Statistical analysis, box-plot tests, and correlation analysis were used to deal with missing values and outliers. Redundant data were eliminated according to the configuration conditions of the model, and an interpolation method was selected to impute the missing data after unifying the data sampling rate. Data processing goes through five steps: (1) correlation analysis, calculating the correlation of various meteorological data (rainfall, wind speed, temperature, minimum pressure, etc.) and waterlogging depth; (2) data screening, using domain knowledge to set thresholds to identify abnormal data; (3) resampling, accounting for different sensors having different working mechanisms, meaning the sampling time interval is different by unifying the sampling interval through the resampling function in Python to prepare for model training; (4) data interpolation, using data interpolation to complete the data after resampling to make the time series continuous and fit the real situation; (5) sliding-window segmentation and data integration, according to the mode structure requirements, sliding-window segmenting the time series and inputting them into the model by data integration.

In this paper, an accumulated-rainfall data set (R) and a historical waterlogging-depth data set (D) are used to predict the waterlogging depth in the future. By adjusting the data combination method Φ, a new data set X can be constructed by Eq. (1): 1X=Φ(R,D).

For each input data point xi, the vector combining ri (r∈R) and di (d∈D) is in the set sharding mode; vector ri can be represented by [ri+1ri+2,…,ri+m-1ri+m], and vector di can be represented by [di+1di+2,…,di+n-1di+n]. The sharding mode is realized by adjusting the sliding-window size m and n. The combined input vector xi can be represented as [ri+1ri+2,…,ri+m-1ri+mdi+1di+2,…,di+n-1di+n]. Through the continuous iteration of i, the sliding window can loop through all the training data and combine them into the input data set X (Eqs. 2–9), which is a high-dimensional matrix. 2R=[r1r2r3r4,…,rl-1rl]3ri=[ri+1ri+2,…,ri+m-1ri+m],r∈R4D=[d1d2d3d4,…,dl-1dl]5di=[di+1di+2,…,di+n-1di+n],d∈D6xi=[ridi]=[ri+1ri+2,…,ri+m-1ri+mdi+1di+2,…,di+n-1di+n]7i∈(1,l-m+1),m≥ni∈(1,l-n+1),m<n 8X=r1r2…rmd1d2…dnr2r3…rm+1d2d3…dn+1r3r4…rm+2d3d4…dn+2r4r5…rm+3d4d5…dn+3⋮⋮⋮⋮ri-1ri…rm+i-1di-1di…dn+i-1riri+1…rm+ididi+1…dn+i 9y=y1y2⋮⋮yi-1yi

In the above, y is the label of the model, which stands for the output of the regressor, and it is a vector with the same length as X. There are five training modes under the MSMWP framework.

Through the analysis of data correlation, the maximum correlation coefficient between rainfall and waterlogging was 0.61. It can be concluded that there is an obvious positive correlation between rainfall and waterlogging depth, which proves that it is feasible to use accumulated rainfall to predict waterlogging.

In reality, waterlogging often occurs after raining for a period of time. Therefore, waterlogging has a certain delay characteristic compared with rainfall. The fluctuation of waterlogging is a continuous physical process affected by multiple factors, so the waterlogging depth at the next moment is often the most closely related to the previous one. In multi-R and single-D mode, only one historical waterlogging data point is selected as input.

This situation corresponds to multi-R and single-D mode, in which both rainfall and waterlogging data are taken into consideration as input, but the proportion of rainfall input is reduced while the proportion of waterlogging-depth input is increased.

This mode also covers more rainfall and waterlogging-depth information because it can better extract the characteristics of time series, balance the weight of the two data sets' coupling, and better conform to the law of time change in rainfall waterlogging.

