# Using Lean-and-Green Supersaturated Poly-Factorial Mini Datasets to Profile Energy Consumption Performance for an Apartment Unit

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}(energy status ‘B’) for the apartment—an almost 20% reduction in energy consumption while also achieving upgrading from the original ‘C’ energy status. The proposed approach may aid energy engineers to make general empirical screening predictions in an expedient manner by simultaneously considering the apartment unit’s structural configuration as well as its installed electromechanical systems arrangement.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Technical Description of the Studied Building Apartment Unit

#### 2.1.1. Basic Location and Energy Consumption Status Information for the Building Apartment Unit

^{2}. Based on the national cumulative statistics from the estimations on the issued Energy Performance Certificates, 83.82% of the residential buildings which have been constructed before the year 1980 (55% of the total available building stock) have been awarded an energy consumption status of ‘H’. Even for more contemporary buildings such as the one that will be analyzed in this work, the majority of the energy performance ratings have been categorized as either a ‘C’ or ‘D’ class. The particular apartment unit was certified to require a year-round primary energy consumption of 133.8 kWh/m

^{2}. This was compared against a (theoretical) reference-building energy demand of 129.2 kWh/m

^{2}. The ratio of the two energy consumption estimations (the former over the latter) provides a measure for the energy efficiency. Since the estimated energy efficiency corresponds to the standardized interval (1.00, 1.41), it was awarded a rating status of ‘C’. This study is meaningful because, to reach the desired transitory ‘yellow zone,’ the energy efficiency score should lie within the standardized interval (0.75, 1.00), which corresponds to an energy consumption grade of ‘B’. The top rating (class ‘A+’) is awarded for energy efficiency scores lower than 0.33. However, the ‘green’ status for a residential building is awarded upon certification after attaining at least a ‘B+’ rating, which corresponds to an energy efficiency standardized interval of (0.50, 0.75). Therefore, any recommended improvement interventions should lead, at least, to climbing up to the ‘yellow zone’ scale, before any apartment renovation gains become substantial enough to contribute to the ‘greener outlook’ of the building’s energy performance. Ostensibly, the respective intervention costs, the financial status of the apartment owners/occupants, and any potential government incentives may affect the pace of progress toward reaching an enhanced green building status.

#### 2.1.2. Apartment Unit Structural Details

#### 2.1.3. Electromechanical and Renewable Energy Systems

#### 2.1.4. The Energy Efficiency Certification Software Package TEE-KENAK

#### 2.2. The Statistical Analysis Approach

^{24}= 16,777,216 trials). The experimental tactic of resorting to supersaturated designs befits the condition to drastically compress the trial schedule. Therefore, the 14-run, 24-parameter supersaturated design of Williams [87], with its versatile parameter screening acceleration properties [89], which was implemented for the lean experimental data collection of Rousali and Besseris [91], is determined to be an attractive sampling planner for this work, as well. The adopted supersaturated design class is modified factorial half-fractions [88], which may also include the special case of half-split Plackett–Burman [93] design matrices. The 24 controlling factors which will be accommodated in the 14-run, 24-parameter supersaturated screening design have been tabulated in Table 7.

**X = {**X

_{ij}

**}**, of size n × m, where the number of the supersaturated design explanatory variables is m and the number of supersaturated design recipes is n. Then, the response matrix

**Y**, of size n × r, where the number of responses is r = 1 for this work (energy consumption), and in conjunction with the prescribed condition for design supersaturation in m regressors, i.e., m > n + 1, is written as:

_{o}and β

_{j}, with 1 ≤ j ≤ m, symbolize the coefficients of regression and ${\u03f5}_{i}$ is denoted as the error term for 1 ≤ i ≤ n, which is assumed to be an independent and identically distributed random normal variable. The stepwise regression method uses forward sequences of F-test applications, but the model selection technique alternatives will include assessments which consider: (1) the adjusted coefficient of determination (adj R

^{2}), (2) the Bayesian information criterion [95], and (3) the Mallow’s C

_{p}metric [96] for best subsets regression.

**X**, of size n × m, and the response matrix

**Y**, of size n × r, are prescribed for m > n + 1:

**X**=

**CL**

_{X}

^{T}+

**E**

_{X}

**Y**=

**DL**

_{Y}

^{T}+

**E**

_{Y}

**X**and

**Y**are defined as

**C**and

**D**, respectively, and they are both of size n × p. The orthogonal loading matrices

**L**

_{X}and

**L**

_{Y}correspond to the matrices

**X**and

**Y**, with dimensions m × p and r × p, respectively. The error terms,

**E**

_{X}and

**E**

_{Y}, corresponding to the respective

**X**and

**Y**matrix models, are assumed to be independent and identically distributed random normal variables. The subsequent maximization of the covariance of the matrices

**C**and

**D**permits the decomposition of the matrices

**X**and

**Y**.

#### 2.3. The Computational Aids

_{p}metric. Therefore, the best-subsetting factorial combinations of the supersaturated energy consumption dataset were determined using the linear regression outcomes from two R-packages (v.4.1), ‘leaps()’ (v.3.1) and ‘StepReg()’ (v.1.4.4).

