CN112367208B - Method for establishing Weibull distributed mixed wavelet network flow model - Google Patents

Method for establishing Weibull distributed mixed wavelet network flow model Download PDF

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CN112367208B
CN112367208B CN202011306497.1A CN202011306497A CN112367208B CN 112367208 B CN112367208 B CN 112367208B CN 202011306497 A CN202011306497 A CN 202011306497A CN 112367208 B CN112367208 B CN 112367208B
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coefficient
owm
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杨力
魏德宾
潘成胜
孔志翔
庞涛
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Dalian University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/14—Network analysis or design
    • H04L41/145—Network analysis or design involving simulating, designing, planning or modelling of a network
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/148—Wavelet transforms
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/14—Relay systems
    • H04B7/15—Active relay systems
    • H04B7/185—Space-based or airborne stations; Stations for satellite systems
    • H04B7/1851—Systems using a satellite or space-based relay
    • H04B7/18519—Operations control, administration or maintenance
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/12—Discovery or management of network topologies
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/14—Network analysis or design
    • H04L41/142—Network analysis or design using statistical or mathematical methods

Abstract

The invention discloses a method for establishing a Weibull distributed mixed wavelet network flow model, which comprises the following steps: establishing a wavelet model; establishing an independent wavelet model; establishing a Weibull distributed mixed wavelet network flow model; the w_owm parameter is determined. The invention combines the long correlation characteristic and the multi-fractal characteristic, and on the basis of an independent wavelet model, the scale coefficient is generated by the IWM, and in order to ensure that the flow is non-negative, the MWM is adopted to assign a random coefficient factor. When the random coefficient factor is determined, the invention selects the Weibull distribution function which accords with the actual flow distribution characteristic, and can accurately describe the network flow behavior characteristic. W_OWM performs well in characterization while reconstructing accurate flow. The W_OWM has the advantages that the characteristics of heavy tail, multi-fractal characteristics and long-term correlation are greatly improved, and the distribution similarity of the W_OWM reconstructed flow and the actual flow is high.

Description

Method for establishing Weibull distributed mixed wavelet network flow model
Technical Field
The invention relates to an intelligent network flow model integrating the world, in particular to a method for establishing a Weibull (Weibull) distributed mixed wavelet network flow model.
Background
The intelligent network has heterogeneous flow multiple change and large-span flow characteristic mutation characteristics, and meanwhile, the intelligent network node has storage and calculation capabilities. The built-in buffer function reduces the flow exchange, and the flow is changed from big data to small data, and the flow characteristics are changed in the process. These changes will have an impact on the statistical properties of the network node traffic, such as the attainment characteristics, latency characteristics, packet encapsulation length, etc., thus making the random process statistics and modeling of network traffic challenging. Therefore, how to construct a network traffic model is important for new features of the appearance of the world-wide integrated intelligent network traffic.
The network traffic model is mainly divided into the following three types: traditional network traffic models, self-similar network traffic models, and hybrid traffic models. Self-similarity of network traffic at large time scales and multi-fractal at small time scales are important and well-recognized statistical features. The above characteristics also exist in satellite networks. The conventional multi-fractal time-dimension analysis method cannot describe both long-correlation characteristics and multi-fractal characteristics. The extended multi-fractal Brownian model can match the multi-fractal of traffic at small time scales with long correlation of traffic at large time scales. But the multi-fractal spectrum cannot be accurately described.
Disclosure of Invention
In order to solve the problems in the prior art, the invention designs a method for establishing a Weibull distribution mixed wavelet network flow model with both long-term correlation characteristics and multi-fractal characteristics.
