A DeepAR-Based Modeling Framework for Probabilistic Mid–Long-Term Streamflow Prediction

Abstract

Mid–long-term streamflow prediction (MLSP) plays a critical role in water resource planning amid growing hydroclimatic and anthropogenic uncertainties. Although AI-based models have demonstrated strong performance in MLSP, their capacity to quantify predictive uncertainty remains limited. To address this challenge, a DeepAR-based probabilistic modeling framework is developed, enabling direct estimation of streamflow distribution parameters and flexible selection of output distributions. The framework is applied to two case studies with distinct hydrological characteristics, where combinations of recurrent model structures (GRU and LSTM) and output distributions (Normal, Student’s t, and Gamma) are systematically evaluated. The results indicate that the choice of output distribution is the most critical factor for predictive performance. The Gamma distribution consistently outperformed those using Normal and Student’s t distributions, due to its ability to better capture the skewed, non-negative nature of streamflow data. Notably, the magnitude of performance gain from using the Gamma distribution is itself region-dependent, proving more significant in the basin with higher streamflow skewness. For instance, in the more skewed Upper Wudongde Reservoir area, the model using LSTM structure and Gamma distribution reduces RMSE by over 27% compared to its Normal-distribution counterpart (from 1407.77 m³/s to 1016.54 m³/s). Furthermore, the Gamma-based models yield superior probabilistic forecasts, achieving not only lower CRPS values but also a more effective balance between high reliability (PICP) and forecast sharpness (MPIW). In contrast, the relative performance between GRU and LSTM architectures was found to be less significant and inconsistent across the different basins. These findings highlight that the DeepAR-based framework delivers consistent enhancement in forecasting accuracy by prioritizing the selection of a physically plausible output distribution, thereby providing stronger and more reliable support for practical applications.

1. Introduction

The combined impacts of climate change and human activities are profoundly altering the processes of runoff generation and confluence, resulting in increasing uncertainty in the evolution of water resources [1,2,3,4]. At the same time, socio-economic drivers such as population growth and urbanization are intensifying the demand for water supply [5,6,7,8]. Against this backdrop, mid–long-term streamflow prediction (MLSP), which refers to forecasts at ten-day, monthly, seasonal, and annual scales with lead times ranging from three days to one year, has become increasingly vital for effective water resource management and integrated utilization, as it provides valuable insights into future runoff patterns [9,10,11,12,13,14]. Consequently, MLSP is attracting growing attention in both research and practical applications [11,12,13,14].

Many models have been developed and applied in MLSP to improve the predictive performance and provide information for the comprehensive utilization of water resources [11,12,13,14,15,16,17,18]. These models can be broadly divided into physical-based models, which simulate the streamflow based on the runoff generation and confluence equations, and data-driven models, which directly simulate the relationship between streamflow and predictors including precipitation, temperature, and other factors [13,14,19,20,21,22]. Along with the development of artificial intelligence (AI) methods, the AI-based data-driven models, including support vector regression (SVR), artificial neural network (ANN), gated recurrent unit neural network (GRU), long short-term memory network (LSTM) and so on, can obtain better predictive performance than traditional models and have become predominant in the MLSP [10,20,23,24,25,26,27,28]. For instance, many studies applied SVR models in MLSP, and the results demonstrate that the SVR models can generate more accurate forecasts than linear models and ANN models [29,30,31,32,33]. Xie et al. (2024) compared five AI-based models, and the results demonstrate that the LSTM model outperformed other models in forecasting monthly streamflow in 37 basins [14]. In addition, some hybrid models which incorporate preprocessing methods into the AI methods are developed to integrate the advantages of different base models and have obtained better forecasting skill [19,34,35,36,37].

The proposed AI-based models in MLSP demonstrate significant improvements in predictive performance, but their limited capacity to characterize future water resource uncertainties constrains practical applications [38,39,40,41]. To overcome this issue, many post-processing methods are adopted to produce ensemble forecasting results capable of characterizing predictive uncertainties [40,41,42,43]. For example, Liang et al. (2018) proposes the hydrological uncertainty processor to post-process the deterministic outputs from the SVR model to quantify prediction uncertainties [39]. Mo et al. (2023) applies the generalized autoregressive conditional heteroskedasticity model to identify time-varying forecasting errors to improve the predictive performance [44].

Although the post-processing approach in MLSP demonstrates predictive capability, it exhibits two fundamental limitations: (1) inability to directly generate probability distributions and (2) failure to preserve the inherent statistical characteristics of streamflow. To overcome these constraints in MLSP, this study adopts a DeepAR-based modeling framework. The DeepAR architecture exhibits two key capabilities: (1) direct prediction of the predictand’s distribution parameters and (2) flexible selection of probability distributions for the target variable [45,46]. It has been applied in many studies, and empirical validation across five time series datasets demonstrates its superior performance over other existing state-of-the-art methods [45,47,48,49]. However, the DeepAR’s robustness under heavy-tailed streamflow distributions requires further verification, and the selection criteria for appropriate probability distributions lack systematic guidance. Therefore, the objectives of this study are (1) to develop a DeepAR-based probabilistic forecasting framework for MLSP, (2) to validate modeling framework’s applicability through implementation in two case studies, and (3) to systematically evaluate impacts of base model architecture and distribution type selection on predictive skill.

