Prediction and analysis of domestic water consumption based on optimized grey and Markov model Open Access

Water is a basic natural and strategic resource, the material basis for the survival of human society, and an important element for the sustainable development of the national economy (Horne 2013). With the rapid growth of the world's economic level and the acceleration of urbanization process, water consumption presents a growing trend. The consumption uses of water resources are mainly divided into agricultural water, industrial water, domestic water, ecological water, and so on. The amount of domestic water directly affects the quality of life of local residents. The predicted value of domestic water consumption can be used as an important reference index for water supply decision-making and water conservancy construction investment, and the basis for formulating water supply and drainage planning and national economic planning. It plays a vital role in water resources planning and scheduling management, and is of great significance for China's modernization (Qin et al. 2013).

Although China is rich in total water resources, the per capita annual sustainable freshwater supply is only 2,000 m³, less than one-fourth of the average world level of 8,300 m³ (Manju & Sagar 2017). At present, one-quarter of the world's 400 million people facing severe water shortage live in China. The area of Shaanxi Province in China accounts for 2.14% of the whole country, while the total amount of water resources only accounts for 1.26% of the whole country's water resources. The per capita available water volume is only 736 m³, less than half of the national per capita level, and is below the international minimum water demand standard, making Shaanxi Province an extremely water deficient province in China. Moreover, the spatial and temporal distribution of water resources in Shaanxi Province is seriously uneven, with 70% concentrated in southern Shaanxi, and the water resources in Guanzhong and Northern Shaanxi are very scarce. With the rapid development of the social economy, the water consumption of Shaanxi Province is increasing. The development potential of water resources in some areas has reached the upper limit, and the contradiction between supply and demand of water resources has become increasingly prominent. As the core node of the construction of China's Silk Road Economic Belt and an important hub of Eurasia, Shaanxi Province straddles the Yellow River and Yangtze river basins, and is a key area in China's ecological security strategic pattern. It is very important to predict the water consumption accurately and reasonably plan and utilize the water resources in this area. At the same time, the region is also one of the most sensitive regions to climate change. The shortage of water resources is the core problem that restricts the region's economic and social development and ecological environment security. With the development of social economy and the improvement of people's living standards, the contradiction of water use will be further intensified, and domestic water, as an essential part of water use, needs to be given priority. Domestic water consumption prediction also plays a vital role in the unified management and allocation of water sources, the determination of the construction sequence and scale of water supply projects, and the implementation of drinking water safety measures for humans and livestock.

At present, domestic water consumption is affected by many factors, such as population, water price, economic development, environmental conditions, climate change and so on, which are nonlinear function. Many scholars have done a lot of research on the prediction of regional domestic water consumption, and the research methods are mainly divided into three categories: (a) time series method; (b) structural analysis method; and (c) system method.

The time series method does not consider the factors that affect the change of water consumption, but recognizes the change of water consumption as a ‘black box” system, which mainly depends on historical data and data model to predict. The standard methods of time series analysis are: (a) moving average method (Banihabib & Mousavi-Mirkalaei 2019); (b) Markov method (Gong et al. 2019); and (c) exponential smoothing method (Su et al. 2018). As the time series prediction is based on the accuracy of the prediction model, the model's improper use usually leads to a significant deviation (Firat et al. 2010).

The structural analysis method starts with studying the relationship between water consumption and influencing factors, analyzes the changes of different factors affecting water consumption, and establishes the relationship model between water consumption and influencing factors (Kanakoudis & Gonelas 2014, 2015). This method includes the regression analysis method (Chen et al. 2020; Zhang et al. 2021) and the index analysis method (Tian et al. 2020), which belongs to the category of cycle forecasting method and has a good effect on long-term forecasting. However, it is difficult to discuss the relationship between multi-factor and non-linearity in the system for structural analysis to find the law from the digital trend. The prediction methods are mostly simplified, such as simplifying the model to linear, reducing the number of variables, etc. Therefore, the simulation results are rough and the scope of application is narrow.

