PSI - Issue 50

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

259

3

a

b

1 2 3 4 5 6

δUz, mm

2014 2015 2016 2017 2018 2019 2020 -50 -40 -30 -20 -10 0

Date

Fig. 1. (a) Plan of the building foundation with the layout of the hydraulic level sensors; (b) vertical displacement at sensor locations.

2. Methods 2.1. Exponential Smoothing Models (ETS)

The simplest way to predict a process that develops over time is to approximate the available data, for example, by the least squares method and extrapolate the resulting function to a certain interval in the future. Such a predictive model takes into account all previously accumulated data with equal weights. However, events that happened recently are often more significant for the forecast than those that are far in the past. The possibility of taking into account such a feature of the developing process is provided by exponential smoothing models, in which the significance of the observed values for the forecast increases as it approaches the start of the forecast. Classical works in this direction are those of Brown (1961), Holt (2004), Winters (1960) and others. The smoothing procedure is based on the calculation of exponential moving averages of the smoothed series. In the simplest case, the next predicted term of the time series is written through the previous one in the form. Here, y t is the initial series, ˆ t y and 1 ˆ t y  are current and previous predicted values, 0 < α <1 is the smoothing coefficient. In this case, only the data of the latest time steps are significant for the forecast, since the earlier values are smoothed out more. In more complex predictive models, three components are distinguished in the initial time series: the level, which is the average value of the series, the trend, which is a stable trend observed over a long time, and seasonality, which reflects changes of a periodic nature. For each of these components, equations similar to (1) are written. There are a number of predictive models that combine these components as sums, products, or other combinations of components. For example, review by Hyndman (2018) lists 30 ETS (Error Trend Seasonality) models. Of these, the additive Holt-Winters model ETS(A, A) is the most popular; in this model, the forecast value ˆ t h y  is obtained as the sum of the level l t ,, the trend b t and the seasonality s t : 1 ˆ y y ˆ (1 ) y       t t t (1)

ˆ t h t y 

( 1)        , t t h m k l h b s

(2)

( y s      t m  1 t t      ) (1 )  1 1 t      ( t t y l b    ) (1 )( l  t t 1 ( b l l  t t b  ) (1 )  t s t l ,

)

b



,

(3)

1 1 t

 

(4)

s

.

(5)

t m 