# Absorption and Utilization of Pollutants in Water: A Novel Model for Predicting the Carrying Capacity and Sustainability of Buildings

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Methodologies

**Layers**: They contain input, hidden, and output layers. The full connection layer was used in this paper. Each node was connected with all nodes of the previous layer, which was used to synthesize the previously extracted features.

**Loss function**: It is used to define the error between a single training sample and the real value. In the process of training, the weight needs to be constantly adjusted. Its purpose is to obtain a set of final weights to make the output of the input characteristic data reach the expected value.

**Optimizer**: The principal function of the optimizer is to guide the parameters of the loss function to update the appropriate size in the correct direction in the process of back propagation, so that the updated value continues to approach the global minimum.

**Activation function**: In the model, the input features act on the activation function after a series of weighted summation. Similar to the neuron-based model in the human brain, the activation function ultimately determines whether to transmit the signal and what to transmit to the next neuron. However, it is not the function with excellent characteristics that is the most suitable. Each function needs to be tested continuously in the model so that the most suitable function is selected. The swish function proved to be the most suitable activation function for the model after many experiments, which has several characteristics: (1) it is unbounded above (avoids saturation), (2) bounded below (strong regularization, especially for large negative numbers), (3) and smooth (accessible to learn and less sensitive to it). The swish function is represented as

**Overfitting**: The model performs too well in the training set, resulting in poor performance in the validation data and testing data, that is, the generalization error is relatively large. From the perspective of variance and deviation, overfitting means high variance and low deviation on the training set. Overfitting is usually caused by an amount of data that is too small, inconsistent data distribution between the training set and validation set, model complexity that is too large, poor data quality, overtraining, and so on. In order to reduce the complexity of the model, the regularization method was used in this paper. There are two regularization methods, L1 and L2:

#### 2.2. Data and Preparation

## 3. Results and Discussions

#### 3.1. Model Performance Indicators

- The MAE measures the average error amplitude, the distance between the predicted value $\widehat{{y}_{i}}$, and the real value ${y}_{i}$, and the action range is 0 to positive infinity. It is more robust to outliers on the line. However, the derivative at point 0 is discontinuous, which makes the solution efficiency lower and the convergence speed slower. For smaller loss values, the gradient is as large as that of other interval loss values.
- The MSE measures the square sum of the distance between the predicted value and the real value, and the scope of action is consistent with the MAE. There is something about the fast convergence speed. It can give an appropriate penalty weight to the gradient instead of the same, so that the direction of the gradient update can be more accurate. The disadvantage is that it is very sensitive to outliers, and the direction of the gradient update is easily dominated by outliers, so it is not robust.
- The HUBER function combines the MAE and MSE and takes their advantages. The principle is to use the MSE when the error is close to 0 and to use the MAE when the error is large.
- The HINGE function cannot be optimized using the gradient descent method, but needs a sub-gradient descent method. It is a proxy function based on a 0–1 loss function.
- The RMSE is the square root of the ratio of the square sum of the deviation between the observed value and the real value to the number of observations, which is more sensitive to outliers.
- ${R}^{2}$ is defined as using the mean as the error benchmark to evaluate whether the prediction error is greater than or less than the mean benchmark error. Generally, it is more effective in a linear model. It cannot fully reflect the prediction ability of the model, and cannot represent the good generalization performance of the model. The specific equations are as follows:$$MAE=\frac{{\displaystyle {\sum}_{i=1}^{n}|\widehat{{y}_{i}}-{y}_{i}|}}{2}$$$$MSE=\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}{(\widehat{{y}_{i}}-{y}_{i})}^{2}}$$$$HUBER=\{\begin{array}{cc}\frac{1}{2}{(\widehat{{y}_{i}}-{y}_{i})}^{2}& for\left|\widehat{{y}_{i}}-{y}_{i}\right|\le \delta \\ \delta \left|\widehat{{y}_{i}}-{y}_{i}\right|-\frac{1}{2}{\delta}^{2}& otherwise\end{array}$$$$HINGE=\mathrm{max}(0,1-t\times \widehat{y})$$$$RMSE=\sqrt{\frac{{\displaystyle {\sum}_{i=1}^{n}{(\widehat{{y}_{i}}-{y}_{i})}^{2}}}{n}}$$$${R}^{2}=\frac{{\displaystyle {\sum}_{i}{(\widehat{{y}_{i}}-\overline{y})}^{2}}}{{\displaystyle {\sum}_{i}{({y}_{i}-\overline{y})}^{2}}}$$

#### 3.2. Model Developments

#### 3.3. Evaluation of Model Performance

#### 3.4. Visual Interpretation of Results

#### 3.5. Comparison with Other Machine Learning Methods

#### 3.5.1. Regression

#### 3.5.2. Ensemble Learning

#### 3.5.3. Score Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Distribution of several parameters constructed using KDE modeling technique (the right figure of each is Gaussian KDE): (

