1. Introduction
With the advancement of urbanization and the continuous improvement in people’s quality of life, the proportion of urban building energy consumption (EC) is increasing annually. EC is one of the main causes of the increase in greenhouse gas emissions and the increased inhalation amount, and building EC accounts for a large proportion of the total EC [
1,
2]. Building energy efficiency (EE) and EC are interdependent. Enhancing building EE is key to achieving sustainable building development, which can be realized through reducing building EC. Therefore, improving building EE can decrease building EC, which in turn enhances building EE. As an effective means to reflect the operational characteristics of buildings, building EE prediction is an important tool for assessing the energy-saving measures and energy-portal parts of smart buildings and can provide effective support for the intelligent management of buildings [
3]. Therefore, it is important to explore the mechanism and patterns of building EE and to establish an accurate and effective model for predicting EE in buildings.
In recent years, numerous scholars have presented various techniques for predicting building EC and EE [
4,
5]. Currently, the techniques utilized for building EC prediction consist mainly of engineering methods, statistical methods, and artificial intelligence methods. The engineering approach involves utilizing substantial professional knowledge of building physics during the modeling phase to obtain building EC data for analysis and research. This is achieved by developing an accurate building physics model and combining it with a building operation simulation. Shabunko et al. [
6] employed the EnergyPlus V2018 energy simulation software to simulate the EC of three different residential buildings with a total of four hundred units. They subsequently compared the intensity of the building EC with real EC data and found that the simulation results were consistent with the actual data. The authors concluded that EnergyPlus software is an efficient tool for simulating building EC. Asadi et al. [
7] altered the design variables of building materials, including thickness, appearance, and internal personnel scheduling. To assess the EC of buildings under each design scheme, tens of thousands of simulations were run using the eQUEST V3.65 and DOE-2 V.2 building simulation software. This study employed multiple linear regression equations to predict building EC. The study results indicated that the multiple linear regression model’s EC predictions deviated less than 5% from the simulation software’s results. The outcome precision of these approaches depends heavily on the utilization of specific knowledge in building physics, which is intricate due to the complex system setup and numerous design parameters involved in the modeling phase [
5].
Statistical methods are utilized to construct mathematical models by tabulating historical EC data, following which EC is forecasted and analyzed. Zeng et al. [
8] employed Gaussian process regression to forecast the actual electricity consumption of buildings of various types. The findings revealed that Gaussian process regression is capable of predicting not only the electricity consumption of different inputs, but also that of buildings belonging to diverse categories. Atalay et al. [
9] developed a multiple linear regression model, a time series model, and a discrete grey model for predicting HVAC power consumption in commercial buildings located in Paris, France. The experimental results indicated that the multiple linear regression model outperformed both the time series model and the grey model. The ARIMA model predicts the EC and greenhouse gas emissions of iron and steel enterprises and offers excellent EC prediction performance [
10]. Li et al. [
11] introduced an EC prediction model, IPSO-ANN, where the input variables were established using the dimensionality reduction method of principal component analysis. The results demonstrated that the IPSO-ANN model provides higher prediction accuracy than its counterparts. Although the aforementioned prediction models can accurately predict EC, the random and nonlinear nature of building EC data reveals deficiencies in these methods [
12]. These deficiencies are reflected in three main ways: first, the general statistical model fails to account for the nonlinear relationship between the input and output; second, the predictive performance of extended linear regression relies heavily on the correct selection of the activity function, which poses a major challenge; third, the multicollinearity of the input features may influence the prediction results of the statistical model.
In predicting building EC, machine learning methods, also referred to as data-driven methods, can be more efficient than traditional approaches such as engineering methods. Modeling relies on a range of mathematical and statistical algorithms and a wealth of historical data regarding building EC. Typically, building operational data, weather data, internal load data, and other relevant information are used as the inputs, and the corresponding building EC is used as the output. Models for predicting EC can be obtained through training. At present, the prevalent models include artificial neural networks [
13], support vector machines [
14,
15], random forests [
16,
17], and decision trees (DTs) [
18]. Due to the complexity of building EC behavior and the volatility of energy demand, obtaining accurate and reliable building EC prediction results remains a challenging task. Furthermore, in recent years, the scale and dimensions of building EC data have increased significantly due to continuous hardware upgrades and more frequent data collection. This increase poses a significant challenge to the prediction ability of building EC models. Consequently, an increasing number of researchers are focusing on deep learning algorithms that offer superior learning abilities [
19,
20].
