Rainfall prediction in coastal hilly areas based on VMD–RSA–DNC

Rainfall prediction is closely related to human life and socioeconomic aspects, and accurate rainfall prediction can provide a reliable scientific basis for human travel and life safety, but there are many challenges in establishing efficient rainfall prediction models. The commonly used rainfall prediction tools are long in periodicity and too cumbersome, which are not effective in predicting areas with large weather variations and chance. In recent years data-driven machine learning models have become a hot topic in the field of hydrological forecasting. Vidyarthi & Jain (2023) proposed to integrate two machine learning methods, artificial neural network (ANN) and decision tree, for rainfall prediction. Fahad et al. (2022) proposed a depth forecasting model based on an optimized gated recurrent unit. Bajpai et al. (2023) successfully predicted summer monsoon rainfall using 118 years of historical time-series data. Santos et al. (2023) presented a method for forecasting using wavelet neural networks (WNNs) and showed that for long-term forecasting in the Mahanadi River basin, the WNN method outperforms all other applications. Rojas et al. (2023) analyzed the potential of deep learning and used probabilistic ANNs in four observation locations for postprocessing integrated precipitation forecasts, and the results showed the potential of deep learning models in weather forecasting workflows. Xie et al. (2019) proposed to optimize the input parameters of the wavelet neural network using the genetic algorithm to establish a Global Positioning System (GPS) predictable precipitation model based on the genetic wavelet neural network, which better reflected the variation of precipitation characteristics, and the accuracy of the obtained results was significantly improved compared with both the backpropagation neural network and the wavelet neural network. Lv et al. (2020) developed a coupled variational modal decomposition (VMD)–least squares support vector regression machine model and applied it to monthly runoff prediction at several hydrological stations and obtained better prediction results.

From a comprehensive view of domestic and international research, a single model has limited accuracy in predicting rainfall time series, and rainfall prediction models coupled with many different methods are widely used. Recurrent neural networks are widely used to deal with simulation and prediction problems of rainfall time-series data because of their recurrent structure settings, but their utilization of long-lasting information (long memory) is still somewhat limited. Rainfall time series frequently have a certain correlation between before and after, and to effectively extract the feature information within the monthly rainfall time series, some researchers have introduced data preprocessing means such as time-series signal decomposition algorithms into the rainfall time-series prediction process (Wang et al. 2019). Among them, wavelet decomposition requires preset basis functions with poor adaptiveness (Sun et al. 2018), empirical mode decomposition (EMD) has strong adaptiveness but is prone to problems such as modal blending and endpoint effects, and the VMD algorithm can self-adjust the number of modal decompositions to effectively avoid modal confusion and reduce the complexity of sequence data. Long short-term memory (LSTM), a neural network proposed in recent years, effectively solves the problem of long memory loss by changing the neuron structure based on traditional recurrent neural networks, but its processing of long-range sequences relies on a large amount of computational resources and is ineffective. To solve this problem, a variety of neural networks with external storage mechanisms have been designed, and the most famous one is the differentiable neural computer (DNC) hybrid learning neural network proposed by Graves et al., which is based on the neural turing machine to improve the storage management method. The inclusion of improved temporal memory links allows the DNC to jump read or update memory information, thus combining the advantages of recurrent neural networks and computational processing to improve the problem of memory forgetting in LSTM (Graves et al. 2016). There are several hyperparameters that need to be set manually in the recurrent neural network prediction process, and different combinations of hyperparameters affect the final results of the model (Qiu et al. 2020). In contrast, the selection of hyperparameters for the network structure has not been clearly defined in a series of learning models (Aufa et al. 2020). Therefore, many algorithms for optimizing hyperparameters have been proposed, among which RSA is a new nature-inspired meta-heuristic optimizer proposed in 2021, which has the advantages of less adjustable parameters, strong stability of optimization search, and easy programming implementation compared with other intelligent optimization algorithms (Abualigah et al. 2022). Therefore, this article innovatively uses RSA to optimize the controller of DNC on this basis and then predicts the rainfall data after VMD decomposition, so as to establish a VMD–RSA–DNC model and apply it to rainfall prediction in coastal hilly areas, and to explore the application effect of VMD–RSA–DNC in rainfall prediction.

Variational mode decomposition (VMD) is an adaptive and completely nonrecursive method for modal variation and signal processing (Zuo et al. 2020). The adaptive nature of the EMD method is that it determines the number of modal decompositions of a given sequence according to the actual situation, and the subsequent search and solution process can adaptively match the optimal center frequency and finite bandwidth of each mode. It reduces the nonsmoothness of rainfall time series with high complexity and strong nonlinearity and decomposes a relatively smooth sub-series containing several different frequency scales, which is suitable for nonstationary series.

