# Differential Learning for Outliers: A Case Study of Water Demand Prediction
## 2 Related Work  
Predicting water demand has been an active area of research for several decades. This interest is motivated by the increase in water consumption as a result of urbanization as well as the limited availability of this natural resource. A review of various water demand modeling approaches over the last three decades is provided in [2]. This review highlights the differences between short-term and long-term predictions. In general, the focus of previous research in this area is either on identifying predictive features or developing predictive models for water demand. With respect to predictive features, the correlation between water consumption and weather conditions was confirmed in [1; 3; 4; 5]. In particular, temperature and precipitation have been identified as having a strong influence on water demand. Temperature and rainfall were used in [3]. Temperature and precipitation were used in [4]. In [6], a water demand prediction model for summer using maximum temperature, holiday and previous day's water demand was proposed. Economic factors such as water price and household income were also identified as key features in water demand prediction in [2]. An analysis of the spatio-temporal patterns of water consumption across the United States [7] concluded that urban areas are more efficient in terms of water usage. Higher efficiency of water usage was also observed in counties with higher income and education levels. Different models were also proposed for water demand prediction. These include regression, time-series clustering, support vector regression and neural networks. In [3; 4], linear regression models were used. Seasonal variations in hourly and daily water demand patterns are investigated in [8; 9]. In [8], time-series clustering and support vector regression (SVR) were used for hourly water demand prediction in Milan, Italy. Patterns for the type of day (e.g., weekday, weekend or holiday) and specific periods of the year (e.g., spring, summer, fall or winter) were identified. The water supplied up to 6 a.m. in the morning was then used to generate an accurate prediction for water demand for the rest of the day. Different SVRs were trained for each hour of the day and for each pattern. The resulting model was able to predict within a 2.56% and 13.26% standard deviation from the observed water demand in the best and worst case, respectively. Despite the fine granularity of this model and its predictive accuracy for normal water demand days, the study indicates that high prediction errors are associated with outlier days. This particular aspect of water demand prediction is the focus of the model proposed in this paper. An artificial neural network was used in [6] to predict water demand in Seoul, South Korea. The neural network model for the summer months consisted of one input layer with 3 input nodes, 3 hidden layers with 5 hidden nodes each and 1 output node. The resulting neural network outperformed the regression model yielding smaller prediction errors. Other neural network models for water demand forecasting with varying architectures have been proposed in [10; 11]. Despite the extent of the above studies in the water sector, very few address issues related to outliers. In [12], a model for leakage detection in water meters is developed by using a cumulative sum approach. A k-means clustering approach is used in [13] to detect outliers in automated meter reading systems due to leaks, meter breakdowns and meter frauds. A clustering approach for differentiating among different consumption levels is introduced in [14]. A classifier for high and low consumption households is presented in [15]. The first stage of this model is an unsupervised clustering algorithm based on self-organizing maps and the second stage is a neural network classifier. Outliers [16] are anomalies or rare events that are not well represented in the data set used to train the model. Novelties [17] are another type of rare events.