This paper proposes a new training mode for waterlogging prediction. Not only is the prediction value related to the past rainfall and the rainfall at the current time point, but also the subsequent change in rainfall will largely affect waterlogging depth. This mode makes up for the lack of future rainfall information in mode (4) and can better reflect the dominant role of accumulated rainfall. In real applications, real-time rainfall forecast data will be added. Due to the lack of rainfall forecast data for this area, sliding-window rainfall data are used as an approximation. There are two main reasons for this. Firstly, the expanded part (15–30 min) only accounts for 12.5 %–25 % of the sliding-window rainfall (2 h or longer) and has little effect on the whole of it. Second, rainfall forecasts, especially short-term forecasts of heavy rains, are now more than 90 % accurate. It is important to note that the article does not consider only the multi-D input because this mode building of the input matrix X contains only waterlogging-depth information changes over time and does not consider the size of the accumulated rainfall. In this mode, with the extension of prediction time, the prediction ability of the model decreases rapidly, so it is not suitable for long-term warning. In the latter part of this paper, the results of this mode are discussed.

The prediction of future data based on historical data is here defined as a regression problem. We adopt the sliding window to slice the time series data into cycles. Traversal is performed in order of the data index to preserve the characteristics of continuous changes in the time dimension of the data. In this paper, eight types of regression algorithms are selected, which can simultaneously perform one-dimensional and multidimensional regression output. They are linear regression (LR), tree regression (TR), random forest regression (RFR), k-nearest neighbors regression (KNN), ridge regression (RR), kernel ridge regression (KRR), lasso regression (LaR), and elastic net (ETN). The above eight methods are frequently employed in the field of time series prediction. As a simple regression method, linear regression has good applicability although it is sensitive to outliers. On the foundation of general linear regression, the objective function of ridge regression adds L2 regularization, which provides the best fitting error and makes the parameters as simple as possible, giving the model excellent generalizability. Yu and Liong (2007) realized hydrological time series prediction using a ridge regression algorithm based on feature space. Additionally, the kernel ridge regression approach was effectively used for the prediction of monthly mean precipitation (Ali et al., 2020). Shen et al. (2021) took human action prediction by electroencephalogram (EEG) signals as an example to study multivariate time series prediction based on elastic net, and high-order fuzzy cognitive map normalization of lasso regression is achieved by applying L1 regularization to the loss function. Wang et al. (2018) used lasso regression to accurately predict stock market fluctuations. A tree regression model was created to analyze and predict time series of air pollution, since tree regression can describe complex nonlinear data (Gocheva-Ilieva et al., 2019). Wu et al. (2017) used a random forest regression algorithm to analyze the time series of weekly influenza-like incidence and made good findings. Martínez et al. (2017) proposed a time series prediction method using a KNN algorithm.

In this paper, the evaluation is mainly divided into two stages: the test stage and prediction verification stage. The indicators in the test stage mainly include the following three categories: R2score, mean absolute error (MAE), and root mean square error (RMSE). In the verification stage of actual values, a time series of a specific length is taken to carry out the evaluation in two parts. Firstly, in order to test the model's ability to predict the variation trend of waterlogging depth, time series covering water rising, platform, and falling are intercepted. Secondly, by comparing the predicted value with the actual value, the absolute percent error (APE) is used to calculate the model's ability of correct prediction, namely accuracy (ACC). However, it is worth noting that the APE cannot be completely evaluated by the model, so absolute error (AE) is needed to supplement the evaluation because when the waterlogging depth is low, a large APE may correspond to a small AE.

Different mode settings can greatly affect the training results of the model, and training strategies are also crucial. This paper compares three training strategies, which are recursive, single-output coupling, and multi-output. The optimal prediction strategy is selected by comparing their performance applied to the test set of waterlogging prediction and the actual-value test.

