论文原文：点击此处
论文年份：2019
论文被引：10(2020/08/06/) 43(2022/03/26)

文章目录

A Comparative Study of Time Series Forecasting Methods for Short Term Electric Energy Consumption Prediction in Smart Buildings
1. Introduction
2. Related Works
3. Materials and Methods
- 3.1. Data
- 3.2. Methods
4. Results
5. Conclusions and Future Work

A Comparative Study of Time Series Forecasting Methods for Short Term Electric Energy Consumption Prediction in Smart Buildings

Abstract: Smart buildings are equipped with sensors that allow monitoring a range of building systems including heating and air conditioning, lighting and the general electric energy consumption. Thees data can then be stored and analyzed. The ability to use historical data regarding electric energy consumption could allow improving the energy efficiency of such buildings, as well as help to spot problems related to wasting of energy . This problem is even more important when considering that buildings are some of the largest consumers of energy . In this paper, we are interested in forecasting the energy consumption of smart buildings, and, to this aim, we propose a comparative study of different forecasting strategies that can be used to this aim. To do this, we used the data regarding the electric consumption registered by thirteen buildings located in a university campus in the south of Spain. The empirical comparison of the selected methods on the different data showed that some methods are more suitable than others for this kind of problem. In particular, we show that strategies based on Machine Learning approaches seem to be more suitable for this task.

摘要：智能建筑配备有传感器，可以监控一系列建筑系统，包括供暖和空调、照明和一般电能消耗。然后可以存储和分析这些数据。使用有关电能消耗的历史数据的能力可以提高此类建筑的能源效率，并有助于发现与能源浪费有关的问题。当考虑到建筑物是最大的能源消耗者时，这个问题更为重要。本文对智能建筑的能耗进行了预测，并针对这一目标提出了不同预测策略的比较研究。为了做到这一点，我们使用了西班牙南部一所大学校园内13栋建筑的用电量数据。对所选方法在不同数据上的实证比较表明，有些方法比其他方法更适合于这类问题。特别地，我们发现基于机器学习方法的策略似乎更适合这项任务。

Keywords: time series forecasting; electric energy consumption forecasting; machine learning

1. Introduction

The rapid economic growth and the increasing population in developing countries are causing the global energy consumption to increase. Despite efficiency gains over the last decades, it is estimated that by 2040 the global energy demand will likely increase nearly 25 percent [1]. According to the International Energy Agency (IEA) [2], developing regions are expected to experience large demand growth between 2017 and 2040, especially countries from Asia and Africa. Under the studied scenario, CO2emissions from energy will rise around 10% (from 32.6 to 35.9 gigatons). While in developed countries CO2emissions are expected to drop by 23%, in developing regions they are supposed to rise by 27%.

发展中国家经济的快速增长和人口的不断增加，导致了全球能源消费的增加。尽管在过去的几十年里能源效率有所提高，但据估计，到2040年，全球能源需求可能会增长近25%[1]。根据国际能源署（IEA）[2]，发展中地区预计在2017年至2040年期间将经历大量需求增长，尤其是亚洲和非洲国家。在所研究的情景下，来自能源的二氧化碳排放量将增加10%左右（从326亿吨增加到359亿吨）。而在发达国家，二氧化碳排放量预计将下降23%，而在发展中国家，二氧化碳排放量预计将上升27%。

Energy consumption has a significant impact on the environment. CO2emissions have been identified as responsible of most of the progressive warming of the Earth [3]. Furthermore, current predictions show that this growing trend will continue. Therefore, the reduction of CO2 emissions has become the focus of international political, economic and environmental research. To lower the energy consumption and help protect the environment, new energy efficiency policies are being implemented. For example, the current energy plan developed by the European Commission requires European Union’s countries adopt a set of measures to reach an energy efficiency of at least 20% [4]. Despite the fact that many industries contribute to such emissions, the building and building construction sectors are responsible for 36% of total energy consumption and nearly 40% of global CO2emissions [5].

能源消耗对环境有重大影响。二氧化碳排放被认为是造成地球逐渐变暖的主要原因[3]。此外，目前的预测显示，这种增长趋势将继续下去。因此，二氧化碳减排已成为国际政治和经济研究的焦点。为了降低能源消耗和保护环境，新的能源效率政策正在实施。例如，目前由欧盟委员会制定的能源计划要求欧盟各国采取一系列措施，以达到至少20%的能源效率[4]。尽管许多行业都对此类排放做出了贡献，但建筑和建筑施工部门占总能耗的36%，占全球二氧化碳排放量的近40%[5]。

In an efficient scenario, several building sub-sectors, such as space heating and cooling, water heating, lighting, etc., have been identified for potential energy savings [5]. In this context, the development of energy consumption prediction models are growing in importance for decision-making to implement energy policies. The forecasting problem is generally divided into three categories, based on the prediction horizon: short-term, medium-term and long-term load forecasting. Short-term forecasting involves prediction horizons going from one hour up to a week while medium-term refers to predictions from one moth up to a year. Finally , long-term predictions are characterized by a prediction horizon of more than a year [6]. Most works address short-term prediction horizon since it achieves higher accuracy than the other horizons.

在一种有效的方案中，已经确定了多个建筑子行业，例如空间供暖和制冷，热水，照明等，以节省能源[5]。在这种情况下，能源消耗预测模型的开发对于实施能源政策的决策越来越重要。根据预测范围，预测问题通常分为三类：短期，中期和长期负荷预测。短期预测涉及从一小时到一周的预测范围，而中期预测是指从一月到一年的预测范围。最后，长期预测的特征是预测期超过一年[6]。大多数工作都针对短期预测范围，因为它比其他范围具有更高的准确性。

Nowadays, more and more buildings are equipped with sensors that can measure various aspects of the functioning of a building, including the electric energy consumption registered. Such buildings are called smart buildings. Such measurement can be stored, and thus can provide valuable historical data, which can be organized as time series, and thus could be used to predict future energy demands of the buildings.

如今，越来越多的建筑物配备了传感器，可以测量建筑物功能的各个方面，包括所记录的电能消耗。这样的建筑物称为智能建筑物。这样的测量可以被存储，并且因此可以提供有价值的历史数据，其可以被组织为时间序列，并且因此可以被用于预测建筑物的未来能量需求。

Traditionally , time series have been tackled using conventional methods, such as statistical analysis, smoothing techniques and exponential smoothing and regression-based approaches. For example, Sen et al. [7] applied the Auto Regressive Integrated Moving A verage (ARIMA) approach for forecasting energy consumption and greenhouse gas emission in Indian pig iron manufacturing organizations. In [8], Chujai et al. analyzed the household electric consumption using the Autoregressive Moving Average (ARMA) and ARIMA models. However, despite their popularity , these models are not able to capture complex interactions from non-linear data. In this case, Machine Learning (ML) approaches have arisen as a suitable approach to handle such complexity . For instance, Bonetto and Rossi [9] applied Support V ector Machines and Neural Networks to energy consumption in households.However, despite the good performance achieved with Machine Learning techniques, they may get stuck in a local optimum, a phenomenon that negatively affects their performances. To overcome this problem, in recent works, many authors are exploring new approaches. In Divina et al. [10], Divina et al. proposed to apply an ensemble approach, while, in [11], Meira and Oliveira introduced a bagging approach of conventional methods (ARIMA and exponential smoothing methods).

