A Systematic Analysis for Energy Performance Predictions in Residential Buildings Using Ensemble Learning

Abstract

Energy being a precious resource needs to be mindfully utilized, so that efficiency is achieved and its wastage is curbed. Globally, multi-storeyed buildings are the biggest energy consumers. A large portion of energy within a building is consumed to maintain the desired temperature for the comfort of occupants. For this purpose, heating load and cooling load requirements of the building need to be met. These requirements should be minimized to reduce energy consumption and optimize energy usage. Some characteristics of buildings greatly affect the heating load and cooling load requirements. This paper presented a systematic approach for analysing various factors of a building playing a vital role in energy consumption, followed by the algorithmic approaches of traditional machine learning and modern ensemble learning for energy consumption prediction in residential buildings. The results revealed that ensemble techniques outperform machine learning techniques with an appreciable margin. The accuracy of predicting heating load and cooling load, respectively, with multiple linear regression was 88.59% and 85.26%, with support vector regression was 82.38% and 89.32%, with K-nearest neighbours was 91.91% and 94.47%. The accuracy achieved with ensemble techniques was comparatively better—99.74% and 94.79% with random forests, 99.73% and 96.22% with gradient boosting machines, 99.75% and 95.94% with extreme gradient boosting.

1 Introduction

While designing smart buildings, optimal measures should be taken so that energy is used efficiently to safeguard the environment [1]. Studies done by researchers all over the world show that the highest percentage of energy is consumed by the multi-storey buildings [2,3,4,5]. Buildings consume about 40% of the total energy consumed in the world. After buildings, the second major energy consumer is industry which is reported for 32% energy consumption. The third major area is transport with 28% energy consumption. These studies motivated to devise solutions for energy optimization in buildings. Further studies show that within buildings, heating, ventilation and air conditioning (HVAC) system is one of the major energy consumers [6, 7]. HVAC consumes energy to maintain the desired temperature within a building and control humidity. It is responsible for meeting the heating load and cooling load of a building. Heating load can be defined as the amount of heat energy that is required to be added to a certain space for maintaining a desired temperature. Cooling load, on the other hand, is the amount of heat energy to be removed from a certain space to keep the temperature within desired limits. These two are related to the thermal load of the building. When the building is cold, the thermal load is converted into heating load and when the building is hot, the thermal load is converted into cooling load [8]. The heating and cooling loads of a building directly affect its energy performance. It requires analysis of factors that affect the heating and cooling loads. Studies reveal that various characteristics of a building and its structure affect heating and cooling loads to a major extent [9, 10]. Predicting energy consumption in buildings gives insight on the future demand of energy, and if more energy is being consumed than expected, appropriate measures can be adopted to stabilize energy use.

This paper focusses on several important features of buildings for, e.g. relative compactness, surface area, wall area, roof area, overall height, orientation, glazing area and glazing area distribution. The feature selections techniques were employed to derive the relevant features for predicting the heating load and cooling load. Further, three machine learning algorithms namely—multiple linear regression, K-nearest neighbours and support vector regression, and three ensemble learning algorithms namely random forests, gradient boosting machines, extreme gradient boosting were used for creating models. Additionally, these models have been evaluated using various performance measures for, e.g. RMSE, MSE, MAE, R squared and accuracy.

The organization structure of the paper is as follows: Sect. 1 introduces the problem.

Literature review is presented in Sect. 2. Section 3 describes the state-of-the-art machine learning techniques and ensemble techniques. The methodology adopted for the work is described in Sect. 4. The experiments performed and the results obtained are presented in Sect. 5. Finally, conclusions revealed in this research are mentioned in Sect. 6.

2 Literature Survey

Several researchers worldwide worked in the field of energy consumption and optimization in buildings. An important aspect in this domain is the analysis of energy performance gap. In this context, a study on German households [11] introduces prebound effect, in which the occupants actually consume 30% less energy than as calculated in the standard ratings. This gap can be due to incorrect assumptions made during energy ratings. In another study [12], common usages of the term rebound effect have been reviewed. Rebound effect is used in the context where actual energy consumption exceeds the calculated ratings. Some of the important researches in the field of energy using various individual machine learning and ensemble techniques are analysed and shown in Table 1.

Table 1 Survey of literature

3 Machine Learning

In this research, the applied machine learning techniques belong to two different categories.

3.1 Traditional Machine Learning Techniques

Three traditional machine learning techniques have been applied in our research work namely—multiple linear regression, K-nearest neighbours and support vector machines.

