Optimizing Time Series Models for Water Demand Forecasting ^†

Abstract

This study focuses on optimizing time series forecasting models for water demand in a North Italian city as part of the Battle of the Water Demand Forecast (BWDF) challenge. It aims to accurately predict water demands across ten district-metered areas (DMAs) using historical data and weather information over a one-week horizon. The methodology encompasses data preprocessing, including missing data imputation, feature engineering, and novel normalization techniques, followed by the development and hyperparameter optimization of various data-driven models such as random forest, XGB, LSTM, and Prophet. Extensive cross-validation tests assess each model’s performance, revealing that our refined approach markedly enhances forecast accuracy, demonstrating the importance of model and parameter selection for effective water demand forecasting.

1. Introduction

In the realm of water distribution systems (WDS), ensuring a reliable and efficient water supply stands as a critical challenge, further amplified by global trends such as population growth, urbanization, and climate change. Estimating near-future water demands is crucial for improving WDS management, as it supports various applications such as pump control, maintenance scheduling, leakage identification, and even cyberattack protection [1]. Due to its significant importance, the domain of short-term water demand forecasting has been researched extensively in the past decades through various methods including statistics-based, machine learning, and deep learning approaches [2]. In recent years, advances in automatic meter reading (AMR) and advanced metering infrastructure (AMI) have revolutionized the processes of data collection, storage, and analysis. Consequently, more advanced forecasting methods have been introduced, which incorporate a wider range of data sources and place greater emphasis on accuracy, anomaly detection, and real-time robustness [3]. Despite the comprehensive research conducted on water demand forecasting, selecting the optimal models and fine-tuning their performance continue to present significant challenges. Factors such as local consumption patterns, climate variability, and specific WDS characteristics require a tailored approach to model selection and parameter optimization.

Battle of the Water Demand Forecast

The Battle of the Water Demand Forecast (BWDF) poses the challenge of forecasting the weekly demands of ten district-metered areas (DMAs) in a city in North Italy [4]. The dataset provided includes historic demand data for each DMA, derived from mass balance calculated based on flow measurements, alongside four weather parameters (rainfall depth, air temperature, windspeed, and air humidity). Additionally, key characteristics of each DMA are provided such as the consumption sector (residential, commercial, industrial) and the number of consumers. The battle unfolds over four test periods, with a new batch of data released after each period, such that teams have historical data up to the start of the next test period. The objective is to predict the most accurate forecasts according to three criteria:

• i1—Mean absolute error (MAE) of the first 24 h of the test period.
• i2—Max absolute error (MaxAE) of the first 24 h of the test period.
• i3—MAE of the period between the 24th hour and the rest of the test period.

2. Methodology

The proposed methodology follows a comprehensive data-driven workflow, encompassing data preprocessing, feature engineering, hyperparameter optimization, and prediction. The method includes several distinctive features aimed at improving forecast accuracy. The different DMAs vary in demand magnitudes, periodic patterns (daily only, combined daily and weekly), long-term trends, and seasonality; hence, a different model was adjusted for each DMA. Furthermore, acknowledging the distinct accuracy requirements between the initial 24 h period and the remainder of the week, we implemented two specialized forecasting models for each DMA. This approach is rooted in the understanding that models specifically tailored to different time horizons can significantly enhance overall forecasting accuracy. Accordingly, the proposed forecasting architecture was designed to tailor individual models per each DMA and forecasting horizon, resulting in 20 distinct models that were developed and optimized throughout the following stages.

2.1. Outlier Detection

The preprocessing stage incorporated outlier detection and filtering using one or more of the following methods: z-score, interquartile range (IQR), and rolling IQR. Additionally, the outlier filtering process included the removal of consecutive identical records, as this often indicates a malfunctioning or “stuck” sensor. Identified outliers were deleted and subsequently addressed during the next phase of missing data completion.

2.2. Missing Data Completion

The dataset provided contained approximately 5.5% missing values. To address this issue and ensure the integrity of our forecasting model, a k-NN imputation technique was employed. This method allowed us to accurately estimate and fill in missing data by leveraging the similarity between historical demands and weather observations, thus preserving the underlying structure and patterns in the dataset.