Energy efficiency prediction is a typical machine learning problem. Single prediction models are widely used due to their rapid calculation speed and ease of implementation. However, there are still some shortcomings. First, the multiple linear regression method, which is fast and easy to implement, is inadequate for complex problems. Second, support vector machines are better at balancing prediction accuracy and calculation speed, but determining the optimal kernel function is challenging. Furthermore, in comparison to the previous two approaches, artificial neural networks can handle nonlinear issues and have relatively high forecasting accuracy; however, they require significant historical data for model training and the determination of numerous parameters. Traditional optimization algorithms, such as the back propagation (BP) algorithm, are sensitive to initial conditions and easily fall into local optima [
21]. However, evolutionary algorithms differ from traditional algorithms; they are population-based intelligent search methods with self-adaptability, self-adjustment, and parallel search capabilities. Evolutionary algorithms are capable of solving functions that are nondifferentiable, nonmicroscopic, or discontinuous, even for high-dimensional decision variables. Additionally, they can find approximations of optimal values for uncertain functions after several iterations. The common evolutionary algorithms include the genetic algorithm (GA) [
22], the states of matter search (SMS) algorithm [
23], particle swarm optimization (PSO) [
24,
25], the cuckoo search (CS) algorithm [
26], the firefly algorithm (FA) [
27], the gravitational search algorithm (GSA) [
28], and differential evolution (DE) [
29]. The “no free lunch” theorem has been proven, which states that there is no universally applicable optimization algorithm or learning method when searching for the optimal solution or learning the best model. In other words, there is no one method that performs best in all situations for different problems. This theorem emphasizes the importance of algorithm selection and design, as the performance of an algorithm depends on the characteristics of the problem [
30,
31]. Therefore, to address the task of EE prediction, we designed a variant algorithm based on DE. In DE, the best individuals are used to generate the next generation via mutation, recombination, crossover, and selection operations. This evolutionary mechanism exhibits superior search performance in comparison to the conventional evolutionary algorithm. Conventional DE performs optimally in addressing optimization issues featuring relatively simple function structures, but falls short in addressing complex optimization issues. This is particularly applicable in optimization models that have complex structures, where the decision variables have high dimensionality and the objective function has multiple local optima. In such situations, DE converges slowly and may experience stagnation during the search process.
By considering the energy field and relevant data features in detail, we attempt to apply an improved dendritic neuron model to energy-related fields. In our previous work, the effectiveness of the dendritic neuron model was established in various areas, including medical diagnosis [
32,
33], classification [
21,
34,
35,
36], time series prediction [
37,
38,
39,
40], and multiobjective optimization [
41,
42]. This paper proposes the evolutionary dendritic neural regression (EDNR) model for predicting building EE. As the weight space involved in EDNR is complex and has a large landscape, we further propose a complex network-based differential evolution (CNDE) algorithm as a global optimization algorithm for training EDNR. The proposed CNDE algorithm can increase individual diversity during the optimization search process while maintaining computational efficiency. This enables DE to achieve the global optimum, which enhances the accuracy of EDNR in building energy efficiency prediction. The extensive experimental results demonstrate that EDNR possesses considerable potential for predicting building EE, leading to satisfactory results in the prediction of both heating and cooling loads. Using simulated datasets, we propose EDNR, which effectively captures the nonlinear relationships between features and labels. Thus, this work aims to provide more competitive prediction methods for the field of building EE, while further expanding the interdisciplinary research between machine learning and other domains.
In summary, the primary innovations and contributions of this study are as follows: (a) A new method for residential building EE prediction is proposed and applied for the first time. (b) A population guidance strategy based on the scale-free property of complex networks is proposed to enhance the optimization capability of DE. (c) The proposed CNDE algorithm can achieve a better balance between exploration and exploitation in the search process, and the experimental results show that EDNR has better accuracy and stability in predicting EE.