2. Materials and Methods

2.1. DeepAR Model

DeepAR is a probabilistic forecasting framework developed by Amazon Research, which combines recurrent neural networks (RNNs) with parametric probability distributions to generate time-series predictions [45]. Unlike traditional point-forecasting models, DeepAR directly outputs the parameters of user-specified distributions (e.g., Gaussian for real-valued data, Negative Binomial for positive count data, and Beta for data in the unit interval), enabling native uncertainty quantification.

Let $z_{\mathrm{t}}$ denote the streamflow value at time t, and ${\mathit{x}}_{\mathit{t}}$ represent the vector of predictor variables at time t. The DeepAR model estimates the conditional distribution of future streamflow values:

$$P\left({\mathbf{z}}_{{\mathit{t}}_0+1:{\mathit{t}}_0+\mathit{L}}\right|{\mathbf{z}}_{{\mathit{t}}_0-\mathit{H}+1:{\mathit{t}}_0},{\mathit{x}}_{{\mathit{t}}_0-\mathit{H}+1:{\mathit{t}}_0+\mathit{L}})$$

(1)

where $t_0$ is the forecast initialization time, $H$ is the length of the target time series at prediction time, and $L$ is the prediction length.

For a trained DeepAR model, the distribution parameters ${\mathit{\theta }}_{\mathit{t}}$ at time t are computed as a function of the hidden state ${\mathit{h}}_{\mathit{t}}$ and model parameters $\mathit{\Theta }$:

$${\mathit{\theta }}_{\mathit{t}}=\mathrm{f}({\mathit{h}}_{\mathit{t}},\mathit{\Theta })$$

(2)

where $\mathrm{f}(·)$ is a function used to map the hidden state to distribution parameters and the hidden state ${\mathit{h}}_{\mathit{t}}$ evolves recursively via

$${\mathit{h}}_{\mathit{t}}=\mathrm{g}({\mathit{h}}_{\mathit{t}-1},z_{\mathrm{t}-1},{\mathit{x}}_{\mathit{t}},\mathit{\Theta })$$

(3)

where $\mathrm{g}(·)$ is a nonlinear transition function implemented as a multi-layer RNN (LSTM or GRU) parametrized by $\mathit{\Theta }$, and ${\mathit{h}}_{\mathit{t}-1}$ is the hidden state from the previous time step.

Then, the simulated or forecasted streamflow value at time t can be sampled by

$$\widetilde{z_t}~p(·|\mathrm{f}({\mathit{h}}_{\mathit{t}},\mathit{\Theta }))$$

(4)

2.1.1. Training

Given a streamflow time series ${\left\{{\mathbf{z}}_{\mathbf{t}}\right\}}_{\mathrm{t}=\mathrm{1,2},\dots ,\mathrm{T}}$ and associated predictor variables ${\mathit{x}}_{\mathit{t}}$, the DeepAR model parameters $\mathit{\Theta }$, including the parameters of both $\mathrm{f}\left(·\right)$ and $\mathrm{g}(·)$, can be learned by maximizing the log-likelihood, as shown below:

$$\mathcal{L}=\sum _{t_0}\sum _{t=t_0+1}^{t_0+L}\mathit{log}p\left(z_t\right|f({\mathit{h}}_{\mathit{t}},\mathit{\Theta }))$$

(5)

2.1.2. Prediction

Given an observed streamflow sequence ${\mathbf{z}}_{{\mathit{t}}_0-\mathit{H}+1:{\mathit{t}}_0}$ and corresponding predictor variables ${\mathit{x}}_{{\mathit{t}}_0-\mathit{H}+1:{\mathit{t}}_0+\mathit{L}}$, the trained DeepAR model generates probabilistic forecasts for future streamflow values ${\mathit{z}}_{{\mathit{t}}_0+1:{\mathit{t}}_0+\mathit{L}}$ through the following procedure:

(1)

The hidden state ${\mathit{h}}_{{\mathit{t}}_0}$ is obtained by recursively processing the historical streamflow ${\mathit{z}}_{{\mathit{t}}_0-\mathit{H}+1:{\mathit{t}}_0}$ and predictors ${\mathit{x}}_{{\mathit{t}}_0-\mathit{H}+1:{\mathit{t}}_0}$ through the RNN transition function in Equation (3);

(2)

Initial conditions are set as $\stackrel{~}{{\mathit{h}}_{{\mathit{t}}_0}}={\mathit{h}}_{{\mathit{t}}_0}$ and $\stackrel{~}{z_{{\mathrm{t}}_0}}=z_{{\mathrm{t}}_0}$;

(3)

For each subsequent time step $t=t_0+1$ to $t_0+L$, the hidden state $\widetilde{{\mathit{h}}_{\mathit{t}}}$ is updated using $\mathrm{g}(\widetilde{{\mathit{h}}_{\mathit{t}-1}},\widetilde{z_{\mathrm{t}-1}},{\mathit{x}}_{\mathit{t}},\mathit{\Theta })$, and the forecast $\widetilde{z_t}$ is sampled from $p(·|\mathrm{f}(\widetilde{{\mathit{h}}_{\mathit{t}}},\mathit{\Theta }))$;

(4)

Step (3) is repeated N times to produce ensemble forecasts ${\left\{{\mathbf{z}}_{\mathbf{i},{\mathit{t}}_0+1:{\mathit{t}}_0+\mathit{L}}\right\}}_{\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{N}}$, providing a Monte Carlo approximation of the predictive distribution.

