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The prediction 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. Holt-Winters model is suitable to predict the data series of deep horizontal displacement of slope soil, which show both trend growth and seasonal fluctuation. Firstly, this paper selected the data set as the original data for empirical analysis which is deep horizontal displacement of soil after pretreatment from the specific slope monitoring project, then used the Holt-winters’ damped model to perform data mining, finally, compared with the traditional prediction methods including the neural-network model and the k-nearest neighbor classification. The results show that the damped Holt-winters model has the highest prediction accuracy.

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

It can be seen that no matter the collapse and landslide of natural slope, or the instability of artificial slope 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 and nonlinear complex system [

In slope deformation monitoring, the value of deep horizontal displacement of soil is a non-stationary time series that changes with seasonal cycles and has a certain increasing (or decreasing) trend. The traditional prediction methods of stationary time series, such as neural-network model and IBK model of the k-nearest neighbor classification, are not ideal for the prediction results of this kind of value [

The basic idea of Holt-winters model is to analyze and study the time series with linear trend, seasonal variation and random fluctuation, and combine them with exponential smoothing method to estimate the long-term trend, trend increment and seasonal fluctuation respectively, and then establish a prediction model to extrapolate the predicted value [

The predictive extrapolation formula of Holt-winters’ multiplication model is:

y ′ t + h = ( a t + h b t ) s t + h − m (1)

The calculation formulas of a, b and s are as follows:

a t = α ( y t / s t − m ) + ( 1 − α ) ( a t − 1 + b t − 1 ) (2)

b t = β ( a t − a t − 1 ) + ( 1 − β ) b t − 1 (3)

s t = γ [ y t / ( a t − 1 − b t − 1 ) ] + ( 1 − γ ) s t − m (4)

In order to predict a smaller time period data sets, on the basis of the multiplication model, add extension coefficient ϕ; Furthermore, the predictive extrapolation formula of the damped Holt-winters model is obtained through the transformation of exponential smoothing method, as follows:

y ′ t + h = [ a t + ( ϕ + ϕ 2 + ⋯ + ϕ h ) b t ] s t + h − m (5)

where, a stands for intercept; b is the trend; s represents the seasonal factor of the damped model; t represents the predicted time, and t = 1 , 2 , ⋯ , m ; h is the number of smooth phases, h > 0; m represents the cycle length, for example, 4 for the quarterly data and 12 for the monthly data. Since the experimental data interval in this paper is hour, 24 data are collected in a cycle, so m is 24.

The calculation formulas of a, b and s are as follows:

a t = α ( y t / s t − m ) + ( 1 − α ) ( a t − 1 + ϕ b t − 1 ) (6)

b t = β ( a t − a t − 1 ) + ( 1 − β ) ϕ b t − 1 (7)

s t = γ [ y t / ( a t − 1 − ϕ b t − 1 ) ] + ( 1 − γ ) s t − m (8)

In the formula, α, β and γ represent damped factors, also known as smoothing coefficient, and take values between 0 and 1.

The data of deep horizontal displacement of slope soil in this paper are derived from the SQL sever database of Sichuan Shengtuo detection Co., LTD., and the related slope engineering project is the ladder trough landslide monitoring project of Shuangma village, Shiguan township, Maoxian county. SQL-server database is a scalable, high-performance, computable for distributed servers relational data-base management system 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 prediction error, a continuous and stable period of data collection was selected for the prediction experiment.

Randomly selecting 500 - 650 data of a deep horizontal displacement monitoring point in the monitoring project, there are two reasons for choosing between 500 and 650. One is that if too little data is selected, the information in historical data cannot be fully extracted. On the contrary, the horizontal displacement with a long interval will have a small impact on the later prediction of horizontal displacement, resulting in unnecessary errors. The other is to look up many relevant literature about Holt-winters’ model. The number of original sequences for empirical analysis of quarterly data (m = 4) is between 50 and 100, and the number of monthly data (m = 12) is between 250 and 350. Therefore, the number of original sequences for empirical analysis of time-scale data (m = 24) is between 500 and 650.

In the end, this paper selects the “X-axis” direction from 0 points on August 15, 2018 to 23 points on September 10, 2018, and all the integral point data from 0 points on August 21, 2018 to 23 points on September 10, 2018, and constructs the “X” data set and the “Y” data set, respectively, with a data quantity of 648 and 504, as shown in

It can be seen from _{t}} of “X” data set and {Y_{t}} of “Y” data set are obvious time series with deterministic trend and periodicity.

In order to verify whether the data has the significance of predictive research, Origin-Pro data analysis software was used for the discrete analysis of data sets of “X” and “Y” to obtain the correlation coefficient and fitting degree values of data at adjacent moments, as shown in

It can be seen from

By using the WEKA data mining software forecast function, first of all, let α, β and γ carry out equivalent iterative operation within the interval of [0.01, 1]. The statistical prediction error = |the actual value − the predicted value |/actual value × 100%. It is concluded that when α = 0.03, β = 0.03 and γ = 0.03, the average prediction error of “X” data set is the smallest, which is 1.69%. When α = 0.11, β = 0.11 and γ = 0.11, the average prediction error of “Y” data set is the smallest, which is 3.16%.

