Doctor of Engineering, Academician of Russian Academy of Natural History and Russian Academy of Natural Sciences, Volga State University of Technology, Russia
Received Date: November 14, 2016 Accepted Date: November 28, 2016 Published Date: November 30, 2016
Citation: Mazurkin PM (2016) Identification of Wave Regularities According to Statistical Data of Parameters of 24 Pulsars. J Phys Math 7: 206. doi: 10.4172/2090-0902.1000206
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 these measurements of six parameters at 24 pulsars. 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. On the remains it is possible to receive a set of microfluctuations and to bring identification to an error of measurements. 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 pulsars. On the revealed equations it is possible to carry out the amplitude-frequency analysis. Quality of basic data is estimated by rank distributions of values of parameters of pulsars. Thus ranging of values of parameters on a preference preorder vector "better→worse" is carried out in the beginning, the rating of pulsars is formed further.
Wavelet; Identification; Pulsars; Parameters; The relations; 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 [1-17] 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:
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 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.
The pulsars found in the Einstein @Home project are given in article . The data selected for statistical modeling are provided in Table 1. In Table 1 symbols with preference vectors are accepted: P: spin periods; P Epochepochal period of spin; DM: Dispersion Measure; S1400: Flux Densities; D: Estimated Distance; S: Significance are given for reproducibility reasons.
|PSR||P (s)||P Epoch (MJD)||DM (pc cm-3)||S1400 (mJy)||D (kpc)||S|
|J l 817-1938+||2.046838||55991.8||519.6||0.1||8.6||16.9|
Table 1: Parameters of 24 pulsars.
In total from data  it was succeeded to allocate six factors having quantitative values. 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]. In the beginning we will consider rank distributions of values of each factor on a vector of preference and we will determine a rating of pulsars by the sum of ranks, and then we will carry out the factorial analysis of all 62 – 6=30 binary relations.
Any factors have an accurate vector orientation. The person understands an orientation of changes therefore only two options of a vector of preference are possible:
?) Better it is less (yes better, the symbol ↓on a vector "better→worse");
?) Better it is more (and it is good, therefore in the In function=RANG(P1;P$1:P$24;1) for the first indicator P in the program Excel environment the following symbols are accepted: P1: identifier of a column and the first line; P$1: the first line of the ranged column; P$24: the last line of the ranged column according to the Table 1; 0∨1: ranging on decrease (0) or to increase (1). a symbol ↑).
Ranks change from zero therefore it is necessary from results of ranging in the program Excel environment to subtract unit can be clearly understood from Table 2 and Figure 1.
|PSR||RP||P (s)||RPE||P Epoch (MJD)||RDM||DM (pc cm-3)||RS1400||S1400 (mJy)||RD||D (kpc)||RS||S|
|J l 619-42||19||1.023152||16||51975.6||3||172||7||0.6||20||3.7||0||35.4|
|J l 817-1938+||22||2.046838||21||55991.8||20||519.6||23||0.1||1||8.6||12||16.9|
Table 2: Rank distributions of six parameters of 24 pulsars.
The interrelation of a factor from most on rank distribution proves to be good quality or quality of basic data and it serves for check of their reliability.
The analysis of good quality of basic data is made on coefficient of correlation r of the equation
y=f (R=0,1,2,3,...) of rank distribution of a factor on the general formula
Y=Y0 exp(±aRb) ()
where Y - the ranged parameter, Y0- initial value of parameter, a - activity of exponential growth or death of values of parameter; b - intensity of growth or recession.
On decrease of coefficient of correlation a rating of factors on quality of measurements the following: 1: S1400; 2: D; 3: DM; 4: S; 5: P and 6: P Spoch. Thus it appeared that it is convenient to use ranks instead of factors as remove a mathematical problem of "curse of dimensionality", for example, at a rating on a set of diverse indicators.
The general vector of heuristic preference "better→worse" leads all considered factors to one "denominator" that allows to estimate the sum of ranks (Table 3), even without mathematical justification, ratings of subjects and objects (in our example among 24 pulsars).
|PSR||Ranks of values of factors||∑R||Place I|
|J l 619-42||19||16||3||7||20||0||65||12|
|J l 817-1938+||22||21||20||23||1||12||99||24|
|Place I F||4||4||4||1||3||2||-||-|
Table 3: Ranks of values of six parameters and rating among 24 pulsars.
