ARIMA-Based Forecasting of Wastewater Flow Across Short to Long Time Horizons

Abstract

Improving urban wastewater treatment efficiency and quality is urgent for most cities. The accurate wastewater flowrate forecast of a wastewater treatment plant (WWTP) is crucial for cutting energy use and reducing pollution. In this study, two hybrid models are proposed: ARIMA–Markov and ARIMA–LSTM–Transformer. Using 5 min-interval inlet flowrate data from a WWTP in 2024, the two models were verified and compared. Forecasts for 1 day, 7 days, and 2 months ahead were made, and model accuracies were compared. Ten repetitions with the same dataset assess stability, and ARIMA–LSTM–Transformer, with better performance, were selected. Then, the Whale Optimization Algorithm (WOA), Particle Swarm Optimization (PSO) algorithm, and Sparrow Search Algorithm (SSA) were used for optimization, with the WOA excelling in accuracy and stability. Experimental results show that compared to the single model Transformer, WOA–ARIMA–LSTM–Transformer did better in forecasting wastewater flowrate. The combined model enables efficient and accurate wastewater flowrate forecasting, highlighting the combined model’s application potential.

1. Introduction

With the acceleration of urbanization and the deterioration of existing urban drainage pipelines [1], stable and efficient WWTP operation has become challenging due to the dynamic variations in WWTP wastewater influenced by pipeline leakage, the intrusion of rainwater and groundwater, and so on. An accurate wastewater flowrate forecast is crucial for the optimization of WWTP operation [2].

However, traditional methods often rely on extensive historical data and empirical judgement [3], which make timely and accurate forecasts difficult for complex irregular wastewater flowrate changes. Especially during extreme weather events or major festivals [4], traditional forecasting methods cannot effectively and quickly capture changes in wastewater flow, resulting in the risk of overloading in the WWTP. Therefore, the development of reliable wastewater flowrate forecast models has become a top priority for today’s WWTP [5].

The current forecasting models of wastewater flow are mainly based on traditional statistical models and deep learning models [6]. Traditional statistical models include Autoregressive Integrated Moving Average (ARIMA), Prophet, Markov, etc. ARIMA is a statistical model widely used in time series analysis and forecasting, and it can effectively capture the trend, seasonality, and noise characteristics in time series data [7]. Maleki et al. [8] applied ARIMA to the inflow characteristic time series of WWTP to create models for short-term (7 days in advance) forecasting. Bień et al. [9] mixed ARIMA with the Long Short-Term Memory (LSTM) model, greatly improving the forecast accuracy of sludge production in sewage treatment plants. The MAPE (Mean Absolute Percentage Error) of the hybrid model is as high as 9.4%. The Prophet model can take into account holiday effects on the basis of trends, seasonality, and other factors. The Markov model is a mathematical model based on Markov properties, which mainly describe stochastic processes. Li et al. [10] used a discrete hidden Markov model to accurately predict water quality.

In addition, deep learning models, such as LSTM and Transformer, have gradually become the mainstream forecast method in recent years. As a special recurrent neural network (RNN), LSTM overcomes the gradient vanishing and gradient explosion problems of traditional RNNs when processing long sequence data and can effectively capture long-term dependencies in sequences that are suitable for dealing with nonlinear problems. Nitzan Farhi et al. [11] proposed a novel machine learning method based on LSTM architecture that utilizes measurements from bioreactors sampled every minute and combines them with climate measurements to greatly improve forecast accuracy. When predicting nitrogen concentration and nitrate concentration, the AUC increased by 1% and 5%, respectively.

However, LSTM is highly complex. It takes a long time to train and requires a large amount of data [12]. J. Ali et al. [13] used the Transformer and ensemble models to predict the ammonium nitrogen levels in rivers and achieved good results. The coefficient of determination (R²) and the relatively high Nash–Sutcliffe Efficiency (NSE) value reached 0.97 and 0.90, respectively. However, Transformer struggles to model local temporal relationships and lacks a physical explanation. In wastewater flow forecasting, a large number of external input features (such as weather data, holidays, and watershed conditions) are used, and the high-dimensional nature of this data may affect the training efficiency and forecast accuracy of the Transformer model.