“X” data set | “Y” data set | |
---|---|---|

Pearson’s r | 0.9440 | 0.9668 |

R-Square (COD) | 0.8913 | 0.9347 |

The influence of smoothing coefficients α, β and γ on the prediction error was analyzed again (the system default values of α, β and γ were all 0.2). It can be found from

Direction | Coefficient of size | Mean prediction error |
---|---|---|

“X axis” | α = 0.2, β = 0.03, γ = 0.03 | 1.53% |

α = 0.03, β = 0.2, γ = 0.03 | 2.44% | |

α = 0.03, β = 0.03, γ = 0.2 | 1.84% | |

“Y axis” | α = 0.2, β = 0.11, γ = 0.11 | 3.11% |

α = 0.11, β = 0.2, γ = 0.11 | 3.34% | |

α = 0.11, β = 0.11, γ = 0.2 | 3.39% |

error changes greatly, so it is possible to improve the prediction accuracy by changing β and γ.

According to the above analysis results, for the “X” data set, it is determined that α and γ are invariants and β are variables. The analysis and prediction results show that when β = 0.04, the minimum prediction error is about 1.68%, as shown in

To sum up, when α = 0.02, β = 0.03 and γ = 0.03, the average prediction error of the direction of “X axis” is the smallest, which is 1.53%. The predicted results are shown in

In

As can be seen from

When α = 0.11, β = 0.23 and γ = 0.05, the average prediction error of the direction of “Y axis” is the smallest, which is 3.07%. The predicted results are shown in

In

damped Holt-winters model had a high short-term prediction accuracy for “Y” dataset. According to the 95% confidence of the actual value, the upper and lower limits of the prediction were calculated and the prediction effect diagram of the prediction interval was drawn, as shown in

As can be seen from

The predicted results of “X” and “Y” data sets were compared and analyzed, as shown in

As can be seen intuitively from

In order to further verify the reliability and feasibility of experimental results,

Time serial number | Actual value/10^{−}^{2} mm | Predictive value/10^{−2} mm | Prediction error/% |
---|---|---|---|

1 | 56.25 | 57.10 | 1.51 |

2 | 56.55 | 59.40 | 5.05 |

3 | 58.29 | 57.65 | 1.09 |

4 | 57.13 | 57.53 | 0.70 |

5 | 57.82 | 58.43 | 1.05 |

6 | 58.23 | 58.87 | 1.09 |

7 | 58.27 | 56.77 | 2.57 |

8 | 59.42 | 57.99 | 2.40 |

9 | 58.96 | 57.65 | 2.22 |

10 | 57.48 | 57.48 | 0.00 |

11 | 57.48 | 56.61 | 1.51 |

12 | 55.94 | 56.22 | 0.51 |

13 | 55.79 | 56.22 | 0.77 |

14 | 57.52 | 56.82 | 1.22 |

15 | 57.71 | 56.44 | 2.21 |

16 | 56.40 | 56.25 | 0.26 |

17 | 58.62 | 57.56 | 1.80 |

18 | 57.71 | 58.45 | 1.28 |

19 | 59.27 | 59.38 | 0.19 |

20 | 56.63 | 57.83 | 2.11 |

21 | 57.98 | 57.70 | 0.48 |

22 | 56.72 | 57.25 | 0.94 |

23 | 56.88 | 55.52 | 2.38 |

24 | 55.69 | 57.51 | 3.27 |

Note: the predicted time is “2018-09-10-2018-09-10 23”. The actual value comes from the SQL sever data-base of Sichuan Shengtuo detection co., LTD.

Multilayer-Perceptron model and IBK model were used for prediction, and compared with the prediction results of the damped Holt-winters model. Curves were drawn, as shown in

It can be intuitively seen from

From the prediction error data can be seen in

Time serial number | Actual value/10^{−2} mm | Predictive value/10^{−2} mm | Prediction error/% |
---|---|---|---|

1 | −31.05 | −32.43 | 4.44 |

2 | −30.03 | −33.26 | 10.76 |

3 | −31.92 | −31.66 | 0.82 |

4 | −32.02 | −33.56 | 4.80 |

5 | −31.97 | −33.20 | 3.85 |

6 | −32.77 | −33.47 | 2.15 |

7 | −32.89 | −32.74 | 0.46 |

8 | −33.05 | −33.48 | 1.31 |

9 | −32.22 | −33.12 | 2.81 |

10 | −32.49 | −33.11 | 1.90 |

11 | −31.43 | −32.56 | 3.60 |

12 | −29.60 | −31.67 | 6.99 |

13 | −30.85 | −31.67 | 2.67 |

14 | −33.06 | −32.01 | 3.18 |

15 | −32.67 | −32.40 | 0.82 |

16 | −31.09 | −32.76 | 5.37 |

17 | −33.39 | −33.19 | 0.60 |

18 | −32.41 | −32.16 | 0.78 |

19 | −34.01 | −32.77 | 3.64 |

20 | −31.02 | −31.03 | 0.04 |

21 | −32.15 | −31.81 | 1.05 |

22 | −30.27 | −29.64 | 2.08 |

23 | −30.20 | −29.09 | 3.69 |

24 | −29.53 | −31.28 | 5.94 |

Note: the predicted time is “2018-09-10-2018-09-10 23”. The actual value comes from the SQL sever data-base of Sichuan Shengtuo detection co., LTD.