2.1.3. Likelihood Model

The likelihood $p(·|\mathrm{f}({\mathit{h}}_{\mathit{t}},\mathit{\Theta }))$, which defines the target distribution, must be carefully selected to match the statistical characteristics of predictand. While Salinas et al. (2020) [45] recommended Gaussian distributions for real-valued data and Negative Binomial distributions for positive count data, these choices prove suboptimal for streamflow forecasting due to (1) heavy-tailed characteristics of streamflow, and (2) the strictly positive nature of streamflow values. Therefore, to better characterize the statistical properties of streamflow, this study employs Gamma distribution. This choice is substantiated by its strong physical basis, as the Gamma function can be derived from conceptual watershed runoff models, distinguishing it from alternative distributions, which are applied primarily for their statistical fitting capabilities [50]. Its proven ability to accurately represent diverse streamflow regimes has also led to its widespread acceptance and recommendation in hydrological frequency analysis [51]. In addition to the physically grounded Gamma distribution, this study also employs Student’s t-distribution, which is suitable for extreme events, and conventional Gaussian distribution, which has been widely proposed.

For Student’s t-distribution, the likelihood and function $\mathrm{f}(·)$ are as below:

$$\begin{gathered} p_{ST}\left(z|\mu ,\sigma ,\nu \right)={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu +1}{2}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)\sqrt{\nu \pi }\sigma }}{\left(1+{\displaystyle \frac{{\left(z-\mu \right)}^2}{\nu {\sigma }^2}}\right)}^{-{\displaystyle \frac{\nu +1}{2}}} \\ \mu \left({\mathit{h}}_{\mathit{t}}\right)={{\mathit{w}}_{\mathit{\mu }}^{\mathit{T}}\mathit{h}}_{\mathit{t}}+b_{\mu } \\ \sigma \left({\mathit{h}}_{\mathit{t}}\right)=\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}\left({{\mathit{w}}_{\mathit{\sigma }}^{\mathit{T}}\mathit{h}}_{\mathit{t}}+b_{\sigma }\right) \\ \nu \left({\mathit{h}}_{\mathit{t}}\right)=2+\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}\left({{\mathit{w}}_{\mathit{\nu }}^{\mathit{T}}\mathit{h}}_{\mathit{t}}+b_{\nu }\right) \end{gathered}$$

(6)

where the softplus activation function $\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}\left(x\right)=\log (1+e^x)$ is applied to enforce positivity constraints on the distribution parameters.

For Gamma distribution, the likelihood and function $\mathrm{f}(·)$ are as below:

$$\begin{gathered} p_{Gamma}\left(z|\alpha ,\beta \right)={\displaystyle \frac{{\beta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}}{z}^{\alpha -1}{e}^{-\beta z} \\ \alpha \left({\mathit{h}}_{\mathit{t}}\right)=\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}\left({{\mathit{w}}_{\mathit{\alpha }}^{\mathit{T}}\mathit{h}}_{\mathit{t}}+b_{\alpha }\right) \\ \beta \left({\mathit{h}}_{\mathit{t}}\right)=\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}\left({{\mathit{w}}_{\mathit{\beta }}^{\mathit{T}}\mathit{h}}_{\mathit{t}}+b_{\beta }\right) \end{gathered}$$

(7)

2.2. DeepAR-Based Modeling Framework

This study develops a DeepAR-based probabilistic forecasting framework for MLSP (Figure 1), which consists of four main steps: (1) data preparing, involving data collection and optimal distribution selection; (2) data splitting, which spites data into training, validation and test datasets; (3) model calibration, involving model training and selection; and (4) model evaluation, involving model performance assessment [52,53].

2.2.1. Data Preparing

Data preparing and analysis are proposed to collect streamflow and corresponding predictors (precipitation), and select optimal distribution for the streamflow. To select the most appropriate distribution from these candidates, both the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are employed to evaluate the goodness-of-fit between the observed data and theoretical distributions. Then, the optimal distribution is selected by minimizing both AIC and BIC values, which are computed as

$$\begin{gathered} \mathrm{A}\mathrm{I}\mathrm{C}=-2\mathrm{l}\mathrm{n}\left(Li\right)+2k \\ \mathrm{A}\mathrm{I}\mathrm{C}=-2\mathrm{l}\mathrm{n}\left(Li\right)+k\mathrm{l}\mathrm{n}\left(n\right) \end{gathered}$$

(8)

where Li is the maximized likelihood value, k denotes the parameter number, and n is the length of streamflow time series.

2.2.2. Data Splitting

Data splitting is an important step in the data-driven modeling process, through which the available data is divided into training, validation, and test datasets [52,53,54]. In this study, the data after a specific time point is first separated as the test set to ensure no overlap between the test data and other data. Subsequently, the remaining data is randomly shuffled and split into training and validation datasets in a specific ratio.

2.2.3. Model Calibration

Model calibration is used to optimize model architecture and parameters in order for the model to represent the underlying relationships between predictors and predictand. First, the optimal streamflow distribution obtained by minimizing the AIC/BIC metrics is adopted to define the model’s output distribution type. Furthermore, taking into account the specific architecture of the DeepAR model, this study utilizes two types of RNN structures (i.e., LSTM and GRU), the detailed structures and gating equations of which can be seen in Demir (2025) [55]. Then, multiple model variants are generated by varying input conditions, and parameters are optimized using a training dataset by the ADAM optimizer, which is selected for its proven efficiency and robust performance in handling complex models through adaptive learning rates [56,57]. Finally, the model variants are used in a validation dataset and compared to select the best model in terms of their predictive performance.

