Comparative analysis of two decomposition methods for urban domestic water usage patterns Open Access

The rapid growth of the urban population and economy has led to an increasing demand for water for urban living. At the same time, climate change has led to glacier retreat, reduced river flow, and shrinking lakes, further exacerbating freshwater shortages (Gosling 2018). The scarcity of water resources and the imbalance between supply and demand have become key factors limiting urban economic development. Therefore, effective water resource management and accurate water quantity forecasting are crucial.

There are two main paths to achieving efficient water resource management: one is the optimal use of water resources, and the other is the assessment and prediction of water availability. Fortunately, the Earth's atmosphere contains a large amount of water in the form of vapor and droplets. It is estimated that atmospheric water amounts to about 129,000 billion tons, which is six times the volume of water in all lakes on Earth. As part of the global hydrological cycle, atmospheric water can be replenished by solar energy, which is an inexhaustible resource (Sun et al. 2023). Currently, various fog water collection technologies, such as surface cooling, fog nets, and adsorption methods, have been developed to extract water from the air. However, existing technologies have low efficiency in extracting water from the atmosphere, greatly limiting their application in fog water collection. Thus, scientific planning and management of water resources are therefore particularly important. In China, Wang et al. (2021) employed a combined method of the Grey model and the Markov model to forecast residential water usage, demonstrating higher accuracy compared to traditional models; the convergence, panel spatial econometric, and threshold regression models are adopted to examine the convergence and to identify the factors affecting water usage. Internationally, Rasifaghihi et al. (2020) considered climatic variables in predicting urban water usage, while Potter et al. (2022) analyzed factors affecting residential water usage forecasts in central Texas. However, in regions with only short-term water usage data, it is challenging to accurately distinguish the impact of seasonal and climatic variations on water usage. Zanfei et al. (2022) addressed this issue by developing a short-term prediction model using a deep learning approach based on the long short-term memory (LSTM) model to forecast urban water demand. Urban domestic water usage exhibits complex variability influenced by seasons, climate, policies, and other factors, with unique regularities and cycles at different time scales. Currently, data clustering and regionalization methods are increasingly being applied in water resource management to address complex hydrological data analysis and regional water resource issues. For example, Ahani et al. (2020) discussed a watershed regionalization method based on rough set theory. The application of this method to watersheds in Iran shows that by combining canonical correlation analysis and clustering analysis, the homogeneity of regional divisions can be improved, and it is suitable for simultaneously regionalizing both monitored and unmonitored watersheds. Although these methods have made some progress in watershed management, their application in urban water usage forecasting remains limited. Given the complexity and variability of urban domestic water use, analyzing its patterns through traditional surveys is often time-consuming and intricate. The temporal and spatial patterns of urban water usage are influenced by multiple factors. Water demand varies significantly across different urban areas, with residential, commercial, and industrial zones exhibiting distinct water usage characteristics. Climate conditions play a crucial role in determining water demand, with arid or humid climates having a significant impact on water usage. In densely populated areas, water usage tends to be higher, while regions with higher levels of economic activity often experience more complex and fluctuating water demand. Existing research predominantly relies on conventional statistical and forecasting methods, which struggle to fully capture the dynamic characteristics and regularities of water usage changes. Therefore, analyzing the patterns of water usage changes with appropriate methods before making predictions is of great significance for water quantity prediction and rational allocation of water resources.