3.1.1 Multiple Linear Regression

It is although quite similar to linear regression but there is one significant difference. In MLR model, one response variable B is dependent upon multiple independent variables A1, A2, A3 … An. The relationship between predictor variable and response variable can be expressed in the form of conditional expectation as shown in Eq. (1).

$$ E\left( {Y|X} \right) = \beta_{0} + \beta_{i} X_{i} $$

(1)

$\beta_{i}$ is the slope that depicts the change in response variable Y when the predictor variable j is varied by one unit and other predictors are kept constant. The complexity of results interpretation increases in this model as a result of the correlation between different independent variables [5]. The concept of MLR is graphically shown in Fig. 1 [35].

Fig. 1

Multiple linear regression

3.1.2 K-Nearest Neighbours

Also known as “Lazy Learner”, this technique takes into consideration “k” number of closest instances in the training dataset to predict the value of the unknown instance. These k instances are found by applying a certain distance metric such that they are the k-nearest neighbours of the unknown instance [36]. The value returned is obtained after averaging the values of k-nearest neighbours [37].

If D is a dataset consisting of x_i training instances and the value of an unknown instance p is to be predicted, the distance between p and x_i can be obtained with Eq. (2).

$$ d\left( {p,x_{i} } \right) = \mathop \sum \limits_{f \in F} w_{f} \delta \left( {p_{f} ,x_{if} } \right) $$

(2)

where $\delta \left( {p_{f} ,x_{if} } \right) = \left| {p_{f} - x_{if} } \right|$ for continuous attribute.

Graphical representation of KNN regression can be seen in Fig. 2 [38]. In our experiments, the value of k is taken as 4.

Fig. 2

K-nearest neighbours [38]

3.1.3 Support Vector Regression

It aims at finding a function f(x) which allows deviation to a certain extent ε from the obtained target values y_i in training data samples. It should also ensure maximum flatness. A linear function [39] can be described as:

$$ f\left( x \right) = w,\,x + b\quad {\text{with}}\quad w \in {\mathbf{\mathcal{X}}},\;\;b \in {\mathbb{R}} $$

(3)

where ${\mathcal{X}}$ represents input pattern space such that ${\mathcal{X}} = {\mathbb{R}}^{\text{d}}$.

Figure 3 [40] shows the graphical representation of SVR. The legend in the figure represents the results of various SVR kernel functions applied on a sample dataset of 40 random numbers. Values on x-axis represent data points, values on y-axis are target points. A radial basis function (RBF) kernel can be described as:

$$ K\left( {X_{1} , X_{2} } \right) = \exp \left( { - \gamma \left| {\left| {X_{1} - X_{2} } \right|} \right|^{2} } \right) $$

(4)

where $\left| {\left| {X_{1} - X_{2} } \right|} \right|$ is the Euclidean distance between points X₁ and X₂.

Fig. 3

Support vector regression [40]

3.2 Ensemble Techniques

The basic Ensemble technique is to integrate the results of individual machine learning models, such that the prediction results exhibit improvement in terms of accuracy and robustness. Bagging and Boosting are two popular ensemble methods. The ensemble techniques used in this paper are explained as follows.

3.2.1 Random Forests

It is a tree-based ensemble technique that can be applied for both classification as well as regression. Some of the features which make random forests appealing are: prediction efficiency, suitability for highly multi-dimensional problems, missing values handling, outlier removal, etc. [41, 42].

In regression using random forests, to predict a continuous variable, the trees are grown depending on θ in such a manner that h (x, θ) takes on numeric values.

$$ \begin{aligned} & {\text{Where}}\;\theta :\;{\text{A}}\;{\text{random}}\;{\text{vector}} \\ & h\left( {x, \, \theta } \right) :\;{\text{Tree}}\;{\text{predictor}} \\ \end{aligned} $$

The values of the response variable are numeric, and it is assumed that the training sample is drawn independently from the distribution X of random vector Y.

The RF predictor is created by taking the mean over k of the trees

$$ \left\{ {h\left( {x,\theta_{k} } \right)} \right\} $$

(5)

The mean square generalization error for a numeric predictor h(x) is given by

$$ E_{X,Y} \left( {Y - h\left( X \right)} \right)^{2} $$

(6)

For infinite numbers of trees in the forest, the RF predictor is defined as

$$ E_{X,Y} \left( {Y - av_{k} h\left( {X,\theta_{k} } \right)} \right)^{2} \to E_{X,Y} \left( {Y - E_{\theta } h\left( {X,\theta } \right)} \right)^{2} $$

(7)

The schematic diagram of Random Forests is shown in Fig. 4.

