Several studies [6,7,8] have noted the significant impact of various water consumption factors including previous water demand, number of family members, age of family members, garden size, frequency of irrigation, and the water consumption of agriculture.

Previous water consumption data have been considered as the key to estimating future consumption in numerous studies. To manage water consumption effectively, the data of each institution’s water consumption must be collected [9,10]. Creating a suitable model for Taiwanese domestic water consumption requires identifying the major impact factors, thus step-by-step filtering was used in this study to select the major impact factors. Moreover, to avoid multicollinearity problems, all factors were considered in the regression models.

Numerous studies have employed linear and nonlinear regression to establish water consumption models. Some based on linear regression have included rainfall, air temperature, family income, and the cost of water as independent variables. Regression models have also been used to establish models for related topics such as the water utility market structure [11,12,13,14]. A typical linear regression model of water consumption is expressed as (1) y = c + w 1 x 1 + w 2 x 2 + ⋯ + w P x P where y is the unit water consumption; w_i is weights; x_i is an impact factor of water consumption; and c is constant. As the model is linear, it is easy to estimate its advantages and disadvantages; however, the true relationships between water consumption and impact factors are not linear, but more complex. Hence, a model using one dependent variable and multiple predictive variables does not yield accurate forecasts. Therefore, nonlinear regression can also be employed (2) y = c · x 1 C 1 · x 2 C 2 ⋯ x P C P where c_i is the weight of regression. For rapid and convenient calculation, Equation (2) can be reformulated through logarithmic conversion (3) ( y + δ ) = c · ( x 1 + δ ) c 1 · ( x 2 + δ ) c 2 · ( x 3 + δ ) c 3 or (4) ( y + δ ) = c ′ + c 1 · log ( x 1 + δ ) + c 2 · log ( v 2 + δ ) + ⋯ + c P · log ( v P + δ ) where c ′ = log ( c ) .

Errors are common when traditional forecast methods such as time extrapolation are used. Although widely used in the early 20th century, time extrapolation is rarely used in current studies. ANNs are fast and flexible methods for effectively forecasting domestic water demand [15].

ANNs have been used for estimation models and forecasting in numerous fields. An advantage of ANNs is that they can correlate large and complex datasets [16,17]. An ANN was previously used to develop and assess a drinking water quality model, and a multilayer perceptron ANN was required in the hydrological modelling [18].

Over the past few decades, there has been a dramatic increase in the published research on sustainable water consumption, with most studies focusing on different industrial contexts. Few studies have discussed water consumption by individual government institutions. Despite the adoption of recent policies in Taiwan aimed at actively promoting water conservation, water demand has not substantially decreased as water consumption efficiency has not been enhanced (Table 1).

This paper reports the results of a five-phase study that explored the theoretical basis for the estimation model, thus establishing a framework, collecting data, analyzing simulation results, and deriving conclusions. The subjects considered were government institutions located on Taiwan Island, the Penghu Islands, the Kinmen Islands, and the Matsu Islands, all of which have water supplied by faucet. Our data consisted of 2611 units taken from government institution-reported water consumption data since 2006. As there are numerous categories of government institutions in the original database, the categories were divided into 6 primary categories and 47 minor categories (Table 2). Twenty-two independent variables were adopted in this study (Table 3).

The original database was sufficiently large to guarantee the accuracy of outlier effect models and data analysis. The quartile outlier method was adopted in this study. Furthermore, linear regression, nonlinear regression, and ANN models were developed by outlier effect models. To accord and compare these models, stepwise regression was used to select an independent variable. Each variable was also chosen to carry out the regression with other variables one by one. The advantage of this approach was that it avoided the problem of multicollinearity in each independent variable, thus preventing unstable regression parameters.

