# Airborne Particulate Matter Modeling: A Comparison of Three Methods Using a Topology Performance Approach

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Particulate Matter

#### 1.2. Deep Neural Network

## 2. Elman Recurrent Neural Network

**Figure 1.**Elman recurrent network base diagram [36].

- ${x}_{t}$: input vector.
- ${h}_{t}$: hidden layer vector.
- ${y}_{t}$: output vector.
- W, U and b: parameter matrices and vector.
- ${\sigma}_{h}$ and ${\sigma}_{y}$: activation functions.

## 3. LSTM Recurrent Neural Network

- Input gates control when new information can enter the memory.
- Forget gates control when part of the information is forgotten, allowing the cell to discriminate between important and excessive data, thus leaving room for new data.
- Output gates control when it is used as the result of the memories in the cell.

- ${x}_{t}$: input vector.
- ${f}_{t}$: forget gate activation vector.
- ${i}_{t}$: input/update gate activation vector.
- W: matrix weights implemented.
- ${o}_{t}$: hidden state vector, better known as the LSTM unit output vector.

## 4. GRU Recurrent Neural Network

- ${x}_{t}$: input vector.
- ${h}_{t}$: output vector.
- ${\widehat{h}}_{t}$: candidate gate vector.
- ${r}_{t}$: reset gate vector.
- ${z}_{t}$: update gate vector.
- W, U, and b: parameter matrices and vector.Activation functions
- ${\sigma}_{g}$: sigmoid function.
- ${\varphi}_{h}$: hyperbolic tangent.

## 5. Case Study

#### 5.1. Study Area

#### 5.2. PM Data

#### 5.3. Hardware and Software Used

## 6. Proposed Methods

#### 6.1. Preprocessing Data

#### 6.2. Construction Model

- Input layer: 50 neurons.
- Hidden layer: 250 neurons.
- Output Layer: 1 neuron.

- Bach size. Defines the number of samples with which the network will work before modifying its internal parameters. Since the data are continuous records in a time series, it is necessary to divide them into sections. The network can recognize sections of the behavior using a smaller amount of memory than would be required to process all the records in a single iteration [43]. Another point of interest is the execution time, which with a small batch size tends to increase [44]. It is important to find a balance point between the execution time and the memory used; in other words, with a small batch size, the execution time increases but the error obtained is lower when working with a large batch size and consequently a shorter execution time. The batch size to be used depends on the amount of data used during the training of the network, so that the network recognizes sections continuously and is able to model the continuous behavior to predict a future behavior [43,44,45]. Essentially, the batch size allows the network to divide the problem into sections whose output variables are compared with the expected variables, obtaining an error. From this error, the network is updated and the model is improved [46]. Most of the values used corresponded to the parameter of ${2}^{n}$, as shown in the literature [18]. In contrast, 96 corresponds to the multiple of 12 h represented for the prediction of exceedances exposed in [6,10].
- Optimizer. Methods by which the algorithms are used to readjust their internal parameters, thus improving the learning of the implemented algorithm. For the present work, we used the optimizers Adam and Adamax, which are widely used with time series data, because they employ a stochastic gradient method based on the adaptive estimation of first and second-order moments [15,31,47].
- Number of epochs. Defines the number of times the algorithm will use the entire training data set. During each epoch, each data sample will have the opportunity to update the internal parameters of the network [31,44], in this case, 12 epochs were used, since by using a more significant number of epochs, the network evaluation error remains practically constant with only an occasional, slight variation [6].

