Mathematical Problems in Engineering

Volume 2019, Article ID 9765468, 11 pages

https://doi.org/10.1155/2019/9765468

## Committee Machines for Hourly Water Demand Forecasting in Water Supply Systems

^{1}School of Technology, Universidade Estadual de Campinas, Campinas, Brazil^{2}LHC-School of Civil Engineering, Universidade Estadual de Campinas, Campinas, Brazil^{3}Institute for Manufacturing, Department of Engineering, University of Cambridge, UK^{4}Fluing-Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, Valencia, Spain

Correspondence should be addressed to Joaquín Izquierdo; se.vpu@reiuqzij

Received 30 August 2018; Accepted 10 December 2018; Published 8 January 2019

Academic Editor: Javier Martinez Torres

Copyright © 2019 Julia K. Ambrosio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Prediction models have become essential for the improvement of decision-making processes in public management and, particularly, for water supply utilities. Accurate estimation often needs to solve multimeasurement, mixed-mode, and space-time problems, typical of many engineering applications. As a result, accurate estimation of real world variables is still one of the major problems in mathematical approximation. Several individual techniques have shown very good estimation abilities. However, none of them are free from drawbacks. This paper faces the challenge of creating accurate water demand predictive models at urban scale by using so-called committee machines, which are ensemble frameworks of single machine learning models. The proposal is able to combine models of varied nature. Specifically, this paper analyzes combinations of such techniques as multilayer perceptrons, support vector machines, extreme learning machines, random forests, adaptive neural fuzzy inference systems, and the group method for data handling. Analyses are checked on two water demand datasets from Franca (Brazil). As an ensemble tool, the combined response of a committee machine outperforms any single constituent model.

#### 1. Introduction

More than half of the world’s current population live in cities, with a growth of 1,500 million people in the last 20 years, and the United Nations predicts that this trend will continue. This population size scenario, together with the continuously under-stress natural resources, makes it paramount to count on accurate and efficient methods for estimating urban water demand [1]. This is nowadays possible as there are a huge quantity of available data and suitable big data tools to deal with it. However, there are still several open challenges regarding the study and analysis of big data. Not only may new mathematical techniques and machine learning paradigms improve the quality of estimation techniques but also efforts to obtain a better understanding of what the methods do in detail and which effects are caused by changing their parameters can give an insight into how several models could work together. Among them, approaches related to optimal and automatic tuning processes for the models’ hyperparameters [2, 3] should be highlighted. For short-term water demand forecasting processes, machine learning techniques such as artificial neural networks (ANNs) and support vector machines (SVMs) are widely applied [4].

Machine learning methods are able to map highly nonlinear spaces and to accurately estimate the ensuing output space. The results tend to be strongly linked to the preestablished definitions of their architectures (hyperparameters for each method), which are usually defined by the users. In this sense, models may require much time to be built with no guarantees of being optimal. Another interesting paradigm investigates the so-called ensemble learning [5] of several individual methodologies (machine learning methods or, plainly, machines) to generate a single combined model or committee machine. The latter, an approach not enough explored yet, should reach better predictive performance than that obtained from any of the constituent algorithms by themselves [6].

Water demand forecasting has been explored through machine learning techniques. For the short and long term [8] proposed a model of water demand forecasting for summer peak consumption, using and comparing multiple linear regression, time series analysis, and ANNs. Prediction of water demand using a dynamic ANN model was proposed by Ghiassi et al. [9]. The authors modeled water demand data using the DAN2 method, reaching good results and showing that predictions do not depend on the explicit inclusion of weather variables. The study was applied in monthly, weekly, and daily models, obtaining a prediction accuracy of 99%, and for the hourly models obtained a precision above 97%. The model was also compared with autoregressive integrated moving average models and traditional ANN models.