2.2.4. Model Evaluation

Model evaluation is used to assess the predictive performance of the selected model over an independent dataset (i.e., test dataset) [58]. The deterministic and probabilistic predictions are generated, and two types of metrics are proposed to evaluate the predictive performance [22,53,59,60]. The performance of the deterministic predictions is evaluated using the following three metrics: the root mean square error (RMSE) and mean absolute error (MAE), which reflect the overall prediction accuracy, and the Nash–Sutcliffe efficiency (NSE) coefficient, which indicates the overall model effectiveness. The performance of the probabilistic predictions is evaluated using the following three metrics: (1) continuous ranked probability score (CRPS), which measures the overall probabilistic predictive performance; (2) prediction interval coverage probability (PICP), which measures the proportion of observations that fall within their predicted probability intervals; and (3) mean prediction interval width (MPIW), which indicates the average width of the prediction intervals and thus the sharpness of the forecast. These six metrics can be calculated according to the following equations:

$$\begin{gathered} \mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(\widehat{Q_i}-Q_i\right)}^2} \\ \mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|\widehat{Q_i}-Q_i| \\ \mathrm{N}\mathrm{S}\mathrm{E}=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(\widehat{Q_i}-Q_i\right)}^2}{{\sum }_{i=1}^n{\left(Q_i-\frac{1}{n}{\sum }_{i=1}^n{Q}_i\right)}^2}} \\ CRPS={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\int {\left(F_i\left(x\right)-H\left(x-Q_i\right)\right)}^{2}dx \\ H\left(x-Q_i\right)=\left\{\begin{array}{c}\begin{array}{cc}0& x<Q_i\end{array}\\ \begin{array}{cc}1& x\ge Q_i\end{array}\end{array}\right. \\ \mathrm{P}\mathrm{I}\mathrm{C}\mathrm{P}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{c}_i\times 100\% c_i=\left\{\begin{array}{c}\begin{array}{cc}1& L_i\le Q_i\le U_i\end{array}\\ \begin{array}{cc}0& otherwise \end{array}\end{array}\right. \\ \mathrm{M}\mathrm{P}\mathrm{I}\mathrm{W}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{U}_i-L_i \end{gathered}$$

(9)

where n is the sample size in the test dataset, i is the sample index, $Q_i$ is the observed streamflow, $\widehat{Q_i}$ is the ensemble mean prediction, $L_i$ and $U_i$ are the lower and upper bounds of the prediction interval, and $F_i\left(\right)$ is the cumulative distribution function (CDF) of the probabilistic forecast.

2.3. Case Study and Data

The performance of the DeepAR-based modeling framework is examined in two basins upstream of the Wudongde (WDDR) and Sanxia (SXR) reservoirs, as illustrated in Figure 2. The available data and statistical characteristics for the two basins is presented in

Table 1. Available data.

Study Area	Upper WDDR Area		Upper SXR Area
Variable	Naturalized Streamflow	Areal Mean Precipitation	Naturalized Streamflow	Areal Mean Precipitation
Temporal coverage and scale	January 1980 to September 2022 10-day time scale
Min	557 m3/s	0.00 mm	2910 m3/s	0.10 mm
Max	18,600 m3/s	90.83 mm	56,500 m3/s	91.41 mm
Average	3819.97 m3/s	17.59 mm	13,432.22 m3/s	22.69 mm
Standard deviation	3247.06 m3/s	18.98 mm	10,016.31 m3/s	20.36 mm
Coefficient of variation	0.85	1.07	0.75	0.84
Skewness	1.35	0.22	1.21	−0.26
Kurtosis	1.23	1.08	0.98	0.89

, including ten-day naturalized streamflow (Figure 3) and areal mean precipitation records spanning January 1980 to September 2022. To ensure data compatibility with neural network requirements, all variables are first normalized before model processing and then inversely transformed to their original scales. For the streamflow variable, the data is standardized using the Z-score method and then shifted by a constant to ensure all values were above 0, as required by the Gamma distribution. For the precipitation variable, min–max normalization is used to scale its values into the range 0–1.

2.4. Experiment Setup

The overall experimental setup, based on the DeepAR-based modeling framework, is illustrated in Figure 4. Three candidate distribution types—Gaussian, Gamma, Student’s t-distribution—are provided to account for streamflow characteristics. Then, the data after January 2018 is separated as the test dataset, while the other data is randomly split into training and validation datasets at an 8:2 ratio, following the recommendations in previous studies [52,53]. This splitting method effectively prevents data leakage between the test dataset and the training/validation datasets. The resulting training/validation and test datasets are visualized in Figure 3. As shown in Figure 3, the characteristics of the streamflow in the test dataset are largely consistent with those in the training/validation datasets for both the Upper WDDR and Upper SXR areas. Additionally, it is evident that the streamflow in neither basin exhibits a statistically significant increasing or decreasing linear trend over time. The coefficients of determination R² with time are 0.000 for both areas, with corresponding p-values of 0.876 and 0.681, respectively.

During the model calibration phase, different models for predicting streamflow in the next 18 ten-day periods, are trained based on varying input conditions that incorporate combinations of three precipitation input scenarios (temporal lags: [0], [0, 1] or [0, 1, 2]) and two streamflow input scenarios (temporal lags: [1], [1, 2]), with the model training hyperparameters detailed in

Table A1. The hyperparameters.