Hyperparameter optimization is essential to improve model accuracy [14]. Currently, prominent optimization algorithms include WOA, PSO, and SSA. Cui et al. [15] proposed a new load forecast model that integrates the WOA to optimize the hyperparameters of an improved LSTM model, achieving excellent forecast results. Du et al. [16] proposed a model that combines LSTM and kernel density estimation (KDE) using the PSO algorithm to optimize KDE’s hyperparameters, proving the superiority of the PSO optimization model through comparison with numerous hybrid models. Zhang et al. [17] utilized the SSA to optimize Bidirectional Long Short-Term Memory (BiLSTM) and formed the VMD (Variational Mode Decomposition) –SSA–BiLSTM coupled model for predicting the monthly runoff of the Yellow River. Compared with the single BiLSTM model, the R² increased by 0.53059, which greatly improved the accuracy.

The ARIMA–Markov model and the ARIMA–LSTM–Transformer model were developed for wastewater flow forecasting during different periods of time. After comparing the two models, the ARIMA–LSTM–Transformer model was chosen. It was then joined by WOA, PSO, and SSA to optimize hyperparameters, and WOA demonstrated the best effect. Thereafter, the improved WOA–ARIMA–LSTM–Transformer model was created to achieve efficient and accurate wastewater flow forecasting during different periods of time.

In this study, the ARIMA–LSTM–Transformer model was chosen based on the complementarity of the advantages of each model: ARIMA is good at capturing linear trends in time series, but it cannot effectively deal with nonlinear features, and it performs poorly in long-term forecasting. Thanks to its memory unit and gating mechanism, LSTM can capture long-term dependencies more accurately, but the stability of the model is poor, so we further entered the Transformer module. Transformer excels at modeling long-term dependencies and complex nonlinear relationships. However, it is more dependent on the amount of data and computing resources. By integrating these three types of models, the shortcomings of each can be alleviated to a certain extent, and the accuracy and stability of sewage flow prediction can be comprehensively improved.

The paper is divided into five different parts: 1. Introduction; 2. Materials and Methods, including missing value handling, outlier identification, and the ARIMA–Markov model and ARIMA–LSTM–Transformer model; 3. Results and Discussion, including model validation, ablation experiments, and a stability test; 4. Algorithm Optimization, including parameter selection, accuracy testing, and a stability test; and 5. Conclusions.

In this study, Jiawen Ye contributed to the conceptualization, data curation, investigation, methodology, original draft writing, visualization, review and editing, and funding acquisition. Xulai Meng performed the formal analysis, data curation, methodology, original draft writing, and validation. Haiying Wang contributed to the conceptualization, methodology, supervision, funding acquisition, and review and editing. Qingdao Zhou and Siwei An contributed to the data curation, methodology, and original draft writing. Tong An participated in the investigation and in the review and editing. Pooria Ghorbani Bam and Diego Rosso were involved in the revision.

2. Materials and Methods

2.1. Missing Value Handling

Due to reasons such as monitoring equipment failures and data transmission failures, there may be some missing data in the dataset used to train the model [18]. According to statistics, approximately 0.421% of the data in this dataset (See Appendix A) are missing values. The prevailing methodologies for addressing such missing values encompass a range of approaches, including mean, median, or mode interpolation; Lagrange’s interpolation [19]; random forest (RF) [20]; and analogous techniques. Although imputation methods such as mean, median, or mode are easy to implement, they distort the original data distribution and fail to capture the underlying trends in the data. RF is an ensemble learning model composed of multiple decision trees, so it has many parameters and a long training time from synthesizing the results of multiple decision trees. Due to the large amount of data used in this study, using RF would be computationally expensive and difficult to adjust. Consequently, the improved Lagrange’s interpolation method is employed for imputation. The Lagrange interpolation method constructs a function through known discrete data points, enabling the function to pass through all the points successively. It can effectively fit the distribution of a set of data. However, when the data set is too large, the Lagrange interpolation method also has a high computational cost and may exhibit oscillation, leading to poor filling accuracy [21]. Therefore, the dataset was divided into 2108 subsets, with each subset containing 50 data points.