2.3. Feature Engineering

A thorough feature engineering process was undertaken, involving the meticulous selection and transformation of relevant variables that significantly influence water demands. The process included the construction of time variables such as hour of the day, day of the week, week of the year, and month. Next, the time variables were converted to periodic waveforms (sin and cos functions) to capture the cyclic nature of these features (hour 23 should be similar to hour 0). Two more time-related features that were introduced are “is special” and “is DST”, which indicate whether a data record is a holiday and whether it is daylight savings time or not. Lagged features were also constructed by shifting previous time steps of the demand and weather variables to capture temporal dependencies and trends. Lastly, moving statistics (moving average and standard deviation) and the decomposition of the data to its trend, seasonal, and noise components were examined, although not necessarily used for all models.

2.4. Data Normalization

A cornerstone of the presented methodology is data normalization. Besides standard normalization methods like standardization and min–max, two temporal dynamic normalization techniques were employed and found effective. The first one was moving window standardization (MWS) and the second was periodic window standardization (PWS). These methods adopt time-dependent means and standard deviations (STDs) for data standardization instead of considering the entire dataset parameters. PWS uses the mean and STD of a corresponding fixed period, which in our case is one week. It ensures that the normalization process respects the temporal boundaries and characteristics unique to each week, enhancing model sensitivity to long-term seasonal and cyclical trends. The MWS dynamically adjusts the normalization parameters based on a sliding window, offering a tailored approach that adapts to recent changes and fluctuations in the data. This adaptability makes the model more responsive to short-term variations and anomalies, potentially improving forecasting accuracy in rapidly changing conditions.

2.5. Models

Multiple data-driven models were developed, including SARIMA, random forest (RF), XGBoost (XGB), a long short-term memory (LSTM) network, and Prophet. Notably, an advanced version of the XGB model was specifically designed for multi-series forecasting. This approach allows the model to forecast for one DMA while incorporating information from other DMAs, leveraging cross-correlations between areas to enhance predictive accuracy. In the Multi-XGB model, each DMA is associated with a cluster of other DMAs to be used as supporting information. Accordingly, several DMA clustering alternatives were tested to find the optimal information base for each DMA.

2.6. Optimization and Cross-Validation

Next, model configurations underwent optimization, including the selection of the prediction algorithm, normalization method, and hyperparameters. To this end, an extensive random search was employed, significantly enhanced by parallelizing numerous experiments, to explore the vast space of potential configurations efficiently. The process was repeated for each test period to fine-tune the configurations, ensuring they were optimally adjusted to the specific period conditions. Overall, a total of 13.8 million train-predict experiments were conducted with a total run time of 440 million seconds, meaning almost 14 years of computations. The best-performing models were identified based on the BWDF metrics. These selected models were then rigorously tested through cross-validation, ensuring their robustness and generalizability. The models that consistently demonstrated superior performance across a diverse set of validation periods were ultimately chosen.

3. Results

The optimal models yielded by the above process indicate the superiority of XGB and its multi-series version over the other algorithms that were tested (SARIMA, RF, LSTM, and Prophet). Notably, the temporal dynamic normalization methods, PWS and MWS, were found to improve prediction accuracy compared to standard normalization techniques in most cases. Among the 80 models developed (two for each DMA for each test period), 45 used PWS and 11 used MWS. To demonstrate the performance of the proposed approach, the selected models were validated against four validation periods, corresponding to the last week leading up to each real test period. The results are depicted in Figure 1 along with the averaged metrics of i1, i2, i3, and mean absolute percentage error (MAPE) over the four periods. Most DMAs have a MAPE of less than 5%. Additionally, the MAEs of the short and long terms (metrics i1 and i3) are in the same magnitude for all DMAs, which underscores the robustness of the proposed approach against accuracy degradation on the longer horizon.

4. Summary

This study presents a method to predict weekly water demands within the competitive framework of the BWDF. The method employs a set of tailored models for each DMA and respective forecasting horizon. The results underscore the robustness and adaptability of the proposed approach, showcasing its capacity to deliver accurate predictions across diverse DMAs and varying conditions. The achieved accuracy levels meet the requirement for most WDS management applications, illustrating the practical viability of our method for improving WDS efficiency and reliability. While basic forecasting can nowadays be readily produced using numerous available time series analysis packages, the value of a systematic approach cannot be overstated. It refines accuracy beyond the baseline predictions and facilitates ongoing adjustments of models to adapt to dynamic conditions.