The remainder of this paper is organized as follows:
Section 2 introduces the relevant work on EDNR and CNDE in detail;
Section 3 describes the experimental setup, parameter settings, experimental data, evaluation metrics, experimental results, and discussion; finally,
Section 4 provides a summary and discusses future work.
3. Experimental Studies
To validate the predictive ability of the proposed EDNR model for building EE, we conducted comprehensive comparative experiments, including DNR training with different heuristic algorithms to enhance its performance, and popular machine learning models were chosen as competitors. The experimental findings demonstrate that the proposed CNDE algorithm can train a DNR model effectively, and the resulting EDNR model excels in predicting both heating and cooling loads for building EE. The experiments described in this section had the following components: description of the dataset, experimental setup, normalization of the data, parameter configuration and discussion, interpretation of the performance evaluation metrics, comparison and analysis of the experimental results, and computational complexity analysis. Finally, a statistical analysis of the experimental results was performed. To confirm the efficacy and superiority of the proposed method, we selected eight widely used intelligent algorithms as competitors to CNDE for training DNR models. Additionally, eight competitive machine learning models were selected for comparative experiments with EDNR. To prevent the possibility of experimental error, we conducted the experiments for each set of methods 30 times independently and calculated the average of the results. The experiments were conducted on a personal computer equipped with a 13th Gen Intel(R) Core(TM) i7-13770F 2.10 GHz CPU, with 32 GB of memory, and the MATLAB R2020a software.
3.1. Dataset Description and Normalization
In this study, we used a standard dataset created by Angeliki Xifara, a British engineer, and processed by the Centre for Industrial and Applied Mathematics at the University of Oxford, U.K., to study and model the heating and cooling load requirements (energy efficiency) of buildings. This dataset is mainly used to study or model the heating and cooling loads of buildings. The researchers used Ecotect to simulate the energy profiles of 12 different building shape characteristics, such as building orientation, height, roof area, glazing distribution, and size. A total of 768 building samples were obtained from the simulation, where each data sample set contains eight features and two real values of EE loads and all attributes are positive integers or decimals.
Data normalization is a crucial step in data preprocessing and has significant implications for both machine learning and time similarity searches. Normalization aims to provide equal weight to all data attributes, simplifying comparisons and aggregations between attributes. Since the input range of the EDNR synaptic layer is between 0 and 1, applying a normalization operation to the raw data is necessary, which can not only reduce the computational complexity, but also aid in algorithm convergence. The formula for normalization is as follows:
where
and
represent the maximum and minimum values of the data, respectively, and
denotes the normalized data. Notably, since the soma layer of the EDNR model outputs within the interval of [0,1], the final objective of the experiment is to determine the predicted and actual values. Hence, we need to perform inverse normalization on the EDNR output and subsequently calculate the error between the predicted and actual values.
3.2. Parameter Settings
For EDNR to achieve outstanding performance, we must initially determine and evaluate the most appropriate parameter combinations for CNDE. In
Section 2.2.2, we generated a complex network model with an initial node value of 3, as illustrated in
Figure 3. Consequently, the initial node
is a critical parameter that directly influences the topology of the final network model. Therefore, to obtain the optimal
value while ensuring that the other parameters remain the same, we conducted two sets of experiments on heating and cooling loads. Each set consisted of 20 independent experiments, and the experimental results are summarized in
Table 1. The optimal values of
are 5 and 6 for dealing with heating and cooling load problems, respectively.
For our proposed CNDE optimization algorithm, in addition to the initialization parameter
used in constructing complex network models, the population size
is crucial in determining computational accuracy. Thus, it is essential to determine an appropriate
value that balances both computational cost and optimization performance. In general, larger
values tend to result in better optimization performance, albeit at the expense of higher computational costs and slower convergence rates. Conversely, reducing
can decrease the optimization ability. According to the prediction task and considering both the computational cost and the fairness of the algorithmic competition, we set the
values of all population-based optimization algorithms to 50, with the maximum number of function evaluations being
in our experiments. For EDNR, the number
k of the synaptic layer,
M of the branch number, and the threshold
of the cell body layer are crucial customized parameters. To identify enhanced parameter combinations, we conducted 16 sets of experiments employing the
orthogonal array method and established two distinct parameter combinations based on different prediction targets, and the results are shown in
Table 2. Additionally,
Table 3 lists the significant parameter settings of all algorithms and models involved in the experiments, which were derived from relevant literature.