传统上，时间序列已使用常规方法解决，例如统计分析，平滑技术以及指数平滑和基于回归的方法。例如，Sen等[7]应用自动回归综合移动平均（ARIMA）方法来预测印度生铁制造组织的能耗和温室气体排放。在[8]中，Chujiai等人。使用自回归移动平均（ARMA）和ARIMA模型分析了家庭用电量。但是，尽管它们很流行，但是这些模型无法从非线性数据中捕获复杂的交互作用。在这种情况下，机器学习（ML）方法已成为处理此类复杂性的合适方法。例如，Bonetto和Rossi [9]将支持向量机和神经网络应用于家庭能源消耗。然而，尽管机器学习技术取得了良好的性能，但它们仍可能陷入局部最优状态，这种现象会对他们的家庭产生负面影响。为了克服这个问题，在最近的著作中，许多作者正在探索新的方法。在Divina等人中。 [10]，Divina等。因此，在[11]中，Meira和Oliveira提出了一种常规方法（ARIMA和指数平滑方法）的装袋方法。

In this work, we propose a comparative empirical evaluation of different time series forecasting strategies. In particular, we consider both statistical and ML based approaches to this problem. To validate the techniques, we applied them on a dataset regarding the electricity consumption registered by thirteen buildings located at the Pablo de Olavide (UPO) University campus in Seville, Spain, collected over five and a half years. In the experiments conducted, we aimed at predicting the electric energy consumption with a one day horizon. Predictions were based on historical data, and another objective of the experimentation proposed was to assess an optimal value of the amount of historical data that should be used for this kind of problem. Therefore, we can summarize the contributions of this work as follows:

在这项工作中，我们提出了不同时间序列预测策略的比较经验评估。特别是，我们同时考虑了基于统计和基于机器学习的方法来解决此问题。为了验证该技术，我们将它们应用到了一个数据集，该数据集涉及西班牙塞维利亚的Pablo de Olavide（UPO）大学校园中的13座建筑物记录的用电情况，收集了五年半的时间。在进行的实验中，我们旨在预测一天内的电能消耗。预测基于历史数据，提出的实验的另一个目标是评估应用于此类问题的历史数据量的最佳值。因此，我们可以将这项工作的贡献总结如下：

Analysis and comparison of the performance of statistical and ML based strategies;
分析和比较基于统计和机器学习策略的性能；
Establishing the size of the historical window to be used to optimize the predictions; and
建立用于优化预测的历史窗口的大小；和
Analysis of the electricity consumption data collected from the smart buildings considered.
分析从考虑的智能建筑收集的用电量数据。

Results obtained show that Machine Learning approaches could achieve better results, in particular methods based on bagging and boosting ensemble schemes obtained the best results. Moreover, we found that using historical data relating to more than seven days yielded better predictions.

获得的结果表明，机器学习方法可以获得更好的结果，特别是基于装袋和增强合奏方案的方法获得了最佳结果。此外，我们发现使用超过7天的历史数据可以得出更好的预测。

The rest of the paper is organized as follows. In Section 2, some related works are introduced. Then, the data and the techniques used are described in Section 3. In Section 4, the analysis and discussion of the experiments conducted are presented. Finally , the conclusions and future works are presented in Section 5.

本文的其余部分安排如下。在第2节中，介绍了一些相关的作品。然后，在第3节中描述了使用的数据和技术。在第4节中，介绍了进行的实验的分析和讨论。最后，第5节介绍了结论和未来的工作。

2. Related Works

In the scientific community, there is a growing interest in addressing the forecasting problem in energy-related problems such as electricity consumption, loading and demand for various reasons. For example, as pointed out in [12], to create a healthy and sustainable economy , it is necessary to measure the socio-economic and environmental impact of energy production. Another important aspect is the identification of the different electricity consumption behaviors, which may allow adopting policies according to demand response scenarios [13]. Being able to predict future energy demands would provide several concrete benefits, at both economic and social levels. For example, the ability to balance fluctuations in renewable energy generation would facilitate a greater penetration of renewable resources into the electricity system [14]. On the other hand, we may also improve the economic efficiency by applying real-time prices and reducing costs connected to the generation capacity requirements, reducing at the same time the related CO2emissions [15,16].

在科学界，出于各种原因，人们越来越关注解决与能源有关的问题（如电力消耗，负荷和需求）中的预测问题。例如，如[12]中指出的那样，要建立健康，可持续的经济，就必须衡量能源生产的社会经济和环境影响。另一个重要方面是识别不同的用电行为，这可以允许根据需求响应方案采取政策[13]。能够预测未来的能源需求将在经济和社会层面上带来一些具体的好处。例如，平衡可再生能源发电波动的能力将促进可再生资源更多地渗透到电力系统中[14]。另一方面，我们也可以通过应用实时价格并降低与发电能力要求相关的成本来提高经济效率，同时减少相关的二氧化碳排放量[15,16]。

In the literature, time series analysis is the most popular approach for forecasting demands [17]. Among the most widely used time series forecasting methods, we can find the autoregressive moving average (ARMA) and the autoregressive integrated moving average (ARIMA). ARMA and ARIMA are statistical approaches that have been widely applied to short-term forecasting problem. For example, Abdel-Aal and Al-Garni [18] used, in an early work, ARIMA to forecast monthly electric consumption in the Eastern Province of Saudi Arabia. Chujai et al. [8] compared ARIMA with ARMA on household electric power consumption. ARIMA model was also applied by Shirpa and Shashadri [19] to develop a short-term electric load forecasting model of Karnataka State, India.

在文献中，时间序列分析是预测需求的最流行方法[17]。在最广泛使用的时间序列预测方法中，我们可以找到自回归移动平均值（ARMA）和自回归综合移动平均值（ARIMA）。 ARMA和ARIMA是已广泛应用于短期预测问题的统计方法。例如，Abdel-Aal和Al-Garni [18]在早期工作中使用ARIMA预测了沙特阿拉伯东部省的每月用电量。 Chujai等。 [8]在家庭用电量上比较了ARIMA和ARMA。 Shirpa和Shashadri [19]还应用了ARIMA模型来开发印度卡纳塔克邦的短期电力负荷预测模型。

Various extensions of ARMA and ARIMA have been proposed to include different aspects affecting a time series: the Auto Regressive Integrated Moving Average with eXternal (or eXogenous) input (ARIMAX), which add explanatory variables; the Seasonal Autoregressive Integrated Moving Average (SARIMA), which supports the direct modelling of the seasonal component of the series; or the Multiplicative Seasonal Autoregressive Integrated Moving Average (MSARIMA), which incorporates independent explanatory variables to SARIMA. The ARIMAX was used by Newsham and Birt in [20] to forecast several hours ahead the power demand for an office building with the objective of obtaining a better response to utility signals and reducing the overall energy use. MSARIMA was applied in [21] by Rallapalli and Ghosh to forecast monthly peak electricity demand from five different regions of India with the aim of achieving a better load management.

已提出ARMA和ARIMA的各种扩展，以包括影响时间序列的不同方面：具有eXternal (or eXogenous) input (ARIMAX)的自动回归综合移动平均线，它添加了解释变量；季节性自回归综合移动平均线（SARIMA），它支持对该系列的季节性成分进行直接建模；或乘积季节自回归综合移动平均线（MSARIMA），它将独立的解释变量合并到SARIMA中。 Newsham和Birt在[20]中使用ARIMAX预测办公楼的电力需求提前几个小时，目的是获得对公用事业信号的更好响应并减少总体能源消耗。 Rallapalli和Ghosh在[21]中使用MSARIMA来预测印度五个不同地区的每月峰值电力需求，目的是实现更好的负荷管理。

Regression based techniques represent another branch of classical methods used in time series forecasting. In particular, linear regression has been extensively applied. For instance, Schrock and Claridge [22] used a linear regression approach to investigate the hourly and daily electrical consumption pattern of a supermarket located 100 miles north-northwest of Houston, T exas. Nowotarski et al. [23] applied a family of regression models trained on different subsets of variables. More specifically, data generated from the Global Energy Forecasting Competition 2014 and ISO New England were used. In [24], Samarasinghe and Al-Hawani compared multiple linear regression with Gaussian Processes on power consumption data from 2008 to 2010 to forecast the values in the next 24 h in Norway . In a more recent work, Rahman et al. [25] proposed a high-precision methodology that included multiple linear regression and simple regression model along with other techniques to forecast the total energy consumption in India.