In order to achieve the above object, the technical scheme of the present invention is as follows: a method for establishing a Weibull distributed mixed wavelet network flow model comprises the following steps:
A. modeling wavelet
Analyzing the self-similarity and fractal characteristics of the flow of the world integrated intelligent network by adopting a wavelet analysis method, and analyzing and observing the signals from the thickness and the fineness in a weighted sum form after a basic signal is subjected to telescopic translation transformation;
definition of random signalsIs the following successive wavelet transform, CWT:
wherein W is a,b Wavelet coefficients. Psi (t) is wavelet mother function, orthogonal base psi a,b And (t) is obtained by scaling and translating b of the size a from the psi (t), and the reconstructed signal x (t) is as follows:
i.e. the wavelet transform is reversible. The reconstructed signal x (t) is decomposed by inverse wavelet transform into linear combinations, which are mutually orthogonal wavelet basis functions. The wavelet coefficients represent all the information of the reconstructed signal x (t). The scaling factor a and the shifting factor b in the continuous wavelet transform are discretized by the discrete wavelet transform. The random signal was subjected to multi-scale analysis by varying the magnification. The reconstructed signal x (t) passes through a high-pass wavelet mother function psi (t) and a low-pass scale function phi (t), and then the wavelet coefficient W with a scaling factor j and a shifting factor k is calculated j,k And scale factor U j,k And (5) solving.
Reconstructing the random signal by inverse wavelet discrete transformation, expressed as:
wherein U is J0,k Represents the scale factor, phi, of the scale factor J0 and the translation factor k J0,k Representing a scale function with a scaling factor of J0 and a shifting factor of k. The wavelet orthogonal basis is formed by a Haar wavelet function and a scale function, and the scale coefficient and the wavelet of the Haar waveletThe relationship of the wave coefficients is as follows:
wherein U is j+1,2k Scale factor of j+1 and 2k, U j+1,2k+1 Scale coefficients representing a scaling factor of j+1 and a shifting factor of 2k+1 are processed in such a way that the random signal X (k) is processed, and the sequence length of the random signal X (k) is assumed to be 2 n Haar wavelet transform decomposes to finest scale factor U n,k ，U n,k The finest scale factor representing the scaling factor n and the scaling factor k, for the random signal X (k) and the finest scale factor relationship is represented as follows:
X(k)＝2 -n/2 U n,k ,k＝0,1,...,2 n -1 (8)
B. establishing independent wavelet model
The coarsest scale factor U of the independent wavelet model, IWM 0,0 And wavelet coefficient W j,k All adopt recursive mode to calculate and obtain the scale function U of the fine part j,k The specific calculation steps are as follows:
b1, calculating the coarsest scale coefficient U 0,0 。
B2, calculating Gaussian distribution obeying mean value zero under each scale jWavelet coefficients W of (2) j,k The wavelet coefficient variance in scale j is +.>And (3) representing.
B3, calculating the scale coefficient U under each scale j j,k And wavelet coefficient W j,k Iteratively calculating a scale factor U under a scale j+1 by the method (7) j+1,k 。
Turning to step B2 until the finest scale j=n is calculated to be 2 n Scale factor.
Obtaining the scale coefficient U under all scales j through calculation j,k Then to IWM signal X iwm (k) And (5) reconstructing.
C. Establishing a multi-fractal wavelet model
In the wavelet inverse transformation iteration process, in order to ensure the coarsest scale coefficient U 0,0 > 0, and satisfy |w j,k |＜U j,k Ensuring that each scale factor is positive in the iterative process. Introducing random signal factor A into multi-fractal wavelet model (MWM) j,k And causing:
W j,k ＝A j,k ×U j,k (9)
wherein A is j,k Is of value of [ -1,1]Independent random variables on the same ensure that each scale factor is positive in the iterative process.
The MWM calculation steps are as follows:
c1, j=0, distributed by β over the interval [0,1]Calculating to obtain the coarsest scale coefficient U 0,0 ；
Under C2 and scale j, random multiplication factor A j,k Generated from the beta distribution by the formula W j,k ＝A j,k ×U j,k Calculation of W j,k ,k＝0,1,...,2 j -1；
C3, on scale j, wavelet inverse transformation is performed by using U j,k And W is j,k U of the dimension j+1 is calculated j+1,2k And U j+1，2k+1 ,k＝0,1,...,2 j -1；
C4, increasing j, turning to step C2 until the finest scale j=n is found 2 n Coefficient of individual scale U j,k 。
D. Establishing a Weibull distribution mixed wavelet network flow model
D1, mixed wavelet network flow model calculation
The mixed wavelet network flow model, namely W_OWM, ensures that the flow is nonnegative by setting random multiplication factors and selecting Weibull distribution which accords with the actual flow distribution characteristics, and the long-related characteristics and the multi-fractal of the network flow are accurately and comprehensively simulated by the Weibull distribution.