From data of Table 3 it is visible that on the first place on an indicator I there is J1227- 6208b pulsar, on the second - J1652-48b and on the third J1748-3009b. And among the factors for the indicator IF on the first place is a factor in S1400, the second S and in third place D. And the three factors P, P Epoch and DM took fourth place. The rating of pulsars (Figure 2) changes under the law of exponential growth
From the remains in Figure 2 it is visible that at bigger quantity of pulsars in addition to the equation (8) also wave components on model are possible (1).
From six factors of all are possible 62 – 6=30 binary relations. Six distributions are rank. Correlation matrix, including and rank distributions, it is given in Table 4.
|Influencing factors x||Dependent factors (indicators y)|
|P (s)||P Epoch (MJD)||DM (pc cm-3)||S1400 (mJy)||D (kpc)||S|
|Spin periods P (s)||0.4978||0.0608||0.8640||0.2688||0.4270/0.8024|
|P Epoch (MJD)||0.4741||0.1904||0.2097||0.0652||0.3687|
|Dispersion measure DM (pc cm3)||0.1039/0.6948||0.1904||0.0987/0.9846||0.8103/0.9169||0.6445/0.8734|
|Flux densities S1400 (mJy)||0.6069||0.3223||0.4914||0.2089||0.2462|
|Estimated distance D (kpc)||0.1602/0.7742||0.0648||0.8086||0.2188||0.4270/0.6403|
|Significance are given for reproducibility reasons S||0.4335||0.3951/0.9145||0.4800/0.8602||0.2455/0.8576||0.4287/0.9163|
Table 4: Correlation matrix of the binary relations between six factors of 24 pulsars.
The minimum narrowness of factorial communication is observed at the mathematical DM=f(P) function, and the maximum coefficient of correlation at S1400=f(DM) ratio with the accounting of wave indignations of parameters of pulsars.
From data of Table 4 it is visible that each measured factor can be considered in two roles:
First, as the influencing variable; secondly, as dependent indicator. Thus the method of the factorial analysis offered by us allows not thinking a priori of ratios between separate parameters of the studied system (in our example system from 24 pulsars). As a result the psychological barrier at re- searchers gets off. Many binary relations for the researcher will be unexpected. Therefore as our practice showed, the factorial analysis the unique equation of type (1) allows finding unexpected solutions in the field of research. If some factorial communications are unusual and thus are highly adequate, new technical solutions, and often at the level of inventions of world novelty are shown . Thus repeated identification (1) on a single binary relation we called wavelet analysis [1,6,7,11,13-17].
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. Rank distributions, as a rule, are accepted in the form of not wave steady laws [2-5,8-10,12].
The square correlation matrix received after the analysis the rank distributions and the bi- nary relations between all six variables accepted on basic data from Table 1 is given in Table 5.
|Influencing factors x||Dependent factors (indicators y)|| Sum
|P Epoch (MJD)||0.4741||0.8697||0.1904||0.2097||0.0652||0.3687||2.1778||6|
|DM (pc cm-3)||0.1039||0.1904||0.9875||0.0987||0.8103||0.6445||2.8353||4|
|Place of Indicator I y||4||6||2||5||3||1||-||0.4615|
Table 5: Rating of factors on the determined relations (trends).
Here all 36 relations are considered. Besides, summation in the lines and columns two ratings on decrease of this sum turned out: first, a rating of factors as the influencing variables; secondly, rating of factors as dependent indicators. We will define the second rating (Table 6) taking into account identification of wave indignations (1).
|Indicators y||Asymmetric wavelet yi=a1ixa2iexp(-a3ixa4i)cos(π x/a5i+a6ixa7i)-a8i||Correl. coeffic. r|
|Amplitude (half) of fluctuation||Fluctuation half-cycle||Shift|
|The influence P (s)|
|S1400 (Figure 3)||2050.7218||0||7.92517||0.096413||0||0||0||0||0.8640|
|S (Figure 4)||1.53821e6||0||11.67158||0.0081309||0||0||0||0||0.8024|
|The influence DM (pc cm-3)|
|S1400 (Figure 5)||2.04947e-9||3.72097||0.00040479||1.41209||0||0||0||0||0.9846|
|D (Figure 6)||3.34608||0||-0.0015439||1||0||0||0||0||0.9169|
|S (Figure 7)||29.79584||0||-2.61436e-5||1.43411||0||0||0||0||0.8734|
|The influence D (kpc)|
|The influence S|
|P Epoch (Figure 9)||71154.646||0||0.041508||1||0||0||0||0||0.9145|
|DM (Figure 10)||2.95376||2.76075||0.17533||1||0||0||0||0||0.8602|
|S1400 (Figure 11)||7.88199e-6||4.31945||0.00024563||2.84411||0||0||0||0||0.8576|
|D (Figure 12)||0.014339||2.62626||0.018621||1.49231||0||0||0||0||0.9163|
Table 6: Parameters of the strong binary relations at correlation coefficient r ≥ 0.7
The coefficient of a correlative variation for 24 pulsars is equal 21.6276/62=0.6008. This criterion is applied when comparing various systems, for example, of different groups of pulsars with each other.