基于回归的技术代表了时间序列预测中使用的经典方法的另一个分支。特别地，线性回归已被广泛应用。例如，Schrock和Claridge [22]使用线性回归方法研究了位于德克萨斯州休斯顿西北100英里处的一家超市的每小时和每天的用电模式。 Nowotarski等。 [23]应用了一系列对变量的不同子集进行训练的回归模型。更具体地说，使用了2014年全球能源预测竞赛和ISO新英格兰的数据。在[24]中，Samarasinghe和Al-Hawani在2008年至2010年的能耗数据上将多元线性回归与高斯过程进行了比较，以预测挪威未来24小时内的数值。在最近的工作中，Rahman等人。 [25]提出了一种高精度方法，其中包括多元线性回归和简单回归模型以及其他技术来预测印度的总能源消耗。

Other forecasting approaches are based on Machine Learning techniques. Within these approaches, the use of Artificial Neural Networks (ANN) have been extensively and successfully applied. In an early work, Park et al [26] used an ANN to induce the relationship among past, current and future temperatures and loads in the electric load forecasting problem. The model was tested on hourly temperature and load data from the Seattle and Tacoma areas in the period from 1 November 1988 to 30 January 1989, with the objective of predicting 1-h and 24-h ahead values. In another early work, Nizami and Ai-Garni [27] proposed a two-layered fed-forward ANN to study how weather-related features may affect the prediction of monthly electric energy consumption. This approach was tested on data collected from the Eastern Province of the Saudi Arabia between August 1987 and July 1993. In more recent works, ANNs are used to tackle the problem of short-term electrical load forecasting, e.g., [28–31]).

其他预测方法基于机器学习技术。在这些方法中，人工神经网络（ANN）的使用已得到广泛成功的应用。在早期工作中，Park等人[26]使用人工神经网络在电力负荷预测问题中引入了过去，当前和将来的温度与负荷之间的关系。该模型在1988年11月1日至1989年1月30日期间对西雅图和塔科马地区的小时温度和负荷数据进行了测试，目的是预测1小时和24小时提前值。在另一项早期工作中，Nizami和Ai-Garni [27]提出了两层前馈神经网络，以研究与天气有关的特征如何影响每月电能消耗的预测。该方法已在1987年8月至1993年7月期间从沙特阿拉伯东部省收集的数据中进行了测试。在最近的工作中，人工神经网络用于解决短期电力负荷预测问题，例如[28-31]）。

In addition to ANNs, other Machine Learning techniques have been successfully applied to the problem of energy consumption forecasting. For example, Support Vector Regression (SVR) was applied by Jain et al. [32] to develop an energy forecasting model for a multi-family residential building in New York City . The data used in this work span from 27 August 2012 to 19 December 2012. The model was tested using three different temporal granularities: daily , hourly and 10 min. SVR was also applied by Liu et al. [33] to show its effectiveness in predicting hourly energy consumption of buildings. The popular K-Nearest Neighbour (k-NN) technique was used in [34] to explore the performance of the model for short-term electric energy demand. Other approaches were based on Deep Learning [35,36], Random Forest [37] and Evolutionary Algorithms [38]. Finally , it is worth noting that many of these techniques have been adapted within the big data paradigm. For example, Talavera-Llames et al. [39] adapted the strategy proposed in [34] by Lora et al.

除人工神经网络外，其他机器学习技术已成功应用于能耗预测问题。例如，Jain等人应用了支持向量回归（SVR）。 [32]为纽约市的多户住宅建筑开发能源预测模型。该工作中使用的数据跨度为2012年8月27日至2012年12月19日。模型使用三种不同的时间粒度进行了测试：每日，每小时和10分钟。 Liu等人也应用了SVR。 [33]显示其在预测建筑物每小时能耗方面的有效性。在[34]中使用了流行的K最近邻（k-NN）技术来探索短期电能需求模型的性能。其他方法基于深度学习[35,36]，随机森林[37]和进化算法[38]。最后，值得注意的是，其中许多技术已在大数据范例中进行了调整。例如，Talavera-Llames等。 [39]修改了Lora等人在[34]中提出的策略。

Another strategy that has been used is to combine different strategies to get advantage from the different approaches. For instance, Geng et al. [40] proposed a hybridization of SVR with Chaotic Sequence and Simulated Annealing to forecasting monthly electric load data from northeast China and New York City . Both datasets ranges from January 2004 to October 2005. Fan et al. [41] proposed a novel hybrid method that combines SVR, differential empirical mode decomposition (DEMD) strategy and auto regression (AR) for electric load forecasting. The resulting strategy was tested on two datasets. The first consisting of half-hourly electric load data from the New South Wales while the second one of hourly electric load data from the New York Independent System Operator.

已使用的另一种策略是组合不同的策略，以从不同的方法中获得优势。例如，耿等[40]提出了一种SVR与混沌序列(Chaotic Sequence)和模拟退火(Simulated Annealing)的混合方法，以预测来自中国东北和纽约市的每月电力负荷数据。这两个数据集的范围都从2004年1月到2005年10月。 [41]提出了一种新的混合方法，该方法结合了SVR，差分经验模式分解（DEMD）策略和自回归（AR）进行电力负荷预测。产生的策略在两个数据集上进行了测试。第一个由来自新南威尔士州的半小时电力负荷数据组成，第二个由纽约独立系统运营商的每小时电力负荷数据组成。

Lately , ensembles approaches have also received considerable attention. Jetcheva et al. [42] proposed an ensemble model based on ANNs for day-ahead electricity load and generation forecasting. The model was tested on data from commercial and industrial sites collected over a period of 10 months ranging from July 2011 to April 2012. In another work, Khairalla et al. [43] introduced a stacking multi-learning ensemble scheme to address the problem of short-term energy consumption forecasting. The proposal was tested on the Global Oil Consumption dataset collected over 52 years (1965–2016). Another example of a stacking ensemble approach can be found in [10]. In this case, Divina et al. applied such an approach to the global short-term energy consumption forecasting problem in Spain.

最近，集成方法也受到了相当大的关注。Jetcheva等 [42]提出了基于神经网络的集成模型，用于日前用电负荷和发电量预测。该模型在从2011年7月至2012年4月的10个月内从商业和工业现场收集的数据上进行了测试。 [43]介绍了一种堆叠式多学习集成方案，以解决短期能耗预测问题。该提案已在52年（1965年至2016年）收集的全球石油消费数据集中进行了测试。堆叠集成方法的另一个例子可以在[10]中找到。在这种情况下，Divina等人将这种方法应用于西班牙的全球短期能耗预测问题。

To get a deeper insight on this topic, we refer the reader to [44], where an exhaustive review of Machine Learning techniques for time series forecasting is presented. Reviews of conventional and Artificial Intelligence methods are presented in [15,45].

为了获得对该主题的更深入的了解，我们将读者引向[44]，其中详尽地介绍了用于时间序列预测的机器学习技术。 [15,45]中介绍了传统方法和人工智能方法。

3. Materials and Methods

3.1. Data

The time series used in this paper refer to data collected by different sensors installed in 13 buildings located on a university campus in the south of Spain. The selected buildings are equipped with sensors that are able to collect the energy consumption every 15 min. However, for the study presented in this paper, a daily consumption resolution was used. In particular, the data used in this paper cover the daily electric energy consumption registered for the buildings over a period of five and a half years, namely from 1 March 2012 to 31 October 2017.

本文中使用的时间序列是指由安装在西班牙南部大学校园内13座建筑物中的不同传感器收集的数据。选定的建筑物配备了传感器，能够每15分钟收集一次能耗。但是，对于本文提出的研究，使用了每日消费量解决方案。特别是，本文使用的数据涵盖了五年半（即2012年3月1日至2017年10月31日）内建筑物的每日电能消耗。

The campus where the buildings are located consists of 33 buildings. We used only 13 buildings since not all buildings are equipped with sensors for registering the electricity consumption.