This paper proposes the use of committee machines for the creation of predictive models for urban water demand. A committee machine mixes different-nature methodologies. The aim is to take advantage of each component’s strengths, while avoiding its weaknesses when combined with other machine learning methods not necessarily based on the same algorithm [10]. For instance, a committee machine can reduce the influence of an accurate but not robust model by boosting the influence of a more robust algorithm for certain model scenarios. A combination of individual methods was proposed by Huang et al. [11]. The authors combined models, including wavelet transform and mean least squares partial-autoregressive moving average (KPLS-ARMA) to analyze the nonstationary behaviour of an annual urban series of water demand. The combined method proposed by the authors obtained improved accurate forecast of the city’s urban water demand. Arandia et al. [12], also combined methods to predict short-term water demands; in this case, the authors combined seasonally integrated, self-correcting seasonal moving average (SARIMA) models with data assimilation. In their case study, forecasts were compared with actual volumes of water produced by the local utility.

The present paper compares several combinations of models that have already shown good performance by themselves on forecasting urban water demand. This is the case of multilayer perceptrons (MLPs) [13], support vector machines (SVMs) [14], extreme learning machines (ELMs) [15], random forests (RFs) [16], adaptive neural fuzzy inference systems (ANFIS) [17], and the group method for data handling (GMDH) [18]. For each combination, the committee machine integrates independent tuned machines together with some predefined ensemble rules to build the final model. After training, the performance of the models is analyzed through error indices, such as the mean square root error (RMSE) and the mean absolute error (MAE).

To justify the use of the previously mentioned machine learning techniques, let us mention that techniques such as ANFIS have already been used for online identification in control systems and for predicting future values in chaotic time series [17]. RFs have also been used, for example by Cheng et al. [19], in a demand prediction study focused on the energy sector; the authors use an ensemble method based on the RF technique. Still within the scope of demand forecast in energy systems, Majumder et al. [20] proposed a study of solar energy forecast, through a decomposition in a hybrid empirical mode (EMD) and ELM. For the forecasting study of the demand, it is also possible to use the method for data handling (GMDH) [21]. The GMDH provided better results and performance than the obtained by applying SVMs.

There are a number of antecedents of proposals similar to committee machines on the creation of predictive models for various water engineering applications. This is the case of Barzegar et al. [22], which compares the accuracy of three neural computing techniques, such as MLPs, radial basis function neural networks (RBFNNs), and generalized regression neural networks (GRNNs) in the prediction of the groundwater salinity of the simple confined aquifer of Tabriz. The committee machine created by combining MLPs, RBFNNs, and GRNNs is showed to perform better than any of the individual techniques alone for predicting groundwater salinity. Lima et al. [23] generate a final committee machine by grouping a set of SVMs. Nadiri et al. [24] use supervised intelligent committee machines (SICMs) by combining several SVM models and neurodiffuse (NF) and gene expression programming (GEP) methods. Their aim was to evaluate groundwater vulnerability indexes of an aquifer. On urban water demand, Candelieri [25] uses SVMs in two stages, one for clustering and the other for forecasting short-term water demand. Brentan et al. [26] use a hybrid methodology combining SVMs with Fourier time series to approach near real-time urban water demand models. Combinations of self-organizing maps and RFs are also investigated in Brentan et al. [27].

In addition to the most used ANNs, other forms can also be used to predict water demand, as done by Guo et al. [28]. The authors proposed a comparison between a gated recurrent unit network (GRUN), a conventional ANN model, and a SARIMA model. The models aimed at predicting water demand for 24 hour horizon with a time interval of 15 min. In the case studied, the GRUN model obtained better performance compared to ANN and SARIMA. This type of approach shows the importance of studying not only models working individually but also together, as in the case of the ensemble methods.