Hyperparameters	Value
Number of layers	2
Number of cells	40
Dropout rate	0.2
Time features	True
Batch size	32
Epochs	200
Learning rate	0.001

. The optimal input combination is then selected based on a comparative evaluation of all model variants’ predictive performance (i.e., RMSE) on the validation dataset.

After data preparing and model calibration, the final model combining the optimal input configuration and the probability distribution output is established and applied to produce deterministic predictions and probabilistic predictions with 100 members in this study. In order to evaluate the impact of output distribution and RNN structure, alternative models with different probability distribution outputs and different RNN structures (LSTM and GRU), named GRU-N, GRU-S, GRU-G, LSTM-N, LSTM-S, LSTM-G, are established and compared in terms of their predictive performance on the testing dataset. In the “GRU-N” naming structure, the former denotes the RNN structure (including GRU and LSTM), while the latter denotes the model output distribution (including Normal distribution N, Student’s t-distribution S, and Gamma distribution G).

3. Results

Following the modeling framework, the results are presented in three sections: optimal probability distribution selection (Section 3.1), input configuration optimization (Section 3.2), and testing performance evaluation (Section 3.3).

3.1. Optimal Probability Distribution Selection

The distribution fitting results for both study areas over the data excluding the test dataset are presented in Figure 5, while the corresponding AIC and BIC values are listed in

Table 2. AIC and BIC values of three distributions for two study areas.

Study Area	Distribution	AIC	BIC
Upper WDDR area	Normal	25,335.35	25,345.73
	Student’s t	25,223.80	25,239.38
	Gamma	24,044.36	24,059.95
Upper SXR area	Normal	28,283.95	28,294.34
	Student’s t	28,285.95	28,301.54
	Gamma	27,308.55	27,324.14

Note: The bold values indicate the minimum AIC/BIC.

. The results reveals that the Gamma distribution provides superior statistical performance compared to the Normal and Student’s t-distributions. For the Upper WDDR area, the Gamma distribution achieves the lowest AIC (27,321.09) and BIC (27,337.03) values, substantially outperforming the Normal distribution (AIC: 28,574.36, BIC: 28,584.99) and Student’s t-distribution (AIC: 28,455.62, BIC: 28,471.57). Similarly, in the Upper SXR area, the Gamma distribution exhibits superior performance, with AIC (30,848.71) and BIC (30,864.66) values considerably lower than the alternative distributions. The probability density plots also illustrate that the Gamma distribution more accurately captures the right-skewed characteristics and tail behavior of the streamflow data in both study areas.

3.2. Input Configuration Optimization

The comparative analysis of different input configurations reveals consistent performance patterns across both study areas, which can be shown in

Table 3. Average RMSE on the validation dataset of different models.

Input Configuration		Average RMSE on Validation Dataset of Different Models for Upper WDDR Area (m3/s)	Average RMSE on Validation Dataset of Different Models for Upper SXR Area (m3/s)
Lags of precipitation	0	1277.07	3634.55
	0, 1	1237.00	3616.81
	0, 1, 2	1199.21	3481.18
Lags of streamflow	1	1245.33	3577.72
	1, 2	1230.19	3577.31

. For the Upper WDDR area, the precipitation input scenario with temporal lags [0, 1, 2] achieves the lowest RMSE (1199.21 m³/s), indicating superior predictive accuracy when incorporating precipitation data from the current time step and two previous time steps. In contrast, the precipitation-only scenario with no temporal lag [0] shows the poorest performance with the highest RMSE (1277.07 m³/s). Similarly, in the Upper SXR area, the [0, 1, 2] precipitation configuration demonstrates optimal performance, with an RMSE value of 3481.18 m³/s, while the [0] configuration shows the highest RMSE (3634.35 m³/s). The performance gradient follows a consistent pattern across both areas, where increased temporal lag information progressively improves model accuracy.

Regarding streamflow input configurations, the comparison between temporal lags [1] and [1, 2] shows that incorporating additional historical streamflow information ([1, 2]) yields only marginal improvements, with RMSE decreasing modestly from 1245.33 to 1230.19 m³/s in the Upper WDDR area and from 3577.72 to 3577.31 m³/s in the Upper SXR area. This suggests that the contribution of additional streamflow lag information is relatively limited compared to the substantial performance gains observed with precipitation temporal lags.

3.3. Testing Performance Evaluation

The predictive performance metrics of the six models (i.e., GRU-N, GRU-S, GRU-G, LSTM-N, LSTM-S, and LSTM-G) are presented in

Table 4. Predictive performance on the testing dataset of six models with different RNN structures and different probability distribution outputs.

Study Area	Evaluation Metrics	Model
		GRU-N	GRU-S	GRU-G	LSTM-N	LSTM-S	LSTM-G
Upper WDDR area	RMSE (m3/s)	1356.74	1282.09	1098.98	1407.77	1331.08	1016.54
	MAE (m3/s)	817.92	775.23	699.40	844.78	816.79	643.99
	NSE	0.82	0.84	0.88	0.81	0.83	0.89
	CRPS (m3/s)	608.89	578.95	517.54	637.95	620.08	473.26
	PICP (%)	90.08	84.81	92.93	87.09	76.84	93.15
	MPIW (m3/s)	3980.52	3585.95	3330.88	3778.34	3228.71	3485.17
Upper SXR area	RMSE (m3/s)	4217.12	4143.33	4057.33	4091.22	4296.16	4047.15
	MAE (m3/s)	2346.57	2289.88	2222.94	2370.06	2487.38	2312.61
	NSE	0.86	0.87	0.87	0.87	0.86	0.87
	CRPS (m3/s)	1776.92	1716.52	1654.24	1771.43	1870.94	1717.93
	PICP (%)	91.77	91.45	96.46	93.11	87.27	96.54
	MPIW (m3/s)	10,015.73	10,669.20	11,048.15	11,218.44	9156.87	11,464.91

Note: The bold values indicate the best predictive performance in terms of the specific evaluation metric.