#### 2.4. The Methodological Outline

- (1)
- Gather the required building apartment unit structural layout designs, along with the information for the installed electromechanical and renewable energy equipment information.
- (2)
- Determine which featured characteristics will be investigated for the selected apartment unit.
- (3)
- Determine the range values for the featured apartment unit characteristics and code them into controlling factor levels.
- (4)
- Select an appropriate supersaturated screening design to accommodate the large number of controlling factors from steps 2 and 3.
- (5)
- Execute the supersaturated plan runs by inputting each time trial recipe information (from step 4) into the TEE-KENAK software package.
- (6)
- Record the energy consumption (real and reference) estimates from each supersaturated trial run.
- (7)
- Prepare the response table and response graph for the energy consumption estimates.
- (8)
- Conduct stepwise regression analysis and evaluate the model summary results.
- (9)
- Determine the active controlling factors from step 8 and suggest a possible solution for the factorial settings.
- (10)
- Confirm the energy consumption performance improvement by inputting the optimal solution into the TEE-KENAK software package.
- (11)
- Assess and discuss the overall solution using other known methods such as PLS, entropic, and hierarchical clustering comparisons on key descriptive estimators of the energy consumption response.

## 3. Results

^{2}. Meanwhile, the corresponding energy classification ratings varied from status categories of ‘B’ (0.75–1.0 ratio to the reference building’s energy consumption) to ‘Z’ (2.27–2.73 times the reference building’s energy consumption). In Table 9, the energy demands and consumption details are indicatively tabulated for the first trial of the supersaturated trial-design schedule; they result from loading the software platform TEE KENAK 1.31.1.19 with the input from the prescribed recipe. From the ensuing response table (Table 10), it is observed that the factorial variability of the energy consumption performance declines from 102.8 (F4) to 2.11 (F16) kWh/m

^{2}. The two leading factors which contributed to the magnitude of the variability are: (1) the energy source for heating system’s power generation (F4), and (2) the thermal insulation of the roof (F20). This behavior becomes more transparent in the response graph (Figure 3 (MATLAB R2022b)), where the optimal settings are identified at the lower levels of both factors. Factor F4, adjusted at the ‘gas’ setting, reduces the energy consumption estimation down to as low as 171.07 kWh/m

^{2}(an intra-factorial difference of 102.8 kWh/m

^{2}). Similarly, the ‘insulated roof’ option of factor F20 reduces the energy consumption to 179.87 kWh/m

^{2}(an intra-factorial difference of 85.2 kWh/m

^{2}). From the response graph, it is apparent that factors such as F2, F3, F5, F17, F21, and F22 might also be statistically assessed for their contributing effects to the overall improved performance of the energy consumption response.

^{2}). Adding the last influence, F17, to the model corrected the prediction by only 0.041%; the factor F17 was found to be statistically significant at a level of 0.05. Overall, the four-factor prognostication is also statistically significant at a Bonferroni-corrected familywise error rate of 0.05. The two statistically stronger factors, F4 and F20, contributed 60% and 29% to the total variation, respectively. The Durbin–Watson statistic was estimated at a value of d = 3.17 (>dU = 2.296); it does not provide any hint that the successive error terms might be positively autocorrelated. Moreover, the alternative estimation, 4-d (= 0.833), is within the critical value interval (0.505, 2.296), as computed for the test parameters n = 14, k’ = 5, and α = 0.05. Hence, the test for the presence of a negative autocorrelation is inconclusive. The normal P–P plot (IBM SPSS v.29) of the regression-analysis standardized residuals (Figure 4) does not reveal any detectable abnormalities. The model coefficient and collinearity statistics (IBM SPSS v.29) are listed in Table 12. The unstandardized/standardized coefficients of the four active factors are statistically significant to at least an error rate of 0.01. Moreover, the variance inflation factor (VIF) has been estimated to a maximum value of 1.35. Thus, there seem to be no apparent multicollinearity tendencies across effects, with respect to the two leading factors (F4 and F20) in particular.

^{2}(energy status ‘C’), and was to be contrasted against a reference building estimation of 129.2 kWh/m

^{2}. After completing the screening/optimization work, the recommended factorial settings from Table 13 were input to the TEE KENAK 1.31.1.19 software program to confirm any accruing energy savings. The improved solution delivered an energy consumption projection of 110.4 kWh/m

^{2}(energy status ‘B’) for the apartment, which was to be contrasted against a reference building estimation of 125.9 kWh/m

^{2}. This is an almost 18% reduction in energy consumption, which may be considered satisfactory given the fact that only a subset of the total available variables in the TEE KENAK 1.31.1.19 software program was actually studied in this paradigm.