• (1)
  Construction of the variational problem: The constraints of the model are to ensure that the sum of bandwidths of all IMFs is minimum and that the sum of individual Intrinsic Modal Function (IMF)s is equal to the input signal. Assuming that the original signal f is decomposed into k components, the decomposition sequence is guaranteed to be a finite bandwidth of modal components with central frequency, while the sum of estimated bandwidths of each mode is minimized, and the constraint is that the sum of all modes is equal to the original signal. The corresponding constrained variational expression is given as follows:

  

  (1)

  

  (2)

  where ‘K’ is the number of modes to be decomposed (positive integer), and are the decomposed modal component and the center frequency, respectively. is the Dirac function, is the convolution operator, and is the phase volume representation of the central frequency on the complex plane.
• (2)
  Solution of the variational model: To find the optimal solution for the aforementioned variational model, the constrained variational model is transformed into an unconstrained variational model by constructing an augmented LaGrange expression using a quadratic penalty factor and a LaGrange multiplier ⁠, which is expressed as follows:

  

  (3)

  where can be used to guarantee the reconfiguration accuracy of the signal to impose a limit on the bandwidth. BY using the alternating direction multiplier method, ⁠, ⁠, and are updated alternately until the convergence condition is satisfied.

The RSA is inspired by the social behavior of crocodiles in nature and contains two main mechanisms: encirclement mechanism and hunting mechanism. RSA is designed to simulate the crocodile's four location update strategies: high walking, belly crawling, hunting coordination, and hunting cooperation. Traditional grid search and random search are computationally intensive, while RSA imitates the social behavior of crocodiles in nature. The RSA algorithm has the advantages of less adjustable parameters, strong stability of the optimization search, and easy programming implementation compared with other intelligent optimization algorithms.

The RSA mathematical description is briefly described as follows:

• (1)
  Initialization: Set the crocodile population size N. The formula for calculating of the initialized crocodile individual location is as follows:

  

  (4)

  where the ‘j’ denotes the dimensional location of the ‘i’ alligator; N denotes the population size; n denotes the problem dimension; UB and LB denote the upper and lower limits of the search space, respectively; and ‘rand’ denotes a uniformly distributed random number between 0 and 1.
• (2)
  Surrounding mechanism (exploration phase): RSA implements the transition between the alligator surrounding (exploration) and hunting (exploitation) mechanisms by dividing the total number of iterations ‘T’ into four parts. The encirclement mechanism mainly performs a high walking, belly crawling strategy to explore a wider search area and find a better solution. The encirclement mechanism position update operator is specified as follows:

  

  (5)

Here, means the ‘j’ position of the ‘i’ alligator in the ‘t + 1’ iteration; denotes the ‘j’ position of the best solution obtained so far; ‘t’ denotes the current number of iterations, ‘T’ denotes the maximum number of iterations; denotes the hunting operator at the ‘j’ position of the ‘i’ alligator, which is described as follows: ⁠, where denotes the upper and lower boundaries of the ‘j’ position, respectively. α denotes the sensitive parameter to control the accuracy of hunting mechanism exploration, taking the value 0.1; ε denotes the very small constant; β denotes a sensitive parameter controlling the exploration accuracy of the envelope mechanism, taking the value 0.1; used to narrow the search area; denotes a random number between ⁠; indicates the random position of the ‘i’ alligator; denotes the probability ratio, described as ⁠; denotes a random integer between −1 and 1; and other parameters have the same meaning as mentioned earlier.

• (3)
  Hunting mechanism (development stage). Crocodiles exploit search during hunting mainly through hunting coordination and hunting cooperation strategies. Unlike the encirclement mechanism, the crocodile hunting mechanism enables them to easily approach the target prey, i.e., the algorithm optimal solution. The hunting mechanism position update operator is described as follows:

  

  (6)

The meaning of the parameters in the aforementioned formula is the same as earlier.

As shown earlier, the DNC is equivalent to adding external memory to the LSTM, which greatly improves the problem of memory forgetting of the LSTM.

This design allows the controller to regulate its output decisions by reinforcing the dependence on the memory storage matrix.

The controller η operates on the data in the memory through the read/write head. The read or write positions are determined by the corresponding weights, and the set of allowed weights at N positions is the nonnegative quadrant of the standard simplex form in R^N:

The addressing mechanism of the DNC is a combination of several things: content-based addressing and dynamic addressing are used when writing data in memory and content-based addressing and temporal memory links are used to acquire locations when reading data in memory:

• (1)

  Content-based addressing

defines the normalized probability distribution over the storage locations. is a constraint vector defined as a standard (N − 1)-simple form:

• (2)

  Dynamic addressing

Dynamic addressing is achieved through the release list ⁠. The release list records the free storage locations in the memory and is updated after the controller makes changes to the data in the memory.

The value of is obtained, which is sorted in the ascending order, and the release list can be updated. records the storage location with the lowest utilization.

• (3)

  Write weights

The VMD algorithm is based on the strict variational theory, and the number of modal decompositions is selected autonomously to make the decomposed components more regular (Zhang et al. 2023), while the RSA can realize the adaptive selection of the embedding dimension of each component in the prediction. Each IMF obtained from the VMD decomposition is a single harmonic, and the sequence complexity is greatly reduced compared to the original rainfall data, thus allowing effective adaptation to the DNC predictions.

