Nonstationarities and At-site Probabilistic Forecasts of Seasonal Precipitation in the East River Basin, China

The GAMLSS model (Rigby and Stasinopoulos 2005) assumes a parametric distribution for the response variable Y, the seasonal precipitation amount in this study, and models the distribution parameters as functions of an explanatory variable, time t _i and the climatic indices—AO _i , NPO _i , PDO _i , and SOI _i . It is assumed that in GAMLSS, random variables y _i , for i = 1,…, n, are mutually independent and are from the same distribution function, F _Y (y _i |θ _i ) with a vector of p distribution parameters, θ _i  = (θ₁, θ₂,…, θ _p ), accounting for location, scale, and shape parameters. Generally, p ≤ 4 in that a 4-parameter distribution family can describe distribution characteristics of different hydrometeorological variables. Assume that g _k (·) shows monotonic relations between θ _k , the explanatory variable X _k , and the random term, that is:where η _k and θ _k are the vectors with the length of n, β _k ^T  = {β_1k, β_2k, …, β _Jk } is the parameter vector with the length of Jk, X _k is the explanatory variable matrix with the length of n × Jk, Z _jk is the fixed design matrix with the length of n × q _jk , and γ _jk is the random variable following the normal distribution. In this study, a cubic spline function was used as the link function between parameters and covariate variables. Five two-parameter functions were selected and are displayed in Table 1: Gumbel (GU), Weibull (WEI), Gamma (GA), Lognormal (LOGNO), and Logistic (LO).

The Akaike information criterion (AIC) technique was used to select the distribution function with the highest goodness-of-fit. To ensure the quality of the fit, a worm plot was applied to estimate the fitting performance of the distribution functions. More details can be found in Rigby and Stasinopoulos (2005).

Two precipitation forecasting practices were used: the track precipitation forecast and the cross-validation precipitation forecast. In track precipitation forecasting, parameter estimation is done using the seasonal precipitation data covering the period of 1959–2000 and the estimated parameter(s) is(are) used to forecast the precipitation in 2001. The forecasted seasonal precipitation in 2001 is added to the seasonal precipitation series of the period of 1959–2000 and parameters are estimated again, based on the newly produced seasonal precipitation series. Then, the seasonal precipitation is forecasted for 2002, using the newly estimated parameters based on the newly produced seasonal precipitation series of the period of 1959–2001. This procedure is repeated until the forecasted seasonal precipitation for 2010 is attained. In cross-validation precipitation forecasting, parameter estimation is done using the seasonal precipitation data covering the calibration period, such as 1959–2000, and then the estimated parameter(s) is(are) used to forecast the seasonal precipitation for the verification period, such as 2001–2010.

In track precipitation forecasting, the naïve model (simple linear regression) is introduced and a comparison of the forecast performances of the seasonal and global models and the naïve model (Mason and Baddour 2008; Gabriele and Francesco 2012) is done. In cross-validation precipitation forecasting, time, t, is taken as the covariate variable for seasonal precipitation forecasting, called the time model. Comparison is done for evaluating the forecasting performances of seasonal and global models as well as the time model.

Eight evaluation criteria were applied to evaluate the forecasting accuracy of seasonal precipitation. Since each method has its own advantages and limitations, it is assumed that these methods can combine to evaluate the forecasting accuracy of the models considered, thus avoiding the uncertainty and unreliability due to limitations of the individual method. These methods (Table 2) are: absolute mean error (AME), mean absolute error (MAE), median absolute error (MdAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), median absolute percentage error (MdAPE), and root mean squared percentage error (RMSPE). It should be noted that AME, MAE, MdAE, and RMSE are absolute error indices and MAPE, MdAPE, and RMSPE are comparative error indices. The geometric reliability index (GRI) was also used to evaluate the performance of models from a geometric perspective.

The AIC method was used to screen the models that satisfactorily simulate seasonal and integrated seasonal precipitation and the related distribution functions (Fig. 2). It can be seen from Fig. 2a that the distribution functions with the highest goodness-of-fit for spring precipitation series (seasonal model) are Gamma and Logistic models for 19 out of the 29 stations. The location parameter, μ, was influenced mainly by PDO and AO (Fig. 3a) and had a linear relation with AO but a nonlinear relation with PDO (Fig. 3a). The scale parameter, δ, had a close relation with AO, PDO, and SOI (Fig. 4a). A significant influence of AO on scale parameter, δ, was observed for 16 out of the 29 stations (Fig. 4a). AO had a linear relation with the scale parameter and a nonlinear relation was detected between δ, and PDO and SOI. In general, spring precipitation in the East River basin was influenced mainly by AO and PDO. Meanwhile, PDO mainly impacted the location parameter, μ, implying that PDO altered the changing magnitude of spring precipitation in the East River basin. However, AO and ENSO mainly influenced the scale parameter, δ, that is the statistical dispersion properties of spring precipitation changes (Figs. 3a and 4a). Besides, Figs. 3a and 4a show that μ and δ for 4 and 3 out of the 29 stations were constant, indicating that spring precipitation changes at these stations were stationary. In this sense, spring precipitation changes at most of the stations were nonstationary.