For this reason, some system research methods based on nonlinear prediction model are produced, such as neural network model (Walker et al. 2015; Zubaidi et al. 2018), system dynamics method (Lin et al. 2019), random forest algorithm (Brentan et al. 2018; Villarin & Rodriguez-Galiano 2019), support vector machine algorithm (Bai et al. 2015) and grey model method (Cui et al. 2013). These models reflect the nonlinear relationship in the process of domestic water consumption prediction to a certain extent. However, they are still deterministic models of point prediction, so it is difficult to predict the future fluctuation range of water consumption accurately. In the actual water use process, there are various uncertain factors (such as economic development, urban population, climate change, urban water use technology, etc.) affecting the prediction, so the impact of data fluctuation on the simulation data should be considered in the prediction. In order to evaluate the prediction effect of the model after considering the uncertainty factors, Duerr et al. (2018) compared the stochastic process model, machine learning model and other models, and concluded that the stochastic process model has a better prediction effect. Although some progress has been made in predicting urban domestic water consumption considering uncertainty, it is still necessary to explore some methods with sufficient information in the probability of prediction value (Ren et al. 2020). Correspondingly, interval prediction establishes a prediction interval under a certain confidence level, which reflects the uncertainty of domestic water consumption and helps decision makers make decisions according to the actual situation (Jiang et al. 2019). Ji et al. (2018, 2019) proposed an algorithm on parallel biological computing to solve water resources allocation and prediction. Deng et al. (2020a, 2020b) used the improved parallel intelligent algorithm to solve a series of complex problems, and achieved good results. The above algorithm provides a good reference for the prediction and optimization of water resources (Wang et al. 2021a, 2021b). However, the occurrence of water resources is affected by many factors, so we should consider the applicability of different calculation models, so as to find a better algorithm with better efficiency and prediction effect (Li et al. 2021). Therefore, it is necessary to study this issue further.

The Grey system is a new theory and technology for control and prediction, which was born in the second half of the 20th century. It has been widely used in agriculture, social economy and other fields. Grey system theory mainly takes “small sample” and “poor information” uncertain systems as the research object, extracts valuable information through the generation and development of some general information, and realizes the correct grasp and description of system operation behaviour and evolution law. Grey prediction predicts the development and change of system behaviour eigenvalues, the prediction of the system with both general and uncertain information; that is, the prediction of the grey process related to time series that change in a particular range. Although the phenomenon shown in the grey process is random and disordered, it is ordered and bounded after all, so the data set has potential rules. It is a theoretical method used to study the uncertainty problems with fewer data and less information, especially GM (1,1) as its most common model.

In the above modeling process, a is called the development coefficient of the grey system, reflecting the development trend of ⁠; u is called grey action, reflecting the change relationship between data.

Compared with the traditional GM (1,1) model, the unbiased GM (1,1) model basically eliminates the inherent deviation of the traditional GM (1,1) model, and its application scope is wider than the traditional GM (1,1) model. In addition, the unbiased GM (1,1) model does not need to be reduced, which simplifies the modeling steps and improves the calculation speed of the model.

In order to reduce the impact of large fluctuation of individual data on the prediction effect, the original data can be processed by a weighted moving average, and then the unbiased grey model is established, which is derived into the sliding unbiased grey model.

Because the traditional grey model is mainly suitable for system objects with short prediction time, fewer data and slight fluctuation, the prediction accuracy is low for the data series with significant random fluctuation. The Markov model is a mathematical method to predict the future state of things by studying the initial probability of different states and the transition probability between states. When the prediction sequence has the characteristics of the Markov chain stationary process and other mean value, the transition probability can reflect the influence degree of random factors, which makes up for the limitations of grey prediction. In this way, the Markov model, based on stochastic process theory, provides a new prediction solution for the series, with a larger fluctuation range.

Posteriori test is based on the statistical situation between the predicted value and the actual value of the model, which is transplanted from the probability prediction method. Based on the residual value, according to the absolute value of the residual error in each period, the probability of the occurrence of the point with smaller residual error and the size of the index related to the prediction error variance are investigated; that is, the mean square error ratio C and the small error probability P are jointly evaluated and calculated. The process is as follows:

• (1)

  Calculate the mean and variance of the original sequence:

• If the original sequence is and the predicted value sequence is ⁠, then

  

  (18)

  

  (19)
• (2)
  The mean and variance of residual error were calculated:

  

  (20)

  

  (21)

  

  (22)
• (3)
  The posterior variance ratio C and the minimum probability error are calculated:

  

  (23)

  

  (24)

The smaller the index C is, the better. It shows that although the historical data is very discrete, the residual value obtained by the model is not very discrete; that is, the closer the model is to the actual value. The larger the index P is, the better. It shows that there are more points where the difference between the residual and the average residual value is less than the given value; that is, the more uniform the distribution of the predicted value is. According to C and P, the accuracy of the prediction model can be comprehensively evaluated, as shown in Table 1.