Y can be divided into two parts, the historical data YHIS and the prediction data YPRE. The time interval of each data set should be uniform. Even if the sensor sampling rate is different, it should be processed uniformly. We define this time interval as τ (Ben Taieb et al., 2012). In order to predict s time steps after the current time t, the total time of prediction is defined as t, where t=τ×s, according to the number of time steps covered by the time span of the desired output variable. Therefore, the set of desired output variables can be expressed as YT, where YT∈YPRE. In a certain span of time T, between YHIS and YPRE, we use a common notation f∗ in Eq. (11) to denote the functional dependency. 10ypre=f(yhis)+bi, where yhis is the inputs of each set; ypre stands for outputs; the functional relationship between them can be expressed as f; and b stands for modeling error, disturbances, or noise. 11YPRE=f∗(YHIS)+∑bi

The intuitive forecasting strategy is the recursive (also called iterated or multi-stage) strategy. The result of the prediction of the first step is embedded into the final element of the input vector for the subsequent prediction, and the result of the prediction of the second step is obtained. In Eq. (12), when s=1, yt,yt-1,…,yt-n can be used to predict y^t+1. When s=2, the previous predicted value y^t+1 is used to replace the first element of the input vector, and the following elements are replaced in turn. The original last element yt-n is removed from the input vector. The model is iterated recursively, and the mixed vector of historical data and forecast data is used as the model input. The algorithm for this method can be expressed as in Algorithm 1. The prediction vector y^pre[p] in the first step can be obtained based on historical data. Then the final element of the input vector Xpre[p][ds-1] is replaced by y^pre[p-1], and the new y^pre is obtained through the input model and so on, moving the value of X forward one bit and adding the new predicted value. 12y^t+s=f^(yt,yt-1,…,yt-n),s=1f^(y^t+s-1,yt+s-2,…,yt+s-n),s=2⋮f^(y^t+s-1,…,y^t+s-n),s∈(2,n]

The single-output coupling strategy is similar to the direct strategy proposed by Hamzacebi et al. (2009). Different machine learning models have been used to implement the direct strategy for multi-step-ahead forecasting tasks, for instance neural networks (Khashei and Bijari, 2010), nearest neighbors (Sorjamaa et al., 2007), and decision trees (Guimarães Santos and Silva, 2014). The strategy consists of forecasting each horizon independently from the others. The biggest difference from the recursive strategy is that single-output coupling does not use any approximated values to compute the forecasts, thus having no accumulation of errors. In this strategy, error in the previous prediction results will not have a great influence on the later prediction results. Each f value is supported by a corresponding model and trained with its own independent data (Eq. 13). When s=1, it is the same as one-step prediction. When s>1, the model makes prediction across the time interval of s steps. Finally, the results of single-output yt+1,yt+2,…,yt+s-1,yt+s are coupled by Eq. (14) into a new forecast time series [yt+1,yt+2,…,yt+s-1,yt+s]. 13yt+s=fs(yt,yt-1,…,yt-n+1)14yt+1=f1(yt,yt-1,…,yt-n+1)+biyt+2=f2(yt,yt-1,…,yt-n+1)+bi⋮⋮yt+s=fs(yt,yt-1,…,yt-n+1)+bi

The two previous strategies (recursive and single-output coupling) may be considered single-output strategies, which neglects the existence of stochastic dependencies between future values and consequently affects the forecast accuracy. The multi-output strategy requires the design of multiple-response modeling techniques. The output is no longer a scalar quantity but a vector of length s. Using only one model, a time series of s time intervals is output (Eq. 15). Compared with single-output coupling strategy, this strategy involves simple operation and fast calculation. The disadvantage is that some regression algorithms such as Bayesian regression, gradient boosting regression trees (GBRTs), and AdaBoost do not support multidimensional output directly. 15[yt+s,yt+s-1,…,yt+2,yt+1]=fm(yt,yt-1,…,yt-n+2,yt-n+1)+bi, where fm is a vector-valued function and bi is a noise vector.

In order to prevent an over-fitting phenomenon or insufficient prediction, it is necessary to test the performance of the framework in actual waterlogging data sets after completing training and testing. N groups of continuous time series are selected for actual-value testing, and the application results with actual data will be discussed.

