Probabilistic Long-term Prediction of Drinking Water Demand (Case Study: Neyshabur City)

Introduction
Probabilistic prediction model is obtained by development of spot prediction model. Unlike definite prediction, probabilistic prediction provides an amplitude of probable amounts for water demand. It is clear that there is more trust to amplitude values in decision making process. Probabilistic prediction of water demand is calculated through the distribution of uncertainty in independent variables by water demand model and production of a probabilistic distribution function for water demand in the point of interest in future. Looking for the probabilistic distribution allocation, Monte Carlo simulation process is used to develop probabilistic prediction of water demand according to the spot model by using Stone-Geary utility function. For each repeat in Mont Carlo simulation, a value of allocated distributions to each descriptive variable is used randomly. In this research both methods of determination of probable density functions consist of analyzing the past information and other experiences are used. Normal and uniform distribution functions were used because of their easiness in determination of parameters whenever their real distribution is not clear or it cannot be determined easily.

Materials and methods
To apply uncertainty on independent variables used in prediction of domestic water demand consist of per capita real income, the real price of water, stock prices of goods and services, the average maximum temperature and the number of literates, the best distribution of their monthly values, resulted from probabilistic distribution function, were used as the entrance of water demand function and the outputs obtained as the monthly water demand. For validation of water demand prediction model, the predicted values of water demand was compared with its real values in a few periods. In this research the LARS-WG micro scale statistical exponential model was used to predict the annually maximum temperature.
Neyshabur city, located in Razavi Khorsan province, was selected as a case study and the statistical society of domestic water branches was investigated. The required information was obtained from its subsidiaries annually function report and statistics of meteorological organization.
Independent variables which were used for water demand estimation are per capita real income, the real price of water, stock prices of goods and services, the average maximum temperature and the number of literates. The best distribution was selected by applying distribution functions on the value of variables, considering results of the Anderson test, selecting the minimum Anderson multiplier and attending to the others studies and experiences.
The final model for long term water demand according to the Ston-Geary utility function is as equation 1:
(1) t=1,…, 120 
MP: is the average price of water (Rial)
I: is the per capita income of consumer (Rial)
PO: is the stock price of goods & services in Razavi khorasan province
Perc: is the amount of per capita consumption (m3)
E: is the number of literates
MT: is the average maximum temperature
U: is the disturbing element


Discussion of results and conclusions
Calculated elasticity of price, income, intersecting, maximum temperature and number of literates by considering their average values in investigated time periods for Neyshabur city are presented in Table 1.

Table 1. Calculated elasticity of water demand for Neyshabur city
Price Income Intersecting Maximum temperature Number of literates
-0.117 0.195 -0.078 0.054 0.402

Table 1 shows if the price of water increases one unit, it will result to decrease of just 0.117 unit in water demand that indicates the possibility of using pricing policies for decreasing water consumption. Relatively low income elasticity (0.195) shows the small share of water in the family income. Negative intersecting elasticity indicates that water is a complementary good. Temperature elasticity is positive (0.054) that notes the tendency of using water by temperature rise. In this research the elasticity for number of literates obtained equal to 0.402. So one percent increase in the number of literates will result to 0.402 percent increase in water demand.
To assess the accuracy of the model the per capita water demand was predicted by using independently observed variables for the period of 1997 to 2008 and the results were compared with observed information that showed a good match. Also results of RMSE and MSE tests for assessing the accuracy of the models was 0.22 and 0.12 respectively that emphasizes on acceptable accuracy of the model.
After finalizing the model the per capita water demand of Neyshabur was predicted by the spot method and summary of results are showed in Table 2.

Table 2. Summary of the per capita water demand prediction by spot method.
Unit Year Percentage
changes Average of yearly changes
2008 2011 2016 2021 2026 2031
M3 51.18 53.35 60.06 66.03 71.93 79.31 48.6% 2%

Finally the per capita water demand was predicted for Neyshabur through the probabilistic model and summary of the result is presented in Figure 1.



Fig.1. Comparison of the observed values and 90% confidence interval of the annual per capita water demand forecast.

As it can be seen in Figure 1 the annual amount of predicted mathematical expectation for the per capita water demand in year 2031 is calculated equal to 80.36 m3 which shows 50% increase in comparison with year 2011. The average annual increase in water demand is about 2% which is consistent with the results of the spot prediction model. For year 2011 the defined confidence interval for water demand has obtained between 48.34 and 58.36 m3 averagely. While for year 2031 this range has grown and broadened to 67.66 and 98.64 m3. Furthermore, probability function shows 90% confidence interval for all probable predictions with considering uncertainty of independent and explanatory variables.