3.3. Evaluation Metrics
The accuracy pertains to the level of resemblance between the predicted and actual values. To demonstrate the dissimilarities in performance between the different methods, we selected four frequently utilized evaluation metrics:
The mean absolute error (MAE) is the average error between the predicted and actual values without considering outliers and can be expressed using the following formula:
The mean absolute percentage error (MAPE) is a relative measure that is sensitive to errors. Its formula is as follows:
The mean squared error (MSE) represents the expected value of the square of the difference between the predicted and actual values. This indicator reflects the accuracy of the prediction method and its sensitivity to errors. The smaller the MSE is, the better the prediction accuracy. The calculation method is as follows:
The root-mean-squared error (RMSE) is the square root of the MSE, and it is calculated arithmetically so that its magnitude aligns with that of the original indicator. The definition is as follows:
where
and
represent the actual and predicted values, respectively, and
n is the sample size. Notably, all the error results were derived from the test datasets, and the experimental training and test sets were randomly divided.
3.4. Performance Comparison
In this section, we present detailed experimental results and comparative analyses. The experimental results are categorized into two parts: prediction results, which were obtained via various optimization algorithms that were employed to train the DNR models, and EDNR results, which were obtained via machine learning models. For the fairness of the experiments, both the experimental training and test sets were randomly selected at a ratio of 7:3. Additionally, each experiment was independently performed 30 times.
3.4.1. Comparison with Different Learning Algorithms
In this study, we present CNDE, a complex-network-model-based DE algorithm. To verify the optimization capabilities of CNDE, we conducted comparative experiments with eight population-based heuristic algorithms: GA, CS, FA, GSA, PSO, SMS, DE, and JADE. To compare each algorithm at its optimal performance, the optimal parameter combination for each algorithm participating in the experiment was selected according to the pertinent literature. The nine aforementioned algorithms were applied as learning algorithms to train the DNR models, and the resulting models were subsequently utilized to predict the EE of buildings. The results of the experiment are displayed in
Table 4 and
Table 5. We highlight the performance of the best performing method in terms of the four evaluation indicators.
Table 4 and
Table 5 illustrate that the CNDE algorithm outperformed the others in forecasting both heating and cooling loads, indicating its superior optimization performance in model training; hence, it effectively aids EDNR in overcoming the issue of local optimization during the search process. In addition, it is essential to consider the convergence speed of the model as a crucial performance indicator of the method.
Figure 5 shows the error convergence curves of all the models. According to
Figure 5, the EDNR model that was obtained via CNDE training exhibited the fastest convergence and the lowest error rate, irrespective of the predicted heating or cooling load. This demonstrates the excellent performance of CNDE in addressing such problems.
To enhance the clarity of the experimental results, error bar charts that display both the result details and interexperiment variability are presented. Furthermore, each algorithm’s stability is demonstrated through the use of a box-and-whiskers graph.
Figure 6 illustrates the results in detail, while
Figure 7 depicts the MAE of each method following the 30 experiments. The dots in the figure correspond to the results of a single experiment. As shown in
Figure 6 and
Figure 7, CNDE, an EDNR learning algorithm, exhibited both reduced error and enhanced stability. Moreover, from
Table 4 and
Table 5 and
Figure 6 and
Figure 7, it can be observed that SMS and JADE also obtained superior performance, indicating their high generalization ability for addressing common optimization problems.
3.4.2. Comparison with Other Machine Learning Models
The preceding sections examined and confirmed the effectiveness and consistency of the CNDE algorithm. To further demonstrate the superior performance of EDNR, in addition to comparing intelligent algorithms as DNR training algorithms internally, we selected eight prevalent machine learning models as competitors for EDNR. These models include the Elman neural network (ENN), multilayer perceptron (MLP), four different kernel function support vector regressions, SVR(L), SVR(P), SVR(R), and SVR(S), DT models, and the original DNR (ODNR) model, which is trained using the conventional BP algorithm.
Table 3 summarizes all the model parameters referenced in the relevant literature.