The time series intercepted with rainfall events were integrated into a new data set, with a total of 12 039 data points. The first 70 % constituted a training set with 8428 points of data, and the last 30 % constituted a testing set with 3611 points of data. In the testing set, eight regression methods were used to test five modes, and the model structure inside each mode was changed by adjusting the input parameters m and n. The testing results of different modes are shown in Tables A1 to A4. The numbers in parentheses represent the ranking of evaluation indicators (MAE, RMSE, R2score) among different modes of the same algorithm. Bold font indicates the top results of the same algorithm with different modes. The italic number indicates that the indicator is in the top 50 % of the best results (bold) between different algorithms. Taking mode (3) – KRR – as an example, in the KRR optimal indicator, RMSE is 0.0051 ranking second; MAE is 0.0013, ranking third; and the R2score is 0.9779, ranking third, so the italic indicator of KRR is 3.

Among the five modes, mode (1) only uses rainfall data to predict waterlogging accumulation and has the worst testing result. When m changes from 6 to 24, the R2score of RR changes from 0.1209 to 0.3954, which is 3.27 times the initial value. In this mode, the larger m is, the more information the model learns and the better the testing performance is. By comparing the predicted value with the actual value, the results of the eight kinds of regression methods all have large noise when the actual waterlogging depth is 0. Even though KRR and LR can achieve good trend prediction at the peak, there are still large noise fluctuations most of the time (Fig. 7). This phenomenon may be caused by the lack of historical waterlogging time series input, so the noise suppression is not good.

In mode (2), the prediction performance of LR and TR becomes better as m increases, which is the same as mode (1). However, RFR, RR, and KRR are not sensitive to parameter changes.

In mode (3), the larger the parameter n is, the better the model may not be. For example, the optimal results of RFR, RR and KRR are obtained when n=6, while the R2score of the three exceeds 0.977, indicating that the early information of waterlogging depth is not helpful to the prediction of waterlogging depth in the future and may cause some interference. For LR, with the same number of parameters, the result of mode (3) is better than that of mode (2). The main reason for this is that mode (3) extracted more historical waterlogging-depth information, which changes by a continuous process in short-term prediction. However, this does not mean that the model has the best performance, because it contains insufficient rainfall information and may not perform well in the practical application of prediction.

Mode (4) coupled multiple rainfall and waterlogging inputs. Overall, the results of mode (4) are better than those of modes (2) and (3) with one-dimensional input (m=1 or n=1). In this mode, the TR and RFR methods achieved the best testing results. TR achieved a 0.9778 R2score and 0.0050 RMSE (m=24, n=3). The R2score and RMSE of RFR reached 0.9803 and 0.0049 respectively (m=6, n=3).

Mode (5) expanded the rainfall input and considered the influence of future rainfall. When parameters are adjusted to m=6 (3:3) and n=6, the LR evaluation indicator (MAE=0, RMSE=0, and R2score=1) is abnormal. This problem also exists in RFR, TR, etc. The main reason for this is that the prediction label has been included in the input time series, so the result of m=6 (3:3) and n=6 in mode (5) should be removed in the discussion. Based on the performance of mode (5) for the test set, the performance of mode (5) will improve after the predicted time is expanded. The reason for this is that mode (5) expands the rainfall input and considers the influence of future rainfall. Especially the rainfall model with short duration and high intensity is more suitable for this mode. As shown in Fig. 8, (a) only 30 min of waterlogging data was used for prediction and (b) 30 min of waterlogging with 90 min of expanded rainfall was used as input. With the increase in prediction time, the difference in prediction performance between them increases gradually. In the prediction of 40 min waterlogging, the R2score of (b) (0.6841) is 2.4 times that of (a) (0.2847) when the TR method is applied. Using the RFR method, the R2score and MAE of (b) (R2score = 0.7428, MAE = 0.0044) also significantly exceeds (a) (R2score = 0.6488, MAE = 0.0053). Especially for the prediction of medium–high values, in the case of a high value (0.30 m) and medium value (0.13 m), the prediction results of (a) are 0.82 m and 0.73 m, and the large error leads to poor accuracy in the prediction of medium- to large-scale waterlogging.