2.2. Outlier Identification

Prior to determining the method for cleaning outliers, it is necessary to ascertain how the data are normally distributed. To this end, a quantile–quantile plot (QQ plot) [22] was constructed (as shown in Figure A1 (see Appendix C)), and the skewness and kurtosis of the data were measured (0.14 and −0.62, respectively). The data points in the image are approximately distributed on both sides of the reference line, although the two sides are slightly divergent. However, due to the large amount of data used in this study, it can be approximated that the data conform to the normal distribution. Common methods for cleaning outliers include the Z-Score method [23], the Median Absolute Deviation method (MAD) [24], and the K-means clustering algorithm [25]. Although MAD is simple to calculate, it is only based on the median and absolute deviation, failing to fully utilize all the information of the data. The result of the K-means clustering algorithm depends on the selection of the initial clustering centers, and its computational complexity is high, making it difficult to handle large-scale data. The Z-Score method, on the other hand, is suitable for normal distributions, is less likely to be affected by extreme values when dealing with large data scales, and is relatively simple to calculate. Therefore, the standard score method (Z-Score method) is used for outlier cleaning.

When the data roughly obey the normal distribution, the probability of $\left|Z_i\right|>3$ is only 0.3%, which is extremely low. The corresponding data are regarded as outliers and are filtered out by Z-Score. The medians of the wastewater flowrate are used to replace them.

2.3. ARIMA–Markov Model

In this study, a forecast model that combines Autoregressive Integrated Moving Average (ARIMA) and Markov structures is developed, known as the ARIMA–Markov model. ARIMA [26] is better at capturing the main linear dependencies, trends, and cyclical components in the time series through historical data and predicting future wastewater flow [27]. Nevertheless, ARIMA by itself faces challenges in handling the non-linear patterns within the data, which makes it hard to predict spikes in wastewater flow caused by heavy rainfall or special events like festivals.

In order to resolve the aforementioned issues, the Markov model [28,29] is hereby proposed. The residuals between the predicted values of ARIMA and the actual observed values are calculated, and the residuals are divided into multiple states according to their value ranges. Thereafter, the Markov state transition modelling is conducted, and the Markov chain is constructed by using the residual state sequence. The transition probability matrix P between the states is then estimated. This modelling can reveal the transition of residuals between different states and thereby capture the potential nonlinear dynamic features in the data. Based on the Markov model, the steady-state distribution is used to correct the residuals, and the corrected residuals are added back to the forecast results of the ARIMA model [Equation (1)]:

$${\widehat{Y}}_t^{adjusted }={\widehat{Y}}_t^{ARIMA }+\mathsf{\Delta }$$

(1)

where ${\widehat{Y}}_t^{adjusted}$ represents the adjusted time series predicted value, ${\widehat{Y}}_t^{ARIMA}$ represents the time series predicted value obtained using ARIMA, and $\mathsf{\Delta }$ represents the residual correction.

This correction takes into account both the major linear trends captured by ARIMA and the predictive bias caused by nonlinear factors and abrupt changes. By taking advantage of both ARIMA in capturing the long-term linear relationship of time series and the characteristics of the Markov model in describing nonlinear state switching and random mutations, a hybrid forecast strategy is formed that takes into account the advantages of both. The experimental results show that the modified forecast results reduce the forecast error to a certain extent. The architecture diagram of ARIMA–LSTM–Transformer is shown in Figure 1.

2.4. ARIMA–LSTM–Transformer Model

A forecast model that combines ARIMA, the Long Short-Term Memory (LSTM) model, and Transformer structures is developed as well, known as the ARIMA–LSTM–Transformer model. The model uses ARIMA to model precipitation data to capture its trending portions. The forecast residuals of ARIMA are then passed on to the LSTM as one of the new features, and then the LSTM can capture more complex time series dependencies [30,31]. Holiday data (see Appendix B) are provided to the LSTM as an additional input feature to enhance the LSTM’s forecast ability for special dates. The LSTM calculates a hidden state based on each time step of the input, capturing local dependencies in the time series. The output of the LSTM is a fixed-length sequence, where the output of each time step contains the “memory” of the current time step and the previous time step, which captures the time dependencies in the sequence. The Transformer Encoder acts as a second layer network and processes the sequence of LSTM outputs. At this point, the output of the LSTM is already a chronological sequence of features, containing the hidden states of each time step. The Transformer Encoder further enhances feature representation through a self-attention mechanism and uses the relationship between different time steps to enhance the features of the LSTM output. The schematic diagram of the multi-head attention mechanism is shown in Figure A2 (see Appendix C).

