Research Article  Open Access
Research on the Grey Verhulst Model Based on Particle Swarm Optimization and Markov Chain to Predict the Settlement of High Fill Subgrade in Xiangli Expressway
Abstract
It is of vital significance to accurately forecast the settlement of high fill subgrade, which is the foundation for disaster prevention and treatment of subgrade. According to the monitoring data of high fill subgrade, a novel model, called PSOMGVM model, based on particle swarm optimization (PSO) and Markov chain is proposed. Firstly, the typical characteristics of settlement curve are analyzed from the aspect of geomechanics theory and based on the grey theory, the grey Verhulst model (GVM) with unequal timeinterval is proposed. Then, according to the theory of Markov chain, the grey Verhulst model is built to revise the relative residuals of the GVM, in which the effects of volatility characteristics can be considered. Finally, the PSOMGVM model based on PSO algorithm and Markov chain is set up, which whitens the parameters of the grey interval. In order to demonstrate the fitness and the ability of the proposed model, five competing models are introduced to predict the settlement of the high fill subgrade of Xiangli Expressway in Yunnan Province. Through the analysis of APE, MAPE, and RMSE, it states that the accuracy and performance of the PSOMGVM model outperform the other five competing models for simulative and predictive periods.
1. Introduction
Along with the continuous promotion of “the Belt and Road,” the expressway of Yunnan Province, which is the bridgehead in Southeast Asia, will usher in a new development opportunity. The complex geological environment and high mountain landform of Yunnan district have resulted in a large number of high fill subgrade [1]. The settlement control of high fill subgrade is vitally important for the safety and operation of the expressway, which is the premise and key for predicting the settlement of high fill subgrade accurately. At present, the prognostic methods of settlement can be divided as follows. (1) The prognostic method is based on the theory of soil mechanics, which includes the layerwise summation method [2], the finite element method [3], the empirical equation method [4] and so on. Generally, the geometric condition, boundary condition, or constitutive model of the above methods is inconsistent with the engineering practice, which will affect the accuracy of prediction results [5]. (2) The prognostic method is based on the monitoring data, which includes the Asaoka method [6], hyperbola method [7], threepoint method [8], settlement difference method [9], genetic algorithm [10], neural network method [11], grey theory method [12], and so on. This kind of method has attracted the attention of many scholars and is widely used in engineering practice at home and abroad.
At present, many scholars have carried out relevant research on prognostic model based on the grey theory and have obtained some beneficial results and valuable achievements. An optimized grey discrete Verhulst model is proposed to predict the settlement of the foundation pit by Zhang et al. [13]. A modified grey Verhulst model method using Fourier series of error residuals is proposed by Ming et al. [14]. A flexible generalization of the grey Verhulst model is put forward by Evans [15]. A grey Verhulst sliding mode controller is proposed for the laboratory antilock braking system by Boopathi et al. [16]. An improved version of Saito’s method together with the grey Verhulst model is presented by Miao et al. [17]. A grey timedelayed Verhulst model used the method of grey differential equations is established by Wang et al. [18]. Wang et al. present a method for physical software time stageeffort prediction based on the grey models GM(1,1) and Verhulst [19]. The grey Verhulst model based on PSO algorithm is adopted to discuss the relationship between CO_{2} emissions and economy by Wang and Li [20]. The grey Verhulst model and inverse Verhulst model are established to predict the mechanical behavior of a minicomposite by Wang et al. [21]. An optimal grey Verhulst model is proposed to predict the concentration of nitrogen in seawater by Hu et al. [22]. The Verhulst model and DGM(1,1) model are employed to simulate the development tendency based on the information decomposing and model combination by Zeng et al. [23]. However, the data fluctuation is not taken into consideration in these papers.
Based on the grey theory, the grey Verhulst model (GVM) with unequal timeinterval is proposed in this paper. Combined with the Markov chain theory, the relative residual sequence of GVM model is predicted. On this basis, the grey Verhulst model (MGVM) based on the Markov chain is established, which can revise the relative residual error of GVM model. Then, the grey Verhulst model based on PSO algorithm and Markov chain (PSOMGVM) is established by whitening Markov state interval parameters. The settlement monitoring data of high fill subgrade of Xiangli Expressway in Yunnan Province is took as an example, and the feasibility of PSOMGVM prognostic model is verified by the comparing and analyzing of different prognostic models.
