=Paper= {{Paper |id=Vol-1490/paper28 |storemode=property |title=Application of fast discrete wavelet transformation on the basis of spline wavelet for loosening correlation of sequence of data in mass service theory |pdfUrl=https://ceur-ws.org/Vol-1490/paper28.pdf |volume=Vol-1490 }} ==Application of fast discrete wavelet transformation on the basis of spline wavelet for loosening correlation of sequence of data in mass service theory== https://ceur-ws.org/Vol-1490/paper28.pdf 
Mathematical Modeling


Application of fast discrete wavelet transformation on the
   basis of spline wavelet for loosening correlation of
         sequence of data in mass service theory

                   Blatov I.A., Gerasimova U.A., Kartashevskiy I.V.

        Povolzhskiy State University of Telecommunications and Informatics, Samara



       Abstract. The task of loosening of correlation of sequence of strongly
       correlated random variables within the mass service theory is set. The algorithm
       of application of spline wavelet for loosening of correlation of sequence of
       strongly correlated random variables is described. Properties of the matrixes
       received as a result of application of transformation algorithm are studied.
       Results of numerical experiment studies are given.

       Keywords: spline wavelets, decorrelation, fast discrete wavelet transformation


       Citation: Blatov I.A., Gerasimova U.A.,Kartashevskiy I.V. Application of fast
       discrete wavelet transformation on the basis of spline wavelet for loosening
       correlation of sequence of data in mass service theory. Proceedings of
       Information Technology and Nanotechnology (ITNT-2015), CEUR Workshop
       Proceedings, 2015; 1490: 242-245. DOI: 10.18287/1613-0073-2015-1490-242-
       245


1. Introduction
   Strong correlation of sequence of random variables can create considerable
problems in the solution of mass service theory. Let 𝑋 = (π‘₯0 , … , π‘₯𝑛 )𝑇 be some vector
with known correlation matrix 𝐴. It is required to analyze the traffic characterized by
vector 𝑋 taking into account it correlation properties.
   The traffic as random process is known to have self-similar properties, which
indirect sign is the existence of heavy residuals, i.e. big redundancy of the appropriate
integral functions of distribution. Therefore in such situation the method of
preliminary execution of some orthogonal transformation determined by matrix
𝑇 = (𝑑𝑖𝑗 ), which purpose is elimination or lowering of correlation of basic data is
often used. The application of up-to-date analysis from the mass service theory
concerning vector 𝑋̃ = 𝑇𝑋, but not vector 𝑋, will proves to be more effective.
   It is possible to eliminate correlation and to receive the best result by means of
application of Karhunen-Loeva transformation. However creation of such basis is a
very resource-capacious task. In this case matrix 𝑇 consist of eigenvectors of matrix
𝐴. The resultant correlation matrix will be of a diagonal type. However this method
has some drawbacks such as: absence of fast algorithms of computation; dependence


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on the structure of matrix 𝐴. Therefore the task pf creation of more available bases in
which the correlation can be, if not eliminated, but weakened essentially, is actual. In
this report the spline wavelet is used.


1.1. Elaboration of the system of semiorthogonal spline wavelets
   Let [π‘Ž, 𝑏] be a random interval, π‘š β‰₯ 1 be integer, 𝑛0 be such an integer that
2𝑛0 < 2π‘š + 1 < 2𝑛0+1 and π‘˜ be such an integer that 2π‘˜ > 2π‘š βˆ’ 1. Let us consider
the family βˆ†= {βˆ†π‘› , 𝑛 = 𝑛0 , 𝑛0 + 1, … } of partitions of the interval [π‘Ž, 𝑏] with the
constant step β„Ž = β„Žπ‘› = (𝑏 βˆ’ π‘Ž)/2𝑛 . Let us define 𝑆(βˆ†π‘› , π‘š, π‘˜) as the combination of
spline wavelets, where π‘š is a power and π‘˜ is a degree. On each partition, we consider
a space of splines 𝐿𝑛 = 𝑆(βˆ†π‘› , π‘š βˆ’ 1,1). Then, for each π‘˜ β‰₯ 𝑛0 , space 𝑆(βˆ†π‘› , π‘š βˆ’ 1,1)
can be represented as direct sum πΏπ‘˜ = 𝐿𝑛0 β¨π‘Šπ‘›0+1 β¨π‘Šπ‘›0+2 ⨁ … β¨π‘Šπ‘˜ , where π‘Šπ‘˜
denotes the orthogonal complement of πΏπ‘˜βˆ’1 up to πΏπ‘˜ space. The desired wavelet basis
is the result of combination of the basis in 𝐿𝑛0 and all the bases in spaces π‘Šπ‘› , 𝑛0 ≀
𝑛 ≀ π‘˜.
   Let 𝑖 β‰₯ 0 be a such a fixed integer that the interval [π‘₯π‘–π‘›βˆ’1 , π‘₯𝑖+2π‘šβˆ’1
                                                                   π‘›βˆ’1    ] lies within
[π‘Ž, 𝑏]. Function is computes according to formula
πœ“π‘–,𝑛 (π‘₯) = βˆ‘2𝑖+3π‘šβˆ’2
            𝑗=2𝑖    𝛼𝑗 πœ™π‘—,π‘›βˆ’1                                                          (1)
where πœ™π‘—,π‘›βˆ’1 normalized B-spline. The 𝛼𝑗 -coefficients are defined according to
(πœ“π‘–,𝑛 (π‘₯), πœ™π‘˜,π‘›βˆ’1 ) = 0, π‘˜ = 𝑖 βˆ’ π‘š + 1, 𝑖 βˆ’ π‘š + 2, … , 𝑖 + 2π‘š βˆ’ 2.
   The combination of elaborated wavelet functions is resulted by shifting of the only
function according to formula πœ“π‘–,𝑛 (π‘₯) = πœ“0,𝑛 (2π‘›βˆ’π‘›0 π‘₯ βˆ’ 𝑖(𝑏 βˆ’ π‘Ž)/2𝑛0βˆ’1 ).




