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Article

Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study

Department of Civil Engineering, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si 17104, Korea
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Author to whom correspondence should be addressed.
Water 2022, 14(22), 3672; https://doi.org/10.3390/w14223672
Received: 17 October 2022 / Revised: 10 November 2022 / Accepted: 11 November 2022 / Published: 14 November 2022

Abstract

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In recent decades, natural calamities such as drought and flood have caused widespread economic and social damage. Climate change and rapid urbanization contribute to the occurrence of natural disasters. In addition, their destructive impact has been altered, posing significant challenges to the efficiency, equity, and sustainability of water resources allocation and management. Uncertainty estimation in hydrology is essential for water resources management. By quantifying the associated uncertainty of reliable hydrological forecasting, an efficient water resources management plan is obtained. Moreover, reliable forecasting provides significant future information to assist risk assessment. Currently, the majority of hydrological forecasts utilize deterministic approaches. Nevertheless, deterministic forecasting models cannot account for the intrinsic uncertainty of forecasted values. Using the Bayesian deep learning approach, this study developed a probabilistic forecasting model that covers the pertinent subproblem of univariate time series models for multi-step ahead daily streamflow forecasting to quantify epistemic and aleatory uncertainty. The new model implements Bayesian sampling in the Long short-term memory (LSTM) neural network by using variational inference to approximate the posterior distribution. The proposed method is verified with three case studies in the USA and three forecasting horizons. LSTM as a point forecasting neural network model and three probabilistic forecasting models, such as LSTM-BNN, BNN, and LSTM with Monte Carlo (MC) dropout (LSTM-MC), were applied for comparison with the proposed model. The results show that the proposed Bayesian long short-term memory (BLSTM) outperforms the other models in terms of forecasting reliability, sharpness, and overall performance. The results reveal that all probabilistic forecasting models outperformed the deterministic model with a lower RMSE value. Furthermore, the uncertainty estimation results show that BLSTM can handle data with higher variation and peak, particularly for long-term multi-step ahead streamflow forecasting, compared to other models.