W_OWM determines scale factors by IWMSetting random multiplication factor A j,k In order to make the signal non-negative, the constraint +.>
Representing the scale coefficients of IWM, deriving W_OWM wavelet coefficients +.>IWM scale factor by calculation>And W_OWM wavelet coefficients +.>The following iterative formula is used:
W_OWM scale factor representing a scale factor of j+1 and a scale factor of 2k,/and a method for producing the same>Representing the W_OWM scale factor with a scaling factor of j+1 and a shifting factor of 2k+1.
Calculating the W_OWM next scale factorSubstituting equation (10) into equation (11), the iterative equation is reduced to:
IWM cannot guarantee IWM scale factorNon-negative, it was found by equation (12) that in order to ensure W_OWM next scale factor +.>And (3) respectively carrying out the following different treatments on the positive scale coefficient and the negative scale coefficient:
if it isTo ensure that the scale factor is positive, the constraint is met according to the iterative formula (11)Thereby introducing random multiplication factor A j,k Is set in the interval [ -1,1]Applying;
if it isMultiplication factor A by simple shift processing j,k Ensuring that the next scale factor is positive.
The specific calculation steps of W_OWM are as follows:
d1, the coarsest scale factor is generated by Weibull distribution
D2, calculating scale factor by IWM
D21, calculating the scale j-1 wavelet coefficient as by specifying Gaussian distributionWherein (1)>Is the wavelet coefficient variance.
D22, by the already calculated scale j-1 scale factor U j-1,k And scale j-1 wavelet coefficient W j-1,k Calculating IWM scale coefficient by using iterative formula
D3, judging
D31, whenIf->Turning to step D311, otherwise turning to step D312;
D311、
turning to step D32;
d312, weibull distribution generation interval [ -1,1]Up-random multiplication factor a j,k The method comprises the following steps:
d32, when
From (12)) It is known that if A j,k ∈[-1,1]Then (1-A) j,k )∈[0,2]Then there isAnd->By adjusting 1+ -A j,k The value range of (2) is satisfied. Because of->By controlling 1+ -A j,k Negative values are used to make the scale factor non-negative. Order (1+ -A) j,k )∈[-2,0]. The method comprises the following specific steps:
d321 forTo make (1+A) j,k )∈[-2,0]Will A j,k The value range of [ -1,1 [ (R-R)]Translate to [ -3,1]Further ensuring the scale factor->Not negative.
D322 forTo make (1-A) j,k )∈[-2,0]Will A j,k The value range of [ -1,1 [ (R-R)]Translate to [1,3 ]]Further ensuring the scale factor->Not negative. The translated random multiplication factor A is used for solving the scale wavelet coefficient j,k Substitution formula->Calculated +.>And->Substituting the iteration formula (11) to calculate the scale coefficient +.>
D4, go to step D2 until 2 is calculated at the finest scale j=n n The individual scale factors end. Each scale factor ensures non-negativity.
E. Determining W_OWM parameters
E1, determination of A j,k
In the calculation of W_OWM, the random multiplication coefficient A j,k Obeys the Weibull distribution, A j,k The probability density function of the weibull distribution, determined by the parameter, is defined as follows:
setting a non-negative random variable T, and if T exists as a probability density distribution function f (T; l, k);
where k > 0 is a shape parameter, λ > 0 is a scale parameter, and the random variable T obeys the Weibull distribution, denoted Weibull (k, λ). Random multiplication factor a j,k E Weibull (k, λ), where the parameters k, λ are unknown, k, λ are determined by least squares estimation.