2. Analyze the Characteristic of “S” Shape Curve during Settlement
According to the theory of soil mechanics, the settlement of soil can be divided into three parts: the instantaneous settlement, the primary consolidation settlement, and the secondary consolidation settlement [24]. The instantaneous settlement occurs at the moment of the imposing of force, which is caused by the shear strain under the condition of constant volume. Bjerrum pointed out that this settlement is lateral yielding [25]. It shows that the relationship curve between the settlement and the time does not pass through the origin, as shown in point of Figure 1. The primary consolidation settlement and the secondary consolidation settlement vary with the load and the time, which is similar to the growth curve described by Verhulst model. A large number of the engineering practices and the monitoring results of land subsidence with longterm reveal the characteristic of “S” shape curve [26, 27]. In fact, the growth curve reflects the process of the occurrence, development, maturity, and ultimate state of the settlement. The characteristic of “S” shape can be divided into four stages, which are shown in Figure 1.
(1) In the elastic stage , the settlement linearly increases with time. At the initial stage of the loading, the soil is in the state of elastic stress. Then, the settlement increases almost linearly with the load.
(2) In the elastoplastic development stage , the settlement rate increases with time. With the increase of load, the plastic zone develops continuously. When the soil stress is increasing, the settlement and its rate are increasing [28].
(3) In the elastoplastic mature stage , the settlement rate decreases with the time. Because the consolidation is not completed and the soil has the characteristics of rheology, the soil settlement will increase with time, but the settlement rate will decrease [29].
(4) In the stable stage , the settlement tends to be stable. In theory, when the time is infinite, the settlement reaches its limit state.
Based on the above analysis, the relationship curve between the settlement and time is “S” shape curve without passing the origin in the whole settlement process.
3. Methodological Framework
3.1. The Grey Verhulst Model with Unequal TimeInterval
3.1.1. The Data Conversion with Unequal TimeInterval
Assume that the original data sequence is
Assuming , if , there is at least one or more timeintervals which satisfy the condition . This data sequence is called the unequal timeinterval data sequence.
For the raw sequence with unequal timeinterval, the following methods can be used to convert them into the data sequence with equal timeinterval.
(1) Calculate the average timeinterval
(2) Calculate the interpolation coefficient of time point
(3) Generate the data sequence with equal timeinterval
(4) The data sequence with equal timeinterval can be obtained as follows:
3.1.2. Grey Verhulst Model
Generally, the cumulative generating operator can be obtained by accumulating the original data.
where .
The average sequence of its accumulated data sequence is as follows:
The grey Verhulst model is as follows:
The whitening equation of grey Verhulst model is as follows:
The variation rule of grey Verhulst prognostic model is consistent with the settlement curve in the whole process. It has the characteristic of “S” shape type, which is similar to the logistic growth model.
The whitening response equation of grey Verhulst model is as follows:
The prognostic value of is obtained by the cumulative subtraction of .
In (8), and are substituted by .
It can be obtained by the least square method.
where
The obtained and are substituted into (11) to get . Then, it is converted into the data sequence with unequal timeinterval as before.
3.2. The MGVM Model Based on Markov Optimization
Given that the prognostic value is affected by many factors, the relative error of prognostic value is volatile, which can be optimized by Markov theory.
The relative error of the prognostic model is
According to , the can be deduced. If can be calculated, the of (17) can be taken as the prognostic value when .
Assuming that the data sequence is , the states of the time are known. If the state of is only related to the state of and it has nothing to do with the previous states before , as shown in (18), this data sequence can be called Markov chain process, which has no aftereffect.
The basic procedure of Markov optimization is as follows. The relative residual error of original sequence can be segmented into the state, which is divided by equidistant in general. The variable of state is denoted as , whose the upper limit is and the lower limit is . The state transfer matrix of the residual sequence is obtained according to the Markov chain, and the prognostic value can be revised through (17).
The onestep state of transition probability matrix is
where
Assuming that the state of is
If , then . If , then , .
The state of is
The probability of the corresponding state is determined according to (21), and the predicted value of the residual error can be calculated.