                         Fig. 1. – Sequence of experiment’s data 𝑋


1.2. Fast discrete wavelet transformation in the space of spline wavelets on the
finite interval
   The direct transformation consist in the search of wavelet coefficients 𝑑0𝑗 and 𝑐𝑖𝑗 .
                                   π‘˜βˆ’π‘›
{𝑑0𝑗 , βˆ’π‘š + 1 ≀ 𝑗 ≀ 2𝑛0βˆ’1 } ⋃ ⋃𝑖=1 0{𝑐𝑖𝑗 , βˆ’π‘š + 1 ≀ 𝑗 ≀ 2𝑛0+π‘–βˆ’1 βˆ’ π‘š}                   (2)

  According to known function 𝑓 = {𝑓𝑖𝑗 }, 0 ≀ 𝑖 ≀ 2π‘˜ βˆ’ 1,1 ≀ 𝑗 ≀ 𝑠.


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     The inverse transformation consist in reconstruction of all values of function
𝑓𝑖𝑗 , 0 ≀ 𝑖 ≀ 2π‘˜ βˆ’ 1,1 ≀ 𝑗 ≀ s by {𝑓𝑖𝑗 } ∈ 𝑆̃(βˆ†π‘˜ , π‘š βˆ’ 1,1) according to the known set
of wavelet coefficients
               𝑛0                 π‘˜βˆ’π‘›      𝑛0 +π‘–βˆ’1
𝑓 = βˆ‘2𝑗=βˆ’π‘š+1
         βˆ’1
             𝑑0𝑗 πœ™π‘—,𝑛0 + βˆ‘π‘–=1 0 βˆ‘2𝑗=βˆ’π‘š+1βˆ’π‘š 𝑐𝑖𝑗 πœ“π‘—,𝑛0+1                                      (3)

           For more details on the a.m. items please consult [1].


2. Numerical experiment
We made an experiment on loosening of correlation of data represented by sequence
π‘Œ consisting of 9999 random values, each represents the time of traffic processing in
the system1.
   To the initial experiments data 𝑋 we applied direct fast discrete transformation on
the basis of the linear spline.




                       Fig. 2. – Sequence of data 𝑋̃(after direct transformation)

   The correlation coefficients were calculated on the basis of the experimental data 𝑋
(for more details see [3]):
          π‘‰π‘˜
π‘…π‘˜ =                                                                                        (4)
           𝐷

where
           1
π‘‰π‘˜ =            βˆ‘π‘›βˆ’π‘˜
                 𝑖=1 (π‘Œπ‘˜ βˆ’ 𝑋)(π‘Œπ‘–+π‘˜ βˆ’ 𝑋)                                                     (5)
       π‘›βˆ’π‘˜
       1
𝐷 = βˆ‘π‘›π‘–=1(π‘Œπ‘– βˆ’ 𝑋)2                                                                          (6)
       𝑛

      1
𝑋 = βˆ‘π‘›π‘–=1 π‘Œπ‘–                                                                                (7)
      𝑛




1
    Problem definition about decorrelations and data for experiment were provided I.V.
    Kartashevsky

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          Fig. 3. – Correlation coefficients for experiment data and for transformed data

   The sums of modules of correlation coefficients are calculated:
    π‘˜βˆ’1

π‘…π‘˜ : βˆ‘|π‘Ÿπ‘– | = 18.963
    𝑖=0

    π‘˜βˆ’1

π‘…Μƒπ‘˜ : βˆ‘|π‘ŸΜƒ|
         𝑖 = 5.344
    𝑖=0

    The correlation is seen to have reduced by more than by 3.5 times. The similar
result was received for square and cubic splines.


3. Summary
   In the conclusion we would like to stress that the application of fast discrete
algorithm of wavelet transformation in the space of spline wavelet allows loosening
the correlation of sequence of strongly correlated random variables. That is confirmed
by the data obtained in the numerical experiments.


References

 1. Blatov IA, Rogova NV. Fast wavelet-transform in the space of discrete polynomial
    semiorthogonal splines. Computational Mathematical and Mathematical Physics, 2013;
    53(5): 727-736.
 2. Kartashevsky IV. Application Lindley equation for correlation traffic processing. Journal
    Electrosvyaz, 2014; 12: 41-42.
 3. Kartashevsky IV. Calculation of correlation coefficients of times intervals in sequence of
    events. Journal Electrosvyaz, 2012; 10: 37-39.
 4. Umnyashkin SV, Kochetkov ME. Analysis of efficiency of using orthogonal transform
    for digital coding of correlation data. Electronic, 1998; 6:79-84.
 5. Myasnikov VV. Effective algorithm of calculation of local discrete wavelet-
    transformation. Computer Optics, 2007; 31(4): 86-94. [in Russian]


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