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The forecasting of deep horizontal displacement of slope soil is an important part of slope deformation monitoring, which has important guiding significance for the prevention of slope safety accidents. The historical data of deep horizontal displacement of slope soil are non-stationary time series with both random growth and time-varying fluctuation, and ARIMA model is suitable for forecasting such time series. In this paper, 648 historical data of deep horizontal displacement of soil were selected as the original sequence of empirical analysis, and forecasted the law and trend of displacement change by establishing ARIMA model. The results of empirical analysis showed that the effect of short-term static forecasting was good, which could provide useful reference for managers of slope quality safety govern slope and analyze the slope stability.

Slope stability has always been an important research content of slope engineering [

It can be seen that for natural slope collapse, landslide, etc., or man-made slope instability caused by human engineering activities, these geological disasters have caused huge losses to economic construction and people’s property [

However, slope engineering is an uncertain, nonlinear and complex system [

The forecasting of deep horizontal displacement series of soil mass refers to the estimation of the displacement value of a certain period or periods in the future by using historical data of the evolution of things, comprehensive consideration of various influencing factors and some scientific research methods. The time series of the known horizontal displacement value of deep soil is { H 1 , H 2 , H 3 , ⋯ , H T } , to forecast the displacement value of H T + 1 , H T + 2 , H T + 3 , ⋯ , H T + m , the formula is defined as:

H ^ T + 1 , H ^ T + 2 , ⋯ , H ^ T + m = f ( H 1 , H 2 , ⋯ , H T ) (1)

Among them, only doing the one-step forecasting, so the H_{T}_{+1} is estimated, which is called single-step forecasting; When H_{T}_{+m} and m > 1 are forecasted, it is called m-step forecasting. It is obvious that m-step forecasting is composed of multiple single-step forecasting, that is, formula (1) can be written as:

H ^ T + 1 = f ( H 1 , H 2 , ⋯ , H T ) H ^ T + 2 = f ( H 1 , H 2 , ⋯ , H T + 1 ) ⋮ H ^ T + m = f ( H 1 , H 2 , ⋯ , H T + m − 1 ) (2)

The purpose of forecasting research is to understand the future development state or trend of things. It is widely concerned because the current state of things may affect the future results. The appropriate forecasting method is applied to the forecasting of deep horizontal displacement of slope soil, and the Suggestions are provided for the slope quality and safety management, which is conducive to the analysis of slope stability and the prevention of slope safety accidents. Therefore, the forecasting of deep horizontal displacement of slope soil is a very important forecasting problem.

The method of time series forecasting is to regard the historical monitoring data of the deep horizontal displacement of slope soil as the time series of changes, and to forecast the trend and rule of the future evolution of things by building a reasonable time series model, see

(MA) and autoregressive sliding average model (ARMA), etc., and their theoretical formula [

The formula of AR(p):

x t = δ + ϕ 1 x t − 1 + ϕ 2 x t − 2 + ⋯ + ϕ p x t − p + u t (3)

The formula of MA(q):

x t = u + u t + θ 1 u t − 1 + θ 2 u t − 2 + ⋯ + θ q u t − q (4)

The formula of ARMA(p, q):

x t = ϕ 1 x t − 1 + ϕ 2 x t − 2 + ⋯ + ϕ p x t − p + δ + u t + θ 1 u t − 1 + θ 2 u t − 2 + ⋯ + θ q u t − q (5)

There are two forecasting methods: static forecasting and dynamic forecasting. Static forecasting refers to the forecasting based on the actual value in the original sequence, which can only be used after obtaining the real data. Dynamic forecasting refers to the forecasting of the first phase from the actual value, and then the first phase value is put into the original time series to forecast the value of the second phase together with it, and so on, but for long-term forecasting, it may produce cumulative error.

In the long-term slope deformation process, the monitored horizontal displacement value of soil depth changes with the seasonal cycle and has a certain increase (or decrease) trend of non-stationary time series. Traditional forecasting methods, such as regression model and gray GM(1,1) model, are applicable to stationary time series, and the forecasting results for such values are not ideal [_{t}} as stationary time series and the mean value is zero, then the first-order autoregressive first-order sliding average model formula of ARMA(1,1) can be expressed as:

H t = ϕ H t − 1 + e t − θ e t − 1 (6)

Therefore, the formula of ARMA(p, q) can be expressed as:

H t − ϕ 1 H t − 1 − ⋯ − ϕ p H t − p = e t − θ 1 e t − 1 − ⋯ − θ q e t − q (7)

Formula (4) can be abbreviated as:

ϕ ( B ) H t = θ ( B ) e t (8)

Among them, e_{t} represents the white noise parameter, t = 1 , 2 , ⋯ , p ; ϕ said regression coefficient; θ represents the regression parameter. In the model ARIMA(p, d, q), d represents the difference order. In slope monitoring projects, time series data of deep horizontal displacement of soil are generally unstable, and there are many ways to make the series stable. Difference is one of them, and the modeling process of ARIMA model is almost the same as that of ARMA.