Based on Markov optimization, the prognostic value of the MGVM model can be determined.
3.3. The PSOMGVM Model Based on PSO Algorithm
When the particle swarm composed of particles is flying at a certain speed in exploring space, the particle will consider its own historical optimum and the historical optimum of all particles. It will constantly update the flight speed and the direction to search for the optimal solution, which is called particle swarm optimization (PSO) algorithm.
The velocity and the position of particles are updated according to (25) and (26).
where
is inertia weighting coefficient, and it can change the search ability of the PSO algorithm. When , it is called the basic PSO algorithm.
In order to improve the search efficiency of the PSO algorithm, the dynamic inertia weight coefficient , the dynamic cognitive acceleration coefficient , and the dynamic social acceleration coefficient are proposed. When the single coefficients , , and are adopted, it often falls into the “local optimization solution”. This strategy can avoid the risk of getting in the “local optimization solution”.
Through the PSO algorithm, the optimization of interval state of the weight coefficient can be obtained. The PSOMGVM prognostic value is as follows:
The calculation flow of the PSOMGVM prognostic model based on the PSO algorithm and the Markov chain is shown in Figure 2.
3.4. Inspection the Accuracy of Prediction
In order to appraise the predictive accuracy of the competing models, the first thing is to choose the appropriate quantitative appraise indices, which can effectively judge whether the predictive accuracy is excellent or inaccurate. In this paper, three classical statistics indicators, namely, absolute percentage error (APE), mean absolute percentage error (MAPE), and root mean squared error (RMSE), are adopted to elaborate the predictive performance of different models. The APE, MAPE, and RMSE are calculated as the following three equations:
Generally, the smaller the APE, RMSE, and MAPE values are, the more accuracy the prognostic model is. According to the Lewis theory [30], the predictive performance can be divided into four grades, which is shown in Table 1.

4. An Example of the Settlement of High Fill Subgrade
In order to demonstrate the simulative and predictive capabilities of the proposed model in Section 3, three competing models, namely, the classical GM(1,1), the PSOMGM(1,1) model based on the optimization of Markov chain and classical PSO algorithm, and the ARIMA model, are introduced to elaborate the forecasting performances. The data points from July 2017 to April 2018 are adopted as the input source data to forecast parameters, build competing models, and proofread simulative performance. Meanwhile, the observation data points from April 2018 to June 2018 are employed as the validation data for verifying the predictive accuracy. Through the analysis of evaluation indices of competing models in Section 3.4, the model with the best performance will be used to predict the settlement of high fill subgrade.
4.1. Data Collection
The Xiangli Expressway of Yunnan Province is a section of the BeijingTibet expressway of the national network G0613. The Xiangli Expressway has the characteristics of huge amount of engineering, complex topographic and geological conditions, and difficult construction. The subgrade of Xiangli Expressway locates in the “V” shaped gully with the thickness of the subgrade fill up to 30m. The high fill subgrade of K34+480 segment was monitored from July 2017 to June 2018, and the observative results are shown in Table 2. According to the monitoring plan, it is monitored once every 2 to 10 days, depending on the construction progress, loading, weather conditions, and so on. The observative data was sorted out according to the month. The maximum timeinterval is 35 days, the shortest timeinterval is 27 days, and the data points with equal timeinterval are 30.91 days.

4.2. Simulation and Validation
In order to predict conveniently for further elaboration, the unequal data points should be reorganized by month. The methods to generate the data points with equal timeinterval are elaborated in Section 3.1.1, as shown in Table 2. Then, the classical GM(1,1) and GVM models are calculated by the using of compiled Matlab program. The parameters and are obtained by the least square method, as shown in Table 3. The simulative data points of the ARIMA model are acquired through the Eviews software, which can be referred to Agung [31]. The simulative and predictive values of competing models can be obtained based on the above methods, as shown in Table 4.


With regard to the simulative values of the MGVM model, the relative errors are calculated by (16). Then, the state intervals of relative error are classified into three parts, namely, , , and . The occurrence numbers of , , and are 3, 1 and 5, respectively. The onestep transition probability matrix is obtained through (19), namely,
The state probability of the simulative values can be calculated by (24), as shown in Table 4.
