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No Time to Lose: Time Series Analysis
Published in Jesús Rogel-Salazar, Advanced Data Science and Analytics with Python, 2020
In the examples of the sunspot and bitcoin datasets, we can see from the lower panels of the correlograms in Figures 1.13 and 1.14 that only the most recent values are really useful in building an autoregression model. A correlogram is a plot showing the correlation statistics.
A comparison between ARIMA, LSTM, ARIMA-LSTM and SSA for cross-border rail freight traffic forecasting: the case of Alpine-Western Balkan Rail Freight Corridor
Published in Transportation Planning and Technology, 2023
Miloš Milenković, Miloš Gligorić, Nebojša Bojović, Zoran Gligorić
In the first step, the ADF test is applied for detecting non-stationarity. The test provides a p-value, which is used to assess the stationarity of the series. A p-value below a certain threshold (commonly 0.05) indicates that the series is stationary, while a p-value above the threshold suggests non-stationarity. In cases where a time series is non-stationary (such as the export and import flow in the border crossings of Sid and Subotica), it needs to be transformed or adjusted to achieve stationarity using techniques like the first order differencing. In the second step, based on a visual plot of ACF and PACF, a general structure of the ARIMA model is proposed. Significant values of correlograms refer to the lag values in the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) that fall outside the confidence bounds (at the 5% level), and indicate significant correlations. These significant values in the correlograms suggest the presence of a pattern or relationship between the observations at different time lags and provide initial guidance to determine the most appropriate AR(I)MA model for a given time series. The third step includes determining the best model based on the criteria of minimum AICc. The values of MAPE and Adjusted R-squared are calculated for selected models. In the last step, Ljung–Box statistics is applied to check if the selected model is correctly specified. The p-value greater than the significance level (0.05) indicates that the model is considered adequately specified in terms of residual autocorrelation.
Dynamics of thermal motion of liquid lead nanoinclusions attached to fixed dislocations coupled in a dislocation node in the aluminum-based matrix
Published in Philosophical Magazine, 2023
It is noted in Section 3.4 that the mutual interaction of all the inclusions can lead to appearance of time correlations in their thermal motions (oscillations). To detect them the dependences of the coefficients of autocorrelation for all the inclusions, and the coefficients of cross-correlation for all pairs of the inclusions on a lag (time delay) τ were obtained. These dependences are known as autocorrelograms and cross-correlograms, respectively [26]. The x-displacements of the inclusions are usually much smaller than their z-displacements then the relative errors of the measurements of the x-displacements are much larger than those of the measurements of the z-displacements. Therefore, the autocorrelation and cross-correlation for the longitudinal thermal oscillations of the inclusions are considered only. Furthermore, since the thermal motions of the inclusion in x- and z directions are coupled then their similarity is expected. The equation is used to obtain the autocorrelograms and cross-correlograms. Here , , τm = Δt·m is the lag, Δt = 0.04 s, and the integer m lies in the interval [0, 500]. The coefficients of autocorrelation Ri(τm) and cross-correlation Rij(τm) are calculated taking in Equation (5) at i = j and i ≠ j, respectively.
Forecasting COVID-19 impact on RWI/ISL container throughput index by using SARIMA models
Published in Maritime Policy & Management, 2021
Kaan Koyuncu, Leyla Tavacioğlu, Neslihan Gökmen, Umut Çelen Arican
The concept of stationary has a great priority of the time series analysis. The concept of stationarity is expressed as the mean and variance of a time series is constant and the covariance between the two values of the series depends not only on the examined time but only on the difference between the two-time series. In order to apply the time series models, the series should be adjusted from the trend and seasonality (time-invariant). Correlograms (ACF and PACF graphs) can show a stationarity pattern or a unit root with significant lags. A more subjective way to evaluate stationarity is using (augmented) Dickey-Fuller (ADF) test statistics (Dickey and Fuller 1979). The null hypothesis is that the series have a unit root. The alternative hypothesis is that the time series is stationary (or trend-stationary).