Fig. 4

Random forests

3.2.2 Gradient Boosting Machines

GBM is also an ensemble learning technique, whose underlying structure is a decision tree. In GBM, additive regression models are created by iteratively fitting a simple base to currently updated pseudo residuals by calculating least squares at every continuous iteration [43]. In gradient boosting a function F*(x) is generated that maps x to y, so that when the joint distribution of all (y, x) values is taken, the expected value of Ψ(y, F(x)) is minimized, where Ψ(y, F(x)) is some specified loss function. This relation is depicted in Eq. (8).

$$ F^{*} \left( x \right) = \arg \hbox{min} E_{y,x}\Psi \left( {y,F\left( x \right)} \right) $$

(8)

where y: The random output or response variable, x = {x₁, x₂, …x_n}: a set of random input variables.

Figure 5 shows the scheme behind gradient boosting machines.

Fig. 5

Gradient boosting machines

3.2.3 Extreme Gradient Boosting

Apart from performance and speed as its key features, this technique has an added feature of Scalability. Several optimizations have been performed on the basic algorithm to ensure the scalability of the model [44]. Figure 6 shows the schematic diagram of XGBoost.

Fig. 6

Extreme gradient boosting

4 Method and Data

This section explains the workflow approach followed in this research. Figure 7 shows the steps of the methodology employed. The methodology starts with the data collections followed by data analysis and pre-processing, data partitioning and model constructions using various machine learning and ensemble learning algorithms. Finally, models have been evaluated on various parameters. Each phase in the process has been explained below.

Fig. 7

Research approach

4.1 Data Set Collection and Preparation

The dataset used in this research is a standard dataset that has been collected from the University of California, Irvine (UCI) repository [45].

This dataset is related to energy efficiency in buildings and consists of eight different characteristics of buildings which act as input variables (X₁,X₂…X₈) and heating load (Y₁) and cooling load (Y₂) of buildings as two output variables.

The detailed description of the parameters of the data used along with symbols and its respective type is given in Table 2.

Table 2 Dataset parameter description

4.2 Data Analysis and Pre-processing

Data pre-processing is a process that consists of checking the dataset for missing values and filling them with appropriate values, detecting and removing any outliers, converting it into a particular form suitable for applying algorithm, attribute selection, etc.

4.2.1 Statistical Analysis

The statistics of input parameters like minimum value, maximum value, mean, standard deviation, and variance were derived and are described in Table 3.

Table 3 Parameter statistics

Probability distribution of all input variables, X₁ to X₈ and both the output variables, Y₁ and Y₂ using histograms is shown in Fig. 8. The distribution graphs show that none of the input and output variables follow Normal distribution.

Fig. 8

Probability density estimates [12]

4.2.2 Feature Selection

Feature selection is an important step in the process of predicting results using machine learning because all the features are generally not equally important for predicting the response value. Some features carry more weights than others for deriving a particular value and are thus more important, whereas some are very less or not at all important in the derivation of results. Such irrelevant features need to be excluded from the input to save training time and computation time. Additionally, applying the algorithm on only important and relevant features may result in more accurate prediction, reducing over fitting. In this research, feature selection has been performed in the following two ways:

4.2.2.1 Filter Feature Selection

It is a univariate method in which statistical techniques are used to derive the relationship between each input variable and the target variable. The features which are strongly related to the response variable are selected as input for algorithm application and the features which are weakly related to the response variable can be eliminated. In our research, Spearman correlation coefficient was calculated to derive the strength of the relationship of several independent variables of the dataset with each of the response variables. Spearman’s method for computing correlation was employed as the distribution of dataset used is non-Gaussian. A zero value for the correlation coefficient means the variables are not correlated, i.e., they are independent. A value closer to 1 indicates a high correlation among variables [46]. High correlation among two variables means one variable varies in accordance with the other; if one increases the other also increases. Similarly, reduction in one variable tends to reduce the other. The values of correlation coefficient between independent variables X₁–X₈ and Y₁ are represented in Fig. 9 and the same with Y₂ are represented in Fig. 10.

