Forecasting freight volume based on wavelet denoising and FG-Markov

ZHU Chang-feng，WANG Qing-rong，LIU Dao-kuan，YE Qian-yun

(1. School of Traffic and Transportation，Lanzhou Jiaotong University，Lanzhou 730070，China；2. School of Electronic and Information Engineering, Lanzhou Jiaotong University，Lanzhou 730070，China；3. China Railway Fourth Survey and Design Institute Group Co., Ltd, Wuhan 430063，China)

Abstract：To eliminate the grey bias and improve ant-jamming performance of the standard grey-Markov forecasting model, a forecasting model based on wavelet packet decomposition and fuzzy grey Markov(FG-Markov)is proposed considering the characteristics of randomness and nonlinearility of freight volume forecasting.Firstly, based on the data analysis ability of wavelet packet to non-stationary random signal, wavelet packet decomposition is used to improve the analysis ability of data signal by decomposing historical freight volume data into wavelet packet component.On this basis, FG-Markov chain is proposed to obtain the transfer probability matrix of wavelet packet coefficients by introducing fuzzy grey variables, and forecast the freight volume by reconstructing wavelet packet coefficients.Finally, an example of Lanzhou railroad hub is carried out in order to testify the validity and applicability of this forecasting model.Compared with neural network model and other forecasting models, the proposed forecasting model can improve the forecasting accuracy under the same conditions.The forecasting accuracy of wavelet packet decomposition and FG-Markov is not only greater than that of any other single forecasting models, but also superior to that of other traditional combinational forecasting models, which can meet the actual requirements of freight volume forecasting.

Key words：freight volume forecasting；fuzzy grey model；wavelet packet；Markov chain

0 Introduction

Freight volume forecasting is a very complicated system engineering that is influenced by multitudinous factors such as freight rate, railway network structure, economic condition, logistics infrastructure, service level, etc., some factors of which are fuzzy or very difficult to quantitatively describe.Moreover, some factors are even randomized, therefore it is very difficult to model or design an optimization algorithm to describe diverse mechanisms of freight volume.

Various studies of freight volume forecasting have been undertaken, including multiple regression method, time-series method[1]，rough set theory[2]，support vector machine[3-4]，and so on.However, these methods above mentioned ignore the compatibility between the influencing factors and the fuzziness.Consequently, it is very difficult to receive hypothetical forecasting result.

Grey system theory is one of the major methods for solving uncertain problems, and a series of encouraging achievements have been obtained[5].However, when the increasing rate of original data series has a obvious nonlinear characteristic, this model will have some limitations[5].Therefore, GM(1,1,λ)model was proposed by introducing parameterλ[6].GM(1,1,λ)model can improve forecasting accuracy to a certain extent, but it considers only one aspect of the influencing factors in the forecasting system.When the forecasting system is a complicated nonlinear system impacted by various factors, this model has many limitations.Based on this, grey Markov forecasting model has been applied in the related fields, but forecasting accuracy is restricted by poor anti-interference ability of grey Markov model.

Some modified nonlinear characteristics for gray models were introduced and the practical influencing of various models was discussed in Ref.[7].To overcome the difficulty of determining structure and weights, genetic optimization algorithm was used by combining encoding with real encoding[8].A forecasting method for fuzzy inference has been proposed and the maximum and minimum loads were forecasted by fuzzy inference strategy[9].Markov chain was introduced to modify the forecasting accuracy[10].The general regression neural network forecasting model was introduced by adaptive training and extrapolation forecasting[11].Nevertheless, these studies above mentioned ignore random factors.

To improve the accuracy of freight volume forecasting, grey forecasting model was improved by some scholars[12]，and the grey forecasting model based on Markov method was put forward[13].The residual GM(n,h)model has been developed to forecast the development trend of the China’s railway freight volume by the introduction of stochastic process of relative error series[14-15].Considering that the railway freight volume and turnover do not possess characteristic of chaotic, the railway freight volume and turnover were forecasted and analyzed with the largest Lyapunov exponent forecasting model[16].The self-adaptive grey forecasting method has been studied by analyzing the existing problems of the basic grey forecasting method[17-18], and the multi-variable grey forecasting model based on grey system theory has been studied[19].

However, these studies ignore fuzziness of influencing factors, and random interference factors has not been considered, therefore, these forecasting methods have certain limitations.

In order to reduce random factors or interference, Zhou et al.used different wavelet theories to describe the forecasting models[20-22].However, most of the existing research focuses on how to improve the forecasting accuracy of the model, whereas problems such as forecasting stability in different states, redundant noise produced by combined model, and satiation points, etc.have not been disscussed.

