## 5 Variability of QBIE Temperature Events
### Internannual Variability and Predictability  
While long-term trends in the activity of QBIE temperature events are not significant, obvious interannual variations are observed (Figures 7a,d,g,j,m,p and 8a,d,g,j,m,p). Here, we first employ the Autoregressive Integrated Moving Average (ARIMA) approach, a method commonly used in econometrics and time series analysis yet seldom applied in meteorology and climate science. This approach is particularly effective for modeling and forecasting nonstationary time series data by extracting their intrinsic temporal evolution patterns [35]. By applying the ARIMA to our century-long dataset of QBIE temperature extremes, we aim to explore the variability and predictability in these events over time. Figure 7: The time series (**a**,**d**,**g**,**j**,**m**,**p**), autocorrelation plots (**b**,**e**,**h**,**k**,**n**,**q**), and partial autocorrelation plots (**c**,**f**,**i**,**l**,**o**,**r**) for annual event counts of six types of QBIE heat waves from 1950 to 2022. In the autocorrelation plots and partial autocorrelation plots, the blue shaded areas represent two standard deviations, with blue points indicating autocorrelation and partial autocorrelation coefficients within two standard deviations and red points indicating coefficients exceeding this range. Through applying white noise tests (Ljung-Box Test, LB Test) [36] and ARIMA modeling analysis (Table 5) to six categories of QBIE event counts, we found that most time series could not be effectively modeled or predicted using the ARIMA model. The LB Test results indicated that eight out of twelve time series do not pass the white noise test. Only event count series of four kinds of QBIE events can be categorized as non-white-noise series, including CDN heat waves, SDT cold surges, SNT cold surges, and ALL cold surges. However, for these four non-white-noise time series, the residuals in ARIMA modeling are rather large, suggesting significant prediction errors when using the ARIMA model (Table 5). Figures 7 and 8 further illustrate the lag-k autocorrelation and partial autocorrelation plots for the interannual event count variability of various types of QBIE heat waves and cold surges. In these plots, the blue shaded areas represent two standard deviations, and the blue and red points indicate autocorrelation coefficients (Figures 7b,e,h,k,n,q and 8b,e,h,k,n,q) and partial autocorrelation coefficients (Figures 7c,f,i,l,o,r and 8c,f,i,l,o,r). Correlation coefficients exceeding two standard deviations, shown as red points, represent significant correlations for the corresponding lags and indicate the potential for short-term climate prediction. The results show that significant correlations are generally absent across QBIE heat wave and cold surge events in six different QBIE categories. In particular, significant autocorrelation coefficients are only observed in CDN heat waves (11-year period, Figure 7e), SNT cold surges (4-year period, Figure 8k), and SDT cold surges (4- and 6-year period, Figure 8q). Meanwhile, significant partial autocorrelation coefficients are only observed in CDN heat waves (11- and 16-year period, Figure 7f), WNT heat waves (2-year period, Figure 7i), SNT cold surges (4-year period, Figure 8l), and SDT cold surges (10-year period, Figure 8r). However, these significant autocorrelations are likely results from the randomness of the time series rather than meaningful patterns, as indicated by their low  
Figure 8: Same as Figure 7, but for QBIE cold surge events. The time series (**a**,**d**,**g**,**j**,**m**,**p**), autocorrelation plots (**b**,**e**,**h**,**k**,**n**,**q**), and partial autocorrelation plots (**c**,**f**,**i**,**l**,**o**,**r**) for annual event counts of six types of QBIE cold surge from 1950 to 2022.  
correlation coefficients. In conclusion, due to the lack of substantial correlations in QBIE time series, it is challenging to model and predict QBIE events using the ARIMA model.