Data-driven models for atmospheric air temperature forecasting at a continental climate region

equations: ACF ¼ PN k t¼1 Xt �x ð Þ Xtþk �X � PN k t¼1 Xt �X ð Þ ð17Þ PACF ¼ ACF Pk 1 j¼1 PACFk 1;jACFk 1 1 Pk 1 j¼1 PACFk 1;jACFk 1 ð18Þ The N is the total number of temperature records, Xt and X is the time series record at time t and the mean of the temperature records, and finally, the k is number of lags in the time series data. lower and upper limits (UP, LO) can be determined at 95% significant level by the following equation: LO; UP ¼ � 1:96 ffiffiffiffi N p ð19Þ The ACF declines more slowly concerning the daily scale, which means that the time series data is not stationary. It is challenging to select the most effective lags for daily scale using ACF because of seasonality. The PACF, similar to the ACF, shows the association between two rec- ords that the shorter delays between those observations do not describe. For instance, the par- tial autocorrelation coefficient for the third lag in the daily scale temperature is only a correlation that the previous short lags (lag two and lag one) have not explicitly explained. Therefore, the PACF is more suitable for selecting the input lags for predicting the short scale of time series than the ACF. Table 3 shows the input combinations used in this study. It is worth mentioning that 70% of the data is used to train the suggested models, and 30% of the data included the end of the time series of the data utilized for the testing phase and checking the models’ performances (see Fig 3). The following steps summarize the primary process of developing the models for forecasting short-and mid-term air temperature.

Table 3. Air temperature input design for Baker and Crary stations. Model Input groups Output Scale Training data records Testing data records M1 Tt−1 Tt Daily 5624 2411 M2 Tt−1, Tt−3 Tt Daily 5622 2411 M3 Tt−1, Tt−3, Tt−4 Tt Daily 5621 2411 M4 Tt−1, Tt−3, Tt−4, Tt−5 Tt Daily 5620 2411 M5 Tt−1, Tt−3, Tt−4, Tt−5, Tt−6 Tt Daily 5619 2411 M6 Tt−1, Tt−3, Tt−4, Tt−5, Tt−6, Tt−7 Tt Daily 5618 2411 M7 Tt−1, Tt−3, Tt−4, Tt−5, Tt−6, Tt−7, Tt−8 Tt Daily 5617 2411 M8 Tt−1, Tt−3, Tt−4, Tt−5, Tt−6, Tt−7, Tt−8, Tt−9 Tt Daily 5616 2411 M9 Tt−1, Tt−3, Tt−4, Tt−5, Tt−6, Tt−7, Tt−8, Tt−9, Tt−10 Tt Daily 5615 2411 M10 TWt−1 TWt Weekly 803 344 M11 TWt−1, TWt−2 TWt Weekly 802 344 M12 TWt−1, TWt−2, TWt−3 TWt Weekly 801 344 M13 TWt−1, TWt−2, TWt−3, TWt−4 TWt Weekly 800 344 M14 TWt−1, TWt−2, TWt−3, TWt−4, TWt−5 TWt Weekly 799 344 M15 TWt−1, TWt−2, TWt−3, TWt−4, TWt−5, TWt−6 TWt Weekly 798 344 M16 TWt−1, TWt−2, TWt−3, TWt−4, TWt−5, TWt−6, TWt−7 TWt Weekly 797 344 M17 TWt−1, TWt−2, TWt−3, TWt−4, TWt−5, TWt−6, TWt−7, TWt−8 TWt Weekly 796 344 https://doi.org/10.1371/journal.pone.0277079.t003 PLOS ONE Air temperature forecasting: Mutli-time scale PLOS ONE | https://doi.org/10.1371/journal.pone.0277079 November 3, 2022 10 / 31