Set S 1 ,S 2 ,...,S n The actual flow of Weibull (k, lambda) is subject to independent co-distributed samples, and the least squares estimation of parameters k, lambda:
thus, the random multiplication factor is determined as
E2, determination of
Definition of U by scale factor j,k ＝∫X(t)φ j,k The (t) dt and Harr wavelet functions are:
coarsest scale factorIs represented by the physical meaning of signal sequence X 1 ,X 2 ,...,X n And (5) an average value. Also because the actual flow signal X.epsilon.Weibull (k, lambda), then ∈>Determined by the expectations of the Weibull (k, λ) distribution, namely:
wherein Γ is a gamma function, thereby combining the parameter estimates obtained from equations (18) - (19)Determine->
Compared with the prior art, the invention has the beneficial effects that:
the invention combines the long correlation characteristic and the multi-fractal characteristic, and on the basis of an independent wavelet model, the scale coefficient is generated by the IWM, and in order to ensure that the flow is non-negative, the MWM is adopted to assign a random coefficient factor. When the random coefficient factor is determined, the invention selects the Weibull distribution function which accords with the actual flow distribution characteristic, and can accurately describe the network flow behavior characteristic. W_OWM performs well in characterization while reconstructing accurate flow. The W_OWM has the advantages that the characteristics of heavy tail, multi-fractal characteristics and long-term correlation are greatly improved, and the distribution similarity of the W_OWM reconstructed flow and the actual flow is high.
Drawings
FIG. 1 is a flow chart of the W_OWM algorithm.
Fig. 2 is a schematic diagram of a topology of a space-earth integrated space-based access network.
FIG. 3 is a W_OWM reconstruction tag traffic statistical probability density curve.
Fig. 4 is a Q-Q plot of traffic versus actual traffic after MWM (selected beta profile) reconstruction.
Fig. 5 is a Q-Q diagram of traffic versus actual traffic after w_owm (select weibull distribution) reconstruction.
Fig. 6 is a graph of a multi-fractal spectrum versus label traffic modeling.
Fig. 7 is a flow model autocorrelation coefficient comparison graph.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
As shown in fig. 1, a method for establishing a weibull distributed hybrid wavelet network traffic model includes the following steps:
step1: the number of iteration layers n is set. In order to reconstruct the integrated intelligent network flow signal completely, the number of iterative layers N is accurately set, and for the network flow sequence x (t) with the length of N, the number of iterative layers is set
Step2: generating the coarsest scale factor U 0,0 . Scale factor tableLocal average value of the world-wide integrated network flow signal under the scale, and the coarsest scale coefficient U is calculated by the (21) 0,0 Representing the mean value of the whole network traffic sequence, U is determined by the expectations of the Weibull distribution 0,0 ；
Step3: generating random multiplication factor A from formulas (16) - (19) j,k Calculating wavelet coefficients W j,k . Random multiplication factor A j,k Specifying compliance with Weibull distribution, with A j,k E Weibull (k, lambda), combined with Step 2's distribution parameters k, lambda, produce a random multiplication factor A j,k Where k=0, …,2 j -1; from formula W j,k ＝A j,k ×U j,k Calculating to obtain wavelet coefficient W j,k ；
Step4: calculating the following scale factor: at the scale j, a scale coefficient U is calculated j,k And wavelet coefficient W j,k Substituting formula (12) to iteratively calculate the next scale factor U j+1,2k And U j+1,2k+1 ；
Step5: iteratively calculating all scale coefficients, replacing the scale j by the scale j+1, and repeating Step3 and Step4 until the finest scale j=n is obtained, wherein 2 is obtained n The scale coefficient is up to;
step6: reconstructing the flow signal by using the calculated finest scale factor U j,k Reconstructing a signal from the following equation
In order to better embody the complexity and the authenticity of the space-earth integrated network flow, a simulation environment of a space-earth access network in the space-earth integrated intelligent network is built, and specific parameters of the simulation environment are shown in table 1:
table 1 satellite network simulation data
The topology structure of the space-earth integrated intelligent network space-based access network is shown in fig. 2, and mainly comprises polar region boundary dimensions, inter-orbit inter-satellite links and inter-orbit inter-satellite links. The flow time sequence of the intelligent node A is collected through experimental data, sampling is determined to be 400 minutes, the time stamp is in microsecond level, and the data set is called as label flow.
The invention mainly comprises the following parameters: the number of layers, wavelet function, multiplier range, multiplier distribution. The parameter settings in W_OWM are as in Table 2:
table 2 parameter settings in W_OWM
The statistical probability density curve for the w_owm reconstructed tag traffic is shown in fig. 3. Verifying whether the probability density of the reconstructed label traffic accords with the probability density distribution of the actual network traffic, and analyzing the statistical probability density distribution of the reconstructed label traffic, wherein the arrival time distribution of most data packets in the W_OWM reconstructed traffic is between 0 and 0.3s, and the cumulative probability is obviously more than 90%. The reconstructed flow appears in the graph as a distinct heavy tail characteristic. Traffic with arrival times greater than 0.3s is a minority of the figures. From this, the W_OWM reconstruction tag traffic probability density distribution embodies that the world-wide integrated intelligent network traffic has a heavy tail distribution characteristic.
