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 time-interval 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
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 [
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. [
Based on the grey theory, the grey Verhulst model (GVM) with unequal time-interval 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.
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 [
The typical settlement curve of foundation.
(1) In the elastic stage
(2) In the elastoplastic development stage
(3) In the elastoplastic mature stage
(4) In the stable stage
Based on the above analysis, the relationship curve between the settlement
Assume that the original data sequence is
Assuming
For the raw sequence with unequal time-interval, the following methods can be used to convert them into the data sequence with equal time-interval.
(1) Calculate the average time-interval
(2) Calculate the interpolation coefficient of time point
(3) Generate the data sequence
(4) The data sequence with equal time-interval can be obtained as follows:
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
In (
It can be obtained by the least square method.
where
The obtained
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
Assuming that the data sequence is
The basic procedure of Markov optimization is as follows. The relative residual error of original sequence
The one-step state of transition probability matrix is
where
Assuming that the state of
If
The state of
The probability of the corresponding state is determined according to (
Based on Markov optimization, the prognostic value of the MGVM model can be determined.
When the particle swarm composed of
The velocity and the position of particles are updated according to (
where
In order to improve the search efficiency of the PSO algorithm, the dynamic inertia weight coefficient
Through the PSO algorithm, the optimization of interval state of the weight coefficient
The calculation flow of the PSOMGVM prognostic model based on the PSO algorithm and the Markov chain is shown in Figure
The flow-process diagram of the PSOMGVM prognostic model.
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 (
Generally, the smaller the
Appraise criteria of
| <10 | 10-20 | 20-50 | >50 |
---|---|---|---|---|
Predictive ability | Excellent | Good | Reasonable | Inaccurate |
In order to demonstrate the simulative and predictive capabilities of the proposed model in Section
The Xiangli Expressway of Yunnan Province is a section of the Beijing-Tibet 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
The unequal data points are converted into the data points with equal time-interval for settlement of high fill subgrade in Xiangli Expressway [unit: mm].
Date | 2017 | 2018 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6/19 | 7/20 | 8/24 | 9/25 | 10/26 | 11/25 | 12/24 | 1/21 | 2/24 | 3/25 | 4/26 | 5/27 | 6/25 | |
Time interval | 31 | 35 | 32 | 31 | 30 | 29 | 28 | 34 | 29 | 32 | 31 | 29 | |
Original observation data | 22.9 | 25.5 | 21.5 | 18.8 | 16 | 14.4 | 13.5 | 15.5 | 12.6 | 14.3 | 12.4 | 11.5 | |
Data with equal time-interval | 22.8 | 22.5 | 20.8 | 18.7 | 16.5 | 15.4 | 14.9 | 14.1 | 13.4 | 13.8 | 12.4 | 12.3 |
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 time-interval are elaborated in Section
Parameter values for the competing models.
Parameters | | |
---|---|---|
GM(1,1) | 0.0708 | 24.1432 |
GVM | 0.0356 | -0.0013 |
The comparison table of different prognostic models for settlement of high fill subgrade in Xiangli Expressway [unit: mm].
Year | Month | | GM(1,1) | PSOMGM(1,1) | ARIMA | GVM | MGVM | PSOMGVM |
---|---|---|---|---|---|---|---|---|
2017 | Jul. | 22.90 | / | / | / | / | / | / |
Aug. | 25.50 | 24.29 | 24.80 | / | 23.86 | 24.51 | 24.49 | |
Sept. | 21.50 | 20.95 | 21.39 | 22.10 | 20.75 | 21.31 | 21.29 | |
Oct. | 18.80 | 18.93 | 18.01 | 19.72 | 18.93 | 18.56 | 18.73 | |
Nov. | 16.00 | 17.05 | 16.22 | 17.14 | 17.24 | 16.10 | 16.21 | |
Dec. | 14.40 | 15.30 | 14.56 | 14.70 | 15.66 | 14.63 | 14.73 | |
2018 | Jan. | 13.50 | 13.67 | 13.01 | 12.69 | 14.19 | 13.26 | 13.35 |
Feb. | 15.50 | 15.51 | 15.83 | 14.39 | 16.34 | 15.26 | 15.37 | |
Mar. | 12.60 | 12.37 | 12.63 | 12.12 | 13.24 | 12.37 | 12.46 | |
Apr. | 14.30 | 12.76 | 13.77 | 14.00 | 13.89 | 14.27 | 14.26 | |
| ||||||||
2018 | May | 12.40 | 11.53 | 12.44 | 14.82 | 12.13 | 12.35 | 12.45 |
Jun. | 11.50 | 10.00 | 10.79 | 15.38 | 10.74 | 10.93 | 11.02 |
With regard to the simulative values of the MGVM model, the relative errors are calculated by (
The state probability of the simulative values can be calculated by (
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
For the GM(1,1), PSOMGM(1,1), ARIMA, GVM, MGVM, and PSOMGVM models, the minimum
The appraise indices of different prognostic models to predict the settlement of high fill subgrade in Xiangli Expressway.
