# Landslide Forecast by Time Series Modeling and Analysis of High-Dimensional and Non-Stationary Ground Motion Data

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## Abstract

**:**

## 1. Introduction

## 2. Motivational Data on Ground Motion in Landslide

## 3. The Error-Correction Cointegration (ECC) Approach for VAR Time Series

#### 3.1. ECC–VAR(p) Model

`R`function

`ca.jo()`in package

`urca`to perform the cointegration test-estimation of r and $\mathbf{\beta}$. The estimators of r and $\mathbf{\beta}$ are denoted as $\widehat{r}$ and $\widehat{\mathbf{\beta}}=({\widehat{\mathbf{\beta}}}_{1},\cdots ,{\widehat{\mathbf{\beta}}}_{\widehat{r}})$, respectively. Writing ${\mathit{y}}_{t-1}={\widehat{\mathbf{\beta}}}^{\top}{\mathit{z}}_{t-1}$; and by the ECC form (5), the observed vector time series ${\mathit{z}}_{t}$ can be asymptotically characterized by the following ECC($\widehat{r}$)-VAR(p) model:

#### 3.2. Making Statistical Inferences from the ECC–VAR Model

`cajorls()`function in

`R`package

`urca`. The AR order p can be best estimated by a standard model selection criterion such as AIC or BIC, which are commonly used in time series analysis. Details are not pursued here.

#### 3.3. Forecasting Based on the Fitted ECC–VAR Model

## 4. The EDQ Technique for Vector Time Series Dimension Reduction

## 5. Applying the ECC–VAR–EDQ Method to Analyze the InSAR Landslide Data

#### 5.1. Unit Root Test and Cointegration Test for the EDQ Series

`R`package

`urca`, are $-3.43,-2.86$ and $-2.57$, respectively. By the ADF test, the null hypothesis of unit root non-stationarity should be rejected at a significance level if the absolute value of the computed ADF statistic is greater than the absolute value of the corresponding critical value. The ADF unit test results for the 22 times series in $\left\{{\mathit{z}}_{t}\right\}$ and $\{\Delta {\mathit{z}}_{t}\}$ are provided in Table 2, from which we see statistical evidence, at the 1% or 5% significance level, of stationarity for 16 series, and there is no significant evidence to reject the unit-root non-stationarity for the level-($\mathrm{Min},0.6,0.7,0.8,0.9,\mathrm{Max}$) EDQ series in $\left\{{\mathit{z}}_{t}\right\}$.

`ca.jo()`to perform a sequence of cointegration tests based on the trace statistic for the vector time series $\left\{{\mathit{z}}_{t}\right\}$ determined by the 11 EDQ series. The null hypothesis of $r=\mathrm{rank}(\mathbf{\Pi})\le {r}_{0}$ was to be rejected at significance level $\mathbf{\alpha}$ if the trace statistic was greater than the level $\mathbf{\alpha}$ critical value. The cointegration test results for ${r}_{0}=0$ to 10 are provided in Table 3.

`ca.jo()`, the CCA based estimate of the cointegration vectors $\mathbf{\beta}$, with its top 6 rows constituting a $6\times 6$ identity matrix, is found to be

#### 5.2. Estimating and Fitting the ECC(r)–VAR(p) Model for the EDQ Series

`R`function

`cajorls()`to fit an ECC(6)-VAR(2) model for the 11 EDQ series $\left\{{\mathit{z}}_{t}\right\}$ shown in the right panel in Figure 2, giving the following results:

#### 5.3. Landslide Displacement Forecasting

## 6. Probabilistic Landslide Prediction via the ECC–VAR–EDQ Method

#### 6.1. Forecast Intervals for Displacement and Velocity

#### 6.2. Probability of Future Risk of Landslide

#### 6.3. Landslide Prediction for All Locations

`mine_r6.mp4`is provided in Supplementary Materials which displays the aforementioned forecasts of displacement and the red-alert probability for all 1803 locations and at time t from 1601 to 3820. Two screenshots of this video at time $t=3513$ and 3568 are shown in Figure 7 and Figure 8, respectively. The video and the screenshot show that the displacement forecasts capture the spatiotemporal dynamics and trends of the actual slope surface movement at all locations very well, and the red-alert probability forecasts provide timely predictions of the landslides at all locations.

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) Slope surface displacement from 31 May to 21 June with $1083\times 5090$ observations in total. (

**Right**) The 1803 small areas (locations). The 11 locations highlighted in black correspond to the 11 EDQ series at quantile levels 0.0 (min), 0.1, ⋯, 0.9 and 1.0 (maximum).

**Figure 2.**(

**Left**) The 11 log-displacement EDQ series from t = 1 to t = 5090. (

**Right**) The 11 log-displacement EDQ series from the training sample t = 1 to t = 1600.

**Figure 3.**(

**Left**) Log-displacements of the 11 selected EDQ series from $t=1$ to $t=3820$. (

**Right**) Displacements of the 11 selected EDQ series from $t=1$ to $t=3820$. The black lines represent the forecasts and the gray lines represent the actual observations.

**Figure 4.**Slope surface displacement (

**Left**), velocity (

**Middle**) and probability of triggering a red alert (

**Right**) in the Min, 10%, 20% and 30% EDQ series. The forecast displacement and velocity are plotted in black with their estimated 80% forecast intervals, and the corresponding observed data are plotted in gray lines. The red dashed line in each velocity plot is the velocity alert threshold, which is 1 mm/6 min (10 mm/h), and indicates 50% probability of triggering a red alert in the probability plots.

