# Towards Characterization of Indoor Environment in Smart Buildings: Modelling PMV Index Using Neural Network with One Hidden Layer

## Abstract

**:**

## 1. Introduction

#### 1.1. The Novelty and Purpose of the Work

#### 1.2. The Structure of the Paper

## 2. Materials and Methods

#### 2.1. Personalized Thermal Comfort Models

#### The Present Research Gap, Literature Review

#### 2.2. Classic PMV Thermal Comfort Evaluation Model

#### 2.3. Deep Learning or Classic Network Structure

#### 2.4. Data Processing, Network Structure, General Equation, Structure Identification Method

#### 2.4.1. Data Processing—The Mapminmax Method

#### 2.4.2. The Network Structure and its General Equation

- ${\mathrm{norm}}_{\mathrm{mapminmax}}$—data preprocessing operation,
- ${\mathrm{denorm}}_{\mathrm{mapminmax}}$—data postprocessing operation,
- $\mathit{X}$—input data vector,
- ${W}^{\left\{1\right\}}$—matrix of weights of input arguments for the hidden layer,
- ${B}^{\left\{1\right\}}$—column vector of biases for the hidden layer,
- ${s}^{\left\{1\right\}}$—number of neurons in a hidden layer,
- ${\mathit{f}}^{\left\{1\right\}}{}_{\mathrm{activ}}\left(\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}\right)$—hidden layer activation function,
- $\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}$—argument of the hidden layer transfer function, described as:$$\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}={W}^{\left\{1\right\}}\xb7\mathit{X}+{B}^{\left\{1\right\}}$$
- ${W}^{\left\{2\right\}}$—vector of weights of input arguments for the output layer,
- ${\mathrm{b}}_{{1}^{\left\{2\right\}}}$—bias for the output layer,
- ${f}^{\left\{2\right\}}{}_{activ}\left(ar{g}^{\left\{2\right\}}\right)$—output layer activation function,
- $ar{g}^{\left\{2\right\}}$—argument of the output layer transfer function, described as:$$ar{g}^{\left\{2\right\}}={W}^{\left\{2\right\}}\xb7{\mathit{Y}}^{\left\{1\right\}}+{\mathrm{b}}_{1}^{\left\{2\right\}}$$
- ${y}_{\mathrm{NN}}$—output value of the NN.

#### 2.4.3. The Method Identifying Structures with the Best Number of Neurons in a Hidden Layer and the Best Neural Network

- $MainCrit$—main criterion for choosing the best neural network structure,
- ${s}^{\left\{1\right\}}$—number of neurons in the hidden layer,
- $MaxAR{E}_{\mathrm{TEST}}$—Maximum Absolute Relative Error obtained for the testing stage:$$MaxAR{E}_{\mathrm{TEST}}=max\left(\left|\frac{{y}_{i\mathrm{Test}}-{y}_{\mathrm{NN}i\mathrm{Tes}t}}{{y}_{i\mathrm{Test}}}\right|\right)$$
- ${y}_{i\mathrm{Tes}t}$—target for the network in testing stage,
- ${y}_{\mathrm{NN}i\mathrm{Test}}$—output for the network in testing stage.

- $MainCrit$—main criterion for choosing the best neural network structure,
- ${s}_{min}^{\left\{1\right\}}$—the smallest number of neurons in the hidden layer,
- $MaxAR{E}_{\mathrm{TEST}}$—maximum absolute relative error obtained for the testing stage.

- show robustness to changes of initial values of weights and bias in network neurons,
- are characterized by negligible impact of overfitting or underfitting.

#### 2.5. Data for NN and Chosen Learning Specification

#### 2.5.1. Data for NN

- general,
- environment,
- personal characteristics,
- comfort/productivity/satisfaction,
- behaviour,
- personal values,
- model.

- matrix $\mathit{X}$ (inputs) with dimensions of 10,646 × 20, which is a series of input sets of samples assigned to the network learning process,
- the matrix
**Y**(targets) with dimensions of 10,646 × 1, which is a series of PMV index reference output values obtained from the data included in the matrix**X**.