Huang et al. (1998) introduced the empirical mode decomposition (EMD) algorithm in the late 20th century, providing a powerful tool for understanding and analyzing the complexity of urban domestic water usage patterns. This algorithm is distinguished by its recursive signal decomposition. Dragomiretskiy & Zosso (2014) proposed the variational mode decomposition (VMD) algorithm as an adaptive signal processing technique, offering a more robust and flexible approach to analyzing urban water usage patterns. VMD can break complex time series down into a certain number of intrinsic mode functions (IMFs) that EMD cannot identify. Recent studies on the modal decomposition of time series utilizing VMD demonstrate that this method enhances data regularity and offers an effective and robust approach to unveiling implicit patterns in urban water usage data. Wu & Lin (2019) applied VMD to decompose the original wind speed series into sub-series of different frequencies. They combined this with a least squares support vector machine model, optimized by the Bat algorithm, to predict wind speed. This improved the stability of wind power integration into the grid. Wu et al. (2024) used a VMD-informer-dynamic conditional correlation (DCC) photovoltaic power forecasting model. This approach overcame the volatility and uncertainty of photovoltaic power generation, improving prediction accuracy. VMD has been widely applied in wind energy and power sectors, but its use in analyzing urban water usage patterns is rare.

VMD can decompose urban water usage data into multiple scales. It extracts key intrinsic modes and identifies short-term and long-term water usage cycles. This is particularly useful for handling seasonal fluctuations and emergencies, such as droughts or extreme weather. VMD improves water resource forecasting accuracy and provides a scientific basis for water allocation and scheduling. This helps to manage future water shortages more effectively.

In traditional VMD decomposition, the number of modes (K) has some flexibility. It significantly affects the decomposition results. A K-optimization method is proposed to solve issues of under-decomposition or over-decomposition. Fourier transform-based analytical techniques have been employed for nearly two centuries in the study of linear time-invariant systems and smooth signals. The fast Fourier transform (FFT) is widely used in signal processing as an effective algorithm.

Using an optimized VMD multi-scale decomposition method and the traditional FFT method to analyze urban water usage patterns provides a comparison, exploring the advantages and limitations of both in capturing water usage trends. This analysis deepens the understanding of their applicability and effectiveness in specific signal processing, revealing subtle changes in nonlinear and non-stationary signals. Furthermore, it provides important references for efficient urban water usage prediction in the future.

Each data point in the dataset contains the following information: a timestamp, specific date and time of recording water usage, and water usage at that point in time. Data are collected through automated water meter readings to ensure accuracy and reliability. After inspection, the dataset contains no missing values; any outliers are treated as routine emergencies without special pre-processing.

The VMD algorithm is an adaptive and completely non-recursive signal decomposition algorithm. Compared with the EMD proposed by Huang et al. (1998), a completely non-recursive approach avoids common problems such as modal mixing and endpoint effects in EMD. It also improves the accuracy and reliability of the analysis. The VMD algorithm can decompose the original water usage data into a specified number of IMF components, adeptly isolating the temporal change patterns of the original time series. This ability exhibits strong oscillatory characteristics. The core principle of VMD is based on the variational algorithm, which decomposes the original signal by minimizing a predefined cost function. The cost function considers two factors: the bandwidth of each mode and the fidelity of the reconstructed signal to the original signal. This process is implemented through iterative optimization until convergence conditions are reached. Finally, each IMF possesses a relatively narrow bandwidth, and it can indicate a distinct frequency component of the original signal.

In the formula, and denote the solution that minimizes A when condition B is satisfied, and the objective function ensures that each IMF is concentrated within a relatively narrow frequency band. It also ensures that the difference between the reconstructed signal and the original signal is minimal; is the Dirichlet function; is the one-sided spectrum obtained by performing a Hilbert variation to the ⁠, ⁠, and denote the set of modal components and corresponding center frequencies, with a decomposition scale of K, the number of components obtained by decomposition; * denotes the convolution; denotes the sum of all modal components; denotes the gradient squared paradigm.

The principle and process flowchart of the VMD algorithm is shown in Figure 3. The analysis of urban water usage data with the VMD algorithm can stabilize the data and reveal the underlying patterns in the water usage time series. Using virtual prototypes can accurately capture the periodic changes in urban water usage to provide a scientific basis for urban water management and planning. In recent years, with the wide application of the VMD method in signal processing, the potential of VMD in urban water usage analysis has become more and more recognized, and this will provide a powerful tool for solving the complex dynamics behind urban water usage.