, and the Taylor diagram are presented in Figure A1. The results consistently demonstrate that models utilizing a Gamma distribution output (GRU-G, LSTM-G) generally outperform their counterparts that use Gaussian or Student’s t-distributions. In the Upper WDDR area, LSTM-G demonstrates exceptional performance, achieving the lowest RMSE (1016.54 m³/s) and MAE (643.99 m³/s), and the highest NSE (0.89). For probabilistic forecasts, it also proves highly reliable, with a PICP of 93.15% and the lowest CRPS (473.26 m³/s). For the Upper SXR area, the superiority is shared between the Gamma-based models. The GRU-G model achieves the lowest MAE (2222.94 m³/s) and CRPS (1654.24 m³/s). Meanwhile, the LSTM-G model achieves the best RMSE (4047.15 m³/s) and the highest reliability in its probabilistic forecast, with an impressive PICP of 96.54%. The differences between GRU and LSTM architectures are relatively small and show no consistent patterns. In the Upper WDDR area, LSTM-G slightly outperforms GRU-G, while in the Upper SXR area, the differences are marginal. This suggests that RNN architecture choice has minimal impact compared to output distribution selection.

4. Discussion

The predictive performance of the six models (i.e., GRU-N, GRU-S, GRU-G, LSTM-N, LSTM-S, and LSTM-G) is discussed in this section. First, Section 4.1 discusses the deterministic prediction performance of models with different distribution outputs and different RNN structures across various forecast horizons. Then, Section 4.2 provides a comparative analysis of their probabilistic prediction performance. Finally, Section 4.3 demonstrates the overall predictive performance of the models across different forecast horizons.

4.1. Deterministic Prediction Performance of Different Models

The deterministic predictive performance of six models (i.e., GRU-N, GRU-S, GRU-G, LSTM-N, LSTM-S, and LSTM-G) across 18 forecast horizons (10-day periods) for two study areas is presented in

Table 5. RMSE of six models across various forecast horizons.

Forecast Horizon (10-Day Periods)	Upper WDDR Area						Upper SXR Area
	GRU-N	GRU-S	GRU-G	LSTM-N	LSTM-S	LSTM-G	GRU-N	GRU-S	GRU-G	LSTM-N	LSTM-S	LSTM-G
1	876.2	878.3	940.6	892.9	881.5	859.8	3897.6	4024.8	4126.8	3863.2	4022.0	3687.2
2	1035.7	1007.6	1017.9	1083.1	1054.9	958.9	4031.0	4091.6	4104.6	3880.4	4254.2	3841.5
3	1124.9	1085.5	995.7	1164.5	1129.9	913.6	4167.4	4109.5	4221.6	4045.0	4260.8	3938.4
4	1201.3	1157.9	1032.1	1235.9	1190.4	924.8	3982.7	3978.0	3947.0	4011.7	4176.2	4011.2
5	1239.9	1159.2	1006.4	1264.5	1178.6	902.7	4080.1	4002.4	4118.1	4106.4	4272.7	3977.0
6	1270.4	1215.3	1020.5	1312.1	1219.7	912.3	4098.8	4148.2	4043.1	4024.7	4213.8	4068.3
7	1330.8	1247.1	1082.2	1351.1	1263.2	986.2	4145.4	4102.0	4044.4	4104.7	4285.8	4020.6
8	1367.0	1303.7	1119.5	1406.9	1350.6	1051.4	4190.1	4031.1	3977.6	4065.0	4249.4	3936.1
9	1408.0	1340.6	1102.5	1467.4	1390.4	1093.2	4297.1	4161.7	4030.8	4063.2	4283.7	4023.0
10	1453.8	1350.4	1119.3	1492.2	1430.3	1089.7	4312.4	4183.7	4002.9	4155.7	4348.2	4117.6
11	1451.8	1362.3	1116.0	1518.1	1441.8	1068.9	4343.3	4163.7	4020.1	4116.1	4329.2	4064.6
12	1487.0	1394.6	1153.5	1532.0	1458.4	1073.3	4374.3	4184.0	4009.1	4158.2	4295.4	4095.5
13	1483.9	1394.1	1152.8	1545.5	1469.7	1048.0	4328.6	4229.5	4107.5	4140.8	4311.1	4103.9
14	1485.5	1394.9	1167.4	1559.4	1449.5	1055.8	4358.9	4277.4	4034.3	4176.3	4406.5	4258.0
15	1484.5	1390.3	1174.9	1574.5	1454.6	1071.8	4339.0	4217.5	4096.7	4170.8	4371.7	4235.7
16	1494.8	1391.5	1164.6	1565.6	1456.0	1069.7	4331.0	4219.1	4042.1	4119.7	4391.1	4180.6
17	1507.1	1407.0	1177.8	1574.5	1474.5	1068.0	4336.2	4245.0	4098.1	4216.8	4428.7	4178.2
18	1503.5	1430.7	1192.9	1559.5	1468.4	1096.7	4251.0	4194.8	3998.1	4203.0	4411.2	4070.5