#### 2.5.2. Chosen Learning Specification

- $n$—number of sets for each learning stage: training $\left({\mathit{X}}_{i\mathrm{Tr}},{y}_{i\mathrm{Tr}}\right)$, validation $\left({\mathit{X}}_{i\mathrm{Val}},{y}_{i\mathrm{Val}}\right)$, tests $\left({\mathit{X}}_{i\mathrm{Test}},{y}_{i\mathrm{Test}}\right)$,
- ${y}_{i}$—target for the network-reference PMV index value,
- ${y}_{\mathrm{NN}i}$—output of the network respective to the i-th target, which corresponds to estimated PMV index value.

#### 2.6. Assessment of the Applicability of the Network Structure

#### 2.6.1. Robustness Study Methodology

- less than 1, then one should use the indicator that contains the Sum of Absolute Errors made by the network;
- less than 10, then one should select the indicator that contains the average of the Sum of Absolute Errors made by the network. In this case, it is recommended to check how the structure behaves in terms of the selected indicator from the case ${\left|{y}_{i}\right|}_{\mathrm{max}}\le 1$;
- greater than 10, then one needs to choose an indicator that amplifies network Errors by using the exponentiation operation.

#### 2.6.2. Methodology of Overfitting and Underfitting Study

## 3. Results

#### 3.1. Robustness Study of the Examined Neural Network Structures

#### 3.2. Study of Overfitting and Underfitting of the Examined Neural Network Structures

#### 3.3. Identification of the Best Network Structure and the Best

- the fact described in Section 3.1 that the structures with ${\mathrm{s}}^{\left\{1\right\}}=5,10,11,16,19,21-23,26,36$ are sufficiently insensitive to changes in the initial weights and bias of the neural networks for the network training stage;
- the fact described in Section 3.2 that the impact of underfitting or overfitting is acceptable or has negligible significance in the case of the structure for ${s}^{\left\{1\right\}}=5,\text{}10-14,\text{}16,\text{}19-23,\text{}25,\text{}26,\text{}28,\text{}31,\text{}32,\text{}35,\text{}36$.

#### The Best Identified Neural Network and PMV Index Mathematical Model

^{−5}. This means that slight changes in weights and bias during the training stage improved the learning process for the validation stage. This phenomenon can be observed in Figure 12 as well. Figure 13 also shows changes in momentum (Mu) for each successive learning epoch. The figure shows that the value of momentum decreased with the increase of the learning epochs’ numbers, which means that the learning process was carried out correctly [66].

- the dimensions of the matrix are specified; therefore, this information was noted in their subscripts;
- the elements of the matrix are identified numerical values.$$\begin{array}{l}y=denor{m}_{mapminmax}({f}^{\left\{2\right\}}{}_{activ}({W}_{1\times 5}^{\left\{2\right\}}\\ \phantom{\rule{68pt}{0ex}}\xb7\left({\mathit{f}}^{\left\{1\right\}}{}_{activ}\left({W}_{5\text{}\times \text{}20}^{\left\{1\right\}}\xb7nor{m}_{mapminmax}\left(X\right)+{B}_{5\text{}\times \text{}1}^{\left\{1\right\}}\right)\right)+{\mathrm{b}}_{1}^{\left\{2\right\}}))\end{array}$$

## 4. Discussion

## 5. Conclusions and Future Research Program

- The method presented in the article enables filling the gap identified by the scientific community in comfort studies related to energy consumption in buildings.
- There are two approaches to filling the identified gap in the case of NNs with one hidden layer (Section 2.4.3): the first for the best quality fit of the model (Equation (5)), and the second one takes into account the quality of the fit with the minimum complexity of the NN model (Equation (7)).
- When designing the PMV index using NNs with one hidden layer, it is necessary to perform a robustness study (Section 3.1.) along with an overfitting and underfitting study (Section 3.2). Otherwise, there is a high likelihood (in the analyzed case about 80%) that NN will not be usable.
- NNs with one hidden layer enable PMV index modelling with almost perfect quality of model fit as long as the best structure identification method is used (Section 2.4.3 and Section 2.6).
- The use of the identification method (Section 2.4.3 and Section 2.6) compared to similar studies with NNs with one hidden layer gave better results in each case.
- The method presented in the article (Section 2.4.3 and Section 2.6) makes it possible to formulate the equation (Equations (20) and (A1)) characterizing individual thermal comfort for the object under study in terms of its basic functionality.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Histogram of All Input Data and Reference Output Values (Targets) Assigned to the Learning Process with the Division into Learning Stages

**Figure A1.**The histogram of all input data values assigned to the learning process with the division into learning stages.