The ANN used in this study was the backpropagation neural network (BPNN), which is the most classic and general training algorithm. It also effectively solves problems including multilayers, feed-forwards, and supervised learning functions for different industries [19]. A constructive algorithm was used to determine the number of neurons in the hidden layer, which was initially set to one and gradually incremented until the most suitable number was determined [20]. The output was then expressed as (5) Z k = f ( b 0 k + ∑ j = 1 J b j k · f ( a 0 j + ∑ i = 1 I a i j · x i ) ) where f ( · ) is a transfer function; x i is the input; a i j and b j k are the weights; and a 0 j and b 0 k are the bias. The function f ( · ) is a mapping rule for converting input into output. The most commonly adopted nonlinear conversion function in BPNN studies is the binary logistic sigmoid (6) f ( x ) = 1 1 + e − x where f ( x ) = [ 0 , 1 ] . To obtain more optimal BPNN parameters, Z k (output value) and t k (target value) are adjusted through (7) E = 1 2 ∑ k = 1 K ( Z k − t k ) 2

BPNN uses the method of gradient descent to train all the examples during each learning epoch and obtains the weights a i j and b j k . The results obtained during the learning epoch are then fed back into the hidden layer to increase accuracy. Accordingly, (8) ∂ E ∂ b j k = ∂ E ∂ z k · ∂ z k ∂ v k · ∂ v k ∂ b j k where ∂ E / ∂ z k = ( z k − t k ) , z k = f ( v k ) . Thus, ∂ z k / ∂ v k = f ′ ( v k ) .

As v k = b 0 k + ∑ j = 1 J b j k · y j , ∂ v k / ∂ b j k = 1   ( j = 0 ) or y j ( j = 1 , 2 , ⋯ , J ) , Equation (8) can be differentiated as (9) ∂ E ∂ a i j = ( ∑ k = 1 K ∂ E ∂ z k ) · ∂ y j ∂ u j · ∂ u j ∂ a i j where y j = f ( u j ) ; thus, ∂ y j / ∂ u j = f ′ ( u j ) . The weights can be determined using Equations (8) and (9). When gradient descent was used, a common problem was that convergence did not feedback to the whole network, but only a partial network. To increase learning rate and accuracy, a momentum term was added to avoid oscillation during convergence. The mth weight can be expressed as (10) Δ w m = − η · ∂ E ∂ w m + α · Δ w m − 1 where η is the learning rate of the gradient descent method; and α is the momentum factor. To fit the range of the transport function, data were normalized using the max–min mapping method. For a minimum and maximum of the transport function f m i n and f m a x , the minimum and maximum inputs in the database were x m i n and x m a x , respectively (11) x ( s ) = ζ · [ f m i n + x − x m i n x m a x − x m i n   ( f m a x − f m i n ) ] where ζ is the normalized factor. Equation (11) can be reversed as (12) x ^ = x m i n + ( x ^ ( s ) / ζ ) − f m i n f m a x − f m i n × ( x m a x − x m i n ) where x ^ ( s ) and x ^ are estimates of x ( s ) and x, respectively.

A comparison of three methods was adopted, where the R² of ANN was obviously the highest. However, judging which method was more suitable via R² was far from enough. Five model efficiency indices were employed to determine the suitability of each model: the mean absolute deviation (MAD), root mean squared error (RMSE), revised Teil inequality coefficient (RTIC), correlation coefficient (CC), and coefficient of efficiency (CE), defined as (13) M A D = 1 N ∑ i = 1 N | Q i − Q ^ i | where N is the total number of units; Q i is the real water consumption; and Q ^ i is the estimated water consumption. (14) R M S E = ∑ i = 1 N ( Q i − Q ^ ) 2 N (15) R T I C = ∑ i = 1 N ( Q i − Q ^ i ) 2 ∑ i − 1 N ( Q i ) 2 (16) C C = ∑ i = 1 N ( Q i − Q ¯ i ) ( Q ^ i − Q ^ i ¯ ) ∑ i = 1 N ( Q i − Q ¯ i ) 2 · ∑ i = 1 N ( Q ^ i − Q ^ i ¯ ) 2 where Q ¯ i is the mean of Q i ; and Q ^ i ¯ is the mean of Q ^ i . (17) C E = 1 − ∑ i = 1 N ( Q i − Q ^ i ) 2 ∑ i = 1 N ( Q i − Q ¯ i ) 2

Of the five efficiency indices, MAD, RMSE, and RTIC indicated higher efficiency as they approached zero. As CC approached one, the simulated and actual values became more closely correlated, whereas CE approaching one indicated higher precision.