The data of deep horizontal displacement of slope soil in this paper are obtained from the SQLsever database of Sichuan Shengtuo Detection co., LTD. The related slope engineering project is the ladder trough landslide monitoring project of Shuangma village, Shiguan township, Maoxian county. SQLsever database is a high performance and extensible relational database management system designed by Microsoft.

Due to the failure of data acquisition equipment and external environment, data acquisition at some time points is missing. Therefore, in order to reduce the forecasting error, a continuous and stable period of monitoring data was selected for the forecasting experiment.

Randomly selecting 500 - 650 data of a deep horizontal displacement monitoring point in the monitoring project. There are two reasons for the selection of 500 - 650: first, if too little data is taken, the information in the original sequence cannot be fully mined; and if too much data is selected, the sequence interval will be too long, making the early data have little influence on the later forecasting data and causing unnecessary errors. Second, a lot of related time series forecasting literatures are consulted. If the number of original series in the empirical analysis of quarterly data (m = 4, cycle four quarters a year) is between 50 and 100, and the number of monthly data (m = 12, cycle for 1 year and 12 months) is between 250 and 350, the number of original series in the empirical analysis of time data (m = 24, cycle 24 hours a day) is between 500 and 650.

In this paper, from 0 points on August 15, 2018 to 23 points on September 10, 2018, all the whole point data in the direction of “X axis” are selected to form the “H” data set and sequence {H_{t}}. The data amount is 504, as shown in _{t}} is an obvious time series with deterministic trend and periodicity”.

According to the above “three criteria”, AIC, SC and HQ values with smaller values are better. Based on the unit root test results in

It can be seen from

ADF tests the category | AIC | SC | HQ |
---|---|---|---|

Intercept, Trend | −6.3234 | −6.2818 | −6.3073 |

Intercept | −6.3263 | −6.2916 | −6.3129 |

No intercept, No trend | −6.3284 | −6.3006 | −6.3176 |

Note: AIC, also known as AIC information criterion, was put forward by Japanese statistician Hiroji Akaike in 1973 [

t-Statistic | Prob.* | |
---|---|---|

Augmented Dickey-Fuller test statistic | −18.576 | 0.0000 |

Test critical Values: 1% level | −2.569 | |

5% level | −1.941 | |

10% level | −1.616 |

of a unit root exists. And then P-value is small. All these indicate that the first-order difference sequence of the original sequence is stable, so d = 1.

As can be clearly seen from

As can be clearly seen from

rapidly approaches to 0, and the first-order difference sequence is preliminarily judged to be the fourth-order truncation of partial autocorrelation, so the AR(4) model can be tried to fit. When lag = 5, the partial autocorrelation coefficient is at the edge of the confidence band of 2 times the standard deviation, so the AR(5) model can be tried to fit. For lag = 1, the autocorrelation coefficient is not significantly zero, and for lag = 2, 8, it is on the edge of the confidence band of 2 standard deviations, so we can try to fit the MA(1), MA(2) and MA(8) models. Therefore, ARIMA(4,1,1) and ARIMA(5,1,1) can be preliminarily fitted. Finally, it is found that p may be equal to 4 and q may be equal to 1, 2 and 8 according to the model’s order determination.

According to the judgment principle that the P-value is less than the significance level 0.05 and the smaller the result is, the more significant the result is (the model is better), the P-value values of C, AR(p) and MA(q) in

First, according to the “three criteria”, AIC, SC and HQ value are relatively low, and the test statistics of each model in

The formula of the model ARIMA(3,1,1) is derived [

1) Because of W t = α ^ + v t (9)

2) Again,

P-values/(10^{−}^{2}) | ARIMA (3,1,1) | ARIMA (4,1,1) | ARIMA (4,1,2) | ARIMA (4,1,8) |
---|---|---|---|---|

C | 0 | 0 | 0 | 0 |

AR(3) | 0.12 | 0.74 | 0 | 0.10 |

AR(4) | - | 14.96 | 0.04 | 2.88 |

MA(1) | 0 | 0 | 81.92 | 74.08 |

MA(2) | - | - | 0 | 3.99 |

MA(8) | - | - | - | 8.72 |

Note: C stands for constant; AR represents the autoregressive process, and the lag order is denoted in brackets; MA represents the sliding average process, and the brackets indicate the lag order.