1. Introduction

Sustainable water resource management is an essential requirement worldwide, and streamflow forecasting is an essential component of an effective water resource management plan [1]. Accurate streamflow forecasting plays a critical role in many decision-making scenarios related to water resource management such as flood/drought control and mitigation, reservoir management, hydropower generation, sediment transport, and irrigation management [2]. Owing to the complex and nonlinear characteristics associated with streamflow [3], forecasting is challenging. Sustainable water resource management plans are used to meet the requirements of people today and in the future. To support risk-aware decision-making in water resource management, current streamflow forecasting approaches should be improved to estimate forecasting uncertainties and leverage large volumes of data with complex dependencies [4].
Deep learning (DL), a sophisticated and mathematically complex evolution of machine learning (ML) algorithms, has recently received huge attention from researchers and has gradually become the most widely used forecasting approach in hydrology [5,6,7,8,9]. The advantage of DL is its flexibility to learn massive data and the ease of incorporating exogenous covariates [10]. The advantages of DL techniques over traditional ML algorithms for streamflow forecasting have been discussed in many studies [1,11]. However, DL has not been extensively explored in forecasting uncertainty.
Only a few studies have been conducted on probabilistic approaches to streamflow forecasting. Thus, the existing uncertainty was not addressed by most DL approaches. However, deterministic approaches may not be as efficient as probabilistic methods and exhibit suboptimal performance. In general, deterministic forecasting is widely used in hydrology as an input for various water resource management plans. The transition from deterministic to probabilistic forecasting methods with uncertainty quantification is strongly favored in academia and industry. The primary issue in streamflow forecasting is the complex uncertainty rooted in the stochastic characteristics of streamflow time series. Furthermore, probabilistic forecasting has emerged to overcome the shortcomings of conventional deterministic methods and to deal with uncertainty more effectively. The probabilistic approach has recently gained importance because it can extract more valuable information from historical data and quantify the uncertainty of the future by forming a probability distribution over possible outcomes. The probabilistic approach extends beyond single-point forecasting for each time step and can provide a band of likely forecasting intervals above and below the mean forecasted value. Existing deterministic methods report the mean of possible outcomes, and they are unable to reflect the inherent uncertainty that exists in the real world.
Despite the fact that hydrological prediction can be most helpful when given in probabilistic form [12], the use of probabilistic modeling is still a relatively new concept in the field of hydrology [13]. Moreover, the probabilistic approach is crucial to optimal decision-making that reveals the upper and lower bounds between which the uncertain actual future values may exist. Occasionally, decision-making requires more than single-point forecasting; this is where distribution would be beneficial. To make reliable forecasts and to conduct a comprehensive performance evaluation, a probabilistic approach should be considered in streamflow forecasting. Most existing streamflow forecasting methods focus on deterministic forecasting. The application of various machine learning algorithms in deterministic prediction has been investigated in many studies [14,15,16,17,18,19]. Limited research has been conducted on multistep-ahead streamflow predictions [1,20,21,22,23]. Even though considerable efforts have been made to improve the performance of streamflow forecasting models from short- to long-term [24,25] and from single- to multi-step ahead [9,26,27], they are still limited by uncertainties [28,29,30,31].
An effective way to perform probability forecasting in the field of hydrology is to apply the Bayesian approach due to the benefits of uncertainty representation, understanding generalization, and reliable prediction through the lens of probability theory. The Bayesian approach can be classified into four primary groups: Bayesian model averaging (BMA), Bayesian model updating (BMU), Bayesian networks (BN), and Bayesian neural networks (BNN) [32]. A BNN is a type of stochastic artificial neural network that uses a BMU for training and updating the probabilistic distributions of network parameters. Furthermore, BMU and BN have become prevalent, and they have been implemented in various fields such as computer vision, natural language processing, medical diagnostics, autonomous driving, and flood hazard analysis [32]. Han and Coulibaly [33] presented a comprehensive review of Bayesian approaches applied to flood forecasting from 1999 to 2015. The results reveal that probabilistic flood forecasting can reduce uncertainty and provide more accurate and reliable forecasting. Moreover, they mentioned that only a limited number of river basins have been studied from the Bayesian perspective to date. Furthermore, we should determine if the Bayesian approaches are suitable for different watersheds with different sizes and physical and climatic characteristics. Costa and Fernandes [34] developed a Bayesian framework to estimate the extreme flood quantile from a rainfall-runoff model of a dam in California. Xu et al. [35] developed a real-time probabilistic channel flood forecasting model by combining a hydraulic model with the Bayesian approach in the upstream reaches of the Three Gorges Dam on the Yangtze River, China. A state-of-the-art review was provided by Huang et al. [36] to summarize the application of Bayesian inference in system identification and damage assessment for civil infrastructure. Goodarzi et al. [37] proposed a decision-making model using BN to predict heavy precipitation in the Kan Basin. Bayesian neural networks are yet to be applied to probabilistic streamflow forecasting, as aforementioned.
Recent studies on probabilistic prediction in the field of hydrology are summarized in Table 1. As shown in Table 1, a few researchers trained a deterministic model and used the obtained deterministic result to obtain a probabilistic forecasting result to estimate the uncertainty [38]. However, in a few studies, a deterministic layer has been coupled with a probabilistic layer to achieve forecasting uncertainty [39]. Conversely, a few studies have focused on developing a probabilistic model by introducing stochastic components into the network by giving the network either stochastic activation or weights [40,41,42].
The application of probabilistic DL showed superior performance in various fields, including residential net load forecasting [45,46], short-term scheduling in power markets [47], photovoltaic power [48], load forecasting for buildings [49], and electricity consumption [50]. This indicates the wide range of the potential applicability of the probabilistic DL approaches. Univariate streamflow forecasting using conventional data-driven models has been investigated in the previous studies [51,52,53,54]. To the best of the authors’ knowledge, the application of BLSTM in multi-step ahead probabilistic prediction using a retrospective univariate time series has not been applied to streamflow prediction yet. To address the aforementioned research gaps, this study proposed a framework for transforming a deterministic model into a probabilistic model with improved performance. This study developed a Bayesian deep neural network framework to characterize the prognostic uncertainties for probabilistic streamflow forecasting, which investigated both epistemic and aleatoric uncertainties. The motivation of the framework was to transform existing deterministic prediction models into their probabilistic counterparts for better performance in water resources management and decision-making, and to cover newly emerged challenges that humankind encountered primarily due to climate change.
The primary contributions of the study are as follows: For the first time, in streamflow prediction, we introduced the Bayesian LSTM network’s application for multi-step ahead probabilistic forecasting in water resource management. Bayesian theory and LSTM networks were combined to generate probabilistic streamflow forecasts to capture both epistemic and aleatoric uncertainties. This is the first study to exploit Bayesian deep learning for streamflow prediction. Moreover, a comprehensive comparison with a series of state-of-the-art probabilistic prediction methods is conducted. The superior performance of the proposed scheme was demonstrated with respect to both the deterministic and probabilistic forecasting results. Moreover, to demonstrate the superiority of probabilistic forecasting, particularly for water resource management, a comparative analysis was conducted for three case studies with different forecasting horizons and timescales.
The paper is organized as follows: In Section 2, the materials and methods are presented in subsections on Bayesian long-short-term memory (BLSTM) (2.1), experimental setup (2.2), and performance evaluation (2.3). In Section 3, the case study, study area (3.1), and experimental setup (3.2) are detailed. The results are presented in Section 4, with two subsections focusing on the probabilistic forecasting performance assessment (4.1) and the impact of the forecast horizon on probabilistic forecasting performance (4.2). Furthermore, the concluding remarks of this study with directions for future research are discussed in Section 5.

2. Materials and Methods

The proposed Bayesian deep-learning approach for probabilistic streamflow forecasting is presented in detail in the following sections.