Ensemble methods have been as well applied out to engineering related topics. To mention just a few, Johansson et al. [29] propose the use of a parallel model combination for predicting consumption of heating systems. Oliveira et al. [30] present a combination of ten MLP networks with different architectures and parameters. Polikar [31] shows that cluster-based systems may be more beneficial than their individual classifier counterparts. Ensemble methods are also used to predict nonstationary time series. This is the case of the work by Castro et al. [32], in which ANNs are trained with various parameter configurations and then grouped to provide a single solution.

The paper is organized as follows. After the Introduction, next section presents the methodological aspects. Then, a new section describes the case study followed by a section thoroughly describing the obtained results. Finally, the paper is closed by the section of conclusions and the references.

The current paper presents an important step ahead for the implementation of cutting-edge machine learning developments to improve water distribution systems control and management. On the other way round, the work is of interest on expanding the ways and topics on which novel machine learning and data-driven models are applied. The paper creates ensemble models for committee machines based on six machine learning techniques. The paper shows a simple but theoretically sound way of how estimation problems can be solved with an efficient multiresolution technique. Each model is applied to investigate the demand on two water supply areas of a medium-size city in the State of São Paulo in Brazil. Apart from the superior results produced by the committee machines when compared with single models, the paper also opens an in-depth discussion about the influences, limitations, and applicability of each technique and model combination rule.

#### 2. Materials and Methods

Machine learning techniques are able to learn patterns and solve complex problems just by processing (very often) large-size databases. Probably, the most classical machine learning approach is constituted by the artificial neural network (ANN) paradigm, especially MLPs. Over the years, ANNs have evolved towards other approaches for example, SVMs. A SVM maps the model input onto a high-dimensional feature space to ease further computations. ANN structures have also been adapted to new configurations. This is the case of ELMs in which some parameters do not need to be tuned. ANFIS, in its turn, is a variant of ANNs that uses fuzzy inference, which is powerful in approaching hybrid methodologies for parameter tuning. GMDH methods split a problem into manageable pieces where to apply regression techniques to thus produce simpler problems than the original. In a natural way, some versions of GMDH can be considered as variations of ANNs.

This section briefly introduces MLPs, SVMs, ELMs, ANFIS, and GMDH. RFs are also introduced for further combination within committee machines.

##### 2.1. Multilayer Perceptron (MLP)

MLPs have been extensively studied in the literature and applied in several areas, in particular in studies on water supply systems and demand forecasting [33, 34]. These networks are based on interconnections of their calculation units, called perceptrons, organized in various layers. MLPs count on an input layer, and the input data is available to all its nodes. They also have an output layer providing the outcome of the process. Then, one or more hidden (or inner) layers facilitate the internal computations required for reaching optimal estimation for the problem to solve. The outputs of the neurons of one layer are distributed to only the inputs of the neurons of the next layer. Thus, the input signal propagates through the network in a progressive (feedforward) way.

The information processing in a MLP flows in two modes. One is related to its (forward) propagation through the network, layer by layer, until the corresponding output is produced. This stage is directly related to the network performance. The other mode regards the network adaptation. The MLP connection weights are modified using the backpropagation algorithm, which is based on the observed error at comparing the estimated output and the real value available. The error is propagated backwards from the output layer to the input layer, and the connection weights at the hidden layer(s) can be consequently adjusted so as to minimize the difference between the estimated and the real value [35].

Considering a simple MLP with one inner layer with neurons, and neurons in the input layer, the output, , can be written in terms of the input vector, , as

In (1) are the input weights, and the weights of the hidden layer. Each linear combination is processed by an activation function, , responsible for further nonlinear transformations. The sigmoid function is the most popular activation function for MLPs.

##### 2.2. Support Vector Machine (SVM)

Support vector machines (SVMs) were developed focusing on nonlinear separable data problems [36, 37]. To achieve this separation, a SVM finds the ideal hyperplane that maximizes the distance between two groups, thereby minimizing the margin error. To do this, the input data is projected onto a higher dimensional space where the SVM is able to linearly separate the nonlinearly separable data in the original space. The transformation of dimension and data separation is schematized in Figure 1.