According to this paper, the PSOMGVM prognostic model is applicated to predict the settlement of high fill subgrade. The parameter settings of the PSO algorithm are as follows. The number of particle group is 200, the maximum number of iterations is 500, the particle velocity belongs to the interval of , and the particle position belongs to the interval of . The whitening weight coefficient of different states is obtained by the PSO algorithm. The whitening weight coefficients , , and of different states are 0.4866, 0.6917, and 0.6389, respectively.
For the GM(1,1), PSOMGM(1,1), ARIMA, GVM, MGVM, and PSOMGVM models, the minimum APE is 0.67, 0.24, 2.07, 0.67, 0.24, and 0.30 for the simulative period, and the maximum APE is 6.55, 4.21, 7.16, 8.77, 3.89, and 3.95, respectively, as shown in Table 5. For the simulative and predictive periods, the minimum APE of the PSOMGVM model is the second lowest and second lowest, respectively. From Figure 3, it indicates that the APE line of PSOGVM model is more close to the zero lines compared with the other competing models. Based on the APE analyses, it demonstrates that the PSOGVM model has a more excellent performance to predict the settlement of high fill subgrade compared with the other five competing models.
 
Note: the optimal indices of the six competing models are in italic. 
With regard to the appraise indices of the MAPE and RMSE, the minimum MAPE and RMSE are 1.37 and 0.38, respectively, both provided by PSOMGVM model for the simulative period. For the predictive period, the minimum MAPE and RMSE are 2.30 and 0.34, respectively, both calculated by the PSOMGVM model. Therefore, the accuracy and the ability of the PSOMGVM model outperform the other five competing models according to the MAPE and RMSE indices.
In addition, the predictive values of PSOMGVM model are not the same as the observative values to predict the settlement of high fill subgrade from Figure 4. Nevertheless, according to the appraise criteria of MAPE in Table 1, all the competing models, except for the ARIMA model, have excellent performance for the simulative and predictive periods.
4.3. Forecast
Through the analysis of Section 4.2, the performance of the PSOMGVM prognostic model outperforms the other five competing models, which is selected for forecasting the settlement of high fill subgrade. The predictive values for the settlement of high fill subgrade from July 2018 to December 2018 are shown in Table 6.

5. Discussion
As a result, we can draw a conclusion that the PSOMGVM model presented in this paper outperforms the other competing models to predict the settlement of high fill subgrade with higher prediction accuracy. From the GVM model to PSOMGVM model, we employed a gradually progressive optimization process and the comparison results indicated the proposed new model, called PSOMGVM model, based on PSO algorithm has a good fitting effect.
The grey prognostic model was suitable for time series with the feature of “small sample and poor information” [32]. The settlement of high fill subgrade not only has the grey characteristics of poor information and uncertainty, but also has the feature of nonlinearity and complexity [33]. The MGVM model could reflect the fluctuation law of sequence by determining the transfer law of sequence state, which could overcome the defect that the grey model cannot reflect the fluctuation of the data sequence and improve the prediction accuracy greatly. In order to efficiently search for the optimal whitening weight of the state interval, the improved PSO algorithm was introduced in this paper. This PSO algorithm has the advantages of fast searching speed and high efficiency. Nevertheless, it has the disadvantage of easily falling into local optimum, which is improved to a certain extent through the PSOMGVM algorithm proposed in this paper. In fact, to overcome the disadvantage of falling into local optimum, we had done more than ten times calculations and then chose the best whitening weight.
The modeling condition of the grey prognostic model is that the original sequence or construction data sequence by a variety of algorithms must have “quasiexponential law.” From the analysis of Section 2, it can be seen that the settlement of high fill foundation follows the characteristics of exponential law. However, the settlement of some high fill subgrade is not stable, and the grey model is inappropriate. With the passage of time, the grey level of the model will become larger and larger, and the longterm prediction effect of the grey prognostic model will not be well. Since the PSOMGVM model proposed in this paper is based on the Grey theory, it can only be suitable for shortterm and mediumterm prediction, and not appropriate for longterm prediction. In addition, the computational process of the model is more complex, which increases the application difficulty of prediction.