2.1. Bayesian Long Short-Term Memory (BLSTM)

In this study, the Bayesian approach is employed, which is a well-established and thorough approach to fit probabilistic models that capture and distinguish different sources of uncertainties [55]. The BNN is a stochastic artificial neural network (ANN) trained using the Bayesian approach. Probability is defined in terms of the degree of belief in the Bayesian approach; the more likely an outcome is, the higher its degree of belief. The primary idea of the Bayesian approach in deep learning is to replace each weight with a distribution [56]. An LSTM network overcomes the long-term dependency issue of conventional RNNs through additional interactions in its various unit cells. Additionally, LSTM cells (memory cells) are composed of three gates (input, forget, and output) for short-term memory selection and a state vector transmission responsible for long-term memory. Information can be selectively passed during the learning procedure by manipulating the gate settings. The LSTM network is mathematically represented as follows [57]:
i t = σ ( W i . h t 1 , x t + b i ) f t = σ ( W f . h t 1 , x t + b f ) ,
o t = σ ( W o . h t 1 , x t + b o ) ,
h t = o t × tanh ( C t ) ,
C ˜ = tanh W c . h t 1 , x t + b c ,
C t = f t C t 1 + i t C ˜ ,
σ x = sigmoid x = 1 1 + e x   ,
tanh ( x ) = e x e x e x + e x ,
where at time step t, x t is the input vector,   h t is the LSTM output and hidden state (short-term memory), and i t , f t , and o t are the input, forget, and output gates, respectively. W and b are the weight matrix and bias, respectively. C t is the current cell state (long-term memory), and C ˜   is the candidate cell state value. σ is a sigmoid activation function that uses h t 1 and x t to make decisions regarding the input, forget, and output gates [57].
Given the input data, X train = [ x 1 , , x Train ] and their corresponding output labels Y train = [ y 1 , , y Train ] . The primary goal of the Bayesian approach is to identify the parameter W of a function y = f W x that probably generates the desired output [58,59]. In this approach, a prior distribution that represents the prior belief about the neural network parameter distribution before observing the inputs is employed over W to capture epistemic uncertainty. The Bayesian neural network structure is illustrated in Figure 1. Rather than a single network, this method trains a set of networks in which the weight of each network is derived from a shared learning probability distribution [59].
Setting a standard normal distribution as a prior with zero mean, which can bring the benefit of regularization, has been demonstrated as one of the most effective solutions when the prior distribution is difficult to identify. After training the Bayesian deep neural network and observing data, the model likelihood distribution p Y Train | f W should be defined as a normal distribution N f W ( X Train , σ 2 ) and observation noise σ to capture roughly suitable parameters. Based on the Bayesian rule, the posterior p W | X Train , Y Train is employed over the weights to generate samples of predictions rather than the prior distribution. The posterior is calculated as follows [59]:
p W | X Train , Y Train = p Y Train | X Train , W . p W p Y Train | X Train
where p Y Train | X Train is the marginal likelihood probability that cannot be estimated, thereby, the posterior is not tractable without a variational inference to approximate it. With this distribution, suitable parameters given by the input data can be captured, and the output y can be predicted for a new input x by integration [58]:
p y | x , X Train , Y Train = p y | x , W p W | X Train , Y Train dW .
To evaluate the true posterior p W | X Train , Y Train , an approximation variational distribution q θ W , which is parameterized by θ , is required to ensure that the optimal distribution q ˜ θ W can represent the posterior by minimizing the Kullback–Leibler (KL) divergence [60] between the approximation variational and posterior distributions [61]:
KL ( q θ W p W | X Train , Y Train ) = q θ W log q θ W p W | X Train , Y Train dW .
Generally, two methods are available to approximate the posterior distribution: variational inference (VI) and Monte Carlo (MC) dropout [56,61,62]. The study employed the VI to solve the optimization issue analytically. Interested readers can refer to [62] for detailed information on the approximation method. The predictive distribution can be approximated by:
p y | x , X Train , Y Train = p y | x , W q ˜ θ W dW = q ˜ θ y | x .

2.1.1. Epistemic Uncertainty

Mathematically, by simulating the model based on input x, the predictive mean can be estimated with an unbiased estimator, as follows [63]:
E ˜ y 1 T t = 1 T f W t ^ x ,
where E ˜ y is the predictive mean, f W t ^ is the stochastic output of the prediction model, W t ^ represents the sample weights, and T denotes the number of samples at time t. Similar to the estimation of the predictive mean, given that W t ^ ~ q ˜ θ W and p y | f W x = N y ; f w x , σ 2 for σ > 0, the predictive variance can be estimated by an unbiased estimator as follows [63]:
E ˜ y T y 1 T t = 1 T f W t ^ x T f W t ^ x + σ 2 .
The term σ 2 corresponds to inherent noise in the input data. Afterward, the epistemic uncertainty, which represents the uncertainty of the model about its prediction outputs, is captured by the predictive variance that can be approximated as [63]:
Var ˜ y = E ˜ y T y E ˜ y T E ˜ y .

2.1.2. Aleatoric Uncertainty

Aleatoric uncertainty can be divided into homoscedastic and heteroscedastic uncertainties. To capture the aleatoric uncertainty, parameter σ should be tuned. For each input x, in the homoscedastic uncertainty, the observation noise σ is assumed to be constant. In contrast, heteroscedastic uncertainty assumes that observation noise varies with the input. Heteroscedastic models are data-dependent and can be expressed as:
L θ = 1 T train i = 1 T train 1 2 σ x i 2 y i f x i 2 + 1 2 log x i 2 .
Because the maximum posterior is performed to find a single value for parameter θ , this approach does not capture the epistemic uncertainty since it is a property of the model, not the input data.

2.1.3. Combining Aleatoric and Epistemic Uncertainty

Abdar et al. [64] explained that an effective way to combine both uncertainties in a single model is to transform the heteroscedastic model into a Bayesian model by placing a distribution over its weight and bias parameters. Thus, both the predictive mean and variance were derived from the developed prediction model.
y ^ , σ ^ 2 = f M W X ,
where f M W is the prediction model (BLSTM) used in this study, parameterized by the model weight W ^ [55,56]. The Gaussian likelihood is used to model the aleatoric uncertainty, and the final loss function of the prediction model can be expressed as [58]:
L M θ = 1 T train i = 1 T train 1 2 σ x i 2 y i y ^ i 2 + 1 2 log σ ^ i 2 .
Finally, the predictive uncertainty of the prediction model, consisting of both aleatoric and epistemic uncertainties, can be approximated as
Var ˜ y = 1 T sample t = 1 T sample y ^ t 2 1 T sample t = 1 T sample y ^ t 2 + 1 T sample t = 1 T sample σ ^ t 2 ,
where T sample denotes the number of training samples. An example of the Bayesian LSTM cell of the proposed BLSTM network is shown in Figure 2, with a zoomed-in plot of the forget gate at time step t in the first layer.

