of factors of EEE7 and EEE8.


Figure 2: Schedules of models of influence of EEE7 on change of EEE6.


Figure 3: Schedules of models of influence of EEE8 on change of EEE7.


Figure 4: Schedules of models of influence of EEE8 on change of EEE6.


Figure 5: Schedules of models of influence of EEE8 on change of EEE9.


Figure 6: Schedules of models of influence of EEE11 on change of EEE10.


Figure 7: Schedules of models of influence of EEE10 on change of EEE11

Indi-cators y Asymmetric waveletyi= axa2i1iexp(-a3ixa4i)cos(∴πx/(a5i+ axa7i6i)–a8i)   Correl. coeffic. r
Amplitude (half) of fluctuation Fluctuation half-cycle Shift
a1i a2i a3i a4i a5i a6i a7i a8i
The influence ???6 [decimaldegree]
???71 Figure 1 4.23078 0 -0.16452 0.54045 0 0 0 0   0.9732
-0.31020 1.48076 0 0 0 0 0 0
132.72384 0 8.79872?-5 1.76602 4.87479 0.29176 0.56033 0.24849
???81 Figure1 1.66509?6 0 0.31234 0.94742 0 0 0 0   0.9491
7.82146?-5 4.91282 0.018337 1.20027 0 0 0 0
The influence ???7 [decimaldegree]
  ???62 Figure2 64.56655 0 0.00076040 1.01838 0 0 0 0   0.9688
-1.24710?-15 6.59346 0 0 0 0 0 0
-2.35962?-7 5.20107 0.042456 1 12.92183 0.0036148 1.03167 2.65582
-0.00052866 3.02216 0.025670 1 3.80507 9.27492?-5 1 4.71131
The influence ???8 [ma.s.l.]
  ???7 Figure3 -87.70507 0 -8.09363?-5 1.07588 0 0 0 0   0.8611
2.50470?-6 2.55776 0.00026769 1.12617 0 0 0 0
-9.59526?-8 3.19687 0.0017158 0.99993 157.16690 0.28654 0.88567 6.01043
  ???6 Figure4 7.25053 0 -0.00032396 1.09546 0 0 0 0   0.8178
-6.85611?-17 5.03862 0 0 0 0 0 0
-0.00012515 2.24860 0.0020534 1 2922.5439 -0.48413 0.99988 -2.92162
  ???9 Figure5 59.61396 0 -0.14457 0.46269 0 0 0 0   0.7201
-2.90947?-7 3.11597 0 0 0 0 0 0
-0.00062264 2.34099 0.0018640 1.00057 846.23612 -0.042840 0.99814 -0.80809
The influence ???11 [mmw.e.]
???103 Figure6 -28940.548 0 0 0 0 0 0 0   0.7592
270.01817 0.52442 2.61663?-5 0.99761 0 0 0 0
The influence ???10 [mmw.e.]
???113 Figure7 -11303.0999 0 0 0 0 0 0 0   0.7308
9.38697 0.76192 0 0 0 0 0 0

Table 5: Parameters of the strong binary relations at correlation coefficient r≥0.7.

From data of Table 5 it is visible that some regularities required replacement of variables because of negative values of the influencing variables.

From schedules in Figure 1 it is visible that the function EEE7=f(EEE6) supports three members from whom the first two are a difference of two laws: first, law of exponential growth; secondly, indicative law. The third component is wavelet on a formula (1) with very high coefficient of correlation 0.9362. Therefore the main contribution to formation of the binary relation is carried out by wave function with a variable amplitude under the law of exponential death and the variable period of fluctuation. Thus the initial stage of fluctuation is equal 2×4.87479 ≈ 9.75 degrees. In process of increase in width fluctuation calms down because of growth of a half-cycle.

Remains EEE7 after three members of model have significant values at the latitudes of 40-65 degrees. Therefore if it is necessary, the wavelet analysis of microwaves on the remains for this width is possible further. The EEE8 function=f(EEE6) it consists of only the binomial trend. According to Table 5 it contains the sum from two laws: first, law of exponential death; secondly, biotechnical law [2-5]. The remains show small fluctuation which can be identified a formula (1), however adequacy of a wave component will be very small.

Other schedules have similar explanations. Therefore glaciers, as well as other types of natural objects [5], have natural distributions.

The maximum number of members of the statistical model EEE6=f(EEE7) is equal to four that corresponds to computing opportunities of the program CurveExpert-1.40 environment.

For the full wavelet analysis it is necessary to develop the special program environment according to our scenarios of statistical modeling for a supercomputer of a petaflop class. Thus the new program environment for large volumes of the Table of basic data will be universal for science.

The analysis of schedules according to amplitude-frequency characteristics shows that the system of glaciers possesses a certain property of wave adaptation to living conditions on the planet Earth.

For some binary relations the number of members in the general statistical model can exceed 100-120 pieces. In this case there is a possibility of carrying out the fractal analysis for group of wavelets on mega, macro, meso and to micro fluctuations.


Applicability of statistical model (1) to parameters of points of fluctuations of balance at the mass of glaciers of Earth is proved. As a result each binary relation contains a trend and wavelet signals. And the trend is a special case of fluctuation of a wavelet, superlong on the period. As a result the general statistical model represents the plait consisting of a set of lonely waves with variables amplitude and the period of fluctuations. After statistical modeling the factorial analysis allowing making ratings of factors as the influencing parameters and as dependent indicators is carried out.

The offered methodology of identification allows allocating waves of the binary relations between the measured factors at glaciers. Thus for 2528 glaciers of subgroup of EEE finite-dimensional wavelets which can be compared further with heuristic representations of experts are characteristic. The method of identification allows to allocate significant parameters of glaciers and the binary relations between them in which it will be necessary to increase the accuracy of future measurements. Thus from different subgroups we allocated 26 factors, but their joint analysis is disturbed by a disagreement between Tables of data.


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