From the perspective of the comparison of regression methods, the performances of LR, TR, RFR, RR, and KRR are relatively good, which is reflected in the strong generalization ability of the model (Fig. 9). KRR, as ridge regression with a kernel function added, is more suitable for high-dimensional data. In this study, it shows a slightly stronger regression performance than RR. It can be seen from the comparison (Fig. 9a and b) that LR and KRR have strong prediction ability for high values but poor noise suppression for low values and 0 values, and the model fluctuates constantly around the x axis. The prediction of RFR for the highest value is insufficient, but the prediction performance for other high values is better. Its noise control for low values is better. TR has the best noise control effect for a 0 value, but the curve is not smooth or ladder shaped at high values. Of course, this is related to the principle of the algorithm. When applying RFR, selecting the parameter n_estimators which is equal to 100 can solve the problem of TR (Fig. 9c). LR, RFR, KRR, and TR show strong fitting ability in the training set (TR has MAE = 0.0000, RMSE = 0.0000, and R2 score = 1.0000) (Fig. 10); KNN and ETN show relatively poor fitting ability. KNN, LaR, and ETN have weak ability of fitting and generalization and are not suitable for regression prediction of such data (Fig. 11). KNN methods have a negative R2score in mode (1), mode (2), mode (4), and mode (5). For most data sets with eigenvalues of 0, the prediction performance is often poor, which can explain why the results of the KNN method are the worst (Fig. 11a). LaR and ETN are mode-insensitive and have the same results in modes (1) and (2). However, within each mode, as m and n change, the results will be different. The poor results of the LaR method (Fig. 11c) may be because the method is suitable for multi-variable models and the variables are selected by adjusting the λ value to change the compression variable coefficient, but there are fewer variables in this study. Similarly, considering that ETN (Fig. 11b) works well when many features are interconnected, this model has a small number of features and the model with similar basic principles to lasso regression also has poor performance.

To sum up, mode (5) performs better than any other modes, indicating that the short-term prediction of waterlogging considering the change in the future rainfall trend is more realistic. LR seems to have achieved good prediction results in all five modes. However, the factor that cannot be ignored is that the original waterlogging-depth data are sparse and uneven, which must be due to resampling interpolation processing. It is necessary to go through an actual-value test to judge whether the LR method is really applicable to prediction.

Actual-value verification takes a subset of the testing set, so the first 85 % of the full data set is selected as the training set, which can increase the number of training samples and improve the training ability of the model. The time series from 26 May 2020 at 13:00 to 26 May 2020 at 17:30 was selected, lasting 4.5 h (Fig. 12), covering the complete process of waterlogging fluctuation. In this way, the waterlogging-depth prediction ability of the model can be verified. The changes in rainfall and waterlogging in this period are shown in Fig. 12. The time series was divided into six groups on average. The waterlogging-depth changes of 30 min were predicted by six sequential steps.

In order to highlight the prediction performance of each strategy, we set s=6 for 30 min prediction. That is a long forecast for real-time water levels. In practical application, 15 to 20 min prediction may be a common prediction time, and 20 min prediction has fully met the requirement of releasing early warning information in advance and dispatching the nearest traffic and fire personnel to the scene for disposal (Georgiadou et al., 2010). We will make evaluation from multiple indicators such as absolute error (AE), absolute percent error (APE), and time cost. Mode (5) with m=18 (12:6) and n=6 was selected as the basic parameters of the model to evaluate the model performance under each strategy. Figure A1 in the Appendix represents the isometric segmentation of the verification data set with different strategies. The first 30 min of the time series is taken to show the curves between the actual values in each group and the predicted values for each method. KNN, LaR, and ETN perform poorly in the application of prediction and have high values of AE and APE. Therefore, we will not discuss these three poor methods in the analysis of results. The absolute error of the predicted value and actual value of each strategy was also discussed (Fig. A2). We set a tolerance of 0.02 m to exclude reasonable error. APE reflects the accuracy of the model, but at low values it does not independently reflect the performance of the model because even small errors (< 0.01 m) can cause APE to rise (Fig. A3). Therefore, the mean value of AE and APE (MAE and MAPE) was used to evaluate the model accuracy, and the variance in AE and APE (V-AE and V-APE) was used to evaluate the robustness. Since real-time sensor data are used in prediction, continuous calculation is required to update the model. Time cost is adopted as an indicator to evaluate the continuous computing capability of the model.