2.2. Performance Evaluation Metrics

To assess the performance of the prediction models, this study adopted the root mean square error (RMSE) metric for deterministic prediction and three metrics for probabilistic prediction: continuous ranked probability score (CRPS), prediction interval coverage probability (PICP), and mean prediction interval width (MPIW), which are formulated as follows.
1.
Metric for Deterministic Forecasting
To evaluate the accuracy of the deterministic forecasting results, the root-mean-square error (RMSE) was selected as a commonly used hydrological evaluation indicator. The RMSE is defined as follows:
RMSE 1 n i = 1 n ( Y i Y ^ i ) 2 ,
where Y ^ i is the predicted variable, Y i is the observed value, and n is the number of samples.
2.
Metrics for Probabilistic Forecasting
A useful metric to assess the accuracy of probabilistic prediction models is the CRPS. The CRPS expresses the distance between the probabilistic forecast p and the observed value Y i and is defined as
CRPS = + P Y ^ i H Y ^ i Y i 2 d Y ^ i ,
P Y ^ i = Y ^ i p x d x ,
H Y ^ i Y i = 0     for   Y ^ i < Y i   1   for   Y ^ i Y i ,
where p x is the probability density function (PDF), P Y ^ i is the prediction cumulative distribution function (CDF), and H is the Heaviside step function, which equals 0 if Y ^ i < Y i   and equals 1 otherwise.
The mean prediction interval width (MPIW) is an effective representation of sharpness in probabilistic predictions. This metric is defined as
MPIW = 1 n i = 1 n ( Y ^ i u Y ^ i l ) ,
where n is the size of the test set, and Y ^ i u and Y ^ i l denote the upper and lower bounds of the 95% prediction interval, respectively.
Prediction interval coverage probability (PICP) or (PI) is the probability that the target lies within the interval provided by the prediction model. PICP is defined as:
PICP = 1 n i = 1 n c i ,   c i = 1 ,     i f   Y i Y ^ i l , Y ^ i u     0 ,     i f   Y i Y ^ i l , Y ^ i u .
Thus, PICP indicates the frequency with which the prediction interval (PI) captures the observed value, ranging from 0 if all predicted values lie outside the PI and 1 if all predicted values lie inside the PI.

3. Case Study

To evaluate the performance of the probabilistic data-driven models under different conditions, three basins in the United States with different hydroclimatic conditions and drainage areas were selected as study areas, as described in the following section.

3.1. Study Area

The study basins were located in different climate regions of three states across the United States, i.e., IN (Indiana), MN (Minnesota), and CA (California). Figure 3 shows the locations of the three basins. The first case study was conducted in Bartholomew County, IN, the second was conducted in Koochiching County, MN, and third was conducted in Shasta County, CA. The drainage area of the river basins was approximately 1560–4420 km2.
Based on the USGS statewide streamflow–water year 2021 report, the annual mean streamflow was ranked by state from 1 to 92, indicating the maximum and minimum annual flow for all years analyzed. Streamflow rankings were grouped into categories of much below normal, below normal, normal, above normal, and much above normal based on percentiles of flow (<10%, 10–24%, 25–75%, 76–90%, and >90%, respectively) [65]. Much-below-normal streamflow with a rank 84–91 is reported in CA and below-normal streamflow with a rank 70–83 is reported in MN. The annual mean streamflow rank for IN was reported to be normal, with a rank 24–50. Daily historical streamflow data for the three selected case studies were obtained from the United States Geological Survey (USGS) website, (https://waterdata.usgs.gov/nwis) (accessed on 1 February 2022).
The descriptive information of the daily streamflow in the three case studies is presented in Table 2. Details of the case studies, including gauge ID, gauge name, and drainage area, are presented in Table 3. For all catchments, streamflow with a 30-day lag was considered owing to the cross-correlation results.

3.2. Experiment Setup

Before using the data to train the model, data preprocessing began with min-max normalization and log transfer as the initial phase of model development. The input time step was then derived from an autocorrelation analysis of the transformed-streamflow time series. Using a threshold of more than 0.5, which represents a moderate relationship, the past 30 days were selected as input. The autocorrelation analysis results for three case studies are given in the Supplementary File. The datasets for the three case studies were split into three sets: the first set accounted for 60% of the data, and it was used for model training; the second set was used for model validation (20%), and the remaining 20% was used for test purposes. Subsequently, the sliding window technique with a window size of 30 days was used. To demonstrate the superior performance of the Bayesian forecasting approach, probabilistic methods that have been widely used in the literature were employed for comparison. More specifically, LSTM-BNN [66], LSTM Monte-Carlo Dropout (LSTM-MC) [62,67], BNN [68], and deterministic LSTM were implemented in this study. Monte Carlo dropout is a straightforward epistemic uncertainty extension to the neural network. In general, dropout is a technique used to avoid overfitting by randomly dropping units during training. This can be considered the application of random noise in training. When this dropout was performed multiple times, multiple results were obtained. The distribution of the samples represents the uncertainty of the prediction model. The structure of the prediction models along with their graphical scheme are given in the Supplementary File.
The prediction models were developed in Python 3.6.9 with the Keras [69], TensorFlow [70], and PyTorch [71] libraries. The prediction model was implemented by an NVIDIA® GeForce® RTX 2070 SUPER and an Intel® Core i9-10920X central processing unit at 3.5 GHz utilizing 128 GB random access memory. For a fair comparison among the prediction models, a grid search for hyperparameter tuning was used to ensure identical evaluation.