The application of VMD in the analysis of urban water usage data consists of the following steps:

• (1) Decomposition of the original water usage time series data into multiple IMFs. Each IMF represents a major frequency component of the original signal and may correspond to daily water usage fluctuations, seasonal variations, or other cyclical patterns.

• (2) Identify water usage patterns in urban water usage by analyzing the characteristics of each IMF, such as amplitude, frequency, and morphology.

• (3) Trend analysis using IMFs decomposed from VMD in predicting future trends in water usage.

As discussed above, the number of modes K has a certain degree of flexibility, and it significantly impacts the effectiveness of signal decomposition. If the K-value is set too low, it leads to under-decomposition, which means some intrinsic modes present in the signal are not fully decomposed. Conversely, if the K-value is set too high, the signal will be over-decomposed, which can result in the creation of false modes that do not exist in the original signal. In the VMD decomposition process, the principle of energy conservation is followed. It means the sum of the energies of all decomposed signals should equal the energy of the original signal.

In the formula, E represents the signal energy; is the ith element of the original signal ⁠; and n is the number of sampling points.

In the formula, represents the sum of the energies of all components obtained from the current mode K; represents the sum of the energies of all K − 1 components obtained from the previous VMD decomposition. As indicated by Equation (5), the value of is positively correlated with the degree of decomposition. A larger indicates more pronounced over-decomposition by VMD, while a smaller suggests the possibility of under-decomposition. For a non-stationary complex signal, tends to fluctuate around a small value under conditions of under-decomposition or appropriate decomposition. As the parameter K increases, over-decomposition occurs, and will significantly increase. At this point, the turning point's K-value can be considered the optimal mode number for VMD decomposition.

Additionally, the penalty factor is set to 2,000. An excessively large would make the decomposed modes too smooth, and lose some details; an overly small might result in overly complex modes, and introduce noise interference. The noise tolerance is set to 0, meaning the algorithm is required to preserve the original fidelity of the signal as much as possible, strictly decomposing the true signal components. The DC offset is set to 0, which means that the algorithm does not forcefully separate the DC component from the signal but treats it as part of the overall signal. The initialization mode is set to 1 to uniformly initialize the center frequencies ⁠. The convergence tolerance is set to 10⁻⁷ to ensure the accuracy of the decomposition results.

The Fourier transform holds a central position in digital signal processing. It decomposes complex digital signals into simpler components, facilitating the identification and extraction of valuable signal elements. According to the Fourier principle, any continuously recorded time series or signal can be expressed as a combination of sine waves of countless different frequencies. The Fourier transform algorithm is based on this principle: it uses the original signal to calculate the frequency, amplitude, and phase of the sine wave components.

Through FFT processing of urban water usage data, key frequency components can be extracted from the data, and the data reflect changes in water usage patterns over time. Identifying these frequency components is critical for forecasting water demand, optimizing water resource management strategies, and formulating effective water conservation policies. FFT is an efficient method for computing the discrete Fourier transform (DFT). By employing the DFT and its efficient algorithm FFT, we can effectively process and analyze digitized water usage data, capturing patterns of water usage fluctuations ranging from short daily cycles to longer ones.

By splitting the DFT formula into even and odd terms, the values of the last N/2 points can be determined by computing the intermediate values of the first N/2 points. This process continues with even and odd decompositions, leveraging symmetry and periodicity, which reduces the computational complexity from to ⁠.

Using the inverse fast Fourier transform (IFFT), the correctness of the signal processing can be verified.

Higher SNR values indicate that the signal is less affected by noise, resulting in better signal quality. In this study, SNR is employed to compare the performance of the VMD and FFT methods in signal reconstruction. A higher SNR implies that the reconstructed signal retains more of the original signal characteristics with less noise interference.

By substituting the original water usage data into the K-optimization formula, we obtain the optimal decomposition mode number k for this signal, which is 4. The results of decomposing the time series are as follows.

IMF1 primarily fluctuates between 1,500 and 2,000 L, which corresponds to the most fundamental and significant range of water usage. In the original data, this range also has the darkest shading, indicating that the line passes through this range most frequently. This suggests that IMF1 reflects the most basic and prominent water usage trends.