In order to ensure that information can be efficiently transferred in a multi-layer network, Transformer introduces residual connections behind each sub-layer, such as self-attention and feedforward networks [32]. Layer normalization accelerates training and improves model stability by normalizing each input to ensure that the inputs for each layer have a mean value of 0 and a variance of 1. Eventually, this representation is mapped to the final output value (predicted value) through a fully connected layer. The architecture diagram of ARIMA–LSTM–Transformer is shown in Figure 2.

3. Results and Discussion

3.1. Model Validation

3.1.1. ARIMA–Markov’s Maximized Likelihood Function

The maximum likelihood function is a widely used method for parameter estimation in statistics. The core is to find a set of model parameter values under the premise of the given observation data so that the probability of data occurrence under this set of parameters reaches the maximum and can help determine the model configuration that best conforms to the internal laws of the data.

In our study, in order to ensure that the ARIMA model has good stationarity under different prediction periods, and to avoid information loss or model oversimplification due to excessive differences, the difference order d is limited to [0, 3], and the stationarity of the sequence is confirmed by graphical analysis, a unit root test, and ACF/PACF attenuation. The experimental results show that when d = 2, the differential sequence has reached a plateau, so d = 2 is finally selected.

For different values of $p$ and $q$, the maximum likelihood functions were calculated for the three types of time periods needed: short term (4 days), medium term (4 weeks), and long term (8 months). In this study, the training-to-test split was set at 4:1; accordingly, the prediction horizons shown in the table are four times the length of the test set—namely 4 days, 4 weeks, and 8 months. Importantly, all parameter tuning and selection were conducted exclusively on the training set, with the test set reserved solely for final model evaluation. By comparing the magnitudes of log-likelihood values, the highest point of the three-dimensional function graph was selected as the optimal combination of $p$ and $q$. The graph of the maximum likelihood function is shown in Figure 3. The optimal values of $p$ and $q$ corresponding to each period are shown in

Table 1. The optimal values of $p$ and $q$ in each period ($d=2$).

Time Domain	$\mathit{p}$	$\mathit{q}$
4 days	2	4
4 weeks	3	4
8 months	4	4

.

3.1.2. ARIMA–LSTM–Transformer’s Loss Function

The loss function, as a function that measures the difference between the predicted outcome of the model and the observed outcome, provides a quantitative metric for model evaluation [33]. By calculating the loss value, the degree of deviation between the model’s forecast and the real situation was understood, and the smaller the value is, the closer the forecast of the model is to the observed value and thus the better the performance of the model is. The loss function is divided into many categories, among which the mean square error is often used for regression problems, and the formula is as follows [Equation (2)]:

$$Loss={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(2)

where $y_i$ is the observed value of wastewater flow, ${\widehat{y}}_i$ is the predicted value of the wastewater flow, and $n$ is the total number of samples.

Figure 4 shows the change curves of the training loss (loss) and validation loss (val_loss) functions of the ARIMA–LSTM–Transformer model during the training process. From left to right, the first picture shows the validation loss of the test set for 1 day and the training loss of the training set for 4 days. The second picture displays the validation loss of the test set for 1 week and the training loss of the training set for 4 weeks. The third picture presents the validation loss of the test set for 1 month and the training loss of the training set for 4 months. The training loss reflects the forecast error of the model on the training dataset, demonstrating the learning ability of the ARIMA–LSTM–Transformer model for the features of the training data. The validation loss, calculated through a validation dataset independent of the training set, is used to measure the generalization performance of the model on unseen data.

It can be seen that both the training loss and the validation loss are decreasing and tend to stabilize at relatively low values, and the gap between them does not increase significantly. This indicates that the model does not have a serious overfitting problem. Moreover, it can find the optimal hyperparameter values within a finite number of iterations, suggesting that the learning process is effective.