Note: The bold values indicate the miminum RMSE value among six models.

and Figure 6, with the results for RMSE shown as a representative example. It is evident that the forecasting accuracy is generally observed to decline as the forecast horizon increases, regardless of model architecture or output distribution. Among the three distributions, the Gamma distribution consistently results in lower RMSE values, particularly at longer lead times. This suggests a better capacity of the Gamma-based models to capture the positively skewed or heteroscedastic characteristics often found in hydrological data. In the Upper WDDR area, the LSTM-G model achieves the lowest RMSE across most forecast horizons, with values ranging from 859.8 m³/s at the 1st forecast period to 1096.7 m³/s at the 18th. A similar pattern is observed in the Upper SXR area, where LSTM-G yields the minimum RMSE of 3687.7 m³/s at the 1st period and 3998.1 m³/s at the 18th.

Differences between LSTM and GRU structures are also observed but are found to be less consistent and less influential than those resulting from output distribution selection. In the Upper WDDR area, GRU-N and GRU-S outperform their LSTM counterparts, while the advantage shifts to LSTM only when the Gamma distribution is employed. In the Upper SXR area, the superiority of LSTM-G is observed only at shorter forecast horizons, with GRU-G performing better as the forecast horizon increases. Across all configurations, no consistent advantage is associated with either architecture, suggesting that model structure plays a secondary role relative to the output distribution in determining predictive performance.

Notably, the magnitude of performance gains brought by the Gamma distribution varies across regions. In the Upper WDDR area, the use of Gamma distribution yields substantial improvements compared to Normal and Student’s t assumptions, particularly when paired with LSTM. In contrast, in the Upper SXR area, although Gamma-based models still outperform others, the improvement is relatively marginal. For instance, the reduction in RMSE from LSTM-N to LSTM-G at the 18th horizon is only modest (from 4203.0 m³/s to 4070.5 m³/s), whereas the gain is more pronounced in the Upper WDDR area (from 1559.5 m³/s to 1096.7 m³/s). These findings demonstrate that the benefit of applying flexible, non-Gaussian output distributions such as Gamma is region-dependent.

A primary reason for this discrepancy likely lies in the different statistical characteristics of the streamflow in the two basins, as detailed in Table 1. The streamflow in the Upper WDDR area exhibits a significantly higher degree of positive skewness (1.35) and kurtosis (1.23) compared to the Upper SXR area (skewness of 1.21, kurtosis of 0.98). This indicates that the Upper WDDR streamflow distribution has a longer right tail and is more “peaked,” deviating more substantially from a Normal distribution. Consequently, the flexible, skewed shape of the Gamma distribution provides a much better fit for the more asymmetric data of the Upper WDDR area, leading to more significant performance gains. Conversely, the streamflow in the Upper SXR area, while still positively skewed, is statistically closer to a symmetrical distribution. Therefore, the advantage of the Gamma distribution over simpler assumptions like the Gaussian distribution is less pronounced, resulting in only marginal improvements.

In summary, the results highlight the critical role of output distribution in deterministic streamflow forecasting, with the Gamma distribution consistently offering performance benefits, though to varying extents across regions. While model structure has some impact, it exerts less influence than distributional assumptions.

4.2. Probabilistic Prediction Performance of Different Models

The probabilistic predictive performance (i.e., CRPS, PICP, and MPIW) of six models (i.e., GRU-N, GRU-S, GRU-G, LSTM-N, LSTM-S, and LSTM-G) across 18 forecast horizons (10-day periods) for two study areas is presented in Figure 7. Similar to the deterministic results, probabilistic forecast accuracy generally declines with increasing lead time, as reflected by rising CRPS values across all model configurations. Among the output distributions, the Gamma distribution again provides the most substantial improvements in forecast skill. This consistency between deterministic and probabilistic evaluations reinforces the robustness of the Gamma assumption in capturing the inherent characteristics of streamflow data. In the Upper WDDR area, LSTM-G yields the lowest CRPS across nearly all horizons. In the Upper SXR area, LSTM-G yields the lowest CRPS during the first three horizons, while GRU-G shows superior accuracy at longer horizons. This pattern echoes deterministic results, where GRU-G also performed better in long-term forecasting under complex conditions.

Further analysis of the PICP and MPIW reveals deeper insights into the models’ reliability and sharpness (Figure 7c–f). As expected, the MPIW for all models generally increases with the forecast horizon, reflecting the growing uncertainty of longer-term predictions. However, the models based on the Gamma distribution (GRU-G and LSTM-G) consistently achieve higher PICP values, indicating that their prediction intervals more reliably encompass the observed values. Notably, these Gamma-based models manage to maintain relatively narrower interval widths (lower MPIW) for most lead times in the Upper WDDR area and for shorter lead times in the Upper SXR area. In contrast, while models using the Student’s t-distribution (GRU-S, LSTM-S) often produce the narrowest intervals (lowest MPIW), this comes at the cost of poor reliability, as their PICP values are frequently the lowest. This demonstrates their inability to effectively capture the true uncertainty of the streamflow.