The Q-Q diagram is typically used to measure how similar the reconstructed flow is to the distribution. Fig. 4 is a Q-Q diagram of flow versus actual flow after a comparative analysis MWM (selected β distribution) reconstruction, and fig. 5 is a Q-Q diagram of flow versus actual flow after a comparative analysis w_owm (selected weibull distribution) reconstruction. And selecting the label flow as a simulation sample. The distribution of the MWM reconstructed flow and the actual flow can be obtained by comparing: only a small part of the earlier stage is relatively close, and most of the earlier stage deviates from the slope of 45 degrees, so that the MWM reconstructed label flow distribution deviates from the actual flow distribution. In the algorithm coefficient construction process, the MWM adopts beta distribution without considering actual flow characteristics, so that the distribution characteristics of the reconstructed tag flow are different from the actual flow. The Q-Q plot of w_owm reconstructed tag traffic versus actual traffic mostly fits to a 45 degree slope, and is highly coincident in some places. Experimental simulation shows that: the distribution characteristic of the W_OWM reconstruction tag flow accords with the distribution characteristic of the actual flow, in the algorithm coefficient construction process, the W_OWM algorithm adopts Weibull distribution which accords with the heavy tail distribution characteristic, and experimental simulation comparison shows that the W_OWM accords with the flow distribution characteristic of the world-integrated intelligent network more than the MWM.
And selecting a multi-fractal spectrum to conduct scale analysis on the network flow model. The label flow data set is selected as a simulation experiment sample, the MWM and the W_OWM are compared and analyzed to reconstruct the multi-fractal spectrum of the label flow and the label flow, and the simulation result is shown in figure 6. Simulation results show that: the MWM and W_OWM multi-fractal spectrums are attached to the multi-fractal spectrums of the label flow when alpha is smaller than 1, and the result proves that the MWM and W_OWM can accurately describe the multi-fractal of the label flow when alpha is smaller than 1. The multi-fractal spectrum of the MWM model is far away from the multi-fractal spectrum of the label flow when alpha is more than 1. The W_OWM model has multi-fractal spectrum curve near the multi-fractal spectrum curve of the label flow when alpha is more than 1. Through simulation comparison, the effect of W_OWM is superior to MWM when describing multi-fractal spectrum. The multi-fractal spectrum of W_OWM is closer to the multi-fractal spectrum of the actual flow. W_OWM is more accurate in describing the multi-scale nature of the flow.
For the random signal X (t), its autocorrelation coefficient ρ is expressed as:
the autocorrelation coefficients of w_owm and MWM are calculated according to the above equation, and a graph of the result of comparing the two models with the correlation coefficients of the tag traffic is shown in fig. 7. As can be seen from the autocorrelation graphs, the autocorrelation coefficient of the W_OWM is closer to that of the label flow, the W_OWM is better than the MWM, and the long-range correlation of the label flow can be well fitted, so that the W_OWM is more accurate when describing the long-range correlation characteristic.
The present invention is not limited to the present embodiment, and any equivalent concept or modification within the technical scope of the present invention is listed as the protection scope of the present invention.

