1 Introduction

An investigation of precipitation changes is one of the key steps in understanding hydrological response to climate change at the local, regional, and global scales, and is of critical importance for the management of water resources and agricultural development. This is especially important in China (Xia et al. 1997; Zhang et al. 2012b)—the largest agricultural country with the largest population in the world. Precipitation forecasts play a crucial role in the planning of water resource use, agricultural management (Wei et al. 2005; Kisi and Cimen 2012; Ortiz-García et al. 2014), and flood forecasting (Francesco and Rebora 2015).

Multitudes of studies have addressed precipitation forecasting using a variety of techniques. Ortiz-García et al. (2014) predicted daily precipitation using the support vector machine (SVM) and pointed out its excellent performance in comparison with other neural computation-based approaches, such as multilayer perceptron, extreme learning machine, and classical algorithms such as decision trees and K-nearest neighbor classifier. Manzato (2007) applied a neural computation approach to the short-term forecasting of thunderstorm rainfall. Shukla et al. (2011) indicated the importance of El Niño / Southern Oscillation (ENSO) indices in precipitation forecasting. With the help of a neural network, Shukla et al. (2011) forecasted precipitation in India due to the summer monsoon by considering ENSO indices. Nastos et al. (2013) applied an artificial neural network to forecast precipitation in Athens, Greece. Ingsrisawang et al. (2008) compared several machine learning algorithms, such as decision trees (DT), neural networks (ANN), and SVMs for short-term precipitation prediction in Thailand. Lu and Wang (2011) predicted monthly rainfall in China, using an SVM approach with different kernel functions. These forecasts are deterministic (point forecast representing our best guess) or probabilistic (providing the forecast in terms of probability of exceedance that reflects a range of possible values for the variable of interest), and are statistical in nature or based on numerical models.

In the above studies, deterministic forecast has been widely used. However, statistical models have also been widely used in precipitation forecasting (Villarini and Serinaldi 2012; Wang et al. 2013). Advantages of statistical models include the ability to identify changing patterns, the estimation of parameters, uncertainty analysis, and diagnostic checking (Mishra and Desai 2005; Wang et al. 2013). Some statistical models, such as the autoregressive moving average (ARMA) models, can predict the output from the values of the output at previous time points. These models are only suitable for a stationary system (Wang et al. 2013), although it has been stated that seasonal autoregressive integrated moving average (SARIMA) models can model time series that exhibit nonstationary behavior (Box et al. 2015).

The generalized additive model for location, scale, and shape (GAMLSS) developed by Rigby and Stasinopoulos (2005) has been used in this study. It can be devised to cope with nonexponential distribution families and is able to incorporate covariates not only in the position parameter but also in the scale and shape parameters. Thus, GAMLSS allows for a better control and representation of dispersion, skewness, and kurtosis (Rigby and Stasinopoulos 2005; Jones et al. 2013) and provides a suitable framework to model the discrete–continuous distribution of daily precipitation accounting for seasonality and exogenous forcing variables, such as alternative climate states (Francesco and Kilsby 2014; Francesco and Rebora 2015). In this study, the GMALSS model was used with exogenous forcing variables such as the ENSO to simulate and forecast precipitation in the East River basin, China.

The East River basin is a major tributary of the Pearl River basin in South China with a drainage area of 35,340 km2, which accounts for about 5.96% of the Pearl River basin area (Fig. 1). Water resources in the East River basin have been highly developed for a variety of uses, such as municipal and rural water supply, hydropower generation, navigation, irrigation, and suppression of seawater invasion (Chen et al. 2010). In recent years the East River has been supplying water to meet about 80% of Hong Kong’s annual hydrological demand (Chen et al. 2010; Wong et al. 2010). Therefore, the vulnerability of the East River water resource system is of great importance for the sustainable social and economic development of the Pearl River Delta, one of the economically most developed regions in China, and also for the sustainable water supply of Hong Kong (Chen et al. 2013). In this context, precipitation changes play a critical role. Investigations of precipitation structure indicate higher frequencies of heavy precipitation (defined as events with ≥ 50 mm/day precipitation) in the eastern parts of the Pearl River basin and the East River basin in recent decades (Zhang et al. 2012a). Therefore, forecasting of precipitation variations in the East River basin is important for the planning and management of water resources. The influence of large-scale climate oscillations, for example ENSO, on precipitation changes in the East River basin has been established and corroborated (Zhang et al. 2013; Huang et al. 2017). Thus, large-scale climate oscillations are used as exogenous factors in precipitation forecasts.

Fig. 1
figure 1

Location of the study region—the East River basin—and the 29 precipitation stations used in the study

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The objectives of this study are to: (1) screen the models that can forecast the precipitation behavior at a seasonal temporal scale; and (2) forecast precipitation changes while taking account of nonstationarity and uncertainty, including those stemming from climate oscillations. Results of this study will provide a framework for precipitation forecasting in humid regions and guidance for planning and management of water resources and agricultural activities at the regional scale.