4. Result and Discussion

To better clarify the forecasting performance of BLSTM method, the study compares the BLSTM to the LSTM-BNN, BNN, and LSTM-MC in terms of prediction interval uncertainty, sharpness, prediction reliability, and multi-step ahead probabilistic prediction performance. Moreover, LSTM is used as a deterministic model to evaluate the performance of all probabilistic prediction models against the deterministic model. In this section, the predictive ability of the four probabilistic models for 1 day (Scenario I), 7 days (Scenario II), and 30 days (Scenario III) ahead of streamflow prediction is investigated.

4.1. Probabilistic Prediction Performance Assessment

The PICP, MPIW, and CRPS values for the four models in the three case studies during the test period are listed in Table 4. The length of the bar represents the value of the evaluation metrics. In terms of PICP, the higher the value, the better and longer the bar, and vice versa for the other measures. Three major aspects must be considered simultaneously to evaluate probabilistic forecasting performance. PICP refers to the reliability of a model, MPIW refers to the model’s sharpness, and CRPS indicates overall performance. In Scenario I, case study I was considered as an example because the models used the same mechanism to quantify the forecast uncertainty, and the PICP values of the four models were relatively close. Note that the larger the PICP and the smaller the MPIW and CRPS, the better the model performance. We observed that BLSTM showed better performance in handling datasets with high Std and peak streamflow. Case study III had 22,645 samples, which was ~17% and ~34% less than that of case studies I and II, with 27,146 and 34,205 samples, respectively. This difference did not lead to a particular change in the prediction performance of all the models for the first scenario. In this case study, from Scenarios I to III in BLSTM models, PICP decreased ~2% and 4%, respectively. While for case study I, PICP decreased ~25% and 50%, respectively, and for case study II, PICP decreased ~1% and 2%, respectively. Therefore, from the obtained results, we inferred that for single-step ahead prediction, the results were promising for all case studies, and the number of samples and peak streamflow did not affect the prediction performance. This made BNNs extremely data-efficient because they could learn from even a small dataset without overfitting. Furthermore, we predict that more uncertainty is associated with the results, particularly for the case study with a higher peak of streamflow, leading to wider prediction intervals and lower coverage. As expected, all models exhibited better predictive performance during shorter lead times (1–7 days) than during the longer horizon (30 days). Therefore, from the obtained results, we can infer that the probabilistic forecasting model can lead to higher uncertainty and lower accuracy over a longer forecasting horizon.
To further evaluate the results of the four probabilistic models for all scenarios in the three case studies, LSTM as a well-known deterministic model was used to make a comparison in terms of RMSE, as shown in Figure 4. All probabilistic models in all horizons performed well and provided more accurate prediction performance in terms of RMSE than the deterministic LSTM, indicating the superiority of all probabilistic models in comparison with the conventional deterministic model. The range of RMSE indicated that all models were fairly trained, and they showed promising predictability performance in terms of RMSE.
As shown in Figure 5, the probability that the prediction lies within the prediction interval by the LSTM-MC model is higher, followed by the LSTM-BNN, BNN, and BLSTM in the first scenario. The values of PICP indicate the percentage of the observed streamflow data lies within their 95% predictive intervals. In Scenario I, LSTM-MC outperformed BNN in terms of PICP. Moreover, for a longer horizon (7 days and 30 days ahead), due to the gradient vanishing of LSTM and BNNs as non-sequential models, both showed the lowest coverage and the points falling within the interval decreased by increasing the uncertainty in comparison with BLSTM and LSTM-BNN.
The MPIW values of the four models during the test period are shown in Figure 6a. The MPIW was considered an effective representation of sharpness in probabilistic predictions, and referred to the concentration of the predictive distributions. The more concentrated the predictive distributions, the lower the MPIW, the sharper the prediction, and consequently the better the predictive performance. As shown in Figure 6a, BLSTM has the lowest MPIW, which indicates that it is the sharpest predictive model among the other models in all scenarios. The stand-alone BNN and LSTM-MC were slightly different, whereas the LSTM-MC obtained the highest MPIW value among the other models, particularly by increasing the horizon. Compared with the BLSTM model with the narrowest MPIW, LSTM-MC had the worst prediction sharpness over all horizons. The fact that BLSTM presents the best predictive capability indicates the significance of capturing both epistemic and aleatoric uncertainties.
To comprehensively evaluate the probability prediction accuracy and reliability, a comparison among all prediction models in terms of the CRPS is shown in Figure 6b. Overall, BLSTM and LSTM-BNN competed with each other in all case studies and scenarios. However, in the longer horizon, BLSTM outperformed other models and proved its superiority by keeping more points falling within its forecasting interval while keeping the interval as narrow as possible while also increasing the uncertainty for a longer horizon. Therefore, the proposed BLSTM model outperformed the other models in terms of RMSE, MPIW, and CRPS, demonstrating the forecasting accuracy, sharpness, and overall performance of the model.