建筑物所在的校园由33座建筑物组成。我们仅使用了13座建筑物，因为并非所有建筑物都配备了用于记录用电量的传感器。

Table 1 shows a general description of the selected buildings, including the year of construction (and year of refurbishment where applicable), the size and a description of the main purpose of the building.

表1列出了所选建筑物的一般说明，包括建造年份（以及适用时的翻新年份），大小和建筑物的主要用途的说明。

The geographical location of the campus is characterized by extremely high temperatures, with maximum temperatures exceeding 40◦C during the summer months, and by relatively mild winters, with minimum temperatures of approximately 0–2◦C. This climate is termed Mediterranean hot summer climate (Csa) according to the Köppen–Geiger classification. Table 2 shows average temperatures, average maximum and minimum temperatures, the amount of rainfall and average annual wind speed recorded for the years considered in this paper.

校园地理位置的特点是温度极高，夏季最高温度超过40℃，冬季相对温和，最低温度约为0-2℃。根据柯本气候分类法，这种气候被称为地中海夏季炎热（Csa）。表2显示了本文考虑的年份中记录的平均温度，平均最高和最低温度，降雨量和年平均风速。

The raw data collected presented missing values. To overcome this issue, the raw data were preprocessed. For each building, we computed the daily average electric energy consumption in Kwh. However, raw data presented missing values on some days and we proceeded to apply a preprocessing step to handle this issue. The steps are summarized as follows: (a) days with missing values corresponding to periods of more than 8 h (either continuous or discontinuous) were discarded; (b) days with missing values corresponding to a continuous period of time corresponding to more than 4 h were discarded; and © in the rest of the cases, the missing values were approximated by means of a linear regression method. The same approach was used in the case of missing values corresponding to a period of time of less than 8 h for the period from 00:00 to 08:00. In total, 260 days were discarded, thus the final dataset consists of 1810 days. The final dataset is available upon request.

收集的原始数据存在缺失值。为克服此问题，对原始数据进行了预处理。对于每座建筑物，我们计算了每天的平均电能消耗量，以千瓦时为单位。但是，原始数据有时会缺少值，因此我们继续执行预处理步骤来解决此问题。**这些步骤总结如下：（a）丢弃值对应于超过8小时（连续或不连续）的天（连续或不连续）；（b）丢弃与连续4小时以上的时间对应的缺失值的天数；（c）在其余情况下，缺失值是通过线性回归方法估算的。在缺少值的情况下使用相同的方法，该值对应于从00:00到08:00的时间段少于8小时的时间段。总共丢弃了260天，因此最终数据集包括1810天。**最终数据集可应要求提供。

Figure 1 shows the electric consumption and the general trend of each building over the period of time analyzed, scaled using a logarithmic scale in base 10. We used a logarithmic transformation since some significant outliers were present and would have rendered the graphs relative to the original data difficult to read. More specifically, significant outliers are present in buildings 3, 24 and 25. To further analyze such anomalous cases, we checked the temperature registered by Spanish National Agency of Meteorology [46] on such dates. We verified that no sudden changes in the temperatures can justify these anomalous values. Moreover, we verified that there are no records of anomalous operation in the buildings. We decided to maintain such anomalous values to simulate the real functioning of the sensors, and to check in this way the robustness of the techniques to the presence of outliers.

图1显示了经过分析的时间段内每座建筑物的耗电量和总体趋势，使用以10为底的对数比例进行缩放。我们使用对数转换，因为存在一些明显的离群值，并且将相对于原始图绘制了该图。数据难以读取。更具体地说，建筑物3、24和25中存在明显的异常值。为进一步分析此类异常情况，我们检查了西班牙国家气象局[46]在此类日期记录的温度。我们证实，温度的突然变化不能证明这些异常值是合理的。此外，我们验证了建筑物中没有异常操作的记录。我们决定维持这样的异常值，以模拟传感器的实际功能，并以此方式检查技术对异常值的鲁棒性。

Figure 2 shows the distribution of the values in the final dataset for the buildings considered in this paper. From the results this figure we could confirm the presence of the outliers in the buildings highlighted above. However, we can also notice that there are anomalous values in almost all the buildings, even if their values do not exceed by too much the values considered to be normal. These values may be due to a malfunctioning of the sensors as well as extreme peaks of temperatures. We can also noticed that Building 25 presents a wider distribution of values with a higher average. These aspects could be partially explained by the fact that Building 25 hosts the university’s library , which consists of a huge open space. This, and the particular climatic conditions of south of Spain, implies that this building uses more energy for the heating, ventilation, and air conditioning (HV AC) than other buildings that consist of smaller spaces.

图2显示了本文考虑的建筑物的最终数据集中值的分布。根据该图的结果，我们可以确认上面突出显示的建筑物中存在异常值。但是，我们还可以注意到，几乎所有建筑物都存在异常值，即使它们的值没有超出被认为是正常值的太多也是如此。这些值可能是由于传感器故障以及极端的温度峰值引起的。我们还可以注意到，Building 25呈现的值分布范围更广，平均值更高。这些方面可以由25号楼托管大学的图书馆这一事实来部分解释，该图书馆由一个巨大的开放空间组成。这以及西班牙南部的特殊气候条件意味着，与其他由较小空间组成的建筑物相比，该建筑物在供暖，通风和空调（HV AC）方面使用的能源更多。

To complete our study of the time series, we performed an analysis aimed at establishing whether the time series used presented stationarity behavior or not. In particular, we performed the Ljung–Box (LB) statistical tests for stationarity [47]. The p-value returned by this test is, in all cases, lower than 0.01 except for Building 3. To further investigate this case, we present in Figure 3 the the Partial AutoCorrelation Function (PACF) and the AutoCorrelation Function (ACF) pertaining to the data of Building 3. We can notice that in both plots there are no lags that exceed the confidence interval of the ACF and PACF plots, respectively . We can then conclude that Building 3 shows a stationary behavior.

为了完成对时间序列的研究，我们进行了旨在确定所使用的时间序列是否表现出平稳行为的分析。尤其是，我们进行了Ljung-Box（LB）统计测试来检验平稳性[47]。除Building 3外，该测试返回的p值在所有情况下均小于0.01。为进一步研究此情况，我们在图3中给出了与以下内容有关的部分自相关函数（PACF）和自相关函数（ACF）我们可以注意到，在两个图中，没有滞后分别超过ACF和PACF图的置信区间。然后我们可以得出结论，3号楼显示出平稳的行为。

3.2. Methods

In this section, we provide a basic description of the methods used in the experimentation, starting from the classical statistical approaches, namely linear regression and ARIMA. After that, we describe the Machine Learning techniques used in our empirical study.

在本节中，我们将从经典的统计方法（即线性回归和ARIMA）开始，对实验中使用的方法进行基本描述。之后，我们描述了我们的经验研究中使用的机器学习技术。

Linear Regression (LM). Ref. [48] is statistical technique commonly used for time series forecasting. The basic idea behind linear regression is to try to find the relationship between two variables. In its simplest formulation, a linear equation, Y = a+bX, is used to represent the association between the independent variable (X) and the dependent one (Y), i.e., the variables to be predicted. In the case where multiple independent variables are used to determine the value of a dependent variable, as is the case of this work, the process is called multiple linear regression. In this case, the goal is to model the linear relation between multiple independent, or explanatory , variables and a dependent variable using an equation such as Y = a + b1X1+ . . . + bnXn+ e, where e is the residual (difference between the predicted and the observed value) and Xi, 1 ≤ i ≤ n are the n explanatory variables. For this work, we used the implementation provided by the caret [49] R package.