3.2. Ablation Experiments

An ablation experiment is an experimental method commonly used in scientific research [34], especially in machine learning, deep learning, and other fields. The ablation experiment is used to analyze the impact of individual components or factors in the model on the overall model performance and to observe the changes in the model performance by gradually removing different components or features in the model, ultimately providing insight into the specific role and contribution of each component or feature in the model. The experiment was conducted five times for each time interval in order to eliminate the most accurate and least accurate forecasts. The average predicted wastewater flow values of the ARIMA model and ARIMA–Markov model were compared with the observed values (as shown in Figure 5). The residual plot of the predicted flowrates for each model is shown in Figure 6.

It can be seen from the graph that the ARIMA–Markov model predicted values are the closest to the observed values. Nevertheless, to quantify the comparative performance of Markov–ARIMA and the ARIMA models, the evaluation indicators of the models while drawing the images were calculated, and the results are shown in

Table 2. Evaluation indicators of different forecast models (1).

Model	Entity	1 Day	7 Days	2 Months
ARIMA	$MAE$	193.1590	233.2706	274.2393
	$RMSE$	296.5314	343.8097	377.0760
	$R^2$	0.7501	0.7337	0.7118
ARIMA-Markov	$MAE$	149.6994	188.4114	154.6708
	$RMSE$	225.8415	283.7057	214.6911
	$R^2$	0.8550	0.8187	0.9066

.

As can be seen in Table 2, the ARIMA model’s highest $R^2$ is 0.7501 and lowest $MAE$ and $RMSE$ are 193.1590 and 296.5314, respectively, which appear when the forecast period is 1 day. These data indicate that the forecast of a single ARIMA is more accurate for short intervals of time. However, it is difficult to capture the medium- and long-term data patterns, and there is room for progress.

After adding the Markov module to correct the residuals, the forecast accuracy of the short, medium, and long intervals significantly improves, with an average increase of 17.52% for the $R^2$ and an average decrease of 29.67% and 28.81% for $MAE$ and $RMSE$, respectively. Among them, the long-term forecast increases the most significantly, with the $R^2$ side increasing by 27.37%, and the $MAE$ and $RMSE$ decreasing by 43.60% and 43.06%, respectively. The statement above shows that the hybrid model greatly improves the forecast ability of a single ARIMA for long-term data and verifies the accuracy and feasibility of the fusion of ARIMA and Markov.

Another ablation experiment was based on the ARIMA, LSTM, and Transformer models. The experiment was conducted five times for each time interval in order to eliminate the most accurate and least accurate forecasts. The average predicted wastewater flow values of the LSTM, Transformer, Transformer–LSTM, and ARIMA–LSTM–Transformer models were compared with the observed values (as shown in Figure 7). The residual plot of the predicted flowrates for each model is shown in Figure 8.

It can be seen from the graph that the ARIMA–LSTM–Transformer model predicted values are the closest to the observed values. The comparative quantification of ARIMA and ARIMA–Markov was performed here and is tabulated in

Table 3. Evaluation indicators of different forecast models (2).

Model	Entity	1 Day	7 Days	2 Months
Transformer	$MAE$	189.5547	233.8810	189.9022
	$RMSE$	274.5323	304.2115	265.6617
	$R^2$	0.7858	0.7914	0.8546
LSTM	$MAE$	197.9939	247.2158	202.7077
	$RMSE$	285.9588	323.0652	266.9309
	$R^2$	0.7676	0.7648	0.8532
LSTM–Transformer	$MAE$	182.7820	176.8608	181.0977
	$RMSE$	270.8501	251.3609	254.3222
	$R^2$	0.7915	0.8576	0.8668
ARIMA–LSTM–Transformer	$MAE$	178.6055	161.4459	153.2685
	$RMSE$	269.6197	230.9424	210.7759
	$R^2$	0.8257	0.8798	0.9178

.

As can be seen from Table 3, both the single Transformer and LSTM models achieve the highest accuracy when the prediction interval is 2 months. Specifically, the $R^2$ of the Transformer is as high as 0.8546, with the $MAE$ and $RMSE$ being 189.9022 and 265.6617, respectively. This indicates that the forecast of long-term conditions by a single Transformer or LSTM model in this experiment is more accurate, but it is difficult to capture the short-term local data features, and there is room for progress.