**a**) W/B; (

**b**) SC; (

**c**) ST; (

**d**) AT; (

**e**) A/B; (

**f**) SA; and (

**g**) CSR.

**Figure 6.**Loss curves for the dataset: (

**a**) for five functions of training phase; (

**b**) for MSE; (

**c**) for MAE; (

**d**) for RMSE; (

**e**) for HUBER; and (

**f**) for HINGE training and testing phases, respectively.

**Figure 7.**Prediction results of the deep learning model between actual and predicted results of 48 randomly selected tests: (

**a**) dispersion; (

**b**) value.

Input/Output Variables | Symbol | Unit | Category | Statistics | |||
---|---|---|---|---|---|---|---|

Min | Max | Mean | STD | ||||

Water-to-binder ratio | W/B | / | Input | 0.21 | 0.95 | 0.46 | 0.08 |

Sulfate concentration | SC | mg/L | Input | 0.00 | 30.00 | 6.58 | 3.97 |

Sulfate type | ST | / | Input | 1.00 | 5.00 | 1.74 | 1.03 |

Admixture type | AT | / | Input | 0.00 | 30.00 | 3.16 | 3.89 |

Admixture percentage in relation to the total binder | A/B | % | Input | 0.00 | 100.00 | 15.83 | 20.53 |

Service age | SA | month | Input | 1.00 | 120.00 | 12.71 | 16.99 |

Final change rate of compressive strength | CSR | % | Output | −100.00 | 144.08 | −6.18 | 32.18 |

Layer | Neuron Number | Activation Function |
---|---|---|

Input | 6 | / |

Hidden 1 | 20 | Swish |

Hidden 2 | 15 | Swish |

Hidden 3 | 15 | Swish |

Hidden 4 | 10 | / |

Output | 1 | / |

Hyper Parameters | Batch Size | Learning Rate Base | Learning Rate Decay Rate | Regularization Rate | Training Steps | Exponential Moving Average Decay Rate |
---|---|---|---|---|---|---|

Value | 16 | 0.01 | 0.99 | 0.001 | 300,000 | 0.99 |

Numbers/Variables | W/B | SC | ST | AT | A/B | SA | CSR | Prediction |
---|---|---|---|---|---|---|---|---|