Compared to output distribution selection, differences due to model structure are less consistent and generally less influential. While LSTM tends to perform better in the Upper WDDR area, GRU offers competitive or better results in the Upper SXR area when combined with the Gamma distribution. Regional differences are again observed, with CRPS values in the Upper SXR area consistently higher than those in the Upper WDDR area, reflecting greater predictive uncertainty. However, the relative benefit of using the Gamma distribution persists in both regions, though with different magnitudes.

In summary, probabilistic forecasting results confirm key findings from deterministic evaluations—particularly the consistent and robust advantage of using the Gamma distribution across regions and horizons. This advantage is not only reflected in superior accuracy (lower CRPS) but also in a better balance between forecast reliability (PICP) and sharpness (MPIW). However, nuanced differences are observed in the relative contributions of model structure, especially under probabilistic metrics. These findings suggest that while distributional assumptions remain the most critical component for improving streamflow forecast quality, model structure and regional hydrological characteristics jointly shape both deterministic and probabilistic forecasting performance.

4.3. Overall Predictive Performance

The predicted and observed streamflow at the first forecast horizon are shown in Figure 8. It is evident that all models produce narrow prediction intervals that closely follow the observations during non-flood seasons, when streamflow is low and uncertainty is limited. In contrast, flood seasons are characterized by increased variability, leading to wider forecast intervals and larger deviations, particularly around peak flows. The choice of output distribution also significantly affects forecast reliability. For example, models using the Normal distribution often generate overly wide intervals due to their poor fit to the skewed nature of streamflow, which is reflected in their generally higher MPIW values (Figure 7e,f). In contrast, Gamma-based models produce tighter and more stable intervals while maintaining high coverage probability, as evidenced by their superior balance between PICP and MPIW across most horizons (Figure 7c–f). While Gamma-based models generally provide better predictions in most cases, their advantage may not hold in every situation—for example, the GRU-G model performs worst in the Upper WDDR area. Nonetheless, their ability to constrain uncertainty remains evident, and such model-specific variability also underscores the importance of model fusion strategies for improving forecast robustness under diverse application scenarios [21,61,62,63,64].

The predicted and observed streamflow across 18 forecast periods for models using Gamma distribution in both the Upper WDDR and Upper SXR areas are presented in Figure 9. As the forecast horizon increases, streamflow uncertainty becomes more pronounced, and the width of the predictive intervals correspondingly expands, which is quantitatively confirmed by the rising MPIW values shown in Figure 7. This widening of intervals effectively reflects the growing uncertainty associated with longer lead times, particularly during high-flow periods. The use of the Gamma distribution enables the models to better capture the skewed nature of streamflow, resulting in predictive intervals that not only reflect the asymmetry of the data but also reliably encompass the observed hydrographs across seasons and forecast periods, which is validated by the high PICP values (Figure 7). This demonstrates the capability of using suitable output distribution to improve both the reliability and calibration in probabilistic streamflow prediction.

4.4. Limitations and Future Research Directions

Despite the promising performance of the proposed framework in the two selected study areas, there are still several limitations that need to be further studied. On one hand, given that different models exhibit varying performance under different scenarios, future work could explore model fusion or ensemble approaches. This is motivated by the finding that the LSTM-G model performs best at all lead times in the Upper WDDR area and at shorter lead times in the Upper SXR area, while the GRU-G model excels at longer lead times in the Upper SXR area. A model fusion method could leverage complementary model strengths to enhance overall forecast robustness.

On the other hand, the applicability of our modeling framework needs to be validated in basins with different hydrological regimes. The areas used in this study are humid, monsoon-influenced regions, for which precipitation is the primary input driver and the non-negative Gamma distribution is suitable. However, its effectiveness in other basins requires further investigation. For example, in semi-arid regions, streamflow is often intermittent, and evapotranspiration becomes a dominant influencing factor. Therefore, the model would need to incorporate evapotranspiration as an input, and the Gamma distribution, which cannot handle zero values, would likely need to be replaced with more suitable alternatives, such as Zero-Inflated or mixture distribution models. Similarly, in snowmelt-dominated regions, the framework would need to be adapted to include key input variables such as air temperature and snow water equivalent to capture the essential physics of snow accumulation and melt processes.

5. Conclusions

In this study, a DeepAR-based modeling framework is developed to generate probabilistic streamflow forecasts by integrating distribution selection with the DeepAR model. The framework is applied to two case studies to evaluate both deterministic and probabilistic forecasting performance. The influence of model structure and output distribution choice on prediction accuracy is also examined. The main conclusions are as follows.

(1) The proposed DeepAR-based modeling framework effectively identifies the most appropriate output distribution and input configuration, leading to superior deterministic and probabilistic forecasting performance in both case studies. By integrating distribution selection and input optimization, the framework ensures that the final models are well-calibrated to the characteristics of local streamflow dynamics.

(2) Multiple factors influence predictive performance, among which the selection of output distribution has the most significant impact. Notably, the advantage of the Gamma distribution is more pronounced in basins with higher skewness, providing a data-driven explanation for the observed regional performance differences. Meanwhile, differences between RNN model structures (i.e., GRU vs. LSTM) are relatively minor and less consistent.

(3) As forecast lead time increases and during flood seasons, streamflow uncertainty grows substantially. The developed probabilistic forecasting framework is capable of capturing this variation in uncertainty, and models using the Gamma distribution demonstrate superior performance by better representing the skewed nature of streamflow. This contributes to more reliable and better-calibrated forecasts across both typical and high-variability conditions.