Fig. 9

Correlation of input variables with Y₁

Fig. 10

Correlation of input variables with Y₂

4.2.2.2 Feature Importance

As mentioned earlier, features play a very important role in prediction and some features tend to be more important than others. In this context, feature importance graphs were generated to obtain the degree of effectiveness of each of the independent parameters, so that the contribution of each feature in prediction can be known and accordingly selection can be made. Figures 11 and 12 show the feature importance graph generated using random forests and gradient boosting machines, respectively. These techniques showed similar importance of features for both response variables, Y₁ and Y₂. According to random forests, overall height has maximum importance, followed by relative compactness, then surface area, wall area, glazing area, roof area, glazing area distribution and orientation. The sequence of features as derived by gradient boosting machines in the decreasing order of importance is—relative compactness, Surface area, roof area, overall height, glazing area, wall area, orientation and glazing area distribution.

Fig. 11

Feature importance using RF

Fig. 12

Feature importance using GBM

Figures 13 and 14 represent the graphs generated by applying LASSO technique for feature importance for Y₁ and Y₂, respectively. According to LASSO, relative compactness, overall height, glazing area and glazing area distribution are more important for predicting the heating load of a building. Furthermore, for predicting cooling load, the parameters—relative compactness, surface area, overall height, orientation and glazing area—are more important than others. Therefore, for performing experiments, X₁, X₅, X₇ and X₈ have been selected as input parameters for predicting Y₁. Likewise, X₁, X₂, X₅, X₆ and X₇ have been selected for the prediction of Y₂.

Fig. 13

Feature importance for Y₁ using LASSO

Fig. 14

Feature importance for Y₂ using LASSO

4.3 Data Analysis and Pre-processing

Dataset was partitioned according to 70–30% rule into two subsets: training dataset and testing dataset. For partitioning, random sampling without replacement was applied which resulted in 70% training data, on which the algorithms were applied, and the remaining 30% was used for testing the algorithms.

4.4 Model Construction

The model was constructed by applying three machine learning techniques namely—multiple linear regression, K-nearest neighbours and support vector regression. Three ensemble techniques were applied namely—random forests, gradient boosting machines, and extreme gradient boosting. The models were applied on the training dataset and tested using the testing dataset.

4.5 Model Evaluation

The evaluation of the results obtained after applying algorithms was done using five well-known performance measures namely root mean square error, mean square error, mean absolute error, R squared and accuracy. These measures can be calculated by applying following formulae, where Y_i: is the observed value for the ith observation, $\hat{Y}_{i}$: is the predicted value, N: is sample size.

The original dataset consisting of 768 instances has been partitioned into two subsets—70% training dataset and 30% testing dataset, by random sampling. So N = 538 for model construction on training dataset and N = 230 for testing purpose.

Root mean square error Following equation defines the formula for RMSE:

$$ {\text{RMSE}} = \sqrt[2]{{\sum\limits_{i = 1}^{N} {\frac{{\left( {\hat{Y}_{i} - Y_{i} } \right)^{2} }}{N}} }} $$

(9)

Mean square error Following equation defines the formula for MSE:

$$ {\text{MSE}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {Y_{i} - \hat{Y}_{i} } \right)^{2} $$

(10)

Mean absolute error MAE can be defined by the following equation:

$$ {\text{MAE}} = \mathop \sum \limits_{i = 1}^{N} \frac{{\left| {Y_{i} - \hat{Y}_{i} } \right|}}{N} $$

(11)

R Squared R squared can be defined by the following equation:

$$ R^{2} = 1 - \frac{{\sum \left( {Y_{i} - \hat{Y}_{i } } \right)^{2} }}{{\sum \left( {Y_{i} - \bar{Y}} \right)^{2} }} $$

(12)

Accuracy Accuracy of a model can be calculated using the following formula:

$$ {\text{Accuracy}} = \frac{{\left| {V_{\text{A}} - V_{\text{O}} } \right|}}{{V_{\text{A}} }}*100 $$

(13)

where V_A: actual value, V_O: obtained value.

Sample calculations performed on the dataset using the above equations are shown in “Appendix”.

5 Results

All the experiments of the research were performed using Python programming language. Three machine learning algorithms namely MLR, KNN and SVR and three ensemble techniques namely, RF, GBM, and XGBoost have been experimented on the collected dataset. The results of the ML and Ensemble experiments are described in Tables 4 and 5 respectively.