**Figure A2.**The histogram of all reference output values (targets) assigned to the learning process with the division into learning stages.

## Appendix B. Results of the Selection of Network Learning Parameters

^{{1}}= 5, that is, for the structure of the network identified as the best.

**Figure A3.**The learning rate selection results for the NN learning process obtained for the range from 0.005 to 0.5. The best learning rate was obtained for the value 0.01.

**Figure A4.**The results of the momentum selection for the NN learning process obtained for the range from 0.1 to 3. The best momentum was obtained for the value of 0.9.

## Appendix C

**Figure A5.**Sum of Absolute Errors obtained for the neural network structure with a certain number of neurons in a hidden layer. Results obtained for the tests stage.

## Appendix D

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**Figure 1.**The course of Relative Errors made by the network for the best possible structure (Equation (5)) of feedforward type with one hidden layer. Results obtained for the case without learning data selection.

**Figure 2.**The general structure of a feedforward neural network with one hidden layer. All symbols are explained in the text.

**Figure 3.**The algorithm for identification of the best possible value of ${s}^{\left\{1\right\}}$ in the feedforward neural network with one hidden layer model. On the left side: the parent procedure P1; on the right side: the nested procedure P2.

**Figure 4.**An example of original data (targets) distribution for training, validation and tests stage drawn for NN input arguments ${x}_{1}$, ${x}_{2}$.

**Figure 5.**Maximum Absolute Relative Error obtained for the neural network structure with a certain number of neurons in a hidden layer. Results obtained for the validation stage.

**Figure 6.**Sum of Absolute Errors ($SAE$) obtained for the neural network structure with a certain number of neurons in a hidden layer. Results obtained for the validation stage.

**Figure 7.**Mean Absolute Error obtained for the neural network structure with a certain number of neurons in a hidden layer. Results obtained for the validation stage.

**Figure 8.**Sum of Absolute Errors obtained for the neural network structure with a certain number of neurons in a hidden layer. Results obtained for the tests stage.

**Figure 9.**Sum of Absolute Errors obtained for the neural network structure for the range $3>{s}^{\left\{1\right\}}\le 50$. Results obtained for the tests stage.

**Figure 10.**Mean Absolute Error obtained for the neural network structure for the range $3>{s}^{\left\{1\right\}}\le 50$. Results obtained for the tests stage.

**Figure 11.**Maximum Absolute Relative Error obtained for the neural network structure with the following numbers of neurons in a hidden layer ${s}^{\left\{1\right\}}=5,\text{}10,\text{}11,\text{}16,\text{}19,\text{}21-23,\text{}26,\text{}36$. Results obtained for the tests stage.

**Figure 12.**Performance function values obtained during the learning process for the best analyzed neural network.

**Figure 13.**Gradient, momentum, validation check values obtained during the learning process for the best analyzed neural network.

**Figure 14.**Error histograms obtained during the learning process for the best analyzed neural network.

**Figure 15.**Relative Error histograms with 40 bins obtained during the learning process for the best analyzed neural network.

**Figure 16.**Relative Error in accordance with measured sample number obtained during the learning process for the best analyzed neural network.

**Figure 17.**Correlation plots between a network’s outputs and targets for all learning stages separately and combined, obtained for the best analyzed neural network.

**Figure 18.**An example of the outputs of the network ${y}_{\mathrm{NN}i}$ on the background of ${y}_{i}$ (targets) drawn for the input arguments ${x}_{1}$ and ${x}_{2}$ (see Table 1).