Therefore, this study combines the grey prediction model and Markov model to improve the prediction accuracy of GM (1,1) model and reduce the requirements of the model on the stability of the sequence. The algorithm steps can be described as follows:

• (1)

  The sliding unbiased GM (1,1) model of annual water consumption series in Shaanxi Province is constructed by using the grey modeling method. The prediction series of the original series are obtained by using the model.

• (2)

  The residual sequence of the model is obtained from the difference between the absolute practical water quantity and the prediction sequence. It is divided into different state intervals according to the mean and standard deviation to obtain the state sequence.

• (3)

  The state sequence is transformed into a probability transition matrix, the autocorrelation coefficient of the sequence is calculated, and then the weight sequence is obtained. According to the transition probability matrix and weight coefficient sequence of each order, the influence probability of the state over the years on the state of the forecast year is determined.

• (4)

  According to the state transition probability and weight coefficient, the water consumption in the N + 1 year is calculated and predicted. The water consumption data is added to the original data series. The data of the first year in the series is removed to form a grey model with the same metabolic calculation dimension.

• (5)

  Repeat steps (1) to (4) to obtain the annual water consumption forecast data of the following year.

In conclusion, the flow chart of annual domestic water consumption prediction algorithm based on a sliding unbiased grey-Markov model is shown in Figure 1.

Shaanxi Province is located in the hinterland of Northwest China, across the Yellow River and the Yangtze River. It is located between 105°29’ ∼ 111°15'E and 31°42’ ∼ 39°35'N, covering a total area of 205,800 square kilometers. It governs ten cities and one administrative region. The province is long in the South and narrow in the East and the West. There are more than ten secondary and tertiary rivers in the area. The annual average temperature is 7–16 °C, and the annual average precipitation is 686.41 mm. The rainfall is mainly concentrated from July to September, showing the distribution characteristics of more in the South and less in the north. The annual average evaporation is about 1,608 mm, belonging to the continental monsoon climate. The inner drainage system is mainly distributed in the sandy grassland area of Northern Shaanxi, accounting for 2.3% of the province's total area, and the outer drainage system accounts for 97.7% of the total area of the province. Using the data of annual water consumption of residents in Shaanxi Province from 2003 to 2019 (shown in Table 2), the average annual water consumption is 1.485 billion cubic metres. The data of this paper comes from the China National Statistical Yearbook, in which the annual water consumption data from 2003 to 2016 are the original calculation data of the model, and the water consumption data from 2017 to 2019 are the prediction and analysis data of the model. The data analysis software is Matlab R2013b.

• (1)
  In this study, using the annual water consumption data of Shaanxi Province from 2003 to 2019, the cumulative generation sequence is established according to formula (1). In order to obtain the first order linear differential equation of Equation (2), the development coefficient and grey action of GM (1,1) model are obtained by using the least squares principle; according to formula (4), the accumulated prediction sequence is obtained, and then the prediction model of GM (1,1) is obtained by subtraction.

  

  (25)
• (2)
  It is observed that the annual water consumption data of residents in Shaanxi Province shows an exponential trend, so a new cumulative sequence is generated according to formula (6). Then, the parameters and are obtained by using the traditional GM (1,1) model; Using the above two parameters' data and calculating the parameters and according to formula (8), the unbiased GM (1,1) model is obtained by substituting them into formula (9):

  

  (26)
• (3)
  Through formula (11), a new series is obtained by weighted moving average processing of annual water consumption data of residents in Shaanxi Province; similarly, according to the processing method of unbiased grey GM (1,1) model in step (2), a sliding unbiased grey GM (1,1) model is obtained:

  

  (27)

The comparison between the predicted data and the real values calculated by the functional equation in 2017–2019 is shown in Table 3.

At the same time, the simulation and prediction results of the three grey simulation methods are compared with the actual annual water consumption (Figure 2).

It can be seen from Figure 2 and Table 3 that the accuracy of the three grey simulation methods is equivalent, the simulation effect of the improved grey simulation method is improved to a certain extent, and the performance of the sliding unbiased grey model is better. Although the two improved grey methods make the simulation effect of the model improved to a certain extent, they still can not overcome the discomfort of the grey model for nonlinear data. In order to further improve the simulation accuracy, the sliding unbiased grey model is modified by Markov, and the sliding unbiased grey-Markov model is established.