Year | Month | | GM(1,1) | PSOMGM(1,1) | ARIMA | GVM | MGVM | PSOMGVM |
---|---|---|---|---|---|---|---|---|
APE (%) | APE (%) | APE (%) | APE (%) | APE (%) | APE (%) | |||
2017 | Jul. | 22.90 | / | / | / | / | / | / |
Aug. | 25.50 | 4.76 | 2.76 | / | 6.42 | 3.89 | 3.95 | |
Sept. | 21.50 | 2.57 | 0.52 | 2.77 | 3.50 | 0.90 | 0.96 | |
Oct. | 18.80 | 0.67 | 4.21 | 4.89 | 0.67 | 1.29 | 0.39 | |
Nov. | 16.00 | 6.55 | 1.38 | 7.12 | 7.72 | 0.64 | 1.33 | |
Dec. | 14.40 | 6.25 | 1.09 | 2.08 | 8.77 | 1.61 | 2.31 | |
2018 | Jan. | 13.50 | 1.27 | 3.64 | 6.03 | 5.11 | 1.81 | 1.13 |
Feb. | 15.50 | 0.05 | 2.15 | 7.16 | 5.40 | 1.54 | 0.86 | |
Mar. | 12.60 | 1.82 | 0.24 | 3.78 | 5.10 | 1.82 | 1.14 | |
Apr. | 14.30 | 10.79 | 3.73 | 2.07 | 2.86 | 0.24 | 0.30 | |
MAPE (%) | 3.86 | 2.19 | 4.49 | 5.06 | 1.52 | | ||
RMSE | 0.83 | 0.44 | 0.77 | 0.95 | 0.38 | | ||
| ||||||||
2018 | May | 12.40 | 7.04 | 0.31 | 19.50 | 2.17 | 0.43 | 0.41 |
Jun. | 11.50 | 13.03 | 6.15 | 33.72 | 6.65 | 4.99 | 4.18 | |
MAPE (%) | 10.04 | 3.23 | 26.61 | 4.41 | 2.71 | | ||
RMSE | 1.23 | 0.50 | 3.23 | 0.57 | 0.41 | |
Note: the optimal indices of the six competing models are in italic.
Comparison of the APE adopted the six competing models to forecast the settlement of high fill subgrade in Xiangli Expressway.
With regard to the appraise indices of the
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
The observative and predictive settlement values of high fill subgrade in Xiangli Expressway using the competing models.
Through the analysis of Section
The predictive settlement values of high fill subgrade using PSOMGVM prognostic model [unit: mm].
Year | 2018 | |||||
---|---|---|---|---|---|---|
Month | Jul. | Aug. | Sept. | Oct. | Nov. | Dec. |
Predictive values | 10.77 | 10.23 | 9.72 | 9.25 | 8.79 | 8.37 |
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” [
The modeling condition of the grey prognostic model is that the original sequence or construction data sequence by a variety of algorithms must have “quasi-exponential law.” From the analysis of Section
With regard to the research of the grey prognostic model, the following work can be carried out for further research. Firstly, based on the self-evolution 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 long-term forecasting effect of the grey model, further research is needed to develop a novel grey prognostic model for long-term forecasting.
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 time-interval of settlement observation data, the grey Verhulst model with unequal time-interval 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.
Grey Verhulst model
Markov grey Verhulst model
Particle swarm optimization
The grey Verhulst model based on PSO algorithm and Markov chain
Grey model
The grey model based on PSO algorithm and Markov chain
Autoregressive integrated moving average model
Absolute percentage error
Mean absolute percentage error
Root mean squared error
The raw data sequence with unequal time-interval
The data sequence with equal time-interval
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
The upper limit of the
The lower limit of the
The transition probability from state
Occurrences number of state
The transition number from 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
The speed of the
Historic optimal value of the
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
The predictive value of the time
All data included in this study are available upon request by contact with the corresponding author or
The authors declare that there are no conflicts of interest regarding the publication of this article.
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.