In contrast, IMF2, IMF3, and IMF4 fluctuate around zero. IMF2 shows a noticeably larger number of positive values. This might indicate that IMF2 captures more frequent above-average water usage events, reflecting cyclical high-demand periods.

During the first half of July, a significant drop in water usage is observed in IMF1, and it is consistent with the overall trend in the original data. Notably, IMF2 and IMF3 also reflect this decrease, but in a different way: the maximum and minimum values during this period become more concentrated, resulting in a narrower range of fluctuations. This compression suggests that the variability in water usage is temporarily reduced during this time.

In the VMD shown in Table 1, the cycles of the four IMFs vary, reflecting different frequency patterns. The predominant occurrence of periods of 1 day or less could result from the limited volume of data and the absence of distinct long-term trends or cyclical variations. IMF1 has a 1-day period, and it likely corresponds to daily water usage patterns. This could be due to regular, repetitive activities that occur on a daily cycle, such as household or business operations following a consistent schedule. IMF2 has a cycle of 0.50–1.00 days. This is particularly evident in IMF2's frequency spectrum, which displays two distinct peaks. The first peak corresponds to a 1-day period, while the second, higher peak corresponds to a 0.5-day period. It shows more frequent fluctuations, and this suggests it may capture shorter daily peaks, like morning or evening water demand in industrial or commercial activities. IMF3 has a much shorter cycle of 0.20 days. This means there are rapid fluctuations every few hours. These could be caused by equipment or machinery cycling on and off throughout the day. IMF4 has the highest frequency. Its cycle is between 0.10 and 0.13 days, or about every 2–3 h. This reflects the most frequent and short-term changes. These could come from automated systems or mechanical processes that repeat many times a day. Urban water usage is not only influenced by domestic consumption but also by industrial activities and other modern sectors, contributing to its complexity. Ioannou et al. (2021) conducted a clustering of household water consumption at different levels to capture individual household usage behaviors. However, their study did not analyze the overall regional water usage patterns, leaving broader trends and collective consumption dynamics unexplored.

Filtered signals demonstrate the appearance of water usage data after specific frequencies are removed. During the IFFT1 processing, frequency components ranging from 0.0001 to 0.0104 rad/s were eliminated. The operation is designed to remove the low- and medium-frequency components of the signal to analyze its high-frequency characteristics. IFFT2 removes bands from 0.0001 to 0.0002 rad/s. IFFT3 eliminated the lowest frequency band, i.e. 0–0.0001 rad/s. IFFT4 removes frequencies within the bands of 0–0.0045 rad/s. The frequency spectrum of IFFT4 shows no dominant frequencies in this range. Therefore, this segment can be treated as noise and removed. Its removal allows further research to proceed more effectively.

The main cycles of the IFFT with different frequency bands removed are calculated as follows.

The evaluation metrics for the two decomposition methods are as follows.

The time complexity of the FFT algorithm is ⁠, making it highly efficient in computation. In contrast, VMD has a higher time complexity, which depends on the number of iterations and the number of decomposition modes K. The results in Table 3 show that the runtime of FFT is significantly shorter than that of VMD. This indicates FFT's clear advantage in computational efficiency. MSE and SNR results suggest that FFT has a smaller reconstruction error. This implies the original signal is relatively stationary in the current configuration. However, with higher-order decomposition, an increased number of modes, and optimized parameter settings, VMD may demonstrate superior performance when handling nonlinear or non-stationary signals.

To further validate the results of the VMD and the FFT and more specifically understand how these patterns behave at different time scales, we have refined the analysis to analyze water use patterns at specific times, including specific months, specific weeks, and specific days.

The results of the VMD decomposition and FFT decomposition are randomly selected over a period of 2 weeks in the dataset. The selected dates, as illustrated in Figure 9, span from Monday to Sunday in both a July week and a September week.

