PSI - Issue 50

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

260

4

Here, t y is the observed time series, ˆ t h y  is the forecast values for the period h , m is the length of the seasonal return period in the past, k is the integer part of ( h − 1)/ m ; the parameters α, β, γ have values from 0 to 1 and serve to fit the predicted values to the observed ones over a certain period in the past (training interval). The choice of model parameters (α, β, γ, m ) is carried out by minimizing a certain statistical parameter (for example, standard deviation, mean absolute error, etc.), which characterizes the difference between the predicted values and those observed in the training interval. 2.2. Autoregressive Moving Average Models (ARMA) For the analysis of stationary time series, the autoregressive moving average model is often used, Ljung et al. (2014). It generalizes two simpler time series models, the autoregressive model AR(p) and the moving average model MA(q), and consists of the following components: Here, c is a constant, ε is white noise (sequence of independent and identically dist ributed random variables with zero mean), α and β are real numbers representing, respectively, the autoregressive coefficient and moving average coefficient, p and q are the orders of the AR and MA models. The model order is denoted by ARMA( p,q ). If the observed process is non-stationary (there is a pronounced trend or seasonality), the differences in the values of the original time series are analyzed. The resulting series, consisting of differences of a certain order is stationary, and an ARMA model is built for it. Usually the first order of differentiation is sufficient, sometimes the second is used. Such models are called integrated moving average autoregressive models and are written as ARIMA( p, d, q ). In addition to the parameters p and q , which characterize the lag order and the size of the moving average window, they contain the parameter d , which indicates how many times you need to take the difference of the original values to get a stationary series from the original one. If there is a clearly defined seasonal component in the initial data, it can be identified and a similar model can be built for it with coefficients P, D, Q . They express, respectively, the order of autoregression, the order of integration and the order of the moving average for the seasonal component. The seasonal model is written as ARIMA ( p, d, q ) ( P, D, Q ). The choice of a model for a specific observed series consists of the following steps: identification of a trial model; estimation of model parameters and diagnostic verification of model adequacy using statistical methods; using the model for forecasting. The choice of optimal model parameters is ambiguous; there are special procedures for their evaluation, a detailed description of them can be found in Box et al. (2015). Currently, there are a number of software products that provide automated ways to solve these problems. In this work, the R software package for statistical data analysis was used to analyze the accumulated array of experimental data. The procedures for constructing statistical forecasting models were implemented using the R package (version 3.6) and the forecast library, Hyndman (2008). The aim of this work was to test the applicability of the described approach to the analysis of an experimental data in order to compare the data obtained using the chosen model with the data of real observations. 2.3. Tuning a predictive model Based on the accumulated experimental data, several forecast models were built. The ETS-Holt-Winters and STL-ETS(A,N) and ARIMA (p, d, q) (P, D, Q) models were tested. The performed numerical experiments showed that the most adequate results were obtained using the ARIMA model. Therefore, its application is shown below. We will demonstrate the possibilities of applying this approach using the example of making a forecast of foundation settlement based on the ARIMA ( p, d, q ) ( P, D, Q ) model. To select the model parameters, we use data for a 5-year observation period (2014-2019). Let us build a forecast of foundation settlement for a 1-year period with various combinations of model parameters and then compare these data with the observational results for the same 1-year period. As a measure of the discrepancy between the prediction and observation data, we use 2 criteria - 1 1 p q t t        . i t i y  i t i  i i y c      (6)