PSI - Issue 50

I. Shardakov et al. / Procedia Structural Integrity 50 (2023) 257–265 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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the root mean square error (RSME) at the forecast interval and the data discrepancy (EE) at the end of the forecast interval (the average value for the last 10 days). Figure 1 shows the results of forecasting using the ARIMA model, in which the values of the parameters p and q vary from 1 to 3, and the parameter d takes values of 0 and 1. Table 1 shows the values of the standard deviation RMSE and the error EE for some combinations model parameters. The presented results show that the data have a pronounced non-stationary character (the forecast obtained at d = 0 gives significant discrepancies with the observed values). The minimum values of the parameters p and q, sufficient to describe the trend, are equal to 1. To describe the seasonal component, the combination of the parameters P, D and Q (0,1,1) gives a more accurate forecast than (0,1,0). a b c

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-14.0 -13.0 -12.0 -11.0

δUz, mm

δUz, mm

δUz, mm

real_data ARIMA(1,0,1) ARIMA(0,1,1) ARIMA(1,1,1) ARIMA(2,1,2) ARIMA(3,1,3)

real_data ARIMA(1,0,1) ARIMA(0,1,1) ARIMA(1,1,1) ARIMA(2,1,2) ARIMA(3,1,3)

real_data ARIMA(1,0,1) ARIMA(1,1,1) ARIMA(2,1,2) ARIMA(3,1,3)

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2014 2015 2016 2017 2018 2019 2020

2019.0

2019.4

2019.8

2020.2

2019.0

2019.4

2019.8

2020.2

Date

Date

Date

Fig. 2. (a) Basement settlement observation data for a 5-year period; (b) forecast for a 1-year period according to ARIMA (p, d, q)(0,1,0); (c) forecast for a 1-year period according to ARIMA (p, d, q)(0,1,1).

Table 1. Forecast error by ARIMA(p, d, q) (P, D, Q) model with different parameters

Model parameters ( p, d, q ) ( P, D, Q )

(1,0,1) (0,1,0) 0.7287

(1,1,1) (0,1,0) 0.4934

(2,1,2) (0,1,0) 0.4935

(3,1,3) (0,1,0) 0.4934

(4,1,4) (0,1,0) 0.4936

(1,0,1) (0,1,1) 0.5792

(1,1,1) (0,1,1) 0.4101

(2,1,2) (0,1,1) 0.4101

(3,1,3) (0,1,1) 0.3828

(4,1,4) (0,1,1) 0.4102

RMSE,mm

EE, mm

2.128

0.987

1.000

0.995

0.993

1.596

0.345

0.345

0.618

0.345

Figure 1a shows that the initial time series contains sharp changes in values due to random factors, in particular, measurement errors. We tried to smooth the available time series and build a forecast based on the smoothed data. Figure 3 shows the data of the original time series (a) and series, where the data are averaged over an interval of 10 days (b) and 50 days (c), as well as forecasts built on these data. As can be seen from the figure, preliminary smoothing of the initial data does not lead to an improvement in the quality of the forecast. a b c

real_data forecast1 forecast2 forecast3 forecast4

real_data forecast1 forecast2 forecast3 forecast4

real_data forecast1 forecast2 forecast3 forecast4

δUz, mm

δUz, mm

δUz, mm

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2014 2015 2016 2017 2018 2019 2020

2014 2015 2016 2017 2018 2019 2020

2014 2015 2016 2017 2018 2019 2020

Date

Date

Date

Fig.3 . (a) Forecast by ARIMA(1,1,1)(0,1,0) for a 1-year period according to the initial observation data; (b) forecast according to the initial data averaged over a 10-day interval; (c) forecast according to the initial data averaged over a 50-day interval.