FFT has inherent advantages in frequency analysis, and VMD shows more significant advantages in handling nonlinear signals, performing multi-scale analysis, pattern recognition, denoising, and signal reconstruction. Both FFT, as shown in Figure 10, and VMD, as shown in Figure 11, are able to clearly reveal daily and weekly patterns of variation when analyzing different randomly selected weeks. IFFT1 and IMF1 look almost identical, as both capture the fundamental daily fluctuation patterns in the data with high accuracy. For regular, well-defined daily cycles, both methods are effective at decomposing the signal into its core components, leading to similar outputs. However, IFFT2, IFFT3, and IFFT4 begin to show different characteristics compared to IMF2, IMF3, and IMF4, respectively. Overall, the IMFs tend to be smoother, while the IFFTs are slightly more coarse. The difference arises because VMD can handle nonlinearities and noise more effectively, resulting in smoother decompositions, while FFT tends to highlight sharper, more abrupt transitions in the signal. The smooth and regular daily fluctuation variations in IMF1 and IFFT1 reflect the fact that both VMD and FFT can reveal major fluctuations in complex data. However, during periods of lower overall water usage, particularly late at night, the smaller fluctuations represented by IMF3 actually become more pronounced. It makes these variations stand out more when the general water demand is reduced.

Figure 12 is used to show the original water usage data for another week being compared, while Figure 13 and Figure 14 respectively display the FFT and VMD decomposition results of this week. When comparing water usage between the week of July and the week of September, the combination of IFFT4 exhibits no substantial correlation between the weeks at this primary frequency. Additionally, FFT demonstrates limited temporal localization of signals and the capacity to handle non-stationary signals. FFT decomposes the signal based on fixed sinusoidal and cosine functions, lacking adaptability to the local characteristics of the signal. In the case of water usage analysis, this can lead to inaccurate short-term water usage patterns or unusual water events. Moreover, VMD decomposition enhances signal quality in denoising and signal reconstruction by selectively disregarding noise components and utilizing only the IMFs containing useful signal components.

Two full 1-day periods from the dataset were randomly chosen to compare the outcomes of VMD and FFT decompositions. The selected dates are 1 June, as shown in Figure 15, and 1 August, as shown in Figure 16.

When analyzing daily data, both FFT, as shown in Figure 17, and VMD, as shown in Figure 18, allow us to precisely uncover the dynamics of intra-day water usage. For instance, they demonstrate the daily pattern of high daytime and low nighttime peaks of IMF1 and IFFT1, as well as the morning and evening peaks of IMF2 and IFFT3 in the lowest frequency range. Liu et al. (2022) employed multiple algorithms, including ensemble empirical mode decomposition–autoregressive integrated moving average (EEMD-ARIMA), ensemble empirical mode decomposition–backpropagation (EEMA-BP), and ensemble empirical mode decomposition–support vector machine (EEMA-SVM), to forecast hourly water consumption. The dataset they used spanned only one month, lacking sufficient time coverage to capture seasonal variations as well. However, their study did not analyze the underlying patterns and drivers of water usage, and it would limit their ability to effectively identify the primary factors influencing water consumption trends. Cao et al. (2023) used 3 years of data to perform short-term forecasts of the water supply for several treatment plants. Their analysis was limited to a few treatment plants, which might not fully capture the broader dynamics of regional or urban water consumption. They did not delve into the underlying behavioral patterns or causes of water usage, which were important for improving predictive models and decision-making in water resource management.

The morning peak from IMF2, as shown in Figure 18 and Figure 19, is slightly higher than the evening peak, validating the effectiveness of VMD in diurnal data analysis. Although clear patterns for different dates were not discernible through the integrated monitoring frameworks of 1 June and 1 August and framework 4, minor fluctuations could still be conveyed to the high-frequency segment based on the overall pattern of change. The decomposition results of FFT, as shown in Figure 20, do not show significant differences.