线性回归（LM）。参考[48]是通常用于时间序列预测的统计技术。线性回归背后的基本思想是试图找到两个变量之间的关系。在其最简单的表述中，线性方程Y = a + bX用于表示自变量（X）和因变量（Y），即要预测的变量之间的关联。在使用多个自变量确定因变量的值的情况下（如本工作的情况），该过程称为多元线性回归。在这种情况下，目标是使用Y = a + b1X1 + … + bnXn + e 等式对多个独立或说明性变量与因变量之间的线性关系进行建模。其中e为残差（预测值与实测值之差），Xi为1≤i≤n为n个解释变量。对于这项工作，我们使用了 caret [49] R 包提供的实现。

Auto-Regressive Integrated Moving Average (ARIMA). is a classical method based on Box and Jenkins [50] widely used for time series forecasting, as shown in Section 2. ARIMA is an extension of the simpler AutoRegressive Moving Average (ARMA) that includes the concept of integration. We used this method since it can be used to forecast non-stationary time series. In [51], ARIMA is described as a stochastic model building. In particular, this method can be viewed as an iterative strategy that consists of three steps:

自回归综合移动平均线（ARIMA）。是基于Box和Jenkins [50]的经典方法，广泛用于时间序列预测，如第2节所示。ARIMA是对简单自动回归移动平均值（ARMA）的扩展，其中包括积分概念。我们使用此方法，因为它可用于预测非平稳时间序列。在[51]中，ARIMA被描述为一种随机模型。特别地，可以将此方法视为包含三个步骤的迭代策略：

Identification. In this step, the data and all related information are used for selecting a sub-class of model that might best represent the data.
识别。在此步骤中，将数据和所有相关信息用于选择最能代表数据的模型子类。
Estimation. In a second step, the parameters of the model are trained using the data.
估算。第二步，使用数据训练模型的参数。
Diagnostic Checking. Finally , the so-obtained model is validated using the data at disposal and areas where the model can be improved are identified.
诊断检查。最后，使用可获得的数据对如此获得的模型进行验证，并确定可以改进模型的区域。

ARIMA requires setting three parameters, namely the order of auto-regressive (AR) terms §, the degree of differencing (d) and the order of moving-average (MA) terms (q). To estimate such parameters for each dataset and each historical window used, we used the arimapar function, part of the TSPred R package [52]. This function returns the parameters of an automatically fitted ARIMA model. The implementation provided by the caret.ts [53] package (an extension for time series models for caret package) was used.

ARIMA需要设置三个参数，即自回归（AR）项的阶数（p），微分度（d）和移动平均（MA）项的阶数（q）。为了估计每个数据集和每个使用的历史窗口的参数，我们使用arimapar函数，它是TSPred R包的一部分[52]。此函数返回自动拟合的ARIMA模型的参数。使用caret.ts [53]程序包（caret程序包的时间序列模型的扩展）提供的实现。

Evolutionary Algorithms (EAs) for Regression T rees (EVT ree). EAs [54] are optimization techniques based on the basic concepts of Darwinian evolution. EAs evolve a population of candidate solutions over a number of generations by means of the application of genetic operators, such as selection, crossover and mutation. Each candidate solution, i.e., an individual of the population, is assigned a quality score, or fitness. Usually, EAs start from a randomly initialized population, which is then evaluated to assign a fitness to each individual. A number of individuals are then selected, usually based on the fitness. Crossover and mutation operators are used to generate new individuals starting from the selected ones. The so-generated offspring are then inserted into the population. The whole process is repeated until a stopping criterion is met, e.g., when a maximum number of generations have been performed. The particular EA provided by the EVTree [55] R package was used. This algorithm evolves regression trees [56], i.e., trees used to predict a continuous value based on a set of predictor variables.

回归算法（EVT ree）的进化算法（EA）。 EAs [54]是基于达尔文进化论基本概念的优化技术。 EA通过应用遗传算子（例如选择，交叉和突变），经过许多代进化出了候选解决方案。为每个候选解决方案（即总体中的一个个体）分配一个质量得分或适应度。通常，EA从随机初始化的种群开始，然后对其进行评估以为每个个体分配适合度。然后通常基于适合度来选择许多个人。交叉和变异算子用于从选定的个体开始生成新个体。然后将如此生成的后代插入种群。重复整个过程直到满足停止标准为止，例如，当已经执行了最大数量的发电时。使用了EVTree [55] R包提供的特定EA。该算法演化出回归树[56]，即用于基于一组预测变量来预测连续值的树。

Generalized Boosted Regression Models (GBM). [57,58]. This is an ensemble algorithm, where a set of regression trees is trained following a sequential procedure. During this sequential procedure, GBM applies a gradient descend procedure, which is repeated until a given number of trees has been built or when no improvement is detected. The last condition is checked by using the set of current trees to produce predictions. Such predictions are also used to correct the model, so that mistakes obtained in previous iterations can be corrected. A known issue in gradient boosting is overfitting.GBM tackles this problem by using regularization methods, which basically penalize various parts of the algorithm. To produce the final prediction a voting mechanism, among all the tree obtained, is used. We used the GBM implementation provided by the caret R package.

广义增强回归模型（GBM）。 [57,58]。这是一种集成算法，其中按照顺序过程训练一组回归树。在此顺序过程中，GBM应用了梯度下降过程，该过程将重复进行，直到建立了给定数量的树或未检测到任何改进为止。通过使用当前树的集合来检查最后的条件以产生预测。这种预测还用于校正模型，以便可以校正在先前迭代中获得的错误。梯度增强中的一个已知问题是过拟合。GBM通过使用正则化方法解决了这个问题，该方法基本上会惩罚算法的各个部分。为了产生最终预测，在所有获得的树中使用投票机制。我们使用了caret R包提供的GBM实现。

Artificial Neural Networks (ANNs). In a rough analogy of biological learning system, ANNs consist of densely interconnected units, called neurons. Each neuron receives a number of real valued input, e.g., from other neurons of the network, and produces a single real valued output. The output depends on an activation function used in each unit, which introduce non-linearity to the output. However, the activation function is used only if the input received by a unit is higher than a given activation threshold. If this is not the case, then no output is produced. Normally , an ANN consists of different layers of neurons. Among such layers we can distinguish the input and output layers, and in between there may be one or more hidden layers. As said before, there is an activation threshold used in the network, which is determined by weights that are adjusted during a training phase. Several types of ANNs have been proposed. Among these, the simplest, and one of the most widely used one, is the so-called feedforward ANN. In these type of networks, neurons of adjacent layers are connected, and each of these connections is assigned a weight. The information advances from the input layer toward the output layer, which consists of only one unit. The output unit produces the final prediction of the network. The caret package implementation was used.

人工神经网络（ANN）。在生物学学习系统的粗略类比中，人工神经网络由密集互连的单元（称为神经元）组成。每个神经元例如从网络的其他神经元接收多个实值输入，并产生单个实值输出。输出取决于每个单元中使用的激活函数，该函数将非线性引入到输出中。但是，仅当单元接收到的输入高于给定的激活阈值时，才使用激活功能。如果不是这种情况，则不会产生任何输出。通常，人工神经网络由神经元的不同层组成。在这些层之间，我们可以区分输入层和输出层，并且在它们之间可能存在一个或多个隐藏层。如前所述，网络中存在一个激活阈值，该阈值由在训练阶段调整的权重确定。已经提出了几种类型的人工神经网络。其中，最简单，使用最广泛的一种就是所谓的前馈ANN。在这些类型的网络中，连接相邻层的神经元，并为每个连接分配一个权重。信息从输入层向仅包含一个单元的输出层前进。输出单元产生网络的最终预测。使用插入符号包实现。

Random Forests (RF). were first proposed by Breinman and Cutle in [59]. Similar to GBM, RF is also an ensemble approach, where a set of trees are used to produce the final output, using a voting scheme. Each tree is induced from a randomly selected training subset and using also a randomly selected subset of features. This implies that the trees depend on the values of an independently sampled input dataset, using the same distribution for all trees. In the case of regression, the final prediction is represented by the average of the predictions of each induced tree. In addition, for this method, the implementation provided by caret was used.