On the basis of this, we combine VMD, RSA, and DNC to build a rainfall time-series prediction model based on VMD–RSA–DNC. VMD decomposes the rainfall data into several components and then uses DNC to predict each component IMF separately; and train and solve the posterior probability distribution of weight parameters of DNC based on variational inference; and use the training sample mean square error (MSE) as the fitness function of the RSA-optimized DNC controller. The crocodile individual position is initialized, the crocodile population fitness value is calculated, the current optimal crocodile individual position is compared and continuously updated, and whether the termination condition is satisfied is judged. If it is satisfied, output (Li & Cui 2022), which is used as the input layer weights and the implied layer bias matrix of the DNC controller for prediction. At the end of the algorithm, the prediction results of each DNC are reconstructed and superimposed to obtain the final prediction results of the rainfall time series based on the VMD–RSA–DNC model. The model coupling process is shown in Figure 3.

As shown in Figure 5, the suddenness of rainfall in the study area, the short duration, and the large amount of rainfall make it a great challenge to build prediction models. There is often a certain correlation between the rainfall time series before and after, and it is difficult for a single neural network model to achieve a better prediction effect, and it cannot simulate some high-frequency sudden change data well. Therefore, the use of rainfall prediction models coupled with different methods is an important way to improve the prediction accuracy.

The VMD–RSA–DNC model was used to train the rainfall data of Zhoushan, Ningbo, Taizhou, and Wenzhou cities from 2000 to 2018 and validate the rainfall from 2019 to 2020. The results of the training and validation periods using the training sample MSE as the fitness function of the RSA-optimized DNC controller are shown in Table 2.

As can be seen from Figure 8 and Table 1, the predicted values of rainfall data for four cities in eastern Zhejiang for 2019–2020 are basically consistent with the true values. Although the error reaches more than 5% in individual months, the occurrence of extreme weather in individual months can cause this phenomenon. From the aforementioned results, it can also be seen that the maximum and minimum relative errors are divided into 9.62 and 0.17%, and all relative errors are less than 20%.

From Figure 9, it can be seen that the VMD–RSA–DNC model is more accurate in predicting the overall trend of monthly rainfall in all four cities, and the prediction at the extreme values of the samples has a significant advantage compared with the other three models.

Table 2 shows the error comparison of the four cities, and it can be seen that the VMD–RSA–DNC model has the best rainfall prediction results for Zhoushan, Ningbo, Taizhou, and Wenzhou, with the average root-mean-square error (RMSE) of 5.43, the average mean absolute percentage error (MAPE) of 3.59%, and the average Nash–Sutcliffe efficiency (NSE) of 0.95, which is significantly better than the other three models and has higher prediction accuracy.

In summary, the combination of modeling with VMD has greatly improved the learning effect of DNC, while the combination of RSA technology has reduced the time step artificial selection and other aspects in the model prediction process, effectively reducing the uncertainty in the modeling process and making the model achieve good prediction results.

Based on the VMD–RSA–DNC model to carry out the rainfall prediction study of four meteorological stations in eastern Zhejiang Province and by comparing the prediction effects of different models under VMD decomposition and RSA optimization, the following conclusions were obtained:

• (1)

  For rainfall time series with strong abrupt variability in coastal hilly areas, they are subjected to VMD processing, so that the complex rainfall time series, which are not easy to extract regularity, are decomposed into a number of regular and simple signals, and the more regular and useful information obtained from the decomposition is combined with DNC for prediction, which effectively brings into play the advantage that DNC can better perform a priori learning for long- and short-term memory data. There are several hyperparameters that need to be set manually in the prediction process, and the RSA optimization algorithm is used to solve the optimization problem, which reduces the link of artificial selection of model time step, effectively reduces the uncertainty in the modeling process and obtains a better prediction effect; the DNC selects a variant of LSTM as a trainable neural network controller and combines it with a readable and writeable external memory, so as to achieve the effect of the linear combination of short-term and long-term memory predictions, which is a breakthrough solution to the problem of memory forgetting of LSTM and more accurate prediction of rainfall time series.

• (2)

  The maximum and minimum relative errors predicted by the VMD–RSA–DNC model are 9.62 and 0.17%, and all relative errors are less than 20%, with an average RMSE of 5.43, an average MAPE of 3.59%, and an average NSE of 0.95, and compared with the three models EMD–RSA–DNC, VMD–DNC, and VMD–RSA–LSTM, the VMD–RSA–DNC has the best prediction effect.

• (3)

  The prediction effect of this model for the May to June rainy season and the July to September typhoon season in the coastal hilly area needs to be improved. The rainfall prediction data are not only a simple time series but also related to a variety of factors such as the geographical location of the study area. The prediction method proposed in this article is only applied to rainfall prediction in coastal hilly areas, and applying the coupled model to different characteristics of the time series for prediction is the next research direction.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.

This work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (CN) [grant numbers 17A570004].

All authors contributed to the study conception and design. Writing and editing: Xianqi Zhang and Qiuwen Yin; chart editing: Fang Liu; preliminary data collection: Haiyang Li, Haiyang Chen. All authors read and approved the final manuscript.