## 4. Discussion

_{p}metric is utilized to reassess the linear regression results using the best subsets approach (R-packages ‘leaps()’ (v.3.1) and ‘StepReg()’ (v.1.4.4)). The suggested solution (F2, F4, F17, F22) achieves an adjusted R

^{2}value of 96.2% and a corresponding C

_{p}value of 28.1. Adding as many as five extra regressors (F5, F7, F11, F21, and F22) via the best subsetting approach increases the adjusted R

^{2}to a value of 98.9% and, thus, substantially reduces the corresponding C

_{p}value to 10.3. However, a 99% confidence interval estimation for the adjusted R

^{2}in the original solution also includes the latter prediction. It is inferred that the small supersaturated dataset may not allow for discerning the need for additional predictors by relying only on the C

_{p}criterion.

_{EC}/h with h = 2·IQR/$\sqrt[3]{n}$ (IQR = interquartile range, n = number of EC response entries). Inputting the values of range

_{EC}= 215.8 kWh/m

^{2}, IQR = 99.03 kWh/m

^{2}, and n = 14, the number of common bins was computed to be approximately three. Next, the synchronous two-setting discretization was conducted using the function ‘discretize2d()’ (R-package ‘entropy()’ (v.1.3.1)). Then, the empirical (Shannon) mutual information of the setting pairs was computed using the function ‘mi.empirical()’ (R-package ‘entropy’ (v.1.3.1)). The two controlling factors with the two lower mutual information estimations between settings, along with their lower-setting optimal shrinkage intensity estimations (function ‘entropy.shrink()’ from the R-package ‘entropy’ (v.1.3.1)), were found to be: (1) F20 (0.08 nats) with optimal shrinkage intensity lowered at 0.263 at the second level, and (2) F4 (0.202 nats) with optimal shrinkage intensity lowered at 0.263 at the second level.

## 5. Conclusions

^{2}(energy status ‘B’) for the apartment. It accounts for an almost 20% reduction in energy consumption. Moreover, the ‘greener’ status rating has improved from the original ‘C’ status. Future work could involve forecasting the costs of apartment unit renovations and optimized predictions that combine economical and technical parameters, as well as occupant usage trends.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Response graph (MATLAB R2022b) for the poly-factorial profiling of the energy consumption (EC) in kWh/m

^{2}.

**Figure 4.**Normal P–P plot of the regression standardized residuals for the dependent variable EC (IBM SPSS v0.29).

**Figure 5.**Gap statistic performance for profiling optimal clustering size for the summarized supersaturated dataset of Table 17.

**Figure 6.**Cluster quality rating using the silhouette measure of cohesion and separation (IBM SPSS v.29).

**Figure 7.**Dendrogram using median linkage for the 48 factor levels resulting from the summarized four-estimator supersaturated dataset (IBM SPSS v.29).

**Figure 8.**Individually contrasting the clustered supersaturated datasets for their four summarizing estimators: (

**A**) median (M), (

**B**) interquartile range (I), (

**C**) skewness (S), and (

**D**) kurtosis (K).

Room No. | Floor Surface Area (m ^{2}) | Windows Surface Area (m ^{2}) | Ventilation Surface Area (m ^{2}) |
---|---|---|---|

1,2 | 72.6 | 7.6 | 5.08 |

3 | 20.2 | 2.99 | 1.42 |

4 | 18 | 2.99 | 1.26 |

5 | 18 | 2.99 | 1.26 |

6 | 24.3 | 2.99 | 1.7 |

Structural Elements | Side A Surface Area (m ^{2}) | Side B Surface Area (m ^{2}) | Side C Surface Area (m ^{2}) | Side D Surface Area (m ^{2}) |
---|---|---|---|---|

Columns | 16.3 | 21.1 | 12.3 | 16.2 |

Windows | 11.5 | 7.6 | 2.2 | 0 |

Brick wall | 19 | 17.2 | 22.7 | 21 |

Shell Element | Element Coding | Orientation (^{o}) | Surface F (m ^{2}) | k (kcal/m^{2}) |
---|---|---|---|---|

Walls | W1 | 346 | 35.3 | 0.61 |

W2 | 166 | 38.3 | 0.61 | |

W3 | 76 | 35 | 0.58 | |

W4 | 256 | 37.2 | 0.59 | |

Windows | F1 | 346 | 11.5 | 2.6 |

F2 | 166 | 7.6 | 2.6 | |

F3 | 76 | 2.2 | 3 | |

F4 | 256 | |||

S | 167.1 |

Surface Area (m^{2}) | 174.9 |

Volume (m^{3}) | 570.25 |

Concrete Height Level (m) | 3.25 |

Final Height Level (m) | 3.3 |

System | Source | Distribution Network of Thermal Medium | Season | Power (kW) |
---|---|---|---|---|

Heating | Natural Gas | Yes | Winter | 25 |

Cooling | Electricity | No | Summer | 9 |

Hot Water | Solar/Electricity | No | Year-round | 5 |

Panel Angle (^{o}) | Panel Surface Area (m^{2}) | Shade Coefficient | Orientation (^{o}) |
---|---|---|---|

45 | 4 | 0.8 | 180 |

**Table 7.**Controlling factors and their settings for influencing the energy consumption of the residential unit.

Coded | Factors | Natural Gas Boiler(−) | Petroleum Boiler(−) | Natural Gas Boiler(+) | Petroleum Boiler(+) |
---|---|---|---|---|---|