随机森林（RF）。最初由Breinman和Cutle在[59]中提出。与GBM相似，RF也是一种集成方法，其中使用一组树来使用表决方案来产生最终输出。从随机选择的训练子集并还使用特征的随机选择子集来导出每棵树。这意味着树依赖于独立采样的输入数据集的值，并且对所有树使用相同的分布。在回归的情况下，最终预测由每个诱导树的预测平均值表示。此外，对于此方法，使用了caret提供的实现。

Ensemble. This method was proposed by Divina et al. [10]. It is a stacking ensemble scheme. In particular, two layers are used to make a final prediction. In the first layer, EVTree, RF and NN are used, and the predictions made by these three methods are then combined on the top layer by GBM to produce the final output. This method shows excellent performances on a problem regarding the prediction of the global energy demands, this it is interesting to check its validity also in the case of a local energy demand setting, such as the one used in this paper.

集成学习。该方法由Divina等提出。 [10]。这是一个堆栈集成方案。特别是，使用两层进行最终预测。在第一层中，使用EVTree，RF和NN，然后由这三种方法做出的预测由GBM组合在顶层，以产生最终输出。这种方法在有关预测全球能源需求的问题上显示出了优异的性能，这对于在局部能源需求设置（例如本文中使用的设置）的情况下检查其有效性也很有趣。

Recursive Partitioning and Regression T rees (RPart). [60]. This method is based on the CART algorithm, and builds regression trees, using a two states strategy . The resulting model can be represented as a binary tree, which can be easily interpreted. In the first step, the algorithm determines the variable which best splits the data into two groups. For regression, as in this case, the attribute with the largest standard deviation reduction is considered the best and is used for building the decision node. The data are then separated according the the attribute selected, and the process is applied separately to each sub-group determined in the first phase, until the subgroups either get to a minimum size or no improvement can be achieved. In a second step, the obtained full tree is pruned using a cross-validation strategy . Again, the caret implementation was used.

递归分区和回归索引（RPart）。 [60]。该方法基于CART算法，并使用两种状态策略构建回归树。生成的模型可以表示为二叉树，可以轻松地对其进行解释。第一步，算法确定最能将数据分为两组的变量。在这种情况下，对于回归而言，标准偏差减少量最大的属性被认为是最好的，并用于构建决策节点。然后根据选择的属性将数据分离，并将该过程分别应用于在第一阶段中确定的每个子组，直到子组达到最小大小或无法实现任何改进为止。在第二步骤中，使用交叉验证策略修剪获得的完整树。再次，使用插入符号实现。

Extreme Gradient Boosting (XGBoost). [61]. This method is similar to GBM, as it shares with it the principle of gradient boosting for building an ensemble of trees. There are, however, important differences. For instance, XGBoost controls over-fitting (a known issue in GBM) by adopting a more regularized model formalization. This allows XGBoost to generally obtain better performance than GBM. XGBoost has recently received much attention in the data science community , as it has been successfully applied in different domains. This popularity is mostly due to the scalability of the method. In fact, XGBoost can run up to ten times faster than other popular approaches on a single machine. Moreover, it is also capable of scaling to billions of instances in distributed or memory-limited settings. The latter feature is achieved via different systems and algorithmic optimizations. Some of this improvements include a novel tree induction algorithm for managing sparse data and a strategy that makes it possible to handle the weights of the instances in approximate tree learning. Parallel and distributed computing makes learning faster, allowing a faster model exploration. Additionally , XGBoost employs out-of-core computation, which enables to process massive date even of a simple desktop. We used the implementation provided by the caret R package.

极端梯度增强（XGBoost）。 [61]。此方法与GBM相似，因为它与渐变增强的原理共享，以建立树木的整体。但是，存在重要的差异。例如，XGBoost通过采用更规范化的模型形式化来控制过度拟合（GBM中的一个已知问题）。这使得XGBoost通常可以获得比GBM更好的性能。由于XGBoost已成功应用于不同领域，因此最近在数据科学界引起了广泛关注。这种普及主要是由于该方法的可扩展性。实际上，XGBoost在一台计算机上的运行速度比其他流行方法快十倍。此外，它还能够在分布式或受内存限制的设置中扩展到数十亿个实例。后一个功能是通过不同的系统和算法优化来实现的。其中的一些改进包括用于管理稀疏数据的新颖树诱导算法以及可以在近似树学习中处理实例权重的策略。并行和分布式计算使学习更快，从而可以更快地探索模型。此外，XGBoost采用核外计算，即使在简单的桌面上也可以处理大量数据。我们使用了插入caret R包提供的实现。

All the methods used in this study have various configuration parameters. To set the different parameters of the methods, we ran a grid search, exploring the standard parameter values that are used in similar literature.

本研究中使用的所有方法均具有各种配置参数。为了设置方法的不同参数，我们进行了网格搜索，探索了类似文献中使用的标准参数值。

4. Results

In this section, we present the results obtained by the different techniques described in Section 3.2 on the datasets presented in Section 3.1 and draw the main conclusions. The aim of the experimentation performed was two folds. Firstly , we wanted to assess which technique is more suitable for predicting short term electric energy consumption in a local setting, such in smart buildings. Secondly , we wanted to find an optimal value for the amount of historical data that should be used to obtain good forecasts.

在本节中，我们将介绍在第3.1节介绍的数据集上通过3.2节介绍的不同技术获得的结果，并得出主要结论。进行实验的目的是两个方面。首先，我们想评估哪种技术更适合预测局部环境（例如智能建筑）中的短期电能消耗。其次，我们希望找到用于获得良好预测的历史数据量的最佳值。

To measure the performances of the different strategies used, we used the Mean Absolute Error (MAE) and the Root Mean Squared Error (RMSE). The measures are defined as follows:

为了衡量所使用的不同策略的性能，我们使用了平均绝对误差（MAE）和均方根误差（RMSE）。措施定义如下：

where ˆYi and Yi are the predicted and the real values, respectively . These two measures are commonly used when assessing the performances of a technique on time series forecasting (see, for instance [62,63]). The advantage of these two measures is that the average forecast error of a model is expressed in the same units of the variable to be predicted. The two measures can assume values greater than or equal to 0, and lower values are considered better. MAE expresses the absolute error, thus it is easy to understand. RMSE assigns high penalties to large errors, since the prediction errors are squared. It follows that the RMSE can be useful when we want abstain from large forecasting errors.

其中 γi^\hat{\gamma_i}γi^ 和 γi\gamma_iγi 分别是预测值和实际值。在评估时间序列预测技术的性能时，通常使用这两种方法（例如，参见[62,63]）。这两种措施的优点在于，模型的平均预测误差以要预测的变量的相同单位表示。这两个量度可以假定值大于或等于0，并且认为较低的值更好。 MAE表示绝对误差，因此很容易理解。由于预测误差是平方的，因此RMSE对大误差给予高罚分。因此，当我们希望避免较大的预测误差时，RMSE可能会很有用。

T o be used in the proposed experimentation, the original datasets were transformed. The particular preprocessing steps are those used by T orres et al. [36], and are graphically shown in Figure 4. The preprocessing phase can be summarized in two steps. Firstly , we extracted the attribute representing the energy consumption, and as a result a vector of consumption data was obtained. Secondly , a matrix was built using the vector obtained in the first step and according to a historical window, w, and a prediction horizon, h. Notice that w represents the number of historical values that are taken into account so as to induce a predictive model. The obtained model was used to forecast subsequent values (h). In our study , we were interested in predicting the consumption of one day , thus h was set to 1.

为了在建议的实验中使用，对原始数据集进行了转换。特定的预处理步骤是Torres等人使用的步骤。 [36]，并以图形方式显示在图4中。预处理阶段可以分为两个步骤。首先，我们提取了代表能耗的属性，结果获得了能耗数据的向量。其次，使用第一步中获得的向量并根据历史窗口w和预测范围h建立矩阵。请注意，w表示为了推导预测模型而考虑的历史值的数量。所获得的模型用于预测后续值（h）。在我们的研究中，我们对预测一天的消耗量感兴趣，因此h设置为1。

Moreover, since one of the aims of the experimentation was to find an optimal value of w, different values of historical window were tested. Specifically, we used historical windows of 7, 10, 15 and 20 days. We also tested historical window sizes of multiples of 7, which are common in these kinds of problems, since they usually allow capturing seasonal patterns. However, in this case, the results obtained with such values are not better than those obtained with the above mentioned values, and thus are not included in this paper. This was probably due to the high number of days discarded in the preprocessing of raw data (described in Section 3.1).