4.2. Impact of Forecast Horizon in Probabilistic Prediction Performance

The prediction results of all models are compared graphically in Figure 7, Figure 8, Figure 9 and Figure 10 in the form of time series for the entire test set for case study II for all scenarios (forecasting horizon). Considering space limitations, we have avoided adding the results of all case studies and scenarios in the main text, and only the results of case study II are presented. The actual streamflow and forecast value for the test period are represented by black and red curves, respectively. The red band represents the prediction interval, with a 95% confidence level. The probabilistic forecasts generated with the BLSTM model presented the benefits of high prediction coverage of observed streamflow data (PICP) with a tighter prediction width (MPIW) and better overall performance (CRPS), corresponding to reliability, sharpness, and resolution. Furthermore, accurate peak prediction, which is a crucial factor for disaster prevention and water resources management, can be predicted with reasonable magnitudes with the proposed BLSTM. Additionally, with increasing the forecast horizon, BLSTM still showed reliable performance, while other models were incapable of handling such a situation. In forecast horizon 30, massive fluctuations in the prediction results occurred for all the models and case studies. However, most of the prediction results were covered by the 95% interval in Scenario I, followed by Scenario II.
As shown in Figure 11, with an increase in the forecasting horizon from 1 to 30 days, the MPIW and CRPS values increase, and the PICP decreases. This indicates that the accuracy of the prediction models decreases with an increase in the forecasting horizon. The prediction accuracy over longer horizons decreased mainly as a result of the accumulative error issue in multi-step ahead recursive models and the gradient vanishing issue in long sequence time-series forecasting. Nevertheless, the predictive mean values of probabilistic streamflow from the BLSTM model matched the observations better than the other three models. The performance of all models gradually worsened with increasing lead times for the three case studies. As shown in Figure 11, when the overall prediction accuracy was low, MPIW was smaller. The interval width of the forecasting with LSTM-MC increased rapidly with the prediction horizon. The average interval width of the proposed BLSTM was much smaller than that of the other models. Simultaneously, the overall performance in terms of CRPS was higher, proving the superiority of the proposed method for the probability forecasting of daily streamflow, particularly for longer prediction horizons.
For a better and more vivid comparison of all model performances, the time series of all models for all case studies in the three scenarios are shown in Figure 12a–c for the first year of the test period (365 days). We observed that BNN and LSTM-MC underestimated the peak flows with a misleading trend in the first 365 days. In the first scenario, a 95% PI was relatively narrow and constant for all models, indicating that models captured both low and high flow values appropriately with low uncertainty. However, longer horizons in Scenarios II and III were associated with a wider 95% PI, indicating greater model uncertainty.
Furthermore, we observed that all case studies can be effectively covered by the PI. Furthermore, for case study II, which had the lowest peak, BLSTM achieved the best results in all three scenarios. In contrast, case study I, with the highest peak at 1654 m3/s and Std. of 90 m3/s, achieved the worst prediction results for all scenarios. From Scenarios I to II in case study I for BLSTM, LSTM-BNN, BNN, and LSTM-MC, PICP decreased by approximately 25, 38, 48, and 53%, respectively, indicating the best performance of BLSTM in maintaining its predictability in case study I, with the highest peak and Std in the extended horizon prediction. Moreover, by increasing the horizon, prediction performance for case study I dramatically decreased, whereas, in terms of PICP for BLSTM, case studies II and III from Scenarios I to III decreased by approximately 1–2% and 2–4%, respectively. Furthermore, LSTM-MC and BNN achieved the worst overall prediction performance for all the scenarios. The catchment area of case study I was relatively large, and heavy rain was the primary source of streamflow. These two characteristics cause the seasonal and annual variations in streamflow to be greater than those in the other two case studies. In this case study, the streamflow was very stable and small during the dry season, whereas in the rainy season, the streamflow increased steeply and then decreased. This made forecasting challenging and resulted in a higher uncertainty than that in the other case studies. Therefore, for this type of catchment, using more in-situ meteorological predictors, such as precipitation and temperature, along with available high-resolution large-scale hydroclimate data, can improve forecasting accuracy.
The kernel density estimation plots of the daily streamflow prediction of all models for case study II are displayed in Figure 13a–c. As depicted, the kernel density estimation plots are on the top with boxplots, and the data points of the prediction are underneath. The boxes represent the inner quartiles, the vertical lines within the box indicate the median, and the diamonds represent the outliers in each model. As shown in Figure 13, the prediction variance of BLSTM is lower in comparison with the other models, in particular for the Scenario III which is forecasting horizon 30. Moreover, the inter-quartile range of BLSTM is smaller which indicates that the BLSTM prediction results has less dispersion while LSTM-MC has the highest dispersion. The results of this study indicate that BLSTM shows the best overall probabilistic prediction performance.

5. Conclusions

This study proposes BLSTM as a probabilistic prediction model to estimate streamflow uncertainty. For comparison, three probabilistic models and one deterministic model, including LSTM-BNN, BNN, LSTM-MC, and LSTM, are developed under three scenarios: 1 day, 7 days, and 30 days ahead daily streamflow forecasting. The results are compared in terms of reliability, sharpness, and overall performance for three different case studies in the USA. Reliability is measured by computing the PICP, sharpness is measured by computing the MPIW, and overall performance is measured by CRPS. The results show that all probabilistic models outperformed the deterministic model (LSTM). Moreover, among the probabilistic models, BLSTM is superior. The Bayesian LSTM achieves better results with less computing time and is easier to train than those of LSTM-BNN and BNN. The results reveal that the BLSTM network with variational inference achieves the highest accuracy. The fact that BLSTM shows the best predictive performance indicates the significance of capturing temporal dependencies by considering both uncertainties. Moreover, taking advantage of the long- and short-term dependencies and capturing the inherent uncertainty that is inevitable in hydrology provides better prediction results. For longer forecast horizons, models such as the BNN and LSTM-MC perform poorly due to the fact that the former is not an autoregressive model, and both have the gradient-vanishing problem in the long sequence time series. In addition, the issue of cumulative error in multi-step ahead recursive model is inevitable. Future research will investigate an enhanced network structure with a large prediction capacity, such as attention-based and parallel-feed architectures, to handle the long sequence time-series forecasting. In addition to the recursive models, other multi-step ahead prediction strategies, such as direct and hybrid, can be studied to minimize the accumulated error issue for longer horizons forecasting. Moreover, in the relevant future work, meteorological parameters such as precipitation, temperature, and humidity will be included as input to allow the model to detect the complexity necessary to enhance the accuracy of prediction, particularly for the longer horizon, and to evaluate the effect of multivariate input on model uncertainty.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/w14223672/s1, Figure S1: Autocorrelation function plots of transformed-streamflow timeseries; Figure S2: The visualizations of network execution graphs and traces for BLSTM models’ outputs; Figure S3: The visualizations of network execution graphs and traces for LSTM-BNN models’ outputs; Figure S4: The visualizations of network execution graphs and traces for BNN models’ outputs; Figure S5: The visualizations of network execution graphs and traces for LSTM-MC models’ outputs; Figure S6: Prediction results of all models for case study II; Table S1: General structures of deep neural networks.