Table 6 and
Table 7 present the experimental results, which show that EDNR achieved the best performance in both heating and cooling EE prediction. Additionally, the DT model demonstrated robust performance in predicting the heating load, with evaluation metrics similar to those of EDNR. This suggests that DTs can accurately capture the nonlinear relationships between the feature input and output for heating load prediction. Nonetheless, as shown in
Figure 8, the DT model exhibited erratic performance when dealing with cooling loads.
It is obvious from
Figure 8 and
Figure 9 that both SVR(S) and ODNR exhibited poor performance. On the one hand, ODNR is optimized by the BP algorithm, which employs a gradient-based mechanism and is easily affected by the initial point during the search process, eventually leading to local optimization. Consequently, ODNR fails to achieve optimal performance. On the other hand, SVR(S) employs the sigmoid function as its kernel function, thereby rendering it equivalent to a multilayer perceptron neural network. Although this function automatically determines the weights of the hidden layer nodes with respect to the input nodes during the training process, it may increase the complexity of the model, and it may be challenging to understand the balance between model overfitting and underfitting. However, SVR(R) with only one parameter performed better, indicating that SVR(R) is more effective at handling tasks with diverse decision boundaries. During the optimization process of CNDE, the complex node ordering operation ensures population diversity and balances exploration and exploitation. This results in a better qualified EDNR model for EE prediction after training. In summary, numerous experimental results demonstrate that our proposed EDNR model exhibited robust competitiveness in predicting building EE and has achieved satisfactory outcomes.
3.4.3. Computational Complexity
Computational complexity is a crucial measure for evaluating algorithm effectiveness, providing a qualitative description of an algorithm’s runtime.
Figure 10 depicts the independent runtimes for each model. On the one hand, the ENN required the most time to execute both heating and cooling load predictions as a result of its specific algorithmic structure. On the other hand, both SVR and DT required small amounts of time, owing to their typical machine learning models, and efficiently obtained the globally optimal solution while dealing with convex optimization problems, avoiding the curse of dimensionality. From
Figure 10, it can be observed that the DNR model trained by the evolutionary algorithm had a similar time cost, while the ODNR model trained by the BP algorithm took a short time. The heuristic algorithm requires a slightly more intricate computational process and, thus, consumed more time. However, the BP algorithm relies on gradient information and is susceptible to falling into local optima, resulting in premature convergence of the algorithm. Upon closer inspection, it can be found that the EDNR model trained by CNDE required slightly less time than the other evolutionary algorithms. Notably, as shown in
Figure 5, EDNR can converge in relatively few iterations, which means that EDNR actually does not take much time to train; in other words, the unique network-node-guided search mechanism enables it to converge rapidly, resulting in a shorter running time.
3.5. Statistical Analysis
To comprehensively evaluate the performance of EDNR in predicting building EE, we used two statistical tests to analyze the results of 30 independent experiments: the Friedman test and the Wilcoxon rank-sum test [
53,
54]. The significance level
p of the Wilcoxon rank-sum test was set at 0.05.
indicates that EDNR significantly outperformed the competitor. The results of the Friedman test and Wilcoxon rank sum test are summarized in
Table 8.
In
Table 8, the symbols ↑, ↓, and ∘ indicate the relative strengths and weaknesses of the method compared to EDNR. ↓ shows a significant difference, meaning that EDNR had significant advantages compared with the other methods; ∘ represents no significant difference between the two methods; ↑ shows that EDNR performed less effectively than the other methods.
is the number of ↓/∘/↑ in each corresponding method. According to the rankings presented in
Table 8, there is minimal discrepancy in the ranking order between the two sets of techniques for both the heating and cooling load prediction tasks. Notably, EDNR obtained the lowest scores, i.e., the highest rankings, for both types of problems, indicating that EDNR not only demonstrates high accuracy in predicting EE, but also shows model stability. In addition, the
values of EDNR in predicting the EE were
and
, respectively, which fully demonstrates the superior performance of EDNR in addressing building EE forecasting. In fact, these results can be inferred from previous data tables or visualization images. In summary, the two nonparametric statistical analysis methods validate the effectiveness and comprehensiveness of EDNR in assessing heating and cooling loads in buildings.