Figure 2b indicates that Gamma distribution is the best model for summer precipitation changes (seasonal model) for 18 out of the 29 stations. In comparison, different changing properties of summer precipitation were detected when compared to spring precipitation changes. The influence of AO on scale parameter, δ, was linear for 18 out of 22 stations (Fig. 4b). However, the location parameter, μ, was largely free from the influence of climate oscillations (Fig. 3b), implying that the mean value of summer precipitation had stationary changes to a certain degree.

Figure 2c indicates that the Gamma distribution function was also the best model for autumn precipitation changes (seasonal model) for 21 out of the 29 stations. A closer look at Figs. 2c, b, 3c, and 4c indicates similar changes in the properties of autumn precipitation changes when compared to summer precipitation. However, complicated changing characteristics were detected for winter precipitation changes.

Figure 2d shows that the Weibull distribution function was the best choice for modeling winter precipitation changes (seasonal model) for 10 out of the 29 stations in the East River basin. When compared to climatic indices that are related to seasonal precipitation changes in summer and autumn, more climate oscillations had considerable impacts on winter precipitation changes, especially location and scale parameters. Specifically, AO, NPO, PDO, and ENSO influenced the location parameter, μ, of winter precipitation changes with different degrees and magnitudes in both space and time (Fig. 3d). Statistically, the impact of ENSO on the location parameter, μ, was nonlinear for 8 out of the 29 stations but the influence of AO and NPO on μ was respectively mostly linear and linear. The impact of AO and PDO on the scale parameter was linear for 13 and 17 out of the 29 stations, respectively. ENSO had little impact on the scale parameter (Fig. 4d). Based on the spatiotemporal properties of μ and δ, winter precipitation changes were nonstationary. Therefore, a nonstationary model is necessary to describe the properties of seasonal precipitation changes in the East River basin.

In general, stationarity properties of seasonal precipitation changes were different from one season to another and climatic indices that were evidently related to seasonal precipitation changes also differ. Specifically, summer and autumn precipitation changes shared similar changing properties, being influenced mainly by AO. Summer and autumn precipitation changes were roughly stationary; spring and winter precipitation changes were nonstationary. Spring precipitation changes were influenced mainly by PDO. Winter precipitation changes were influenced by more climate oscillations. The impacts of ENSO and PDO on winter precipitation changes were pronounced. ENSO began to influence precipitation changes during autumn, and the influence of ENSO became significant during the winter season.

In addition to the seasonal analysis, the spring, summer, autumn, and winter season precipitation series were integrated into one series and seasonal precipitation changes were analyzed, based on the identification of seasonal components, that is, the global model method (Fig. 5). It can be seen from Fig. 2e that the Weibull distribution function satisfactorily described the integrated seasonal precipitation changes for 20 out of the 29 stations. PDO and ENSO influenced the location parameter, implying that these two climate oscillations impacted the changes in the precipitation amount. The scale parameter, however, was impacted mainly by ENSO.

For concise presentation, Xunwu, Dongyuan, Lingxia, and Shenzhen stations were selected to illustrate the results of the seasonal and global models because these four stations are located along the main stream of the East River. Figure 6 shows the results of seasonal and global models for spring precipitation changes. Due to the length limit of the article, the results of the remaining 25 stations that were also considered in this study are not presented here. The seasonal model better captured the spring precipitation changes than the global model, mirrored by the narrower confidence interval circled by the 25 and 75% percentile curves. The seasonal model captured higher and lower amounts in the precipitation variations. This is particularly important in the analysis of precipitation extremes. This analysis was also corroborated by the worm plots in Fig. 7.

The results of the seasonal and global models for summer precipitation are shown in Fig. 8. The fluctuations of summer precipitation were smaller than the spring precipitation changes. Generally, the global model had better modeling efficacy than the seasonal model (Fig. 8). Stationary changes are shown by the seasonal model at the Dongyuan and Shenzhen stations (Fig. 8b-1, d-1). However, the global model, as shown by the 50% percentile curves, mirrored the average changes of summer precipitation amounts. The confidence interval of the global model defined by the 25 and 75% percentiles was narrower than for the seasonal model. The 5 and 95% percentiles of the global model better captured the changing features of summer precipitation than did the seasonal model, particularly the peak and trough values of summer precipitation variations (for example, Fig. 8d-2). The worm plots also verified these results.

