PSI - Issue 64

478 Mohammad Shamim Miah et al. / Procedia Structural Integrity 64 (2024) 476–483 M.S. Miah and W. Lienhart / Structural Integrity Procedia 00 (2024) 000–000 3 is a laser sensor from Micro-Epsilon model namely optoNCDT 1700-50 that has a measuring range of 50mm with a precision of 3 µ m. In a nutshell, the measured data are almost noise free and the sensor is capable of recording displacement or movement of the positions. However, the main drawback of the aforementioned sensor is that this type of sensor may not be suitable where a large displacement is expected to be occurred. After the tests, the post processing is necessary to clean the data as well as to avoid any offset (if there is any). The data process and analysis contains; data screening and selection, de-trending and filtering, fast Fourier transform, selecting and developing a candidate model, model validation, finally, the prediction. 2.1. Autoregressive integrated model The main use of data-based time-series model is that it doesn’t required any information about the underlying physics of the systems. In addition to that the input signal information also not needed for data based models. Hence, such models are very useful for the real-life problems due to underlying complexity of the physics based model. As it is not so simple to have a physics based model for many complex structures (e.g. rail-track, tunnel or very old structures e.g. historical buildings). In such situation, data measuring via sensors are possible and understanding the underlying mechanics are possible. Furthermore, often in the real-life problems, the input signals are not available or not measured, hence, it is useful to have a time-series model that does not require the input signals. Therefore, herein, time-series data modelling via the use of autoregressive (AR) type models are used where the input signal is missing. The simplest form of AR(p) model with no input is given by, A ( p ) Y ( t )= η ( t ) (1) where Y is the time-series data, p is model order, t is the time-steps, η is the noise. Further, the autoregressive integrated model (AIR) can be defined as shown in Eg. (2). The ARI mode has the capability of dealing with sensors noise which might be beneficial for noisy signals data. Due the early mentioned advantages herein ARI type model is adopted. There many applications in different area of science and engineering of the time-series models as well as in the area of finance. More detail about time-series data modelling and analysis procedure can be be explored via Marple (1987). A ( p ) Y ( t )= 1 1 − p − 1 η ( t ) (2) where A is the model coefficients. 2.2. N-step-ahead forecasting In case of predictions, the current time-step ( n ) is estimated based on the previous time-step ( n − 1) information. Hence, the prediction of the time-series data may not be quite accurate as expected (because the model prediction is heavily relies on measured data quality). However, with proper tuning/optimization the accuracy of the used model can be improved. Though, the predicted information may not be accurate (e.g. tomorrow’s weather) but still it can be quite useful to understand what might pop up next. The simplest form of prediction model can be given in Eq. (3), it can be seen that the prediction of the current n state depends on the previous steps ( n − 1). Y predict ( n ) = λY pre ( n − 1) (3) where n is the time-step, Y predict is the predicted data, Y pre is previous time instance data. In case of forecasting, ones look for the future based on the currently available data. A basic forecasting model is given in Eq. (4) while it can be seen that the forecasting next step ( n + 1) is directly depends on the current steps ( n ). Y fore ( n +1) = λY cur ( n ) (4) where Y fore is the next-step forecasted data, Y cur is current time instance data.