In order to solve the above-mentioned problem, considering that freight volume impacted by many complicated factors such as time sequenced with seasonality obvious tendency, random volatility, etc., a novel model of freight volume forecasting based on wavelet theory and fuzzy grey-Markov(FG-Markov)is put forward to improve forecasting accuracy.This forecasting model based on wavelet denoising and FG-Markov can reduce the noise of interference information by making full use of the data analysis ability of wavelet packet to non-stationary random signal, as well as the strong fitting ability of Markov chain to high frequency signal.The actual application shows that this improved method has higher precision and applicability.

1 Methodology

1.1 Wavelet packet decomposition

SupposingVjandWjare scale space and wavelet packet space, respectively, the Hilbert orthogonal decomposition ofVjandWjcan be described as

(1)

(2)

whereg(k)=(-1)kh(1-k)andZis the set of natural number.Then Eq.(1)can be equivalently expressed as

(3)

where ⊕ is the logical operator and the algorithm can be described as

The wavelet space can be decomposed as

j=0,1…,k=0,1….

(4)

Letn=2t+m, then the wavelet packet function is expressed as

ψj,k,n(t)=2-j/2ψn(2tt-k),

(5)

whereψn(t)=2l/2u2l+m(2lt).It can be seen that the wavelet packet function not only includes scale parameterjand translation parameterk, but also introduces frequency parametern, which can improves time-frequency resolution.

1.2 Peak-type Markov chain

Markov process is a kind of special random moving from one state to another state at each time step.A first-order Markov model is a model of such a system in which probability distribution over next state is assumed to only depend on current state(not on previous ones).This characteristic is appropriate to application in change of freight volume because dynamic change of freight volume also possesses the properties of Markov process under certain conditions.

1)Within a certain region, the freight volumes of different transport modes may be transformed into each other.

2)The mutual conversion process between freight volumes of different transport modes includes many incidents which are difficult to be described precisely by a special function.

3)During study periods, average transfer state of freight volume is relatively stable and accordant with requirements of Markov chain.

Firstly, an original transfer probability matrix needs to be defined before Markov process, and its mathematics expression isP=(Pij), wherePijis probability of the state transition, and it meets

i,j=1,2,…,n.

(6)

According to non-aftereffect of Markov process and probability formulas of Bayes condition, the Markov forecasting model is obtained as

P(n)=P(n-1)Pij,

(7)

whereP(n)is state probability of any time, andP(n-1)is preliminary rate probability.

2 Construction of forecasting model based on wavelet denoising and FG-Markov

2.1 Fuzzy classification

Because of the complexity of freight volume forecasting, forecasting values tend to fluctuate in a certain interval.If we can find this fluctuate interval, then forecasting error caused by uncertainty will be eliminated.Furthermore, if this fluctuate interval can be used for quantification, this interval will be the most reliable results.

The original sample is divided into several equidistant intervals.The diagram of state division is shown in Fig.1.

Fig.1 Diagram of state division

Fig.1 not only reflects the development rulein every stage, but also shows the change trend of freight volume in each stage.

However, due to small sample bias, the forecasting value may belong to different states under the condition of a relatively small number of states.This method will affect forecasting accuracy, and it also has certain limitations.

The fitting curve is obtained by fitting the sample by means of the model, and then the states are classified by using fuzzy theory.

Supposing that the characteristic function between elementxand setAis

(8)

LetUbe a given set,μA∶x→[0,1]is the mapping fromUto a closed interval of[0,1], ?x∈U.If there is only oneμA∈[0,1]corresponding tox, a fuzzy subset is given according to this mapping, whereμAis defined as membership function, andμA(x)is defined as degree of membership.Fuzzy interval and membership functions of triangle are shown in Fig.2.

Fig.2 Fuzzy interval and triangle membership functions

Let the fitting curveγ(k)(k≤n)be the reference line, original data are divided intomstates, and any state can be expressed as

Θi=[Θ1i,Θ2i],i=1,2,…,m,

Θ1i=γ(k)+ai-1,Θ2i=γ(k)+ai,

whereai-1andairepresent the lower and upper limits of relative residual error, respectively.Then forx∈U(Uis the set of residual errors), the membership function of each fuzzy set can be expressed as

(9)

(10)

(11)

The state of the data(fuzzy state vector)can be expressed as vector(μ1(x),μ2(x),…,μm(x)).

2.2 Solution of state transition probability matrix

The fuzzy state vector of each point is calculated according to the membership function, the state of each point is defined according to the principle of the maximum membership degree, and then the state transition matrix is obtained.In other words, for any relative residual errorx∈U(Uis the set of residual errors), there existsk0, which meats

(12)

Asxbelongs toAk0.AndA1,A2,…,Amare fuzzy subsets.State transition probability is obtained as

(13)

wherePijis transformation probability from theith state into thejth state through a step transition;Mijis the number of samples transformed from theith state into thejth state through a step transition; andMiis the number of samples in stateΘi.