**Figure 5.**Slope surface displacement (

**Left**), velocity (

**Middle**) and probability of triggering red alert (

**Right**) for the 40%, 50%, 60% and 70% EDQ series. The forecast displacement and velocity are plotted in black with their estimated 80% forecast intervals, and the corresponding observed data are plotted in gray lines. The red dashed line in each velocity plot is the velocity alert threshold, which is 1 mm/6 min (10 mm/h), and indicates 50% probability of triggering a red alert in the probability plots.

**Figure 6.**Slope surface displacement (

**Left**), velocity (

**Middle**) & probability of triggering red alert (

**Right**) for the 80%, 90% & Max EDQ series. The forecast displacement and velocity are plotted in black with their estimated 80% forecast intervals, and the corresponding observed data are plotted in gray lines. The red dashed line in each velocity plot is the velocity alert threshold, which is 1 mm/6 min (10 mm/h), and indicates 50% probability of triggering a red alert in the probability plots.

**Figure 7.**Screenshot of mine_r6.mp4 at time t = 3513. The

**top**(

**middle**) panel displays the actual (forecast) displacement, and the

**bottom**panel displays the probability forecasts at all locations. The ▴s indicate the 11 EDQ locations. Results were obtained based on the data at $t=1,\cdots ,1600$.

**Figure 8.**Screenshot of mine_r6.mp4 at time t = 3568. The

**top**(

**middle**) panel displays the actual (forecast) displacement, and the

**bottom**panel displays the probability forecasts at all locations. The ▴s indicate the 11 EDQ locations. Results were obtained based on the data at $t=1,\cdots ,1600$.

Quantile Level | Selected Pixel | Quantile Level | Selected Pixel |
---|---|---|---|

Min | 202 | 0.6 | 827 |

0.1 | 1432 | 0.7 | 672 |

0.2 | 1454 | 0.8 | 685 |

0.3 | 1392 | 0.9 | 630 |

0.4 | 1307 | Max | 534 |

0.5 | 995 |

**Table 2.**Unit root test results for the level/first-order difference of the 11 selected EDQ time series (log transformed). Superscript *** indicates rejecting the null hypothesis at $1\%$ level; ** indicates rejecting the null hypothesis at $5\%$ level. All these tests were conducted by including a trend term and up to 13 lags.

Quantile Level p | Pixel ID | Augment Dickey−Fuller(ADF) |
---|---|---|

Min | 202 | −0.9334/−26.8762 *** |

0.1 | 1432 | −2.9837 **/−21.3401 *** |

0.2 | 1454 | −4.6048 ***/−21.5254 *** |

0.3 | 1392 | −3.1837 **/−17.0134 *** |

0.4 | 1307 | −3.3355 **/−17.7037 *** |

0.5 | 995 | −2.8695 **/−16.1240 *** |

0.6 | 827 | −1.7187/−14.2224 *** |

0.7 | 672 | −1.4623/−13.3679 *** |

0.8 | 685 | −1.3172/−13.9375 *** |

0.9 | 630 | −1.5063/−11.7859 *** |

Max | 534 | −1.3808/−17.9545 *** |

**Table 3.**Null hypotheses, trace statistic and level-(10%, 5%, 1%) critical values of cointegration tests.

Hypothesis | Statistic | 10% | 5% | 1% |
---|---|---|---|---|

$r\le 10$ | 0.42 | 6.50 | 8.18 | 11.65 |

$r\le 9$ | 5.82 | 15.66 | 17.95 | 23.52 |

$r\le 8$ | 16.77 | 28.71 | 31.52 | 37.22 |

$r\le 7$ | 35.19 | 45.23 | 48.28 | 55.43 |

$r\le 6$ | 57.56 | 66.49 | 70.60 | 78.87 |

$r\le 5$ | 120.14 | 85.18 | 90.39 | 104.20 |

$r\le 4$ | 253.29 | 118.99 | 124.25 | 136.06 |

$r\le 3$ | 404.85 | 151.38 | 157.11 | 168.92 |

$r\le 2$ | 628.63 | 186.54 | 192.84 | 204.79 |

$r\le 1$ | 998.73 | 226.34 | 232.49 | 246.27 |

$r\le 0$ | 1869.85 | 269.53 | 277.39 | 292.65 |

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**MDPI and ACS Style**

Qian, G.; Tordesillas, A.; Zheng, H.
Landslide Forecast by Time Series Modeling and Analysis of High-Dimensional and Non-Stationary Ground Motion Data. *Forecasting* **2021**, *3*, 850-867.
https://doi.org/10.3390/forecast3040051

**AMA Style**

Qian G, Tordesillas A, Zheng H.
Landslide Forecast by Time Series Modeling and Analysis of High-Dimensional and Non-Stationary Ground Motion Data. *Forecasting*. 2021; 3(4):850-867.
https://doi.org/10.3390/forecast3040051

**Chicago/Turabian Style**

Qian, Guoqi, Antoinette Tordesillas, and Hangfei Zheng.
2021. "Landslide Forecast by Time Series Modeling and Analysis of High-Dimensional and Non-Stationary Ground Motion Data" *Forecasting* 3, no. 4: 850-867.
https://doi.org/10.3390/forecast3040051