2 Data

Monthly precipitation data, covering the period of 1959–2010 and derived from 29 meteorological stations in the East River basin, were analyzed in this study. The locations of these stations are shown in Fig. 1. The dataset was compiled by the Climatic Data Center of the National Meteorological Information Center at the China Meteorological Administration.Footnote 1 The quality of the data was controlled before its release (Zhang et al. 2013). Seasonal average precipitation changes were also analyzed, based on four seasons: winter (December to February of the subsequent year), spring (March to May), summer (June to August), and autumn (September to November).

The climatic indices used in this study for seasonal average precipitation forecasts were the Arctic Oscillation (AO) index, the North Pacific Oscillation (NPO) index, the Pacific Decadal Oscillation (PDO) index, and the Southern Oscillation Index (SOI). Many researchers have found that precipitation changes in the East River basin are significantly influenced by these four climate oscillations (Feng and Li 2011; Zhang et al. 2013, 2014; Ding et al. 2014). Data on the climatic indices were obtained from an Internet source of the NOAA Earth System Research Laboratory.Footnote 2

3 Methodology

Due to the influence of human activities and climate change itself, precipitation changes are considered nonstationary (Li et al. 2016; Chang et al. 2017). Therefore, a nonstationary model should be used with parameters changing along with the explanatory variables. In this study, the GAMLSS model was selected for at-site precipitation forecasting.

3.1 The Generalized Additive Model for Location, Scale, and Shape (GAMLSS)

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:

$$g_{k} (\theta_{k} ) = \eta_{k} = X_{k} \beta_{k} + \sum\limits_{j = 1}^{{J_{k} }} {Z_{jk} \gamma_{jk} }$$
(1)

where η k and θ k are the vectors with the length of n, β T k  = {β1k, β2k, …, 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).

Table 1 The five distribution functions considered in this study
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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).

3.2 At-site Probabilistic Forecast

At-site (rain station) probabilistic forecasting was done using the GAMLSS model, taking time, t, and the climatic indices AO, NPO, PDO, and SOI as covariates. Two forecasting modes were developed: (1) Taking the seasonal precipitation series of each individual season—spring, summer, autumn, and winter—as a sole series, this kind of forecasting was denoted as the seasonal model; and (2) Filtering out each seasonal precipitation component such as spring, summer, autumn, and winter with a seasonal factor, for example, 1, 2, 3, and 4 standing for spring, summer, autumn, and winter, respectively, from each year and forms the annual series of spring, summer, autumn, and winter precipitation series. The forecasts were done on the reorganized annual variations of seasonal precipitation components. This kind of forecasting was denoted as the global model. The seasonal and global models were then compared to determine which model produces better forecasts.

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.

3.3 Evaluation of the At-site Probabilistic Forecast

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.

Table 2 Eight estimation methods for evaluation of modeling accuracy
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4 Results

Based on the aforementioned analysis procedure, we obtained the following interesting and important results and findings.

4.1 Nonstationarity of Seasonal Average Precipitation Changes and Teleconnections with ENSO, AO, PDO, and NPO Events

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.

Fig. 2
figure 2

Spatial distribution of the best models in the East River basin study area in southeastern China based on seasonal and global models with the highest goodness-of-fit

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Fig. 3
figure 3

The climatic indices that are closely related to the location parameter in the seasonal models in the East River basin study area in southeastern China. The numbers in the parentheses denote the number of stations with nonlinear relations between seasonal precipitation and climatic indices; ct means that the location parameter (μ) is constant

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Fig. 4
figure 4

The climatic indices that are closely related to the scale parameter in the seasonal models in the East River basin study area in southeastern China. The numbers in the parentheses denote the number of stations with nonlinear relations between seasonal precipitation and climatic indices; ct means that the scale parameter (δ) is constant

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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.

Fig. 5
figure 5

The climatic indices that are closely related to the location (μ) and scale (δ) parameters in the global model. The numbers in the brackets denote the number of stations with nonlinear relations between integrated seasonal precipitation and climatic indices; ct means that the parameter is constant

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4.2 Seasonal Precipitation Modeling for Spring, Summer, Autumn, and Winter

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.

Fig. 6
figure 6

Modeled spring precipitation by the seasonal model (left) and the global model (right) at four selected stations (Xunwu, Dongyuan, Lingxia, and Shenzhen) in the East River basin in southeastern China. The black dashed line denotes the 50% percentile, the pink region denotes the interval defined by 25 and 75% percentiles, and the greenish-blue region denotes the interval defined by the 5 and 95% percentiles. The grey points denote the observed precipitation data

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Fig. 7
figure 7

The worm plot for the simulation of spring precipitation by the seasonal model (left) and the global model (right) at four selected stations (Xunwu (a-1, a-2), Dongyuan (b-1, b-2), Lingxia (c-1, c-2), and Shenzhen (d-1, d-2)) in the East River basin in southeastern China. The two black dashed lines denote the 95% confidence intervals

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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.

Fig. 8
figure 8

Modeled summer precipitation by the seasonal model (left) and the global model (right) at four selected stations (Xunwu, Dongyuan, Lingxia, and Shenzhen) in the East River basin in southeastern China. The black dashed line denotes the 50% percentile, the pink region stands for the interval defined by the 25 and 75% percentiles, and the greenish-blue region displays the interval defined by the 5 and 95% percentiles. The grey points denote the observed precipitation data

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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.