此外，由于实验的目的之一是找到w的最佳值，因此对历史窗口的不同值进行了测试。具体来说，我们使用了7、10、15和20天的历史窗口。我们还测试了7的倍数的历史窗口大小，这在此类问题中很常见，因为它们通常可以捕获季节性模式。但是，在这种情况下，用这种值获得的结果并不比用上述值获得的结果更好，因此不包括在本文中。这可能是由于在原始数据的预处理中丢弃了很多天（在第3.1节中进行了描述）。

The resulting datasets were split in two groups: training set and test set. In particular, we used 70% of the data as training set, and the remaining 30% as test set. Thus, the methods were used to induce a prediction model using the training set, and the forecasting capabilities of the models were assessed on the test set.

结果数据集分为两组：训练集和测试集。特别是，我们将70％的数据用作训练集，其余30％用作测试集。因此，使用这些方法使用训练集来推导预测模型，并在测试集上评估模型的预测能力。

Table 3 presents the MAE and RMSE obtained on the different buildings for the different historical windows used. In particular, in the last row, the averages results obtained on all the buildings by the different methods are shown. The same results are presented graphically in Figures A1 and A2.

表3列出了针对不同历史窗口在不同建筑物上获得的MAE和RMSE。特别是在最后一行中，显示了通过不同方法在所有建筑物上获得的平均值结果。在图A1和A2中以图形方式显示了相同的结果。

One thing that we can notice is that, generally , ARIMA did not perform well on the datasets considered. In fact, on average, it obtained the worst results, in terms of both MAE and RMSE, on all buildings for all historical window values considered. In particular, in some cases, results achieved by this methods are significantly worse, as is the case, for example, for the results obtained on Buildings 2, 6, 7, 10 and 12.

我们可以注意到的一件事是，一般而言，ARIMA在所考虑的数据集上表现不佳。实际上，就所考虑的所有历史窗口值而言，就MAE和RMSE而言，它平均而言在所有建筑物上均获得了最差的结果。尤其是，在某些情况下，用这种方法获得的结果会很差，例如，在2号楼，6号楼，7号楼，10号楼和12号楼获得的结果就是这种情况。

We can also notice that, in general, better results were achieved when the historical window size used was greater than 7. Higher values of the historical window did not seem to yield better results, thus we can affirm that considering a historical window of more than 20 days would not lead to better predictions.

我们还可以注意到，通常，当使用的历史窗口大小大于7时，可以获得更好的结果。历史窗口的值较高似乎并不能产生更好的结果，因此我们可以肯定的是，考虑将历史窗口设置为大于20天不会带来更好的预测。

Looking more closely to each building, we can notice that the worst predictions were those related to Building 25, on which, for example, the mean absolute error achieved by the different strategies was always higher than 3. This could be due to the different features that this building presents, as can be seen in Table 1 and Figure 2. In the same figure, we can also notice that Buildings 3 and 24 presented noticeable outliers values, but, in this case, the forecasting methods did not seem to be affected by their presence. Thus, it appeared that the techniques used in this study are quite robust to the presence of outliers, and that the worst results obtained on Building 25 might be due more to the nature of the building.

仔细观察每个建筑物，我们可以注意到最糟糕的预测是与25号楼有关的预测，例如，在不同的策略上，其平均绝对误差始终高于3。这可能是由于特征不同如表1和图2所示。在同一图中，我们还可以注意到3号楼和24号楼呈现出明显的离群值，但在这种情况下，预测方法似乎并未受到影响通过他们的存在。因此，似乎本研究中使用的技术对于异常值的存在非常鲁棒，并且25号楼获得的最差结果可能更多是由于建筑物的性质造成的。

Figures 5 and 6 show the results, in terms of MAE and RMSE, obtained by the different methods on all the buildings when different sizes of the historical window were used. By analyzing these results, we can conclude that, in general, the best size for the historical window to use is 10 days. In fact, with this value, the results obtained are generally better, in terms of both MAE and of RMSE. This is an important result, since it may help researchers to set the historical window for other time series forecasting regarding data generated by smart buildings, especially in the case where various days present missing values and cannot be considered for inducing a forecasting model. We can also see that ARIMA failed to produce good forecasts for the datasets used in this study . Another difference between ARIMA and the other methods is that, in general, the historical window of 10 days did not yield better results when used with this method. From the graphs, we can also notice the presence of outliers for both MAE and RMSE. Such values are relative to the results obtained on Building 25, where all methods used in this study achieved the worst prediction results. This was probably due to the characteristics of this building, which is different from the others, as it contains the university’s library . It is a newer building, whose surface is also much bigger than that of the other buildings. Moreover, the average energy consumption is significantly higher than the other buildings, as shown in Figure 2. Outliers relative to the RMSE were also due to the results obtained on Building 3, meaning that the predictions errors were larger on this building. In this case, the building is similar to the other ones considered, in terms of both features and for its usage and average energy consumption. This suggest that a further analysis conducted with the mangers of the buildings is needed for this case to acquire more insights.

图5和图6显示了当使用不同大小的历史窗口时，在所有建筑物上通过不同方法获得的MAE和RMSE结果。通过分析这些结果，我们可以得出结论，通常，使用历史窗口的最佳大小为10天。实际上，使用此值，就MAE和RMSE而言，获得的结果通常更好。这是一个重要的结果，因为它可以帮助研究人员为有关智能建筑生成的数据的其他时间序列预测设置历史窗口，尤其是在各天呈现缺失值且无法考虑引入预测模型的情况下。我们还可以看到ARIMA未能对本研究中使用的数据集产生良好的预测。 ARIMA与其他方法之间的另一个区别是，通常，使用此方法时10天的历史窗口不会产生更好的结果。从图中，我们还可以注意到MAE和RMSE都存在异常值。这些值是相对于25号楼获得的结果而言的，该研究中使用的所有方法均达到了最差的预测结果。这可能是由于该建筑物的特性与其他建筑物不同，因为它包含大学的图书馆。这是一栋较新的建筑物，其表面也比其他建筑物大得多。而且，平均能耗显着高于其他建筑物，如图2所示。相对于RMSE的离群值也归因于3号楼获得的结果，这意味着该建筑物的预测误差更大。在这种情况下，该建筑物在功能，用途和平均能耗方面均与其他建筑物类似。这表明，在这种情况下，需要对建筑物的管理者进行进一步的分析，以获得更多的见解。

In Tables 4 and 5, we reported the average performances of the methods for all historical windows values and for all the buildings, and the average results obtained when a historical window of size 10 was used, since this value was considered to be the best one in the experimentation presented. Moreover, in the tables, we also report results obtained with two simple baseline benchmark forecasting models. In particular, we considered the energy consumption to be the same as the one of the previous day and of the previous week, i.e., of seven days earlier. Results obtained by these two methods are, as expected, not competitive if compared with the other methods. Results are ordered according to the averages results. In this way , we can obtain a sort of ranking of the methods, which is useful to establish the overall performances of the methods.

在表4和表5中，我们报告了所有历史窗口值和所有建筑物的方法的平均性能，以及使用大小为10的历史窗口时获得的平均结果，因为该值被认为是最好的。在提出的实验中。此外，在表中，我们还报告了使用两个简单的基准基准预测模型获得的结果。尤其是，我们认为能耗与前一天和前一周（即7天之前）相同。如预期的那样，通过这两种方法获得的结果与其他方法相比没有竞争力。结果根据平均结果排序。这样，我们可以获得方法的一种排序，这对于确定方法的整体性能很有用。

To assess the significance of the differences of the results obtained by the various techniques, we applied a two-tailed t-test with confidence level of 1%. The result of such test are summarized in Tables 6 and 7, for the MAE and the RMSE, respectively . In these tables, a “s” means that the difference of results between two methods was considered to be significant. For example, in Table 6, we can notice that the average MAE obtained by RF was significantly better than the average MAE achieved by EVTree.