Then state transition probability matrix is expressed as

(14)

State transition probability matrixPdescribes the transition rule of every state.

2.3 Fuzzy Markov residual error modification

According tox(0)={x0(1),x0(2),…,x0(n)}, the forecasting value ofγ(k+1)(k≤n)is obtained by usingGM(1,1,λ)model.On this basis, residual errorδ(n)of timenand membership degreeμA(δ(n))can be expressed as

F(δ(n))=[μA1(δ(n)),μA2(δ(n)),…,μAm(δ(n))].

(15)

Then the forecasting value of residual sequence inn+1 times is still a fuzzy vector as

F(δ(n+1))=F(δ(n))×P=

[μA1(δ(n+1)),μA2(δ(n+1)),…,μAm(δ(n+1))].

(16)

Each component represent thes membership degree which is a relative residual estimation value for each fuzzy state inn+1 times.Let the membership degree be weight, thenδ(n+1)is calculated by using weighted sum method, namely

(17)

Then the forecasting value of stepn+1 is

(18)

If the forecasting value of stepn+1 is added in the sample as the new information, repeat the above mentioned steps, and then the forecasting value of stepn+2(even more)steps can be obtained.

2.4 Construction of forecasting process

The forecasting model based on wavelet denoising and FG-Markov consists of five steps：

Step 1)Decompose the historical data of freight volume by using wavelet packet and then obtain the coefficients of wavelet packet.

Step 2)Obtain the probability transfer matrices of the rising state and falling state of each frequency band by using wavelet packet coefficients.

Step 3)Calculate the change trend of current freight volume values and calculate the state transition probability.

Step 4)Forecast the next wavelet packet coefficient by using state transition probability.

Step 5)Reconstruct the wavelet packet coefficient to get the prediction result of freight volume.

The forecasting process is shown in Fig.3.

Fig.3 Flow chart of forecasting process

3 Example analysis

To verify the precision of the model based on wavelet denoising and FG-Markov, the data of container freight volume in this model are trained based on the historical data from Lanzhou Railway Hub of China from 2009 to 2018.The original data of freight volume are shown in Table 1.

Table 1 Container freight volume of Lanzhou Railway Hub from 2009 to 2018

It can be seen from Table1 that it is the random and adulatory property for container freight volume of Lanzhou Railway Hub from 2009 to 2018, the maximum container freight volume is 1 978 963, while the minimum container freight volume is 1 102 543.The difference value between the maximum container freight volume and minimum container freight volume is 876 420.It is very difficult to find the inherent law of freight volume changes to forecast future freight volume with conventional prediction methods.

Parameterλ(0≤λ≤1)is a decisive factor of influencing fitting precision forGM(1,1,λ)model.The value of parameterλ(0≤λ≤1)is determined by using automatic searching method.

Step 1)Choosing the initial value ofλ, letλ=0.5.

Step 2)Choosing fitting rule, the mapping relation between error and parameterλis determined by using grey theory.After calculation, there isε=εi.

Step 3)In accordance with the following random search in a certain interval as

λi+1=λi+[0.5-rand(0,1)]T,

(19)

whererand(0,1)is uniformly distributed random numbers,Tis the parameter which determines the size of the search interval, generally being(0.1,0.2).

Step 4)Calculatingεi+1, ifεi+1<εi, thenλ=λi+1,i=i+1,ε=εi, or else, return Step 3.

Step 5)Determining the position of point set(λi,εi)in plane coordinate system, and the adjacent point is lined to format diagram.

Step 6)End condition is thatireaches a given value, or volatility is up to a certain number of times.Then, the minimum mean absolute error is chosen as

(20)

Letλ=0,5,T=1, and fluctuation frequency of search reduction process is 500, the relationship between the average absolute error and the value of parametersisλis shown in Fig.4.

Fig.4 Relationship between average absolute error and λ

It can be seen from Fig.3 that this method can quickly obtain the optimal value and has the higher searching efficiency.

The forecasting fitting curve ofGM(1,1.λ)model is obtained as

(21)

The fitting results and the relative residual error are shown in Figs.5 and.6.It can be seen that fitting effect is good.

Fig.5 Fitting results and relative residual error

Fig.6 Relative residual error

It can be seen from Figs.5 and 6 that the fitting results is good.The maximum absolute error is-7.367, while the minimum absolute error is 2.436.

According to the application experience of Markov chain analysis method and amplitude distribution of the relative residual error, the standard of state division is shown in Table 2.