Table 4 Results of ML algorithms

Table 5 Results of ensemble algorithms

5.1 Results of Classical ML Techniques

The results obtained after applying all the three aforementioned classical machine learning algorithms on the dataset are summarized in Table 4. These results are based on the performance measures. The values of RMSE range between 3.13 and 4.22 for both output variables Y₁ and Y₂ after applying MLR and SVR, whereas it is lower, 2.86 for Y₁ and 2.25 for Y₂ when KNN is applied. Correspondingly MSE values for KNN are also lower than MLR and SVR. Similarly, MAE values range between 2.25 and 3.19 using MLR and SVR, and the values are 1.96 and 1.54 using KNN. R Squared values are better in KNN (0.90 and 0.94), as compared to MLR (0.87 and 0.83) and SVR (0.76 and 0.84). KNN results are better than the other two algorithms in terms of accuracy also.

5.2 Results of Ensemble Techniques

Table 5 summarizes the results of the experiments performed by applying Ensemble techniques. As per the results, RMSE is 0.50 for output variable Y₁ for RF and XGBoost and 0.52 for GBM. Y₂ value varies slightly, 2.18 for RF, 1.86 for GBM and 1.93 for XGBoost. Correspondingly MSE value is also same 0.25 for Y₁ with RF and XGBoost and 0.27 with GBM. For Y₂, the values of MSE are 4.78, 3.47 and 3.72 with RF, GBM and XGBoost, respectively. MAE value varies slightly for Y₁ in the range 0.36–0.38 for all three algorithms, whereas the range for Y₂ is 1.25–1.39. R Squared values for Y₁ for all three algorithms are same, 0.99 and for Y₂; they vary from 0.94 to 0.96. Accuracy is also same, above 99% for Y₁ with all three algorithms, whereas accuracy percentage for Y₂ is 94.79%, 96.22% and 95.94% when RF, GBM and XGBoost are applied, respectively.

Figures 15 and 16 show the graphs plotted for results obtained in Table 4 for ML algorithms and Table 5 for ensemble algorithms, respectively. Figure 17 represents combined results for all six algorithms (ML and Ensemble) for both response variables Y₁ and Y₂.

Fig. 15

Graphical representation of Table 4 results

Fig. 16

Graphical representation of Table 5 results

Fig. 17

Combined graph for all six algorithm results

5.3 Comparative Analysis of Machine Learning and Ensemble Learning Algorithms

In this section, the results of experiments are represented graphically based on various performance measures used for results evaluation. The graphs for RMSE, MAE, R squared and accuracy are shown in Figs. 18, 19, 20 and 21, respectively.

Fig. 18

Graph for RMSE for Y₁ and Y₂

Fig. 19

Graph for MAE for Y₁ and Y₂

Fig. 20

Graph for R squared for Y₁ and Y₂

Fig. 21

Graph for accuracy for Y₁ and Y₂

It can be observed from Fig. 18 that the RMSE value is high (more than 3.0) for SVR and MLR algorithms for both the output variables Y₁ and Y₂, comparatively lower (approximately 2.0) for KNN, whereas the error values are extremely low (below 0.5) with all ensemble algorithms—RF, GBM and XGBoost for Y₁ and between 1.8 and 1.9 for Y₂.

Similar pattern can be observed from Fig. 19 for MAE. The values for MLR and SVR algorithms range between 2.32 and 2.63 for both Y₁ and Y₂. The values are lower when KNN is applied, 1.50 for Y₁ and 1.35 for Y₂. The results are better with ensemble algorithms with MAE between 1.19 and 1.27 for Y₂ and even lower values (0.35–0.36) are obtained for Y₁.

For R squared (Fig. 20), higher values, i.e., the values approaching 1 are considered better. In this context, again ensemble techniques have outperformed traditional ML techniques. The lowest values for R squared are obtained for SVR, 0.82 and 0.79 for Y₁ and Y₂, respectively. Slightly higher values are obtained for MLR, 0.88 and 0.83, and even higher for KNN with 0.94 and 0.96. R Squared results obtained with ensemble algorithms are significantly better than traditional algorithms with values ranging from 0.94 and going up to 0.99.

In Fig. 21, the plot for accuracy score also concludes that ensemble techniques perform significantly better than traditional algorithms with accuracy ranging between 96 and 99.76%. Among classical ML techniques, KNN performs better with an accuracy score of 95.12% and 96.47%, while the accuracy score for MLR and SVR ranges between 85.75 and 89.63%.