Tables 3 to 5 show the evaluation results of different strategies. As seen in Table 6, MAE of the Rec strategy is 0.0153 m, which is better than 0.0184 m of the SOC and 0.0165 m of the MO strategies. Of course, the MAE of the three is within the tolerance (0.02 m). Under the Rec strategy, the model replaces the last data point of the input vector with the predicted value every step. The reason for the optimal Rec result is that the fluctuation of the waterlogging is basically a monotonically increasing or decreasing process, and generally there is no fluctuation in a short period of time. In addition, the input vector is long enough to minimize the cumulative error caused by the predicted value deviation. The application of recursive algorithm can better continue the prediction trend of the model, so the MAE result of Rec is the best.

Rec and MO strategies have better performance and robustness. Rec and MO have lower MAPE, corresponding accuracy can reach 86.1 % and 85.7 %. The accuracy of SOC was only 81.4 %. The primary cause of this phenomena is the operating process of SOC, which combines the findings of six distinct models into a final prediction vector. When predicting the s step, there will be a loss of yt+1, yt+2, …, yt+s−1 in the middle, resulting in a large error in the following part. In contrast, the V-AE and V-APE of the SOC strategy are the lowest, indicating poor performance and model robustness.

MO has the best V-APE (0.01036), indicating that the accuracy of this strategy has the smallest fluctuation, and the model has the best robustness. The V-AE (0.00026) of Rec was similar to that of MO (0.00022), but the V-APE of Rec was 69 % larger than that of MO. The reason is that APE fluctuated greatly when TR and RFR were applied under Rec strategy, and V-APE was 3.05 times and 2.49 times of MO strategy respectively. Therefore, we infer that under the same conditions, the decision tree algorithm has smaller fluctuation when applied to multidimensional output and can obtain better robustness.

Considering the time cost, the model must be updated within the 5 min interval for data prediction. This paper used tower workstation with 8 core Intel(R) Xeon(R) W-2123 3.60 GHz CPU, 64.0 GB RAM and NIVDIA Quadro RTX4000 GPU. MO strategy has a faster speed in calculation. The average time cost of Rec and SOC is about 2.54 times that of MO. Under the MO strategy, the output form of the model is a six-dimension vector. The advantage of this strategy is that it only needs to perform one training and prediction process and does not need recursion or multiple model coupling, so it is more convenient to use. Therefore, the calculation time of each regression method is greatly reduced. The time cost of KRR was shortened from 240.39 to 83.28 s. It should be noted that although the KRR algorithm has good model performance and robustness, its time cost is too high, reaching 241.28 s under the Rec strategy, which makes it difficult to meet the requirements of update calculation within 5 min. LR, TR, and RR can all be updated within 3 s due to their simple structures. RFR has a high time cost because it has to traverse all trees. However, it can meet the requirement of the update time (only 65.13 s under MO strategy).

In conclusion, LR, TR, RFR, RR, and KRR are superior to other methods, which is consistent with the results based on the testing set. Models using Rec or MO strategies have better performance and robustness, with average accuracy more than 85 % for predicting waterlogging depth in the next 30 min. For short-term prediction, such as 15 min prediction, the accuracy can reach 93 %, and the robustness of the model will be further improved. As can be seen from Figs. A1 to A3, when s=3, the prediction curves of RFR, LR, KRR, and other methods basically match actual values, and AE and APE of each group are almost within tolerance.