Precision index | ARIMA (3,1,1) | ARIMA (4,1,1) | ARIMA (4,1,2) | ARIMA (4,1,8) |
---|---|---|---|---|

AIC | −6.384 | −6.385 | −6.382 | −6.375 |

SC | −6.349 | −6.343 | −6.333 | −6.285 |

HQ | −6.370 | −6.368 | −6.363 | −6.340 |

F-statistics | 62.787 | 50.921 | 42.410 | 21.887 |

Prob (F-statistics) | 0.000 | 0.000 | 0.000 | 0.000 |

( 1 − ρ ^ 1 L − ρ ^ 2 L 2 − ρ ^ 3 L 2 ) v t = ( 1 + θ ^ L ) e ^ t (10)

1 − ρ ^ 1 L − ρ ^ 2 L 2 − ρ ^ 3 L 2 (11)

3) If you multiply both sides of this equation by 11, you get PI (1):

W t = ( 1 − ρ ^ 1 − ρ ^ 2 − ρ ^ 3 ) α ^ + ρ ^ 1 w t − 1 + ρ ^ 2 w t − 2 + ρ ^ 3 w t − 3 + e ^ t + θ ^ L e ^ t (12)

4) The formula of ARMA(3,1) model is obtained by substituting the coefficient:

W t = 0.000061 + 0.381209 w t − 1 + 0.147211 w t − 2 + 0.127094 w t − 3 + e ^ t − 0.994342 e ^ t (13)

5) Then, making the W t = h t − h t − 1 , and the formula of ARIMA(3,1,1) model is:

( H t − H t − 1 ) + 0.381209 ( H t − 2 − H t − 1 ) + 0.147211 ( H t − 3 − H t − 2 ) + 0.127094 ( H t − 4 − H t − 3 ) = 0.000061 + e ^ t − 0.994342 e ^ t (14)

After determining the estimated ARMA model parameters, diagnostic tests are required to verify the adaptability of the fitting model. In this paper, Eviews time series analysis software is used to establish the autocorrelation graph of the residual error, and the residual error is tested by pure randomness.

As can be seen from

In order to verify the applicability of the fitting model, the model ARIMA(3,1,1) is used for forecasting. The dynamic forecasting was carried out first, and the results are shown in

The forecasted results were stored in the DH sequence to make the dynamic forecasting fitting diagram of DH and DHF, as shown in

Secondly, the static forecasting is carried out, and its short-term forecasting can only be carried out step by step. The forecasting results are shown in

The forecasted values were stored in the DH sequence to make the static forecasting fitting graph for DH and DHF, as shown in

According to the existing forecasting value in DHF, the model ARIMA(3,1,1) formula can be used to deduce the forecasted value of the first phase reversely, and the forecasted value can be integrated into the original sequence, the same process can be used to obtain the forecasted value of the second phase, and so on until the m(m = 24) phase. Therefore, the limitations of the model for long-term forecasting are shown in

where, the average relative error e ¯ is,

e ¯ = 1 n ∑ i = 1 24 | h i − h ′ i | h i ≈ 1.73 %

where, h_{i} and h ′ i represent the actual value and the forecasted value respectively; i represents the number of forecasting periods, i = 1 , 2 , ⋯ , 24 ; n = 24.

As can be seen from

1) ARIMA model is feasible as a short-term forecasting model for deep horizontal displacement of slope soil. It can be seen from the forecasting fitting effect

diagram that the fitting effect is good, indicating that the original sequence contains the information of the “X-axis” direction of the deep horizontal displacement of the slope soil, and it is not difficult to see that the static forecasting effect is far better than the dynamic forecasting.

2) It is easy to generate the cumulative error when the static forecasted value is calculated by the model back-stepping. It is not difficult to see from the forecasting results of ARIMA(3,1,1) in

Time serial number | Actual value/(10^{−2}・mm) | Forecasting value/(10^{−2}・mm) | Absolute error (%) | Relative error (%) |
---|---|---|---|---|