F1 | Automation for hot water | no | yes | ||

F2 | Category of automatic control | A | D | ||

F3 | Number of ceiling fans | 0 | 5 | ||

F4 | Energy source for heating systems | gas | petroleum | ||

F5 | Efficiency of power generation of heating systems | 0.977 | 0.9 | 0.955 | 0.84 |

F6 | Passage of distribution network of heating systems | externally | internally | ||

F7 | Efficiency of terminal units of heating systems | 0.89 | 0.93 | ||

F8 | Type of cooling systems | air cooled | water cooled | ||

F9 | Power of cooling systems (KW) | 6 | 9 | ||

F10 | Efficiency of power generation of cooling systems (EER) | 2.5 | 5.3 | ||

F11 | Efficiency of terminal units of cooling systems | 0.9 | 0.96 | ||

F12 | Recirculation of distribution network (yes or no) | yes | no | ||

F13 | Efficiency of domestic hot water storage system | 1 | 0.98 | 0.98 | 0.93 |

F14 | Type of solar panels | Simple | Vacum | ||

F15 | Surface area of solar panels | 2 | 4 | ||

F16 | Utilization rate of solar radiation for domestic hot water | 0.344 | 0.38 | ||

F17 | Thermal insulation of walls | yes | no | ||

F18 | Installation of awnings | yes | no | ||

F19 | Presence of shutters | yes | no | ||

F20 | Thermal insulation of roof | yes | no | ||

F21 | Type of window’s frame | wooden | metallic | ||

F22 | Air gap between glasses | 6 mm | 12 mm | ||

F23 | Percentage of window frame | 30% | 20% | ||

F24 | Type of exit door | Thermal Insulation | No thermal insulation |

**Table 8.**The response output for energy consumption (EC) in kWh/m

^{2}and its energy classification status.

Run # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Energy Consumption Response | 252.1 | 115.1 | 257.1 | 292 | 299.8 | 268.6 | 227.3 | 144.9 | 157.6 | 247.8 | 230.1 | 105.9 | 321.7 | 194 |

Class | E | B | E | E | Z | E | D | C | C | E | D | B | Z | D |

**Table 9.**Energy demands and consumption results for the first simulation trial according to the supersaturated design schedule.

Energy Demand (kWh/m^{2}) | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Heating | 44.1 | 35.3 | 26.3 | 4.6 | 0 | 0 | 0 | 0 | 0 | 0 | 17.8 | 37.2 | 165.3 |

Cooling | 0 | 0 | 0 | 0 | 1 | 11.7 | 24.3 | 20.4 | 1.5 | 0 | 0 | 0 | 58.9 |

Hot Water | 2.1 | 1.9 | 2.1 | 1.8 | 1.6 | 1.3 | 1.2 | 1.2 | 1.3 | 1.6 | 1.8 | 2 | 19.9 |

Energy Consumption(kWh/m^{2}) | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Total |

Heating | 55.6 | 44.5 | 33.1 | 5.8 | 0 | 0 | 0 | 0 | 0 | 0 | 22.5 | 46.8 | 208.3 |

Cooling | 0 | 0 | 0 | 0 | 0.1 | 1.3 | 2.6 | 2.2 | 0.2 | 0 | 0 | 0 | 6.4 |

Hot Water | 1.7 | 1.5 | 1.5 | 1.2 | 1 | 0.7 | 0.5 | 0.5 | 0.7 | 1 | 1.3 | 1.6 | 13.2 |

Hot Water(from solar) | 0.4 | 0.4 | 0.5 | 0.5 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.5 | 0.4 | 0.3 | 6 |

Total | 57.7 | 46.4 | 35.1 | 7.5 | 1.7 | 2.6 | 3.7 | 3.3 | 1.5 | 1.5 | 24.2 | 48.7 | 233.9 |

Factor/ Setting | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 217.47 | 195.56 | 197.95 | 171.07 | 200.58 | 228.28 | 235.03 | 228.13 | 217.24 | 223.81 | 230.61 | 229.03 |

2 | 227.47 | 249.39 | 246.9 | 273.87 | 244.36 | 216.66 | 209.91 | 216.81 | 227.7 | 221.13 | 214.32 | 215.91 |

Range | 10 | 53.83 | 48.95 | 102.8 | 43.78 | 11.62 | 25.12 | 11.32 | 10.46 | 2.68 | 16.29 | 13.12 |

Rank | 20 | 3 | 5 | 1 | 6 | 15 | 9 | 17 | 19 | 22 | 11 | 13 |

Factor/Setting | F13 | F14 | F15 | F16 | F17 | F18 | F19 | F20 | F21 | F22 | F23 | F24 |

1 | 223.53 | 224.85 | 210.86 | 223.52 | 200.61 | 214.38 | 228.29 | 179.87 | 240.39 | 246.9 | 228.21 | 227.7 |