Claims (1)

1. A method for establishing a Weibull distributed mixed wavelet network flow model is characterized by comprising the following steps: the method comprises the following steps:
A. modeling wavelet
Analyzing the self-similarity and fractal characteristics of the flow of the world integrated intelligent network by adopting a wavelet analysis method, and analyzing and observing the signals from the thickness and the fineness in a weighted sum form after a basic signal is subjected to telescopic translation transformation;
the continuous wavelet transform, CWT, defining the random signal X (t) is:
wherein W is a,b Is a wavelet coefficient; psi (t) is wavelet mother function, orthogonal base psi a,b And (t) is obtained by scaling of the scaling factor a and translation of the translation factor b by the psi (t), and the reconstructed signal x (t) is as follows:
i.e. the wavelet transform is reversible; decomposing the reconstructed signal x (t) into linear combinations by inverse wavelet transformation, wherein the linear combinations are mutually orthogonal wavelet basis functions; the wavelet coefficients represent all the information of the reconstructed signal x (t); the scaling factor a and the shifting factor b in the continuous wavelet transformation are discretized by the discrete wavelet transformation; performing multi-scale analysis on the random signal by changing the magnification; the reconstructed signal x (t) passes through a high-pass wavelet mother function psi (t) and a low-pass scale factor phi (t), and then for a wavelet coefficient W with a scaling factor j and a shifting factor k j,k And scale factor U j,k Solving;
reconstructing the random signal by inverse wavelet discrete transformation, expressed as:
wherein U is J0,k Represents the scale factor, phi, of the scale factor J0 and the translation factor k J0,k A scale factor representing a scale factor of J0 and a scale factor of k; the wavelet orthogonal basis is formed by a Haar wavelet function and a scale coefficient, and the relation between the scale coefficient and the wavelet coefficient of the Haar wavelet is as follows:
wherein U is j+1,2k Scale factor of j+1 and 2k, U j+1,2k+1 Scale coefficients representing a scaling factor of j+1 and a shifting factor of 2k+1 are processed in such a way that the random signal X (k) is processed, and the sequence length of the random signal X (k) is assumed to be 2 n Haar wavelet transform decomposes to finest scale factor U n,k ，U n,k The finest scale factor representing the scaling factor n and the scaling factor k, for the random signal X (k) and the finest scale factor relationship is represented as follows:
X(k)＝2 -n/2 U n,k ,k＝0,1,...,2 n -1 (8)
B. establishing independent wavelet model
The coarsest scale factor U of the independent wavelet model, IWM 0,0 And wavelet coefficient W j,k All adoptRecursively calculating to obtain the scale factor U of the fine part j,k The specific calculation steps are as follows:
b1, calculating the coarsest scale coefficient U 0,0 ；
B2, calculating Gaussian distribution obeying mean value zero under each scale jWavelet coefficients W of (2) j,k The wavelet coefficient variance in scale j is +.>A representation;
b3, calculating the scale coefficient U under each scale j j,k And wavelet coefficient W j,k Iteratively calculating a scale factor U under a scale j+1 by the method (7) j+1,2k And U j+1,2k+1 ；
Turning to step B2 until the finest scale j=n is calculated to be 2 n A scale factor;
obtaining the scale coefficient U under all scales j through calculation j,k Then to IWM signal X iwm (k) Reconstructing;
C. establishing a multi-fractal wavelet model
In the wavelet inverse transformation iteration process, in order to ensure the coarsest scale coefficient U 0,0 > 0, and satisfy |W j,k |＜U j,k Ensuring that each scale coefficient is positive in the iterative process; introducing random multiplication factor A into multi-fractal wavelet model (MWM) j,k And causing:
W j,k ＝A j,k ×U j,k (9)
wherein A is j,k Is of value of [ -1,1]Independent random variables on the two-dimensional variable are used for ensuring that each scale coefficient is positive in the iterative process;
the MWM calculation steps are as follows:
c1, j=0, distributed by β over the interval [0,1]Calculating to obtain the coarsest scale coefficient U 0,0 ；
Under C2 and scale j, random multiplication factor A j,k Generated from the beta distribution by the formula W j,k ＝A j,k ×U j,k Calculation of W j,k ,k＝0,1,…,2 j -1；
C3, on scale j, wavelet inverse transformation is performed by using U j,k And W is j,k U of the dimension j+1 is calculated j+1,2k And U j+1，2k+1 ,k＝0,1,…,2 j -1；
C4, increasing j, turning to step C2 until the finest scale j=n is found 2 n Coefficient of individual scale U j,k ；
D. Establishing a Weibull distribution mixed wavelet network flow model