Figure 9 illustrates the results of the seasonal and global models for autumn precipitation. Stationary changes of autumn precipitation were revealed by the seasonal model with no evident temporal changes of percentiles. The global model satisfactorily depicted the changes in the mean autumn precipitation, as reflected by the 50% percentile curves (Fig. 9). The confidence intervals, defined by the 25 and 75% percentiles and also by the 5 and 95% percentiles, well represented the changes in autumn precipitation, especially precipitation magnitudes. The global model better depicted the peak and trough values of the autumn precipitation than did the seasonal model, as mirrored by the 5 and 95% percentiles. The seasonal model showed stationary changes in autumn precipitation, similar to summer precipitation for some of the stations, and as a result, the seasonal model did capture details of the changing properties of autumn precipitation.

The results of the seasonal and global models for winter precipitation changes are shown in Fig. 10. Winter precipitation had larger fluctuations than had spring, summer, and autumn precipitation. A closer look at Fig. 10 indicates the global model had higher modeling efficacy than the seasonal model. The temporal variations of winter precipitation were simulated well as shown by the 50% percentile. The global model had narrower confidence intervals than the seasonal model as defined by the 25 and 75% percentiles for simulation of the winter precipitation changes. The global model also mirrored the nonstationary changes in winter precipitation.

Figures 6, 7, 8, 9 and 10 show that both the seasonal and global models can model temporal changes in seasonal precipitation at 4 out of the 29 stations. Similar results can be obtained for other stations considered in this study. A similar efficacy in the simulation of the spring precipitation changes was observed for the seasonal and global models. However, the results of these two models are subject to large discrepancies in the modeling of seasonal precipitation changes in summer, autumn, and winter. Stationary changes of summer, autumn, and winter precipitation were detected by the seasonal model, though the precipitation changes were nonstationary in the global model output. In summary, the seasonal model was the best choice for the modeling of spring precipitation, and the global model was the best choice for the modeling of seasonal precipitation changes in summer, autumn, and winter. Thus, the global model should be used for the modeling, simulation, and forecasting of seasonal precipitation changes in the East River basin, China. These results do not indicate, however, which model is the best choice for forecasting regional precipitation changes.

The seasonal precipitation series in spring, summer, autumn, and winter, respectively, were integrated into an entire seasonal precipitation series, and then the seasonal and global models were developed, based on the integrated seasonal precipitation series, to model and forecast precipitation changes using the track precipitation forecasting and cross-validation precipitation forecasting techniques. In this study, the forecasting accuracy was evaluated by eight methods (Table 2). In the track precipitation forecasting practice, the naïve model was used to evaluate the forecasting performance of the seasonal and global models; in the cross-validation precipitation forecasting practice, the time model was used to compare and evaluate the forecasting performance of the seasonal and global models.

To evaluate the overall performances of these models in a region, Fig. 11 displays the performances of the seasonal, global, and naïve models in the track precipitation forecasting practice for the forecasting of precipitation changes during the period of 2006–2010. The seasonal model better forecasted precipitation changes in spring, autumn, and winter than did the global model, but did worse than the global model in forecasting precipitation changes in summer. Based on AME and MAE, the seasonal and global models had the smallest error in their modeling performance. However, MAPE indicated differently, that is, the seasonal and global models better forecasted precipitation changes in spring and summer than in autumn and winter. However, in general, the seasonal and global models forecasted better than the naïve model (Fig. 11). Figure 12 shows the performance of the seasonal, global, and naïve models in the track precipitation forecasting practice for the forecasting of precipitation changes during the period of 2001–2010. These results indicate that the longer the forecasting time interval, the lower the forecasting accuracy.

Figure 13 shows the performance of the seasonal, global, and time models in the cross-validation precipitation forecasting practice for the forecasting of precipitation changes during the period of 2006–2010. In general, the seasonal model better forecasted precipitation in spring, autumn, and winter than the global model; but the global model did better in forecasting precipitation in summer. Based on AME, MAE, MdAE, and RMSE, the forecasting accuracy of the global and seasonal models for winter precipitation was the lowest. MAPE seemed to show the highest efficacy of the global and seasonal models for the forecasting of precipitation in spring and summer. Higher forecasting efficacy was detected for the seasonal model than the time model, but no distinct differences in the forecasting performance were observed between the global and time models. Other evaluation indices, except MAPE and RMSPE, indicated the larger forecasting accuracy of the seasonal model than the time model. Figure 14 shows the forecasting performance of the seasonal, global, and time models in the cross-validation precipitation forecasting practice for the forecasting of precipitation changes during the period of 2001–2010. Similar conclusions were obtained from Fig. 14. However, when compared to Figs. 13, and 14 indicates that the longer the forecasting time, the smaller the advantage of the time model over the seasonal and global models in precipitation forecasting practice—for example, MAPE and RMSPE—indicate that the seasonal model was better than the time model in terms of forecasting accuracy.