Author Contributions

F.G.: conceptualization, methodology, investigation, software, validation, formal analysis, data curation, writing—original draft, writing—review and editing, visualization. D.K.: supervision, validation, writing—review and editing, resources, and funding acquisition. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by (1) the Korea Ministry of Environment (MOE) as “Graduate School specialized in Climate Change” and (2) the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Korea Ministry of Trade, Industry & Energy (MOTIE) grant number [20224000000260].

Data Availability Statement

Data will be made available on request. Daily historical streamflow data for the three selected case studies were obtained from the United States Geological Survey (USGS) website, accessed on 1 February 2022 (https://waterdata.usgs.gov/nwis).

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Structure of the Bayesian neural network.
Figure 1. Structure of the Bayesian neural network.
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Figure 2. Example of the proposed BLSTM network with a zoomed-in plot of the forget gate at time step t in the first layer.
Figure 2. Example of the proposed BLSTM network with a zoomed-in plot of the forget gate at time step t in the first layer.
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Figure 3. Location of 3 study basins in different climate regions across the United States.
Figure 3. Location of 3 study basins in different climate regions across the United States.
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Figure 4. Comparison among prediction models in terms of RMSE.
Figure 4. Comparison among prediction models in terms of RMSE.
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Figure 5. Comparison among prediction models in terms of PICP.
Figure 5. Comparison among prediction models in terms of PICP.
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Figure 6. Comparison among prediction models in terms of (a) MPIW and (b) CRPS.
Figure 6. Comparison among prediction models in terms of (a) MPIW and (b) CRPS.
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Figure 7. Probabilistic streamflow forecasting results obtained by BLSTM for case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
Figure 7. Probabilistic streamflow forecasting results obtained by BLSTM for case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
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Figure 8. Probabilistic streamflow forecasting results obtained by LSTM-BNN for case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
Figure 8. Probabilistic streamflow forecasting results obtained by LSTM-BNN for case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
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Figure 9. Probabilistic streamflow forecasting results obtained by BNN for case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
Figure 9. Probabilistic streamflow forecasting results obtained by BNN for case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
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Figure 10. Probabilistic streamflow forecasting results obtained by LSTM-MC for case study II (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
Figure 10. Probabilistic streamflow forecasting results obtained by LSTM-MC for case study II (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
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Figure 11. Change in probabilistic streamflow forecasting results by increasing horizon for (a) case study I, (b) case study II, and (c) case study III.
Figure 11. Change in probabilistic streamflow forecasting results by increasing horizon for (a) case study I, (b) case study II, and (c) case study III.
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Figure 12. Prediction results of all models with 1, 7, and 30 days ahead forecasting for (a) case study I, (b) case study II, and (c) case study III.
Figure 12. Prediction results of all models with 1, 7, and 30 days ahead forecasting for (a) case study I, (b) case study II, and (c) case study III.
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Figure 13. Kernel density estimation plots of daily streamflow prediction of all models in case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
Figure 13. Kernel density estimation plots of daily streamflow prediction of all models in case study II for (a) forecast horizon 1, (b) forecast horizon 7, and (c) forecast horizon 30.
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Table 1. Overview of recent probabilistic prediction studies in the field of hydrology.
Table 1. Overview of recent probabilistic prediction studies in the field of hydrology.
FieldProbabilistic MethodBase ModelsPosterior Approximation *Evaluation MetricsRef.
VIMCMDeterministicProbabilistic
StreamflowLSTM-HetGPANN, HetGP, GLM, LSTM--NSE, RMSE, MRE, MSLEpercentage of coverage (POC) and the average interval
width (AIW)
[39]
FloodLSTMARIMA--RMSE, MAE[40]
StreamflowLSTM with multiparameter
ensemble and dropout
ensemble
_-PBIAS, MARE, RMSE, NSE, KGEPOC, average