1 | 56.25 | 56.84 | 0.59 | 1.05 |

2 | 56.55 | 56.54 | 0.01 | 0.02 |

3 | 58.29 | 56.72 | 1.57 | 2.69 |

4 | 57.13 | 58.58 | 1.45 | 2.54 |

5 | 57.82 | 57.31 | 0.51 | 0.89 |

6 | 58.23 | 57.96 | 0.27 | 0.47 |

7 | 58.27 | 58.36 | 0.09 | 0.15 |

8 | 59.42 | 58.37 | 1.05 | 1.77 |

9 | 58.96 | 59.50 | 0.54 | 0.92 |

10 | 57.48 | 59.02 | 1.55 | 2.69 |

11 | 57.48 | 57.53 | 0.06 | 0.10 |

12 | 55.94 | 57.52 | 1.59 | 2.84 |

13 | 55.79 | 55.98 | 0.19 | 0.33 |

14 | 57.52 | 55.83 | 1.69 | 2.94 |

15 | 57.71 | 57.55 | 0.16 | 0.28 |

16 | 56.40 | 57.74 | 1.35 | 2.39 |

17 | 58.62 | 56.42 | 2.19 | 3.74 |

18 | 57.71 | 58.64 | 0.93 | 1.60 |

19 | 59.27 | 57.74 | 1.53 | 2.58 |

20 | 56.63 | 59.29 | 2.66 | 4.69 |

21 | 57.98 | 56.65 | 1.32 | 2.29 |

22 | 56.72 | 58.00 | 1.28 | 2.26 |

23 | 56.88 | 56.74 | 0.14 | 0.25 |

24 | 55.69 | 56.90 | 1.21 | 2.17 |

Mean relative error | 1.73 |

Note: the forecasted time is “2018-09-11 00-2018-09-11 23”, corresponding to the time serial number; Absolute-error = |Actual-value - Forecasted-value| * 100%; Relative error = (Absolute-error/Actual-value)*100%. The actual value comes from the SQLsever database of Sichuan Shengtuo detection co., LTD.

forecasting is not conducive to long-term forecasting, but overall, the relative error and average relative error of the forecasting results of the fitting model in this paper show that the absolute error of these 24 periods is very small, and the forecasting effect is relatively ideal.

3) The characteristic of time series forecasting itself is to extract useful information from the historical data of things to forecast their future development or trend. In this paper, other factors affecting the deep horizontal displacement of soil are only reflected by random terms, which is also a defect of the fitting model. In the empirical analysis, other factors affecting the deep horizontal displacement of soil cannot be controlled. Therefore, when selecting data, it is necessary to avoid major fluctuations caused by earthquakes and artificial construction as far as possible, so as to reduce the forecasting error.

Combined with the results of the whole empirical analysis, a theoretical method suitable for the forecasting and management of slope stability is proposed, which is a short-term static forecasting method of deep horizontal displacement of soil based on ARIMA model (ARIMA-soil displacement forecasting method), as shown in

ARIMA-soil displacement forecasting method, relies on the historical data of deep horizontal displacement monitoring of slope soil, builds ARIMA model by analyzing the original sequence, and makes short-term static forecasting. If the analysis results of the first phase forecasting are not harmful to the slope stability, the results of the first phase forecasting will be put into the original sequence for the second phase forecasting, and so on. On the other hand, if the analysis results of a certain period are not favorable to the slope stability, the monitoring should be strengthened and emergency measures should be prepared in advance. The application of this method can provide more basis for slope stability analysis and prevent slope safety accidents.

The forecasting of deep horizontal displacement of slope soil is a challenging

problem, but the forecasting of time series has always been considered as an effective means for statistical forecasting of trend growth (decrease) and time-varying fluctuation series. Because time series forecasting has a good short-term forecasting effect, although this paper only carries out short-term static forecasting of ARIMA model for time series of deep horizontal displacement of soil, theoretically time series can achieve long-term dynamic forecasting. In the empirical analysis of this paper, it can be seen from

Due to the time limitation, this paper only makes empirical analysis on the changes of some historical monitoring data of the deep horizontal displacement of slope soil, and when the sample data are changed, the structural parameters of the fitting model will also change accordingly, which indicates that the model is very sensitive to the changes of samples. At the same time, from the perspective of short-term static forecasting results, the fitting model still has the characteristics of short-term stability for the fluctuation pattern of soil displacement forecasting, but the forecasting accuracy also changes with the change of samples, so the conclusion may lack universality. In general, when using the ARIMA model to forecast the deep horizontal displacement of slope soil, it is better to forecast the historical data with stable development and change, no severe construction and large abnormal fluctuation caused by external factors such as emergencies. Finally, in order to better connect the empirical analysis conclusion with the slope management work, a new applied method of slope quality safety monitoring management, to ARIMA-soil displacement forecasting method is proposed.

In conclusion, this paper uses the ARIMA model and historical displacement value as sequence data to make short-term forecasting of the next 24 displacements, hoping that the results will be helpful for slope managers to strengthen slope safety management and prevent slope accidents. In addition, other forecasting methods can be considered to strengthen the attention to various influencing factors, such as slope structure, weather change, engineering construction, etc. Often these unstable factors also have important application value for the long-term trend of the deep horizontal displacement of slope soil.

The authors declare no conflicts of interest regarding the publication of this paper.

Yan, K., Wu, J.Y., Zhang, Y.Q. and Yang, L. (2019) Short-Term Forecasting Management Research of Deep Horizontal Displacement of Slope Soil Based on ARIMA Model. Open Access Library Journal, 6: e5496. https://doi.org/10.4236/oalib.1105496