2 | 221.41 | 220.1 | 234.1 | 221.41 | 244.3 | 230.43 | 216.66 | 265.07 | 204.56 | 198 | 216.73 | 217.23 |

Range | 2.12 | 4.75 | 23.24 | 2.11 | 43.69 | 16.05 | 11.63 | 85.2 | 35.83 | 48.9 | 11.48 | 10.47 |

Rank | 23 | 21 | 10 | 24 | 7 | 12 | 14 | 2 | 8 | 4 | 16 | 18 |

**Table 11.**Stepwise-regression model summary (IBM SPSS v.29) for selecting statistically strong controlling factors.

Model ^{e} | R | R^{2} | Adjusted R^{2} | Std. Error of the Estimate | Change Statistics | Durbin-Watson | ||||
---|---|---|---|---|---|---|---|---|---|---|

R^{2} Change | F Change | df1 | df2 | Sig. F Change | ||||||

1 | 0.775 ^{a} | 0.600 | 0.567 | 45.30 | 0.600 | 18.024 | 1 | 12 | 0.001 | |

2 | 0.943 ^{b} | 0.889 | 0.868 | 24.97 | 0.288 | 28.485 | 1 | 11 | <0.001 | |

3 | 0.966 ^{c} | 0.933 | 0.912 | 20.38 | 0.044 | 6.516 | 1 | 10 | 0.029 | |

4 | 0.987 ^{d} | 0.974 | 0.962 | 13.41 | 0.041 | 14.100 | 1 | 9 | 0.005 | 3.167 |

^{a}Predictors: (Constant), F4;

^{b}Predictors: (Constant), F4, F20;

^{c}Predictors: (Constant), F4, F20, F2;

^{d}Predictors: (Constant), F4, F20, F2, F17;

^{e}Dependent Variable: EC.

**Table 12.**Stepwise-regression model summary coefficients and collinearity statistics (IBM SPSS v.29) for the statistically active controlling factors.

Model ^{a} | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | 95.0% Confidence Interval for B | Collinearity Statistics | ||||
---|---|---|---|---|---|---|---|---|---|---|

B | Std. Error | Beta | Lower Bound | Upper Bound | Tolerance | VIF | ||||

1 | (Constant) | 222.471 | 12.107 | 18.375 | <0.001 | 196.092 | 248.851 | |||

F4 | 51.400 | 12.107 | 0.775 | 4.245 | 0.001 | 25.021 | 77.779 | 1.000 | 1.000 | |

2 | (Constant) | 222.471 | 6.674 | 33.332 | <0.001 | 207.781 | 237.162 | |||

F4 | 46.258 | 6.744 | 0.697 | 6.860 | <0.001 | 31.416 | 61.101 | 0.980 | 1.021 | |

F20 | 35.992 | 6.744 | 0.543 | 5.337 | <0.001 | 21.149 | 50.834 | 0.980 | 1.021 | |

3 | (Constant) | 222.471 | 5.447 | 40.842 | <0.001 | 210.335 | 234.608 | |||

F4 | 39.014 | 6.192 | 0.588 | 6.300 | <0.001 | 25.217 | 52.811 | 0.774 | 1.292 | |

F20 | 39.285 | 5.653 | 0.592 | 6.950 | <0.001 | 26.690 | 51.880 | 0.929 | 1.077 | |

F2 | 15.806 | 6.192 | 0.238 | 2.553 | 0.029 | 2.009 | 29.603 | 0.774 | 1.292 | |

4 | (Constant) | 222.471 | 3.584 | 62.074 | <0.001 | 214.364 | 230.579 | |||

F4 | 35.781 | 4.164 | 0.539 | 8.593 | <0.001 | 26.361 | 45.201 | 0.741 | 1.350 | |

F20 | 38.207 | 3.730 | 0.576 | 10.242 | <0.001 | 29.769 | 46.646 | 0.923 | 1.083 | |

F2 | 19.039 | 4.164 | 0.287 | 4.572 | 0.001 | 9.619 | 28.459 | 0.741 | 1.350 | |

F17 | 14.007 | 3.730 | 0.211 | 3.755 | 0.005 | 5.569 | 22.446 | 0.923 | 1.083 |

^{a}Dependent Variable: EC.

**Table 13.**Optimal settings for several key controlling factors (Key Settings). Combination solution for all controlling factors (Full Settings).