After the fusion of Transformer and LSTM models, the accuracy is greatly improved compared to the single model, especially in the medium-term forecast; the $R^2$ of the LSTM–Transformer combined model is 8.36% higher than that of the Transformer single model and 12.13% higher than that of the single LSTM model. The LSTM–Transformer hybrid model achieves 24.36% and 17.37% lower MAE and RMSE, respectively, compared to the single Transformer model, and demonstrates 28.46% and 22.19% reductions in $MAE$ and $RMSE$, respectively, relative to the single LSTM model.

However, the accuracy of the hybrid model in short- and long-term forecasts does not significantly improve. Compared to the single Transformer model, the $R^2$ of the hybrid model increases by 0.73% and 1.43% for short- and long-term forecasts, respectively, and the improvement effect is not very obvious.

Considering that the forecast model may have some limits with respect to short-term forecasts, the ARIMA model was added. The ARIMA–LSTM–Transformer hybrid model has good performance in the short-, medium-, and long-term forecasts. Specifically, its short-term ${\mathrm{R}}^2$ is 4.32% higher than that of the LSTM–Transformer model, and its medium-term and long-term forecasts also exhibit outstanding results, with an average ${\mathrm{R}}^2$ improvement of 4.24%.

Overall, the ARIMA–LSTM–Transformer model emerges as the optimal choice—especially for long-term forecasting tasks. Moreover, because the medium- and long-term dataset used in this study features high precipitation amounts, these results further demonstrate the model’s strong resistance to interference and validate the applicability and superiority of integrating ARIMA, LSTM, and Transformer techniques.

Noting the accuracy of the two hybrid models compared to their components longitudinally, two hybrid models were compared horizontally. In the one-day short-term forecast, ARIMA–Markov is more accurate; compared to the ARIMA–LSTM–Transformer model, the $R^2$ increases by 0.0293, and the $MAE$ and $RMSE$ decrease by 28.9061 and 43.7782, respectively, probably because the short-term forecast dataset sample size is smaller. ARIMA directly models the linear relationship between the current value and the recent historical value through the autoregressive term, which is more sensitive to the capture of short-term trends and has natural adaptability to small samples and local dependence.

However, deep learning models such as LSTM and Transformer are good at capturing complex nonlinear relationships in long sequences, which require a large amount of data for training. A small amount of data may lead to overfitting and other situations that affect accuracy. Therefore, in the medium- and long-term forecasts, the accuracy of ARIMA–LSTM–Transformer is higher. Compared to ARIMA–Markov, the average $R^2$ increases by 0.0247, and the $MAE$ and $RMSE$ decrease by 14.1839 and 28.3393, respectively. Overall, the accuracy of ARIMA–LSTM–Transformer is higher.

3.3. Stability Test for ARIMA–LSTM–Transformer Model

Of the above two hybrid models, ARIMA–Markov involves the Markov probability transfer matrix, and there is randomness in the transformation between various states; the ARIMA–LSTM–Transformer model involves the weight matrix and Adam optimization algorithm. Both models involve a certain random process, so the stability of the model is also an important indicator to measure the strength and disadvantage of the forecast model. In order to test the stability of ARIMA–Markov and ARIMA–LSTM–Transformer, the three datasets selected above were used, and each dataset was repeatedly predicted ten times with the exact same parameters. The $MAE$, $RMSE$, and $R^2$ of these ten forecasts were obtained, an array containing ten datasets was formed, and the standard deviation corresponding to the array was calculated in turn to achieve a relatively stable size, as shown in

Table 4. Standard deviation of the ARIMA–Markov and ARIMA–LSTM–Transformer models.

Model	Entity	1 Day a	7 Days a	2 Months a
ARIMA–Markov	$MAE$	28.7916	42.3466	25.1970
	$RMSE$	41.0260	27.3987	29.5438
	$R^2$	0.1019	0.0422	0.0378
ARIMA–LSTM–Transformer	$MAE$	12.9359	5.2432	4.9143
	$RMSE$	16.2493	6.0679	3.8584
	$R^2$	0.0157	0.0123	0.0036

^a The data in the table represent the standard deviations of the arrays obtained from ten forecasts of the corresponding indicators, not the specific values.