为了评估通过各种技术获得的结果差异的重要性，我们应用了置信度为1％的两尾t检验。表6和表7分别汇总了MAE和RMSE的测试结果。在这些表中，“s”表示两种方法之间的结果差异很大。例如，在表6中，我们可以注意到RF获得的平均MAE明显优于EVTree获得的平均MAE。

In the tables, we can see that all methods, but ARIMA, obtained better results when w was set to 10, which also confirmed the result regarding the optimal historical window size reported above. We can see that the first three strategies, i.e., RF, GBM and XGBoost, basically achieved the same results, even if, when the MAE was considered, GMB and XGBoost achieved slightly better results when w was set to 10.

在表中，我们可以看到，当w设置为10时，除ARIMA之外，所有方法均获得了更好的结果，这也证实了上述最佳历史窗口大小的结果。我们可以看到，前三种策略（即RF，GBM和XGBoost）基本上取得了相同的结果，即使在考虑了MAE的情况下，当w设置为10时，GMB和XGBoost取得了更好的结果。

In general, we can notice that, according to both MAE and RMSE, the best performing forecasting methods on the datasets used in this study were RF, GMB and XGBoost. Neural Networks and linear regression also obtained competitive results. It is interesting also to notice that the ensemble scheme proposed in [10] did not perform well on the datasets considered in this study , while the performances of the same methods were very competitive in a global short-term electric energy forecasting scenario. This could mean that a stacking ensemble approach, such as the one used in [10], is suitable for global energy consumption predictions, but that it is not suitable for a localized problem such as the one tackled in this study . ARIMA was confirmed to be the worst performing methods, suggesting that perhaps this method is not adequate for these kinds of problems.

一般而言，我们可以注意到，根据MAE和RMSE，在本研究中使用的数据集上表现最佳的预测方法是RF，GMB和XGBoost。神经网络和线性回归也获得了竞争性结果。还有趣的是，在[10]中提出的集成方案在本研究中考虑的数据集上表现不佳，而在全球短期电能预测情况下，相同方法的性能非常有竞争力。这可能意味着，诸如[10]中所使用的那样的一种集成方法适合于全球能源消耗的预测，但是它不适用于诸如本研究中所解决的局部化问题。 ARIMA被证实是性能最差的方法，这表明该方法可能不足以解决此类问题。

Finally , Figure 7 shows four examples of the observed and forecasted values from ARIMA and RF methods obtained on different buildings over a period of 10 days. Recall that RF is the method that achieved better results, while ARIMA is the worst performing technique. We can notice how RF better approximated the real values, while the errors made by ARIMA were in some cases significant. This fact caused ARIMA to perform poorly on the datasets used in this paper. From these graphs, we can conclude that the main difficulty of the datasets presented is to predict the energy consumption corresponding to the peaks represented in the graphs. However, methods such as RF can generally obtain predictions that are close to the observed values.

最后，图7显示了在10天内从不同建筑物获得的ARIMA和RF方法的观测值和预测值的四个示例。回想一下，RF是获得更好结果的方法，而ARIMA是性能最差的技术。我们可以注意到，RF如何更好地逼近实际值，而ARIMA所产生的误差在某些情况下却很明显。这一事实导致ARIMA在本文使用的数据集上表现不佳。从这些图表中，我们可以得出结论，提出的数据集的主要困难在于预测与图表中表示的峰值相对应的能耗。但是，诸如RF之类的方法通常可以获得接近观测值的预测。

5. Conclusions and Future Work

In this paper, we have presented a comparative study of different time series forecasting techniques for predicting one-day electric energy consumption in non-residential buildings. In particular we were interested in both determining which strategy is more suitable for this kind of problem, and also to establish the optimal value for the historical window to be considered so as to produce the most accurate predictions. For such purpose, we compared nine forecasting methods on time series regarding data collected from thirteen smart buildings located in a university campus in Spain. The datasets used in this paper have never been used before, and are available on request. This represents an added value to the paper, since the datasets could represent a relevant resource for researches in this area.

在本文中，我们对用于预测非住宅建筑物中一天的电能消耗的不同时间序列预测技术进行了比较研究。尤其是我们既对确定哪种策略更适合此类问题感兴趣，又对确定要考虑的历史窗口的最优值以产生最准确的预测感兴趣。为此，我们比较了从西班牙西班牙一所大学校园内的13座智能建筑收集的数据的时间序列的九种预测方法。本文中使用的数据集以前从未使用过，可应要求提供。这代表了论文的附加价值，因为数据集可以代表该领域研究的相关资源。

From the experiments conducted, we can conclude that the best performing methods were Machine Learning based approaches. In particular, the Random Forests, Generalized Boosted Regression Mode and Extreme Gradient Boosting achieved, in general, the best performances on the data used in this work, since they achieved the lowest prediction errors. As such, they could represent the first choice for any further study on similar problems.

从进行的实验中，我们可以得出结论，性能最好的方法是基于机器学习的方法。特别是，随机森林，广义增强回归模式和极端梯度增强通常在这项工作中使用的数据上实现了最佳性能，因为它们实现了最低的预测误差。因此，它们可能是任何进一步研究类似问题的首选。

As far as the optimal historical window value is concerned, we found that, when the size of the window used was higher than seven, the results obtained improved. Moreover, the best results were achieved when the window size was set to 10, i.e., when ten days were considered to predict the electric energy consumption of the next day . This could help researchers facing similar forecasting scenarios.

就最佳历史窗口值而言，我们发现，当使用的窗口大小大于7时，获得的结果将得到改善。此外，当窗口大小设置为10时，即当考虑十天来预测第二天的电能消耗时，可获得最佳结果。这可以帮助研究人员面对类似的预测情况。

As for future work, we are planning to apply a Deep Learning approach to the time series presented in this paper, since this strategy has obtained good results in other time series forecasting problems. A problem with this approach is represented by the setting of the parameters used to build the Deep Neural Network. In fact, a wrong setting can prevent such approach from obtaining good performances. T o overcome this problem, we are also planning to use a Neurovolution approach [64–66]. In this approach, an Evolutionary Algorithm is used to establish a sub-optimal set of hyper parameters for the Deep Neural Network.

至于未来的工作，我们计划将深度学习方法应用于本文介绍的时间序列，因为该策略在其他时间序列预测问题中已经取得了良好的结果。这种方法的问题由用于构建深度神经网络的参数的设置表示。实际上，错误的设置可能会阻止这种方法获得良好的性能。为了克服这个问题，我们还计划使用Neurovolution方法[64-66]。在这种方法中，进化算法用于为深度神经网络建立次优超级参数集。

We are also planning to extend this study to other kinds of time series, for instance time series regarding the water and/or gas usage in smart buildings, and to include in the experimentation weather-related data. We also plan to to extend the experimentation to different buildings located in other regions to compare results obtained in different environmental conditions.

我们还计划将该研究扩展到其他类型的时间序列，例如有关智能建筑中水和/或天然气使用的时间序列，并将其纳入与天气相关的实验数据中。我们还计划将实验扩展到位于其他地区的不同建筑物，以比较在不同环境条件下获得的结果。

Author Contributions: F.D. proposed the concept of this research and design and was involved in the whole
experimentation phase. F.A.G.V . was involved in the visualization and interpretation of results and in the data
preparation, while M.G.T. provided overall guidance and was involved in the experimentation. All authors were
involved in preparing the manuscript.

Funding: This research was funded by Spanish Ministry of Economic and Competitiveness and the European
Regional Development Fund, grant number TIN2015-64776-C3-2-R (MINECO/FEDER).

Conflicts of Interest: The authors declare no conflict of interest

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