From six influencing variables on the first place there was S factor (and according to the Table 5 a factor P). In second place fit factor DM, and in third place - D. Among dependent indicators on the first place there is S1400 factor. On the second place is occupied by the factor P, and the third - S.
At correlation coefficient more than 0.7 binary relations between factors become strong (Table 7). As a rule, the concept of wave indignation of the Universe gives significant increase in adequacy of the revealed regularities on a formula (1). From 11 strong communications only two treat the determined relations.
|Influencing factors x||Dependent factors (indicator y)|| Sum
|P (s)||P Epoch (MJD)||DM (pc cm-3)||S1400 (mJy)||D (kpc)||S|
|P Epoch (MJD)||0.4741||0.8697||0.1904||0.2097||0.0652||0.3687||2.1778||6|
|DM (pc cm-3)||0.6948||0.1904||0.9875||0.9846||0.9169||0.8734||4.6476||2|
|Place of Indicator I y||2||6||4||1||5||3||-||0.6008|
Table 7: Rating of factors on the determined (trends) and the wave relations.
In Tables 6-9 and in Figures 3-12 parameters of models which values are written down in a compact matrix form with five significant figures are given.
|Influencing factors x||Dependent factors (indicator y)|
|P (s)||P Epoch (MJD)||DM (pc cm-3)||S1400 (mJy)||D (kpc)||
|DM (pc cm-3)||0.9846||0.9169||0.8734|
Table 8: Rating of factors on the determined (trends) and the wave relations.
The maximum number of members of statistical model 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 all branches of science.
In Figure 8 influence of D (kpc) on change of two factors of P (s) and DM (pc cm-3) is shown. Thus change of P (s) received fluctuation in the form of a finite-dimensional wavelet.
The analysis of schedules according to amplitude-frequency characteristics shows that the system of pulsars possesses a certain property of wave adaptation.
Consider the possibility of further identification model S1400=f(DM) with the highest adequacy 0.9846 (Table 9 and Figure 13). The price of division of a factor of S1400 according to Table 1 is
equal 0.1 (mJy).Then the measurement error equal to ±0.05 (mJy), and the remainder by point graphics in Figure 13 become smaller this error. The identification process is stopped, wherein the wavelet analysis is completed. Apparently from the schedule of four-membered model in Figure 13, influence of a factor of DM for a factor of S1400 gives three clusters (a plot on abscissa axis):
1) Initial site of DM=0-130 (pc cm-3)
2) Average site of DM=130-300 (pc cm-3)
3) Extreme site of DM=300-800 (pc cm-3).
At each of the sites on the abscissa is finite wavelet signal.
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 pulsars is proved. As a result each binary relation contains a trend and wavelet signals. Moreover, the trend is a special case of very long period oscillations of the wavelet. 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.
Quality control input data can be estimated rank distribution of values of parameters of pulsars and the ability to detect the wave patterns of reporting to the design of the same wavelet signal (1). Thus without modeling, only due to ordering of values of parameters on a preference preorder vector "better→worse", it is possible to make a rating of pulsars.
After statistical modeling carried out factor analysis which allows to make the ratings of factors as influencing parameters and how dependent indicators. For strong factorial relations additionally conducted a wavelet analysis, in which re- identification patterns (1) to ascertain residues below the error of measurement of parameters of pulsars. This set of wavelet signals can then be subjected to fractal analysis.
The offered methodology of identification allows to allocate waves of the binary relations between the measured factors. Thus for 24 pulsars characterized by nit dimensional wavelets, which can then be compared with the heuristic views of specialists. The method of identification allows to allocate the most significant parameters and the binary relations between them at pulsars on which it will be necessary to increase the accuracy of future measurements.