.

From Table 4, it can be concluded that the stability of ARIMA–LSTM–Transformer is better than that of the ARIMA–Markov model in short-term, medium-term, and long-term forecasting, and ARIMA–LSTM–Transformer has the highest stability for predicting long-term time data. From the above, it can be seen that the accuracy of ARIMA–LSTM–Transformer is comparable to that of ARIMA–Markov in short-term forecasting, but the accuracy of the former is better than the ARIMA–Markov model in both medium-term and long-term forecasting. Therefore, the ARIMA–LSTM–Transformer model performs better in wastewater forecasting by combining the two aspects of accuracy and stability.

4. Algorithm Optimization

4.1. Parameter Selection

Although the ARIMA–LSTM–Transformer model has achieved good results, there are many hyperparameters in the model and the amount of computation is large. Thus, how to optimize the hyperparameters is an urgent problem that requires attention. Using optimization algorithms (e.g., SSA) can optimize hyperparameter selection and improve model accuracy [35]. In our study, three algorithm optimization models, WOA [36], PSO [37], and SSA [38], were selected to optimize the ARIMA–LSTM–Transformer model. After experimentation, it was found that WOA optimization works best. The initial values of each optimization algorithm are shown in

Table A3. The initial values of each optimization algorithm.

Algorithm Name	Parameters	Initial Values
WOA	Population size	50
	Maximum number of iterations	250
	Convergence constant	Decreasing linearly from 2 to 0
	Spiral shape parameters	1
PSO	Number of particles	40
	Maximum number of iterations	250
	Inertia weights	Decreasing linearly from 0.9 to 0.4
	Individual and social learning factors (${\mathrm{C}}_1$, ${\mathrm{C}}_2$)	2, 2
SSA	Population size	50
	Maximum number of iterations	250
	Proportion of finders	[0.1, 0.3]
	Proportion of vigilantes	[0.1, 0.2]
	Security thresholds	[0.5, 1]

(see Appendix D), and the WOA optimization frame diagram is shown in Figure 9. It demonstrates the complete process, from the pre-processing of sewage flow data, to the construction of a model for feature extraction, and then to the hyperparameter optimization of WOA.

4.2. Accuracy Testing

As above, forecast periods of one day, seven days, and two months were selected. Then, the corresponding $R^2$, $RMSE$, and $MAE$ were calculated to judge the accuracy of the optimized model. The experiment was repeated five times in each time interval to eliminate the best and worst forecast results. The average wastewater flowrate forecast values of the remaining three experiments were recorded and used to plot the predicted flowrate curve of the model, as shown in Figure 10. The residual plot of the predicted flowrates for each model is shown in Figure 11.

In order to evaluate the performance of each optimization algorithm in the ARIMA–LSTM–Transformer model more roundly and scientifically, the evaluation indicators were calculated, and the results are shown in

Table 5. The evaluation indicators of the optimized ARIMA–LSTM–Transformer model.

Optimization Algorithms	Index	1 Day	7 Days	2 Months
SSA	$MAE$	167.2940	183.4126	147.3382
	$RMSE$	253.5052	270.6702	206.8195
	$R^2$	0.8273	0.8449	0.9216
PSO	$MAE$	168.7508	155.3685	139.6621
	$RMSE$	250.4343	227.5126	187.4444
	$R^2$	0.8317	0.8934	0.9374
WOA	$MAE$	162.7161	133.5755	132.8711
	$RMSE$	246.4888	200.2303	184.8365
	$R^2$	0.8373	0.9196	0.9394

.