Ensemble techniques became popular from last two decades in the area of classification and prediction. The idea behind ensemble methods is that it can be compared to situations in real life, such as when critical decisions has to be taken, often opinions of several experts are taken into account rather than relying on a single judgment. Ensembles have shown to be more accurate in many cases than the individual models. Ideal ensembles consist of models with high accuracy which differ as much as possible. If each model makes different mistakes, then the total error will be reduced, if the models are identical, then a combination is useless since the results remain unchanged. It is evident from the survey of literature performed in Table 1 [13, 14, 17, 18, 23, 25, 26, 29] that ensemble techniques are far better in terms of performance prediction as compared to traditional machine learning algorithms. On similar terms, the results of experiments performed in this research also conclude that the predictions done by Ensemble models resulted in much lower error values RMSE, MSE and MAE, better R squared values and improved accuracy as compared to the traditional machine learning models used.

6 Conclusion and Future Scope

The issue of energy consumption at a fast pace and in large amounts demands solutions in this area, which can help in using the energy efficiently. Globally, buildings are the largest energy consumers, accounting for nearly 40% energy consumption. Therefore, the analysis of various energy-consuming components of a building reveal that the HVAC system consumes a large percentage of the building’s total energy. HVAC needs energy for operation so that it can meet the heating load and cooling load requirements of the building. Heating and cooling loads are largely affected by various attributes of a building. This research shows that relative compactness, surface area, overall height, orientation and glazing area are more important in predicting heating load and cooling load of the buildings. Furthermore, the results of experiments prove that ensemble techniques perform better than traditional machine learning techniques.

In this research, only one dataset is used. In future, we can apply experiments on multiple datasets with large number of instances to better prove the accuracy of models. Apart from the models applied in this research, more advanced models like stacking and voting can be applied for better analysis.

Appendix

1.1 Sample Calculations for Model Evaluation

The sample calculations using formulae in Eqs. 9–13 are described here. Table 6 contains the predicted values, observed values of response variables Y₁ and Y₂ from the dataset and predicted values after applying KNN algorithms. The calculations for model evaluation on the basis of values given in Table 6 have been performed manually on 20 and 100 sample size, respectively, which has been selected in respective order from 1–10 and 1–100.

Table 6 Sample dataset showing all predictor values and predicted values using KNN

Referring to Eqs. 9–13, applying the formulae on observed values and values predicted using KNN, For Y₁ calculated results for samples of initial 20 records,

$$ \begin{aligned} & {\text{RMSE}} = 12.01 \\ & {\text{MSE}} = 144.29 \\ & {\text{MAE}} = 10.2 \\ & R\;{\text{Squared}} = - 5.2 \\ & {\text{Accuracy}} = 43.82\% \\ \end{aligned} $$

Referring to Eqs. 9–13, applying the formulae on observed values and values predicted using KNN, For Y₁ calculated results for samples of initial 100 records,

$$ \begin{aligned} & {\text{RMSE}} = 14.02 \\ & {\text{MSE}} = 196.6 \\ & {\text{MAE}} = 10.9 \\ & R\;{\text{Squared}} = - 1.55 \\ & {\text{Accuracy}} = 48.46\% \\ \end{aligned} $$

The sample calculations using formulae in Eqs. 9–13 are described here. Table 7 contains the predicted values, observed values of response variables Y₁ and Y₂ from the dataset, and predicted values after applying XGBoost algorithms. The calculations for model evaluation on the basis of values given in Table 7 have been performed manually on 20 and 100 sample size, respectively, which has been selected in respective order from 1–20 and 1–100.

Table 7 Sample dataset showing all predictor values and predicted values using XGBoost

Applying the formulae on the values predicted using XGBoost, For Y1 the calculated results for samples of initial 20 records,

$$ \begin{aligned} & {\text{RMSE}} = 11.98 \\ & {\text{MSE}} = 143.59 \\ & {\text{MAE}} = 9.82 \\ & R\;{\text{Squared}} = - 5.17 \\ & {\text{Accuracy}} = 40.68 \\ \end{aligned} $$

Applying the formulae on the values predicted using XGBoost, For Y₁ the calculated results for samples of initial 100 records,

$$ \begin{aligned} & {\text{RMSE}} = 14.25 \\ & {\text{MSE}} = 203.1 \\ & {\text{MAE}} = 11.13 \\ & R\;{\text{Squared}} = - 1.63 \\ & {\text{Accuracy}} = 49.57 \\ \end{aligned} $$