In order to further reduce the calculation error of the model, the Markov model is used to modify the prediction sequence data of the sliding unbiased grey model. Firstly, the error state is divided. According to the prediction data and actual water consumption of the sliding unbiased grey model, the relative error of different years is calculated, as shown in Table 4.

According to the relative error between the simulated data of the sliding unbiased grey model from 2003 to 2016 and the actual water consumption, taking ⁠, ⁠, and as the segmentation points, it is divided into five different state intervals, namely state 1: ⁠; state 2: ⁠; state 3: ⁠; state 4: ⁠; state 5: ⁠.

From step 2, the statistical value of was 26.48, and given the significance level ⁠, the statistical value of was obtained. Because of ⁠, the residual sequence satisfies the Markov property.

The autocorrelation coefficients and weights of each order of residual sequence are calculated according to formulas (14), (16) and (15), (17), as shown in Table 5.

According to the historical data, combined with the transfer probability matrix, the annual domestic water consumption from 2016 to 2019 is predicted.

According to Table 6, the maximum weighted sum is 0.4405, and the corresponding state is 5; that is, the state of the 2017 residual sequence is 5, and the interval is ⁠. Take an interval length as the prediction value, that is, ⁠, and add it to the original prediction sequence, and the prediction value after Markov correction is 16.7318.

The model adopts the equal dimension information processing method, removes the data of the previous 2003, adds the newly predicted data of 2016, and calculates the data of the following year. The simulation data obtained are compared with the original data (Table 7). It can be seen from the above analysis results that the prediction accuracy of the dynamic unbiased grey-Markov model is high.

Using the state transition probability matrix and state division table, the annual domestic water consumption data of Shaanxi Province from 2020 to 2025 can be predicted as shown in Table 8.

It can also be seen from Table 8 that the domestic water consumption in Shaanxi Province will maintain a sustained growth from 2020 to 2025, with an average annual growth of 2.06%. Therefore, Shaanxi provincial government should continue to strengthen the construction of a water-saving society. This paper puts forward the following suggestions on domestic water policy:

• (1)

  Improve the supervision mechanism of domestic water and establish a water resources management system in line with the actual situation of Shaanxi Province.

• (2)

  The government should establish a complete and unified water price system, give full play to the leverage of water price, and limit the behaviour of taking water at will and wasting water resource.

• (3)

  It is necessary to promote scientific and technological innovation, improve the recycling technology of water resources in the whole province, and improve the urban domestic water recovery rate.

• (4)

  Strengthen water-saving education and publicity, and raise the awareness of the whole people's water-saving.

The paper focuses on the estimation of the domestic water consumption in Shaanxi Province of China. In this study, taking the domestic water series data of Shaanxi Province from 2003 to 2019 as the research object, different grey models are established for modeling and comparative analysis, and the sliding unbiased grey model is modified by Markov to improve the accuracy of the model simulation data. Markov prediction has the property of ‘no aftereffect’; that is, when the state of the system at a particular time is known, the state of the system in the future is only related to the current time and has nothing to do with the past time. In short-term and medium-term forecasting of water consumption, the positive and negative of the residual sign is consistent with the hypothesis, which reflects the randomness and no aftereffect of the residual sign. Therefore, it is reasonable to use Markov to predict the sign of residual in this study. Especially when the water consumption of that year has nonlinear fluctuation, the improved grey Markov model can better fit the original data series, and the accuracy is improved more obviously, which also reflects the necessity of Markov correction of the grey model. In addition, in the process of building a grey-Markov model, the method of equal dimension information is used to update the original data in real time, so as to improve the accuracy of the model. Nevertheless, from the simulation results, the simulation accuracy of some years is still not high, GM (1,1), and unbiased grey model method simulation accuracy is limited (Rathnayaka et al. 2016). Next, although the model needs less information, the operation steps are relatively cumbersome, and it is not suitable for long-term domestic water consumption prediction. On the other hand, the partition of the Markov model is the key to determine the simulation effect of the model. In theory, the more detailed the partition is, the better. However, in practice, it is found that too dense a partition will make it difficult to select new data intervals, which increases the error. Therefore, reasonable division and judgment of interval is a direction worthy of attention and improvement in the future.

This paper was supported by the Open Research Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research (grant No. IWHR-SKL-201905).

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