1 | 0.6 | 5.00 | 4 | 0 | 0.00 | 5 | −72.34 | −27.60 |

2 | 0.6 | 5.00 | 4 | 0 | 0.00 | 5 | −62.16 | −27.60 |

3 | 0.50 | 5.00 | 1 | 9 | 29.00 | 12 | −44.23 | −20.49 |

4 | 0.54 | 5.00 | 1 | 1 | 20.00 | 5 | −38.11 | −19.86 |

5 | 0.4 | 5.00 | 2 | 5 | 10.00 | 23 | −28.00 | −24.70 |

6 | 0.54 | 5.00 | 1 | 1 | 20.00 | 5 | −25.70 | −19.86 |

7 | 0.50 | 3.00 | 2 | 8 | 0.05 | 9 | −22.24 | −21.09 |

8 | 0.50 | 5.00 | 4 | 6 | 10.00 | 24 | −22.00 | −27.52 |

9 | 0.50 | 0.50 | 1 | 0 | 0.00 | 4 | −21.00 | −12.45 |

10 | 0.54 | 5.00 | 1 | 1 | 20.00 | 5 | −20.95 | −19.86 |

11 | 0.45 | 10.00 | 1 | 0 | 0.00 | 120 | −18.87 | −22.19 |

12 | 0.46 | 5.00 | 1 | 6 | 40.00 | 4 | −17.50 | −24.05 |

13 | 0.45 | 3.00 | 2 | 2 | 100.00 | 3 | −16.56 | −14.83 |

14 | 0.54 | 5.00 | 1 | 1 | 20.00 | 5 | −16.34 | −19.86 |

15 | 0.55 | 10.00 | 1 | 1 | 20.00 | 12 | −16.00 | −15.50 |

16 | 0.46 | 5.00 | 1 | 6 | 40.00 | 10 | −15.00 | −24.77 |

17 | 0.50 | 5.00 | 4 | 6 | 10.00 | 24 | −14.00 | −27.52 |

18 | 0.45 | 10.00 | 1 | 1 | 20.00 | 12 | −13.00 | −13.39 |

19 | 0.28 | 6.00 | 1 | 0 | 0.00 | 30 | −10.30 | 20.33 |

20 | 0.45 | 3.00 | 2 | 2 | 50.00 | 3 | −9.88 | −14.88 |

21 | 0.45 | 10.00 | 1 | 1 | 20.00 | 12 | −9.70 | −13.39 |

22 | 0.45 | 5.04 | 1 | 0 | 0.00 | 120 | −9.18 | −12.59 |

23 | 0.50 | 0.05 | 1 | 0 | 0.00 | 4 | −7.50 | −4.47 |

24 | 0.40 | 5.00 | 1 | 0 | 0.00 | 2 | −7.29 | −20.26 |

25 | 0.40 | 5.00 | 1 | 0 | 0.00 | 2 | −7.17 | −20.26 |

26 | 0.45 | 1.00 | 1 | 1 | 20.00 | 12 | −7.00 | −6.94 |

27 | 0.45 | 5.00 | 4 | 0 | 0.00 | 12 | −6.52 | 4.51 |

28 | 0.50 | 3.00 | 2 | 1 | 20.00 | 6 | −4.41 | −3.66 |

29 | 0.50 | 3.00 | 2 | 1 | 20.00 | 6 | −4.37 | −3.66 |

30 | 0.50 | 3.00 | 2 | 1 | 20.00 | 6 | −3.45 | −3.66 |

31 | 0.50 | 3.00 | 2 | 1 | 20.00 | 6 | −2.95 | −3.66 |

32 | 0.45 | 5.00 | 4 | 0 | 0.00 | 12 | −2.17 | 4.51 |

33 | 0.60 | 5.00 | 1 | 0 | 0.00 | 6 | −1.27 | 0.95 |

34 | 0.30 | 5.00 | 1 | 0 | 0.00 | 6 | 0.01 | 2.78 |

35 | 0.30 | 5.00 | 1 | 7 | 8.00 | 6 | 1.23 | 1.26 |

36 | 0.50 | 11.70 | 4 | 5 | 40.00 | 10 | 2.00 | 5.48 |

37 | 0.30 | 5.00 | 1 | 5 | 8.00 | 6 | 2.44 | 0.20 |

38 | 0.54 | 5.00 | 1 | 1 | 20.00 | 5 | 6.86 | −19.86 |

39 | 0.52 | 10.00 | 1 | 1 | 27.00 | 5 | 7.53 | 18.85 |

40 | 0.52 | 10.00 | 3 | 1 | 27.00 | 5 | 7.88 | 18.14 |

41 | 0.54 | 5.00 | 1 | 1 | 20.00 | 7 | 9.76 | 20.09 |

42 | 0.45 | 5.00 | 1 | 7 | 8.00 | 6 | 11.67 | 2.20 |

43 | 0.56 | 14.70 | 2 | 0 | 0.00 | 7 | 12.80 | 18.49 |

44 | 0.40 | 5.00 | 4 | 0 | 0.00 | 12 | 15.25 | 7.13 |

45 | 0.5 | 5.00 | 1 | 7 | 10.00 | 18 | 20.00 | 11.45 |

46 | 0.54 | 5.00 | 4 | 1 | 30.00 | 120 | 20.29 | 33.19 |

47 | 0.40 | 5.00 | 4 | 1 | 10.00 | 12 | 24.59 | 22.49 |

48 | 0.40 | 5.00 | 4 | 1 | 30.00 | 12 | 34.04 | 26.99 |

Algorithm | Training Method | Model Definition Space | Optimization Rule | Optimization Rule |
---|---|---|---|---|

Artificial neural networks, logistic regression | Gradient descent | Parameter space | ${\theta}_{t}={\theta}_{t-1}+\Delta {\theta}_{t}$ | $L={\displaystyle \sum _{t}l({y}_{t},f({\theta}_{t}))}$ |

Gradient boosting | Gradient boosting | Function space | ${f}_{t}(x)={f}_{t-1}(x)+\Delta {f}_{t}(x)$ | $L={\displaystyle \sum _{t}l({y}_{t},F({x}_{t}))}$ |

Phase/Model | KNNR | MLPR | DT | ET | XGBT | RF | AB | GB | Bagging |
---|---|---|---|---|---|---|---|---|---|

Total | 0.73 | 0.31 | 0.61 | 0.75 | 0.72 | 0.70 | 0.73 | 0.77 | 0.69 |

Train | 0.23 | 0.12 | 0.35 | 0.29 | 0.52 | 0.50 | 0.18 | 0.36 | 0.42 |

Test | 0.22 | 0.11 | 0.32 | 0.36 | 0.52 | 0.51 | 0.16 | 0.35 | 0.45 |

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**MDPI and ACS Style**

Mei, E.; Yu, K.
Absorption and Utilization of Pollutants in Water: A Novel Model for Predicting the Carrying Capacity and Sustainability of Buildings. *Water* **2023**, *15*, 3152.
https://doi.org/10.3390/w15173152

**AMA Style**

Mei E, Yu K.
Absorption and Utilization of Pollutants in Water: A Novel Model for Predicting the Carrying Capacity and Sustainability of Buildings. *Water*. 2023; 15(17):3152.
https://doi.org/10.3390/w15173152

**Chicago/Turabian Style**

Mei, Enyang, and Kunyang Yu.
2023. "Absorption and Utilization of Pollutants in Water: A Novel Model for Predicting the Carrying Capacity and Sustainability of Buildings" *Water* 15, no. 17: 3152.
https://doi.org/10.3390/w15173152