Fig. 9
figure 9

Modeled autumn precipitation by the seasonal model (left) and the global model (right) at four selected stations (Xunwu, Dongyuan, Lingxia, and Shenzhen) in the East River basin in southeastern China. The black dashed line denotes the 50% percentile, the pink region shows the interval defined by the 25 and 75% percentiles, and the greenish-blue region indicates the interval defined by the 5 and 95% percentiles. The grey points denote the observed precipitation data

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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.

Fig. 10
figure 10

Modeled winter precipitation by the seasonal model (left) and the global model (right) at four selected stations (Xunwu, Dongyuan, Lingxia, and Shenzhen) in the East River basin in southeastern China. The black dashed line denotes 50% percentile, the pink region represents the interval defined by the 25 and 75% percentiles, and the cyanic region occupies the interval defined by the 5 and 95% percentiles. The grey points denote the observed precipitation data

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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.

4.3 Integrated Precipitation Forecasting by Seasonal and Global Models

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.

Fig. 11
figure 11

The eight modeling accuracy indices for the track predictions of seasonal precipitation (from left to right: spring, summer, autumn, and winter) during 2006–2010, using the seasonal, global, and naïve models in the track precipitation forecasting practice for all 29 stations in the East River basin of southeastern China

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Fig. 12
figure 12

The eight modeling accuracy indices for the track predictions of seasonal precipitation (from left to right: spring, summer, autumn and winter) during 2001–2010, using the seasonal, global, and naïve models in the track precipitation forecasting practice using for all 29 stations in the East River basin of southeastern China

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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.

Fig. 13
figure 13

The eight modeling accuracy indices for the seasonal precipitation (from left to right: spring, summer, autumn, and winter) during 2006–2010, using the time, seasonal, and global models in the cross-validation precipitation forecasting practice for all 29 stations in the East River basin of southeastern China

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Fig. 14
figure 14

The eight modeling accuracy indices for the seasonal precipitation (from left to right: spring, summer, autumn, and winter) during 2001–2010, using the time, seasonal, and global models in the cross-validation precipitation forecasting practice using all 29 stations in the East River basin of southeastern China

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5 Conclusions

Precipitation forecasting plays a critical role in the management of water resources and agricultural activities, and that is particularly true for the East River basin in China where water resources have been highly developed and exploited. Hong Kong, for example, meets approximately four-fifths of its current annual water demand by withdrawals from the East River basin. In this study, modeling of precipitation changes is done using GAMLSS-based nonstationary techniques with consideration of the influence of large-scale climate oscillations on precipitation changes in the East River basin. Important conclusions of the study are as follows:

  1. (1)

    Spring precipitation follows the gamma and logistic distributions and exhibits nonstationary changes. The changing magnitude of spring precipitation is influenced mainly by AO and PDO; and the changing dispersion is impacted by AO. The summer and autumn precipitation changes follow the gamma distribution. Summer and autumn precipitation at most of the stations exhibits stationary changes. The changing magnitude of summer and autumn precipitation is effected by NPO and ENSO, respectively. In contrast, the precipitation variability in summer and autumn is influenced mainly by AO. Winter precipitation follows the Weibull distribution and shows nonstationary changes. The causes behind winter precipitation changes are complicated when compared to those in spring, summer, and autumn. More or less, NPO, ENSO, AO, and PDO have evident influences on the changing magnitude of winter precipitation. But the influence of ENSO on the changing magnitude of winter precipitation is significant when compared to NPO, AO, and PDO. Integrating seasonal precipitation series into one whole series may tell a different story. The whole seasonal precipitation series, that is, the precipitation series after integration of the precipitation series of spring, summer, autumn, and winter, follows the Weibull distribution. The location parameter, that is, the changing magnitude, is influenced mainly by PDO and ENSO. However, the precipitation variability is influenced mainly by ENSO alone.

  2. (2)

    Both seasonal and global models depict temporal changes in seasonal precipitation in the East River basin. The seasonal model is the best choice for modeling spring precipitation, and the global model is the best choice for modeling seasonal precipitation changes in summer, autumn, and winter.

  3. (3)

    In both the track precipitation forecasting and cross-validation precipitation forecasting practices, the seasonal model better forecasts precipitation changes in spring, autumn, and winter; however, the global model better forecasts summer precipitation changes. In this sense, the global model performs better than the seasonal model in the simulation of seasonal precipitation changes, however, that is not the case in precipitation forecasting. It is not a good idea to use only one model, either the global model or the seasonal model, in precipitation forecasting. In the track precipitation forecasting and cross-validation precipitation forecasting practices, the seasonal model should be the choice in the forecasting of seasonal precipitation changes. Results of this study help clarify the selection of models in modeling, simulation, and forecasting of seasonal precipitation variations, particularly considering the influence of ENSO regimes. Because spatial dependence cannot be neglected, a regional method that considers the influence of spatial dependence may be of help in improving the performance of the seasonal precipitation forecast, and this will be addressed with considerable consideration in our subsequent research.