Key Settings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Factors | Natural Gas Boiler(−) | Petroleum Boiler(−) | Natural Gas Boiler(+) | Petroleum Boiler(+) | |||||||

1 | Automation for hot water | no | yes | ||||||||

2 | Category of automatic control | A | D | ||||||||

3 | Number of ceiling fans | 0 | 5 | ||||||||

4 | Energy source for heating systems | gas | petroleum | ||||||||

5 | Efficiency of power generation of heating systems | 0.977 | 0.9 | 0.955 | 0.84 | ||||||

6 | Passage of distribution network of heating systems | externally | internally | ||||||||

7 | Efficiency of terminal units of heating systems | 0.89 | 0.93 | ||||||||

8 | Type of cooling systems | Air-cooled | Water-cooled | ||||||||

9 | Power of cooling systems (KW) | 6 | 9 | ||||||||

10 | Efficiency of power generation of cooling systems (EER) | 2.5 | 5.3 | ||||||||

11 | Efficiency of terminal units of cooling systems | 0.9 | 0.96 | ||||||||

12 | Recirculation of distribution network (yes or no) | yes | no | ||||||||

13 | Efficiency of domestic hot water storage system | 1 | 0.98 | 0.98 | 0.93 | ||||||

14 | Type of solar panels | Simple | Vacuum | ||||||||

15 | Surface area of solar panels | 2 | 4 | ||||||||

16 | Utilization rate of solar radiation for domestic hot water | 0.344 | 0.38 | ||||||||

17 | Thermal insulation of walls | yes | no | ||||||||

18 | Installation of awnings | yes | no | ||||||||

19 | Presence of shutters | yes | no | ||||||||

20 | Thermal insulation of roof | yes | no | ||||||||

21 | Type of window’s frame | wooden | metallic | ||||||||

22 | Air gap between glasses | 6 mm | 12 mm | ||||||||

23 | Percentage of window frame | 30% | 20% | ||||||||

24 | Type of exit door | Thermal Insulation | No insulation | ||||||||

Full Settings | |||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

- | - | - | - | - | + | + | + | - | + | + | + |

13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

+ | + | - | + | - | - | + | - | + | + | + | + |

**Table 14.**The proportion of the explained variance for the supersaturated dataset using the PLS method.

X Variance | Cumulative X Variance | Y Variance | Cumulative Y Variance (R^{2}) | Adjusted R^{2} | |
---|---|---|---|---|---|

1 | 0.081 | 0.081 | 0.984 | 0.984 | 0.983 |

2 | 0.063 | 0.144 | 0.014 | 0.998 | 0.998 |

3 | 0.083 | 0.227 | 0.001 | 1.000 | 0.999 |

4 | 0.074 | 0.301 | 0.000 | 1.000 | 1.000 |

5 | 0.033 | 0.333 | 4.570 × 10^{−5} | 1.000 | 1.000 |

**Table 15.**The factorial coefficients and the variable importance (latent factors) in the projection for the supersaturated dataset using the PLS method.

Controlling Factors | Latent Factors * | |||||
---|---|---|---|---|---|---|

PLS Coefficients | 1 | 2 | 3 | 4 | 5 | |

F1 | 3.208 | 0.268 | 0.270 | 0.270 | 0.271 | 0.271 |

F2 | 12.227 | 1.443 | 1.436 | 1.436 | 1.436 | 1.436 |

F3 | 13.633 | 1.315 | 1.308 | 1.307 | 1.307 | 1.307 |

F4 | 26.549 | 2.756 | 2.738 | 2.736 | 2.736 | 2.735 |

F5 | 12.178 | 1.174 | 1.170 | 1.169 | 1.169 | 1.169 |

F6 | −1.606 | 0.312 | 0.313 | 0.317 | 0.318 | 0.318 |

F7 | −7.509 | 0.673 | 0.669 | 0.669 | 0.670 | 0.670 |

F8 | −4.134 | 0.303 | 0.324 | 0.324 | 0.324 | 0.324 |

F9 | 0.856 | 0.280 | 0.312 | 0.313 | 0.313 | 0.313 |

F10 | −1.585 | 0.072 | 0.102 | 0.137 | 0.145 | 0.145 |

F11 | −6.211 | 0.437 | 0.452 | 0.454 | 0.454 | 0.454 |

F12 | 0.563 | 0.352 | 0.469 | 0.470 | 0.470 | 0.470 |

F13 | 0.491 | 0.057 | 0.095 | 0.098 | 0.107 | 0.107 |

F14 | −3.333 | 0.128 | 0.188 | 0.192 | 0.193 | 0.193 |

F15 | 4.502 | 0.623 | 0.636 | 0.636 | 0.636 | 0.636 |

F16 | 0.491 | 0.057 | 0.095 | 0.098 | 0.107 | 0.107 |

F17 | 10.085 | 1.172 | 1.164 | 1.166 | 1.167 | 1.167 |

F18 | 3.653 | 0.427 | 0.427 | 0.428 | 0.428 | 0.428 |

F19 | −2.059 | 0.312 | 0.313 | 0.313 | 0.313 | 0.313 |

F20 | 25.524 | 2.284 | 2.281 | 2.280 | 2.280 | 2.280 |

F21 | −7.199 | 0.961 | 0.968 | 0.967 | 0.967 | 0.967 |

F22 | −13.994 | 1.312 | 1.306 | 1.306 | 1.305 | 1.305 |

F23 | −3.650 | 0.308 | 0.308 | 0.307 | 0.308 | 0.308 |

F24 | −1.843 | 0.281 | 0.284 | 0.284 | 0.284 | 0.284 |

Model ^{a} | Sum of Squares | df | Mean Square | F | Sig. | |
---|---|---|---|---|---|---|