Simultaneously, the low-frequency segment of VMD can promptly detect significant fluctuations with minimal parameter adjustments. The high-frequency and low-frequency segments can individually forecast water usage and subsequently merge for further data processing and regulation. When addressing oscillating time series, virtual prototypes can reduce data complexity and offer more intuitive analysis results. However, despite several parameterization experiments, distinguishing the overall fluctuation trend from small fluctuations remains challenging, hindering subsequent observation and prediction.

One of the main limitations of this study is the limited scope of the dataset. While the stable data helped reduce reconstruction errors, it may not fully represent dynamic water usage patterns across different regions or seasons. Another challenge is the lower computational efficiency of VMD. Although VMD performs well with complex signals, it is less efficient than FFT when applied to large datasets. This shows the need for further optimization of VMD in future studies.

A key lesson learned is the balance between simplicity and computational demand. VMD is easier to set up but requires more time and resources. On the other hand, FFT is faster but struggles with capturing the nonlinear variations that VMD handles well. This suggests that neither method is perfect alone, and both have their strengths depending on the analysis needs.

To improve the refined management of urban water resources, this study applied VMD and FFT to analyze urban water usage patterns and highlighted the strengths and limitations of both methods. Through an examination of daily and cyclical fluctuations, a deeper understanding of water usage dynamics was achieved.

• (1) In terms of accuracy, the dataset used in this study is relatively stable overall, which results in lower reconstruction error for FFT. VMD's potential to capture complex signal characteristics and multi-scale periodicity can be further explored by increasing the number of decomposition modes or optimizing the algorithm to reduce reconstruction error.

• (2) Conducting a comprehensive analysis of a city's domestic water usage data using VMD and FFT can reveal the cyclical changes in the characteristics of the water usage pattern, thereby deepening our comprehension of its dynamic fluctuations. Specifically, the VMD decomposition method is simpler and more efficient in terms of parameter settings, whereas the FFT decomposition requires more complex testing and selection of parameters. However, in terms of computational efficiency, FFT significantly outperforms VMD, making it more suitable for handling large-scale data.

• (3) From an interpretability perspective, VMD identifies fluctuations in daily urban water usage by gradually extracting the IMFs from the signal. These patterns often correspond to the daily habits of residents, such as morning and evening water usage peaks. Morning routines are typically more time-sensitive. This leads people to concentrate their water usage in a short period to quickly complete various daily activities, resulting in a rapid increase in water usage and creating a pronounced morning peak. In contrast, evening activities are more flexible, so water usage is spread out over a longer period. If future studies decompose longer cycles, it may reveal the impact of seasonal or climatic factors on water usage patterns. VMD shows higher sensitivity in analyzing nonlinear characteristics and capturing subtle changes in time series. By identifying peak and abnormal water usage events, water allocation can be optimized, water supply system efficiency can be improved, and waste can be reduced. Decomposing down to daily variation cycles, the high-precision decomposition results can identify the underlying regularities in water usage changes. This provides a reference for sudden events, such as water outages, natural disasters, offering a certain level of early warning capability.

• (4) Combining all IMFs enables the description of overall trends and fluctuations in urban water usage, which can be used to predict future water demand. By understanding the cyclical changes and trends in water usage, more accurate water usage forecasting models can be developed to predict future water demand and contribute to more rational water planning and contingency planning. LSTM, as a deep learning model based on recurrent neural networks, has a significant advantage in capturing long- and short-term dependencies within time series, making it particularly effective for predicting complex dynamic signals. For short-term data, future research could incorporate LSTM for time series prediction after performing VMD decomposition.

These findings provide valuable insights for selecting the appropriate method depending on the specific needs of urban water resource management and predictive modeling. The study also lays the groundwork for future research, potentially incorporating advanced forecasting techniques such as LSTM for further improvements in accuracy.

The research was supported by the Joint Fund Project of the Natural Science Foundation of Anhui Province (Grant No. 2208085US05, Grant No. 2308085US05). Its support is gratefully acknowledged, as it has enabled us to pursue our research goals with greater focus and dedication.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.