Avg/(%) | S^{2} | |
---|---|---|

The prediction error of the “X” data set | 1.53 | 1.85E−04 |

The prediction error of the “Y” data set | 3.07 | 6.20E−04 |

damped Holt-Winters model | Multilayer-Perceptron model | IBK model | |
---|---|---|---|

Max/% | 5.05 | 6.68 | 4.85 |

Min/% | 0.00 | 0.23 | 0.24 |

Avg/% | 1.53 | 2.48 | 1.75 |

damped Holt-Winters model | Multilayer-Perceptron model | IBK model | |
---|---|---|---|

Max/% | 10.76 | 11.51 | 14.76 |

Min/% | 0.04 | 0.33 | 0.06 |

Avg/% | 3.07 | 3.97 | 4.21 |

0.04%, 3.07%, and all to optimal Multilayer Perceptron model, IBK model. The result show the damped Holt-winters model had the highest prediction accuracy and the best effect.

Combined with the above experimental results, a new implementation method of slope safety management is proposed, which is prediction method for deep horizontal displacement of soil, referred to as Pm-DHDS, as shown in

The working principle of this method is as follows: relying on the data set of actual horizontal displacement monitoring in the deep layer of slope soil, short-term prediction is conducted with the damped Holt-winters model, which is also known as the one-stage prediction. If the analysis results of the first-phase prediction are harmless to the slope deformation evolution, the first-phase prediction results will be put into the original time series for the second-phase prediction, and then the long-term prediction may generate cumulative errors. On the contrary, if the predicted analysis results in a certain period are harmful to the slope deformation evolution, the corresponding safety management measures should be implemented in advance, which also provides the basis for the slope stability analysis and better prevents the occurrence of slope safety accidents.

It is a challenging problem to predict the deep horizontal displacement of slope soil, but time series prediction has been considered as an effective method to predict the trend growth and seasonal fluctuation. Because the prediction of time series has a good short-term prediction effect, although this paper only carries out the short-term prediction of time series, in theory, the prediction period of time series can be extended by adding effective original data, and the short-term accuracy will also be improved. The empirical analysis in this paper also shows

that the damped Holt-winters model is feasible as a short-term prediction model for deep horizontal displacement of slope soil, and its prediction accuracy is also the highest compared with the results of traditional methods. Thus, the application of damped Holt-winters model may also be of some reference value for this paper to grasp the implementation opportunity of slope safety management measures, analyze slope stability and prevent slope accidents.

Due to the limited time, this paper only makes an empirical analysis on the change of the actual data of the deep horizontal displacement of slope soil, and the difference in prediction accuracy is not significant, which indicates that the prediction of the same type of samples by the model has a strong stability. Therefore, for the prediction of different types of sample data, the conclusion may lack universality. In order to better connect the empirical analysis conclusion with the management work, the theoretical method of slope safety monitoring management is proposed, which is prediction method for deep horizontal displacement of soil, referred to as Pm-DHDS.

In conclusion, through the damped Holt-winters model, this paper used historical displacement values as serial data to make short-term prediction of the next 24 displacements, hoping to help slope safety managers to strengthen slope management and prevent slope accidents. At the same time, it can be considered to combine with other prediction methods to pay more attention to various factors such as slope structure factors, weather changes, engineering construction, etc. These uncertain factors often have important application value for the long-term trend of deep horizontal displacement of slope soil.

Shortcomings and prospects of this study are:

1) The damped Holt-winters model belongs to short-term prediction model, and its prediction effect will gradually deteriorate with the passage of time [

2) Abnormal data may exist in the experimental data, leading to the reduced reliability of the predicted results. There are two reasons for the abnormal data: one is the data acquisition equipment problems, resulting in abnormal data; Second, the structure of the object itself is abnormal, so that the acquisition data variation. For prediction experiments, the second cause of data anomalies is a reasonable category. The data anomaly caused by the first reason is outside the controllable factors. In future prediction experiments, the algorithm model should be improved as much as possible. If abnormal data can be screened and screened in advance, the reliability of prediction results can be guaranteed.

Bridge nondestructive testing and engineering calculation key laboratory of Sichuan University Open Fund Project (2018QZY01).

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

Yan, K., Wu, J.Y., Zhang, Y.Q., Yang, L., Zhang, Y.X., Yuan, H.Y. and Huang, Y. (2019) Prediction Method of Deep Horizontal Displacement of Slope Soil Based on Damped Holt-Winters Model. Journal of Service Science and Management, 12, 391-406. https://doi.org/10.4236/jssm.2019.123027