1 | Regression | 36,987.440 | 1 | 36,987.440 | 18.024 | 0.001 ^{b} |

Residual | 24,625.969 | 12 | 2052.164 | |||

Total | 61,613.409 | 13 | ||||

2 | Regression | 54,752.927 | 2 | 27,376.463 | 43.895 | <0.001 ^{c} |

Residual | 6860.482 | 11 | 623.680 | |||

Total | 61,613.409 | 13 | ||||

3 | Regression | 57,459.467 | 3 | 19,153.156 | 46.108 | <0.001 ^{d} |

Residual | 41,53.941 | 10 | 415.394 | |||

Total | 61,613.409 | 13 | ||||

4 | Regression | 59,994.975 | 4 | 14,998.744 | 83.407 | <0.001 ^{e} |

Residual | 1618.433 | 9 | 179.826 | |||

Total | 61,613.409 | 13 |

^{a}Dependent Variable: EC;

^{b}Predictors: (Constant), F4;

^{c}Predictors: (Constant), F4, F20;

^{d}Predictors: (Constant), F4, F20, F2;

^{e}Predictors: (Constant), F4, F20, F2, F17.

**Table 17.**Summary statistics of the supersaturated dataset (median (M), interquartile range (I), skewness (S), kurtosis (K)) per factorial setting, and their hierarchical cluster identification.

