# Differential Learning for Outliers: A Case Study of Water Demand Prediction
## 4 Predicting Water Demand
The values of \(IW\) (Equation (7)) obtained for the reduced model are shown in Table 2. These \(IW\)s show a strong correlation between the water demand and two input features: day of the year and the previous day's hidden layer values throughout the year. The dependence on the previous day's hidden layer values captures the autoregressive component of water demand [30]. It is also the main reason for the improved performance of the proposed recurrent neural network model compared to the feed forward neural network model [1]. The dependence of the water demand prediction on the maximum temperature is low for the low demand months of January-April and November-December whereas, the dependence on maximum temperature is high for the high demand months of May-September. This confirms the correlation between maximum temperature and high water demand during the summer months. Moreover, the model's dependence on precipitation is low and indicative of the lesser importance of this feature in the service area. In addition to the original and the reduced features recurrent networks, four variant configurations were investigated. In the first configuration, precipitation was encoded as a binary (i.e., 0 or 1) input feature instead of a scaled input. In the second configuration, the hidden layer node values were propagated throughout the year instead of on a monthly basis. That is, the hidden layer values for 31 January 1997, were used as an input to the 1 February 1997, instead of as an input for the 1 January 1998. Both of these variant configurations failed to improve the accuracy of the prediction. In the third and fourth variant configurations, input recurrence and output recurrence were used in the reduced feature set model instead of the proposed hidden layer recurrence, respectively. Again, both of these \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **DoYr** & **MxTp** & **Precp** & **Bias** & **PHL** \\ \hline January & 29.53 & 2.42 & 0.50 & 26.41 & 41.15 \\ February & 16.39 & 2.57 & 6.64 & 14.08 & 60.32 \\ March & 9.82 & 1.68 & 2.66 & 9.92 & 75.92 \\ April & 10.28 & 6.65 & 6.62 & 20.60 & 55.85 \\ May & 10.97 & 30.79 & 3.67 & 37.33 & 17.24 \\ June & 15.82 & 27.21 & 1.97 & 40.79 & 14.21 \\ July & 15.86 & 39.01 & 2.46 & 26.77 & 15.90 \\ August & 4.83 & 24.73 & 3.02 & 28.33 & 39.09 \\ September & 6.93 & 23.86 & 5.72 & 31.31 & 32.18 \\ October & 16.06 & 14.26 & 4.79 & 20.60 & 44.29 \\ November & 13.68 & 2.08 & 5.80 & 14.13 & 64.31 \\ December & 12.65 & 3.32 & 1.70 & 15.72 & 66.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Importance of weight (\(IW\)) of each input feature in the reduced model, for each month over the period from 2011 to 2015. In addition to the bias node, the features are DoYr: day of the year, MxTp: maximum temperature, Prcp: precipitation and PHL: previous hidden layer. Figure 7: Predicted water demand using the reduced model for 2015.  
configurations under-performed compared to the original recurrent network model and the reduced feature model with hidden layer recurrence. The input layer recurrent network had an average error of 4.16% and the output layer recurrent network produced an average error of 3.65%.