A comparison of Table 3 and Table 5 shows that, compared with the unoptimized ARIMA–LSTM–Transformer model, adding the SSA algorithm does not improve the model’s prediction accuracy: the $R^2$ for the short and long-term forecasts increases by only 0.19% and 0.41%, respectively, while the $R^2$ of medium-term forecasts actually drops by 3.97%. This may be due to the model’s inherent complexity, a mismatch between the model and SSA’s optimization mechanism, the high nonlinearity introduced by including precipitation in the dataset, or external factors such as holidays. Nevertheless, after adding PSO and WOA, the model’s forecasts for the short, medium, and long terms all show some improvement, but WOA yields the greater gains: compared to results before optimization, the average $R^2$ increases by 2.78%, better than the 1.49% of PSO. MAE and RMSE decrease by 12.99% and 11.22%, respectively, again outperforming PSO’s reductions of 5.99% and 6.46%. Consequently, among the three algorithms tested, WOA provides the best optimization effect.

4.3. Stability Test for Optimization Algorithms

In order to test the stability of the optimized ARIMA–LSTM–Transformer, the three datasets selected above were used, and each dataset was repeated ten times with the exact same parameters; the $MAE$, $RMSE$, and $R^2$ of these ten forecasts were obtained and formed into an array containing ten data points. The corresponding standard deviations of the arrays were calculated successively to compare the stability. The calculation results are shown in

Table 6. The standard deviation of the optimized ARIMA–LSTM–Transformer model.

Model	Entity	1 Day a	7 Days a	2 Months a
SSA	$MAE$	6.4156	7.4094	6.2892
	$RMSE$	4.3774	9.3836	9.3708
	$R^2$	0.0267	0.01676	0.0065
PSO	$MAE$	4.3836	16.3154	4.2719
	$RMSE$	2.3340	9.4720	4.1376
	$R^2$	0.0037	0.0207	0.0034
WOA	$MAE$	2.8193	6.1603	2.0831
	$RMSE$	2.7386	4.0882	2.5371
	$R^2$	0.0037	0.0076	0.0019

^a The data in the table represent the standard deviations of the arrays obtained from ten forecasts of the corresponding indicators, not the specific values.

.

As demonstrated in Figure 10, after the addition of the optimization algorithm, except for some individual data points, the stability of the model improves compared to before optimization. Among them, the standard deviations corresponding to $R^2$, $RMSE$, and $MAE$ of the model optimized by the WOA are all the lowest, indicating that the stability of the model after WOA optimization is higher than that of the other two algorithms. Compared to the unoptimized model, after WOA optimization, the standard deviation of $R^2$ decreases by an average of 0.0061, the standard deviation of $RMSE$ decreases by an average of 5.6039, and the standard deviation of $MAE$ decreases by an average of 4.0102. This indicates that the ARIMA–LSTM–Transformer hybrid model optimized by the WOA algorithm has good stability in short-term, medium-term, and long-term forecasts.

5. Conclusions

To solve the complex problem of urban wastewater flowrate forecasting, models based on WOA, ARIMA, LSTM, and Transformer for wastewater flowrate forecasting were combined and achieved good results.

The experimental results show that ARIMA can accurately grasp the local characteristics of short-term wastewater flow changes, capture trends, and pass them on to LSTM as one of the characteristics. On this basis, LSTM can capture more complex temporal dependencies, and features such as holidays were also quantified to achieve multivariate regression analysis. Transformer greatly enhances the forecast accuracy of the hybrid model for medium- and long-term data, and WOA optimizes the number of LSTM layers, Encoder layers, and Decoder layers in LSTM and Transformer, reducing the computational cost and the possibility of overfitting the hybrid model.

After the introduction of the WOA, ARIMA, and LSTM modules, the $R^2$ increased by 10.86%, the $RMSE$ was reduced by 25.21%, and the $MAE$ was reduced by 30.02% compared to a single Transformer, demonstrating a significant improvement. The comparison between single and hybrid models shows the latter to be associated with higher accuracy and stability. However, the hybrid model still has some limitations and may not be able to predict the flowrate due to high summer temperatures, water evaporation, or abnormal wastewater flow due to water and power outages in specific areas of the city, which should be the focus of subsequent research.

Future work includes (1) further optimizing the structure and performance of the model—although the proposed model performed well in the current study, its forecast of wastewater flow in the short term is still not sufficiently accurate, and (2) enhancing the anti-interference ability of the model by incorporating more factors that affect wastewater flow, such as urban population migration, urban water, and power outages.