With regard to the research of the grey prognostic model, the following work can be carried out for further research. Firstly, based on the selfevolution characteristics of the original data sequence, the transformation method, which does not destroy the evolution law of the original sequence, will be studied. In order to realize the data mining and highlight of the original data sequence, we can combine data mining technology for further research. Secondly, the combination of the grey model and artificial intelligence is a vital method to solve uncertain system modeling. The traditional grey model has obvious linear feature. The artificial intelligence can further expand the model construction and relax the restrictions on parameter setting and model construction, which can make up for the shortcomings of the traditional grey model. Thirdly, in order to improve the longterm forecasting effect of the grey model, further research is needed to develop a novel grey prognostic model for longterm forecasting.
6. Conclusion
Based on the characteristics of the soil mechanics and the settlement monitoring data series of high fill subgrade, which has the characteristic of “S” shape, the grey Verhulst model is applicated to predict the settlement of high fill subgrade. The PSOMGVM prognostic model based on the PSO algorithm and the Markov optimization is established. Through the theoretical analysis and practical verification, the following conclusions can be obtained.
(1) The settlement of high fill subgrade can be regarded as a grey system. The relationship between the land subsidence and the time conforms to the law of occurrence, development, maturity, and eventual extinction, which is described by the logistic equation. According to the unequal timeinterval of settlement observation data, the grey Verhulst model with unequal timeinterval is established.
(2) Combined with the grey theory, Markov chain, and PSO theory, the PSOMGVM prognostic model is established, which is applicated to the high fill subgrade of Xiangli Expressway in Yunnan Province. The results show that the average relative error of the PSOMGVM model is 1.03%, which is better than 1.62% of the MGVM model and 4.08% of the GVM model. The PSOMGVM prognostic model has high prediction accuracy. It has significance for predicting postconstruction settlement and final settlement of high fill subgrade.
Nomenclature
GVM(1,1):  Grey Verhulst model 
MGVM:  Markov grey Verhulst model 
PSO:  Particle swarm optimization 
PSOMGVM:  The grey Verhulst model based on PSO algorithm and Markov chain 
GM:  Grey model 
PSOMGM(1,1):  The grey model based on PSO algorithm and Markov chain 
ARIMA:  Autoregressive integrated moving average model 
:  Absolute percentage error 
:  Mean absolute percentage error 
:  Root mean squared error 
:  The raw data sequence with unequal timeinterval 
:  The data sequence with equal timeinterval 
:  Cumulative generating operator of 
:  Number of data in the sequence 
:  The mean consecutive neighbors operator 
:  Development coefficient 
:  Grey action coefficient 
:  Predictive value of the grey Verhulst model 
:  The predictive value of the time based on Markov chain 
:  The upper limit of the state and the state 
:  The lower limit of the state and the state 
:  The transition probability from state to state 
:  Occurrences number of state 
:  The transition number from state to state 
:  The relative error of data sequence 
:  The best position of the particles 
:  The global best position of the particles 
:  Particle speeds 
:  Particle positions 
:  The position of the particle 
:  The speed of the particle, which is limited to the interval 
:  Historic optimal value of the particle 
:  Historic optimal value of all particles in the search space 
:  Pseudorandom number, 
:  Cognitive acceleration coefficient 
:  Social acceleration coefficient 
:  Inertial weighting coefficient 
:  Maximum number of iterations 
:  Number of current iterations, 
:  Maximum inertial weight coefficient, generally taken as 
:  Minimum inertial weight coefficient, generally taken as 
:  Maximum cognitive acceleration coefficient, generally taken as 
:  Minimum cognitive acceleration coefficient, generally taken as 
:  Maximum social acceleration coefficient, generally taken as 
:  Minimum social acceleration coefficient, generally taken as 
:  The prognostic value of settlement 
:  The measured value of settlement 
:  The interval state of weight coefficient 
:  The predictive value of the time based on PSO algorithm and Markov chain. 
Data Availability
All data included in this study are available upon request by contact with the corresponding author or haiming0871@163.com
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This paper was supported by the National Natural Science Foundation of China (Grants nos. 51764020 and 51741410) and the National Key Research and Development Program of China (Project no. 2017YFC0804601). The authors would like to thank them for providing financial support for conducting this research.
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Copyright
Copyright © 2019 Haiming Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.