It can be seen from Table 2 that the upper and lower bounds of the residual state can reflect the characteristic of decline or rise in every stage.It can also help decision-maker make their decisions according to the change law of the influence factors of freight volume.

Table 2 Standard of state division

Because there is no forecasting point in state 1 and state 5, the membership function of each fuzzy set is constructed by using triangle method, and the fuzzy space constructed by all fuzzy states is shown in Fig.7.

Fig.7 Fuzzy space constructed

The expressions of states 2, 3 and 4 are given by

(22)

(23)

(24)

The residual error estimation values of next time which is by using general classification and the above-mentioned fuzzy classification are studied, respectively, and fitting results and the relative residual error are shown as in Table 3.

Table 3 Fitting results and relative residual error

It can be seen from Table 3 that the forecasting value by general classification for timen+1 greatly changes when the correction value of residual error produces little change in timen, but the forecasting value by fuzzy classification for timen+1 little changes.The reason for this phenomenon is that the correction value of the residual error by general classification is seriously affected by the interference deviation, which is magnified under the conditions of the limited number of classifications.Nevertheless, the interference deviation only effects the change of different fuzzy classes under the conditions of fuzzy classification.Consequently, it will not impact too much on the forecasting value.The statistics of state transition frequency through a step transition is shown in Table 4.

Table 4 State transition frequency through a step transition

It can be seen from Table 4 that the state transition frequency through a step transition has viciousness.The maximum frequency of state transition is 5, while the minimum frequency of state transition is 1, which reflects the mechanism of dynamic changes of freight volume, and also reflects the objective reality.

The state transition matrixPis shown as

(25)

The forecasting result for railway network container freight station from 2019 to 2023 are shown in Table 5.

Table 5 Forecasting result for railway network container freight station from 2019 to 2023

Meanwhile, in order to verify validity, objectivity and applicability of this model above mentioned, other forecasting model such asGM(1,1), SPA forecasting model time-series analysis model, artificial neural network model and multiple regression model, etc.are employed Lanzhou North Station and the corresponding forecasting resultsare shown in Fig.8.

Fig.8 Forecasting results

It can be seen from Fig.8 that the data fitting results of wavelet denoising and FG-Markov forecasting model are superior to that of the other forecasting methods.Meanwhile, the forecasting accuracy can be circulated as

(26)

Comparative result of forecasting accuracy of each model are shown in Fig.9.

It can be seen from Fig.9 that the corresponding forecasting accuracy values are 82.3%, 86.4%, 87.5%, 88.4%, 88.9% and 96.1%, respectively, for these models mentioned above.It can be seen that the accuracy values of wavelet and FG-markov forecasting model are obvious higher than that of other forecasting methods, which indicates that this forecasting model is of practical value and can serve as the same kind problem.It shows the development rules of things and effectively improves the forecasting accuracy.

Fig.9 Comparative results of forecasting accuracy values by different methods

4 Discussion

1)Freight volume forecasting is influenced by multitudinous factors.It is very difficult to find the inherent law of freight volume changes.The existing studies ignore the compatibility between the influencing factors and the fuzziness.

2)GM(1,1,λ)model is improved.However, when the forecasting system is a complicated nonlinear system impacted by various factors, this model also has many limitations.Markov is a kind of stochastic process with good properties, and has apractical application value.Considering the randomness of freight volume forecasting, Markov and GM(1,1,λ)are combined to improve forecasting accuracy.The improved forecasting model can well reflect the mechanism of dynamic changes of freight volume.This method has certain significance in the study of similar problems.The standard of state division plays an important role in the dynamic forecasting of freight volume.Different standards of the state division should be adopted according to specific forecasting problems.

3)Considering the fuzziness of freight volume forecasting, the fuzzy theory is employed to analyze fluctuation trend of forecasting.Nevertheless, the existing studies ignore the compatibility between influencing factors and the fuzziness.Furthermore, if the freight volume is affected by abnormal factors such as emergency, political factors, adjustment of transportation structure, etc., the forecasting accuracy of the model will be affected.

4)The coupling forecasting model based on wavelet denosing, and FG-Markov forecasting is one of the efficient ways to improve forecasting accuracy.Nevertheless,λis the decisive factor of influencing fitting precision.In order to obtain higher forecasting accuracy, parametersλshould be improved.It is a major subject in the next phase.

5 Conclusion

The general forecasting model cannot deal with mutation factor, even it brings error to the forecasting value.However, historical wave information is considered by Markov state transition in this model, and forecasting accuracy is improved.Moreover, wavelet denosing and FG-Markov forecasting model is based on the statistical analysis of historical data.Consequently, the more the historical statistical data, the more reliable the forecasting accuracy.In addition, the standard of state division is directly related to forecasting accuracy.