The mixed wavelet network flow model, namely W_OWM, ensures that the flow is nonnegative by setting random multiplication factors and selecting Weibull distribution which accords with the actual flow distribution characteristics, and the long-related characteristics and the multi-fractal of the network flow are accurately and comprehensively simulated by the Weibull distribution;
W_OWM determines scale factors by IWMSetting random multiplication factor A j,k In order for the signal to be non-negative, the condition is limited
Representing the scale coefficients of IWM, deriving W_OWM wavelet coefficients +.>IWM scale factor obtained by calculationAnd W_OWM wavelet coefficients +.>The following iterative formula is used:
W_OWM scale factor representing a scale factor of j+1 and a scale factor of 2k,/and a method for producing the same>A W_OWM scale factor with a telescoping factor of j+1 and a translating factor of 2k+1 is represented;
calculating the W_OWM next scale factorSubstituting equation (10) into equation (11), the iterative equation is reduced to:
IWM cannot guarantee IWM scale factorNon-negative, found by equation (12) to ensure W_OWM next scale factorAnd (3) respectively carrying out the following different treatments on the positive scale coefficient and the negative scale coefficient:
if it isTo ensure that the scale factor is positive, it is full according to iterative formula (11)Foot limitation condition->Thereby introducing random multiplication factor A j,k Is set in the interval [ -1,1]Applying;
if it isRandom multiplication factor A by simple shift processing j,k Ensuring that the next scale factor is positive;
the specific calculation steps of W_OWM are as follows:
d1, the coarsest scale factor is generated by Weibull distribution
D2, calculating scale factor by IWM
D21, calculating the scale j-1 wavelet coefficient as by specifying Gaussian distributionWherein (1)>Is the wavelet coefficient variance;
d22, by the already calculated scale j-1 scale factor U j-1,k And scale j-1 wavelet coefficient W j-1,k Calculating IWM scale coefficient by using iterative formula
D3, judging
D31, whenIf->Turning to step D311, otherwise turning to step D312;
D311、
turning to step D32;
d312, weibull distribution generation interval [ -1,1]Up-random multiplication factor a j,k The method comprises the following steps:
d32, when
As is known from formula (12), if A j,k ∈[-1,1]Then (1-A) j,k )∈[0,2]Then there isAnd->By adjusting 1+ -A j,k The value range of (2) meets the requirement; because of->By controlling 1+ -A j,k Negative values to make the scale factor non-negative; order (1+ -A) j,k )∈[-2,0]The method comprises the steps of carrying out a first treatment on the surface of the The method comprises the following specific steps:
d321 forTo make (1+A) j,k )∈[-2,0]Will A j,k The value range of [ -1,1 [ (R-R)]Translate to [ -3,1]Further ensuring the scale factor->Non-negative;
d322 forTo make (1-A) j,k )∈[-2,0]Will A j,k The value range of [ -1,1 [ (R-R)]Translate to [1,3 ]]Further ensuring the scale factor->Non-negative; the translated random multiplication factor A is used for solving the scale wavelet coefficient j,k Substitution formula->Calculated +.>And->Substituting the coefficient into an iterative formula (11) to calculate a scale coefficient
D4, go to step D2 until 2 is calculated at the finest scale j=n n Ending the scale coefficients; each scale factor ensures non-negativity;
E. determining W_OWM parameters
E1, determination of A j,k
In the calculation of W_OWM, the random multiplication factor A j,k Obeys the Weibull distribution, A j,k The probability density function of the weibull distribution, determined by the parameter, is defined as follows:
setting a non-negative random variable T, and if T exists as a probability density distribution function f (T; l, k);
wherein k > 0 is a shape parameter, lambda > 0 is a scale parameter, and the random variable T obeys Weibull distribution and is marked as Weibull (k, lambda); then the random multiplication factor a j,k E Weibull (k, λ), where the parameters k, λ are unknown, k, λ are determined by least squares estimation;
set S 1 ,S 2 ,...,S n To obey the actual flow independent co-distributed samples of Weibull (k, λ), the least squares estimation of the parameters k, λ:
thus, the random multiplication factor is determined as
E2, determination of
Definition of U by scale factor j,k ＝∫X(t)φ j,k The (t) dt and Harr wavelet functions are:
coarsest scale factorIs represented by the physical meaning of signal sequence X 1 ,X 2 ,...,X n The average value; also because the actual flow signal X.epsilon.Weibull (k, lambda), then ∈>Determined by the expectations of the Weibull (k, λ) distribution, namely:
wherein Γ is a gamma function, thereby combining the parameter estimates obtained from equations (18) - (19)Determine->
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