Input Argument | Category | Name | Type | Units (if Applicable) | Range of the Variable |
---|---|---|---|---|---|

${x}_{1}$ | Environment | Indoor ambient temp. | Continuous | °C | [16.76, 25.79] |

${x}_{2}$ | Environment | Indoor relative humidity | Continuous | % | [14.25, 72.57] |

${x}_{3}$ | Environment | Indoor air velocity | Continuous | m/s | [0.026, 0.031] |

${x}_{4}$ | Environment | Indoor mean radiant temp. | Continuous | °C | [16.76, 25.79] |

${x}_{5}$ | Environment | Indoor CO2 | Continuous | ppm | [1.2, 876.7] |

${x}_{6}$ | Environment | Outdoor ambient temp. | Continuous | °C | [–11, 35] |

${x}_{7}$ | Environment | Outdoor relative humidity | Continuous | % | [21, 100] |

${x}_{8}$ | Environment | Outdoor air velocity | Continuous | m/s | [0, 12.5] |

${x}_{9}$ | Personal characteristics | Clothing level | Continuous | CLO | [0.22, 0.99] |

${x}_{10}$ | Personal characteristics | Clothing level (+ chair) | Continuous | CLO | [0.32, 1.09] |

${x}_{11}$ | Personal characteristics | Gender | Discrete | -- | 2 |

${x}_{12}$ | Personal characteristics | Age | Discrete | Years | 32 |

${x}_{13}$ | Personal characteristics | Office type | Discrete | -- | 3 |

${x}_{14}$ | Personal characteristics | Floor number | Discrete | -- | 1 |

${x}_{15}$ | Behavior | Current thermostat cooling setpoint | Continuous | °C | [15.95, 24.13] |

${x}_{16}$ | Behavior | Base thermostat cooling setpoint | Continuous | °C | [23.88, 25.55] |

${x}_{17}$ | Behavior | Current thermostat heating setpoint | Continuous | °C | [22.50, 26.66] |

${x}_{18}$ | Behavior | Base thermostat heating setpoint | Continuous | °C | [15.55, 24.44] |

${x}_{19}$ | General | Occupancy 1 | Discrete | -- | [0, 1] |

${x}_{20}$ | General | Occupancy 2 | Discrete | -- | [0, 1] |

Output Value | Category | Name | Type | Units (If Applicable) |
---|---|---|---|---|

$y$ | MODEL | Predicted Mean Vote (PMV) | Continuous | Limited to [−3,3] |

Learning Parameter | Value |
---|---|

performance function goal | 0 |

minimum performance gradient | 10^{−10} |

maximum validation failures | 12 |

maximum number of epochs to train | 100,000 |

learning rate | 0.01 |

momentum | 0.9 |

**Table 4.**Maximum Absolute Relative Error obtained for the tests stage for neural network structures with ${s}^{\left\{1\right\}}=5$.

$\mathit{M}\mathit{a}\mathit{x}\mathit{A}\mathit{R}{\mathit{E}}_{\mathit{T}\mathit{E}\mathit{S}\mathit{T}}$ | |
---|---|

${s}^{\left\{1\right\}}$ | 5 |

$approach1$ | 0.106 |

$approach2$ | 0.029 |

$approach3$ | 0.025 |

$approach4$ | 0.043 |

$\mathit{a}\mathit{p}\mathit{p}\mathit{r}\mathit{o}\mathit{a}\mathit{c}\mathit{h}\mathbf{5}$ | 0.018 |

$approach6$ | 0.070 |

$approach7$ | 0.047 |

$approach8$ | 0.024 |

$approach9$ | 0.031 |

$approach10$ | 0.072 |

${R}^{2}$ | Value |

Training stage | 0.99998 |

Validation stage | 0.99999 |

Testing stage | 0.99999 |

All data | 0.99998 |

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## Share and Cite

**MDPI and ACS Style**

Dudzik, M.
Towards Characterization of Indoor Environment in Smart Buildings: Modelling PMV Index Using Neural Network with One Hidden Layer. *Sustainability* **2020**, *12*, 6749.
https://doi.org/10.3390/su12176749

**AMA Style**

Dudzik M.
Towards Characterization of Indoor Environment in Smart Buildings: Modelling PMV Index Using Neural Network with One Hidden Layer. *Sustainability*. 2020; 12(17):6749.
https://doi.org/10.3390/su12176749

**Chicago/Turabian Style**

Dudzik, Marek.
2020. "Towards Characterization of Indoor Environment in Smart Buildings: Modelling PMV Index Using Neural Network with One Hidden Layer" *Sustainability* 12, no. 17: 6749.
https://doi.org/10.3390/su12176749