Factor | Level | M | I | S | K | Cluster ID |
---|---|---|---|---|---|---|

F1 | 1 | 227.3 | 111.0 | 0.11 | −1.06 | 1 |

2 | 252.1 | 176.3 | −0.81 | −0.95 | 1 | |

F2 | 1 | 194.0 | 152.9 | 0.15 | −1.48 | 1 |

2 | 252.1 | 61.9 | −0.95 | 2.04 | 2 | |

F3 | 1 | 227.3 | 141.4 | −0.19 | −1.71 | 1 |

2 | 252.1 | 105.8 | −0.67 | −0.04 | 1 | |

F4 | 1 | 157.6 | 111.6 | 0.35 | −1.38 | 1 |

2 | 268.6 | 52.0 | 0.19 | −1.02 | 2 | |

F5 | 1 | 194.0 | 107.2 | 0.12 | −1.59 | 1 |

2 | 257.1 | 72.5 | −1.36 | 2.59 | 2 | |

F6 | 1 | 230.1 | 63.1 | −0.57 | 1.59 | 2 |

2 | 247.8 | 147.1 | −0.36 | −2.03 | 1 | |

F7 | 1 | 230.1 | 98.0 | −0.48 | −0.26 | 1 |

2 | 247.8 | 152.9 | −0.15 | −1.74 | 1 | |

F8 | 1 | 257.1 | 134.4 | −0.52 | −1.01 | 1 |

2 | 230.1 | 107.2 | −0.67 | −0.44 | 1 | |

F9 | 1 | 227.3 | 99.5 | −0.14 | −1.31 | 1 |

2 | 252.1 | 184.1 | −0.72 | −1.12 | 1 | |

F10 | 1 | 230.1 | 142.2 | −0.11 | −1.03 | 1 |

2 | 252.1 | 123.7 | −1.01 | −0.43 | 1 | |

F11 | 1 | 247.8 | 154.9 | −0.55 | −0.91 | 1 |

2 | 230.1 | 111.0 | −0.64 | −0.55 | 1 | |

F12 | 1 | 252.1 | 147.1 | −0.86 | −0.88 | 1 |

2 | 227.3 | 99.5 | −0.17 | −0.04 | 1 | |

F13 | 1 | 230.1 | 111.0 | 0.24 | −0.76 | 1 |

2 | 252.1 | 176.3 | −0.84 | −1.09 | 1 | |

F14 | 1 | 252.1 | 154.9 | −0.30 | −2.11 | 1 |

2 | 230.1 | 63.1 | −1.61 | 2.72 | 2 | |

F15 | 1 | 227.3 | 107.2 | 0.00 | 0.00 | 1 |

2 | 257.1 | 134.4 | −1.07 | −0.32 | 1 | |

F16 | 1 | 230.1 | 111.0 | 0.24 | −0.76 | 1 |

2 | 252.1 | 176.3 | −0.84 | −1.09 | 1 | |

F17 | 1 | 194.0 | 176.3 | 0.35 | −1.55 | 1 |

2 | 252.1 | 41.3 | −1.27 | 2.74 | 2 | |

F18 | 1 | 247.8 | 112.2 | −0.59 | −0.86 | 1 |

2 | 230.1 | 134.4 | −0.51 | −0.70 | 1 | |

F19 | 1 | 252.1 | 142.2 | −0.33 | −1.20 | 1 |

2 | 230.1 | 123.7 | −0.87 | −0.41 | 1 | |

F20 | 1 | 157.6 | 132.1 | 0.16 | −2.25 | 1 |

2 | 268.6 | 72.5 | −0.46 | −0.59 | 2 | |

F21 | 1 | 252.1 | 98.0 | −0.55 | −0.56 | 1 |

2 | 227.3 | 152.9 | 0.04 | −1.69 | 1 | |

F22 | 1 | 247.8 | 64.7 | −0.36 | 0.88 | 2 |

2 | 194.0 | 152.9 | 0.06 | −2.07 | 1 | |

F23 | 1 | 247.8 | 74.6 | −1.31 | 1.86 | 2 |

2 | 230.1 | 154.9 | −0.06 | −1.73 | 1 | |

F24 | 1 | 257.1 | 147.1 | −0.41 | −1.46 | 1 |

2 | 230.1 | 94.5 | −0.82 | 0.30 | 1 |

**Table 18.**Spearman’s ρ correlation coefficients and their respective 95% confidence intervals for median(M), interquartile range (I), skewness (S), and kurtosis (K) of the supersaturated dataset.

Spearman’s ρ | Significance(2-tailed) | 95% Confidence Intervals (2-tailed) ^{a,b} | ||
---|---|---|---|---|

Lower | Upper | |||

M–I | −0.054 | 0.716 | −0.341 | 0.242 |

M–S | −0.579 | <0.001 | −0.745 | −0.346 |

M–K | 0.280 | 0.054 | −0.013 | 0.529 |

I–S | 0.190 | 0.195 | −0.108 | 0.457 |

I–K | −0.723 | <0.001 | −0.838 | −0.546 |

S–K | −0.627 | <0.001 | −0.777 | −0.410 |

^{a}Estimation is based on Fisher’s r-to-z transformation.

^{b}Estimation of standard error is based on the formula proposed by Fieller, Hartley, and Pearson.

**Table 19.**Auto-clustering of the summarized supersaturated datasets (Table 17) using the Schwarz’s Bayesian Criterion (BIC) (IBM SPSS v.29).

Number of Clusters | Schwarz’s Bayesian Criterion (BIC) | BIC Change ^{a} | Ratio of BIC Changes ^{b} | Ratio of Distance Measures ^{c} |
---|---|---|---|---|

1 | 162.043 | |||

2 | 154.178 | −7.865 | 1.000 | 1.378 |

3 | 156.959 | 2.781 | −0.354 | 1.864 |

4 | 172.807 | 15.849 | −2.015 | 1.365 |

5 | 192.696 | 19.889 | −2.529 | 1.456 |

6 | 216.057 | 23.361 | −2.970 | 2.013 |

7 | 243.246 | 27.189 | −3.457 | 1.072 |

8 | 270.690 | 27.444 | −3.489 | 1.090 |

9 | 298.424 | 27.734 | −3.526 | 1.082 |

10 | 326.404 | 27.980 | −3.558 | 1.355 |

11 | 355.168 | 28.764 | −3.657 | 1.551 |

12 | 384.715 | 29.547 | −3.757 | 1.022 |

13 | 414.294 | 29.578 | −3.761 | 1.074 |

14 | 443.968 | 29.674 | −3.773 | 1.126 |

15 | 473.787 | 29.819 | −3.791 | 1.214 |

^{a}The changes are from the previous number of clusters in the table.

^{b}The ratios of changes are relative to the change for the two-cluster solution.

^{c}The ratios of distance measures are based on the current number of clusters against the previous number of clusters.

**Table 20.**Hierarchical clustering and combined statistics for the four individual summarizing estimators (IBM SPSS v.29) from Table 17.

HIERARCHICAL | M | I | S | K | |
---|---|---|---|---|---|

1 | N | 39 | 39 | 39 | 39 |

Mean | 231.782 | 133.336 | −0.3441 | −1.0326 | |

Std. Error of Mean | 4.0194 | 4.1793 | 0.06563 | 0.10261 | |

2 | N | 9 | 9 | 9 | 9 |

Mean | 250.478 | 62.856 | −0.8556 | 1.4233 | |

Std. Error of Mean | 4.6410 | 3.5568 | 0.19677 | 0.46687 | |

Total | N | 48 | 48 | 48 | 48 |

Mean | 235.288 | 120.121 | −0.4400 | −0.5721 | |

Std. Error of Mean | 3.5261 | 5.2895 | 0.07009 | 0.18282 |

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## Share and Cite

**MDPI and ACS Style**

Zarkadas, S.; Besseris, G.
Using Lean-and-Green Supersaturated Poly-Factorial Mini Datasets to Profile Energy Consumption Performance for an Apartment Unit. *Processes* **2023**, *11*, 1825.
https://doi.org/10.3390/pr11061825

**AMA Style**

Zarkadas S, Besseris G.
Using Lean-and-Green Supersaturated Poly-Factorial Mini Datasets to Profile Energy Consumption Performance for an Apartment Unit. *Processes*. 2023; 11(6):1825.
https://doi.org/10.3390/pr11061825

**Chicago/Turabian Style**

Zarkadas, Spyridon, and George Besseris.
2023. "Using Lean-and-Green Supersaturated Poly-Factorial Mini Datasets to Profile Energy Consumption Performance for an Apartment Unit" *Processes* 11, no. 6: 1825.
https://doi.org/10.3390/pr11061825