width (AW), average
interval score (AIS)
[41]
StreamflowVariational Bayesian Long Short-Term Memory network (VB-LSTM)Bayesian model Averaging (BMA)-MAECRPS[42]
RunoffXGBoost (XGB)
and Gaussian process regression (GPR) with Bayesian optimization algorithm (BOA)
GBR, LGB, CNN, LSTM, ANN, SVR, QR, GPR—combined with GPR--RMSE, MAPE, R2Coverage probability, Mean width percentage, Suitability metric, CRPS[38]
RunoffB-spline quantile regression model combined with kernel density estimationQR, QRNN--RMSE, R2, QrPICP, PINAW, CRPS[43]
StreamflowBayesian LSTM model physics-based hydrologic model (Precipitation-Runoff Modeling System)-NSE, RMSE-observations standard deviation ratio (RSR)[44]
* The tick mark (✓) denotes the application of Posterior Approximation.
Table 2. Descriptive information of daily historical streamflow data for three case studies.
Table 2. Descriptive information of daily historical streamflow data for three case studies.
CriteriaCase Study 1Case Study 2Case Study 3
No. Samples 27,14634,20522,645
Mean (m3/s)583031
Std (m3/s)905142
Min (m3/s)2.50.63
25% (m3/s)1349
50% (m3/s)311120
75% (m3/s)643236
Max (m3/s)16547021274
Table 3. Details of the selected case studies.
Table 3. Details of the selected case studies.
Case Study. No.Station IDG-NameElev. (m)Drainage Area (km2)Lon. (°)Lat. (°)Period
103364000EAST FORK WHITE RIVER AT
COLUMBUS, IN
183.8442185°55′32″39°12′00″1948–2022
205131500LITTLE FORK RIVER AT
LITTLEFORK, MN
330.3440393°32′57″48°23′45″1928–2022
311368000MCCLOUD R AB SHASTA LK CA335.31564122°13′07″40°57′30″1945–2007
Table 4. Summary of prediction performance results for three case studies and three scenarios.
Table 4. Summary of prediction performance results for three case studies and three scenarios.
Forecast Horizon = 1Forecast Horizon = 7Forecast Horizon = 30
ModelMetricCase ICase IICase IIICase ICase IICase IIICase ICase IICase III
BLSTMPICP0.950 Water 14 03672 i0010.964 Water 14 03672 i0020.956 Water 14 03672 i0030.709 Water 14 03672 i0040.958 Water 14 03672 i0050.941 Water 14 03672 i0060.477 Water 14 03672 i0070.943 Water 14 03672 i0080.921 Water 14 03672 i009
MPIW0.021 Water 14 03672 i0100.006 Water 14 03672 i0110.016 Water 14 03672 i0120.023 Water 14 03672 i0130.024 Water 14 03672 i0140.023 Water 14 03672 i0150.032 Water 14 03672 i0160.042 Water 14 03672 i0170.028 Water 14 03672 i018
CRPS0.087 Water 14 03672 i0190.035 Water 14 03672 i0200.066 Water 14 03672 i0210.375 Water 14 03672 i0220.212 Water 14 03672 i0230.214 Water 14 03672 i0240.576 Water 14 03672 i0250.437 Water 14 03672 i0260.337 Water 14 03672 i027
LSTM-BNNPICP0.957 Water 14 03672 i0280.967 Water 14 03672 i0290.971 Water 14 03672 i0300.591 Water 14 03672 i0310.962 Water 14 03672 i0320.956 Water 14 03672 i0330.371 Water 14 03672 i0340.953 Water 14 03672 i0350.942 Water 14 03672 i036
MPIW0.023 Water 14 03672 i0370.008 Water 14 03672 i0380.018 Water 14 03672 i0390.035 Water 14 03672 i0400.037 Water 14 03672 i0410.032 Water 14 03672 i0420.039 Water 14 03672 i0430.076 Water 14 03672 i0440.040 Water 14 03672 i045
CRPS0.086 Water 14 03672 i0460.034 Water 14 03672 i0470.070 Water 14 03672 i0480.400 Water 14 03672 i0490.226 Water 14 03672 i0500.237 Water 14 03672 i0510.615 Water 14 03672 i0520.457 Water 14 03672 i0530.367 Water 14 03672 i054
BNNPICP0.955 Water 14 03672 i0550.953 Water 14 03672 i0560.961 Water 14 03672 i0570.496 Water 14 03672 i0580.779 Water 14 03672 i0590.870 Water 14 03672 i0600.281 Water 14 03672 i0610.630 Water 14 03672 i0620.865 Water 14 03672 i063
MPIW0.027 Water 14 03672 i0640.009 Water 14 03672 i0650.022 Water 14 03672 i0660.045 Water 14 03672 i0670.050 Water 14 03672 i0680.040 Water 14 03672 i0690.053 Water 14 03672 i0700.141 Water 14 03672 i0710.047 Water 14 03672 i072
CRPS0.101 Water 14 03672 i0730.041 Water 14 03672 i0740.083 Water 14 03672 i0750.412 Water 14 03672 i0760.240 Water 14 03672 i0770.262 Water 14 03672 i0780.634 Water 14 03672 i0790.461 Water 14 03672 i0800.371 Water 14 03672 i081
LSTM-MCPICP0.973 Water 14 03672 i0820.994 Water 14 03672 i0830.981 Water 14 03672 i0840.454 Water 14 03672 i0850.948 Water 14 03672 i0860.913 Water 14 03672 i0870.268 Water 14 03672 i0880.945 Water 14 03672 i0890.909 Water 14 03672 i090
MPIW0.046 Water 14 03672 i0910.040 Water 14 03672 i0920.027 Water 14 03672 i0930.049 Water 14 03672 i0940.050 Water 14 03672 i0950.032 Water 14 03672 i0960.057 Water 14 03672 i0970.076 Water 14 03672 i0980.039 Water 14 03672 i099
CRPS0.109 Water 14 03672 i1000.071 Water 14 03672 i1010.106 Water 14 03672 i1020.414 Water 14 03672 i1030.251 Water 14 03672 i1040.248 Water 14 03672 i1050.636 Water 14 03672 i1060.545 Water 14 03672 i1070.376 Water 14 03672 i108
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Ghobadi, F.; Kang, D. Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study. Water 2022, 14, 3672. https://doi.org/10.3390/w14223672

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Ghobadi, Fatemeh, and Doosun Kang. 2022. "Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study" Water 14, no. 22: 3672. https://doi.org/10.3390/w14223672

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