#### 6.3. Model Evaluation

## 7. Results and Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Base neuron LSTM [37].

**Figure 3.**Base neuron GRU [39].

**Figure 4.**Map of the distribution of stations belonging to RAMA [41].

**Figure 8.**CC variability obtained from modeling ${\mathrm{PM}}_{2.5}$ particle. (

**a**) Elman neural network using Adam optimizer. (

**b**) Elman neural network using Adamax optimizer. (

**c**) GRU neural network using Adam optimizer. (

**d**) GRU neural network using Adamax optimizer. (

**e**) LSTM neural network using Adam optimizer. (

**f**) LSTM neural network using Adamax optimizer.

**Figure 9.**CC variability in ${\mathrm{PM}}_{10}$ particle behavior modeling. (

**a**) Elman neural network using Adam optimizer. (

**b**) Elman neural network using Adamax optimizer. (

**c**) GRU neural network using Adam optimizer. (

**d**) GRU neural network using Adamax optimizer. (

**e**) LSTM neural network using Adam optimizer. (

**f**) LSTM neural network using Adamax optimizer.

**Figure 10.**RMSE variability obtained from modeling ${\mathrm{PM}}_{2.5}$ particles. (

**a**) Elman neural network using Adam optimizer. (

**b**) Elman neural network using Adamax optimizer. (

**c**) GRU neural network using Adam optimizer. (

**d**) GRU neural network using Adamax optimizer. (

**e**) LSTM neural network using Adam optimizer. (

**f**) LSTM neural network using Adamax optimizer.

**Figure 11.**RMSE variability obtained from modeling ${\mathrm{PM}}_{10}$ particles. (

**a**) Elman neural network using Adam optimizer. (

**b**) Elman neural network using Adamax optimizer. (

**c**) GRU neural network using Adam optimizer. (

**d**) GRU neural network using Adamax optimizer. (

**e**) LSTM neural network using Adam optimizer. (

**f**) LSTM neural network using Adamax optimizer.

**Figure 12.**Processing time variability in modeling particle behavior ${\mathrm{PM}}_{2.5}$. (

**a**) Elman neural network using Adam optimizer. (

**b**) Elman neural network using Adamax optimizer. (

**c**) GRU neural network using Adam optimizer. (

**d**) GRU neural network using Adamax optimizer. (

**e**) LSTM neural network using Adam optimizer. (

**f**) LSTM neural network using Adamax optimizer.

**Figure 13.**Processing time variability in modeling particle behavior ${\mathrm{PM}}_{10}$. (

**a**) Elman neural network using Adam optimizer. (

**b**) Elman neural network using Adamax optimizer. (

**c**) GRU neural network using Adam optimizer. (

**d**) GRU neural network using Adamax optimizer. (

**e**) LSTM neural network using Adam optimizer. (

**f**) LSTM neural network using Adamax optimizer.

**Table 1.**Classification of particulate matter [4].

Representation | Particle Size | Designation |
---|---|---|

${\mathrm{PM}}_{10}$ | ≤10 µm | Coarse material |

${\mathrm{PM}}_{2.5}$ | ≤2.5 µm | Fine material |

${\mathrm{PM}}_{10}$ | ${\mathrm{PM}}_{2.5}$ | ||
---|---|---|---|

MER | TLA | MER | TLA |

11.02% | 13.41% | 13.33% | 14.73% |

Hyperparameters | Changes |
---|---|

Bach size | 96,128,256,512 |

Optimizer | Adam, Adamax |

Number of epochs | 12 |

Neural Network | Optimizer | Batch Size | CC | RMSE | Times(s) |
---|---|---|---|---|---|

Elman | Adam | 512 | 0.7573 | 9.6801 | 224.16 |

GRU | Adamax | 96 | 0.7588 | 9.6332 | 4421.91 |

LSTM | Adamax | 128 | 0.7575 | 9.6449 | 2056.16 |

Neural Network | Optimizer | Batch Size | CC | RMSE | Times(s) |
---|---|---|---|---|---|

Elman | Adam | 512 | 0.7977 | 16.6869 | 256.72 |

GRU | Adam | 96 | 0.8004 | 16.2505 | 2544.90 |

LSTM | Adamax | 128 | 0.7992 | 16.2492 | 2536.40 |

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**MDPI and ACS Style**

Ramírez-Montañez, J.A.; Aceves-Fernández, M.A.; Pedraza-Ortega, J.C.; Gorrostieta-Hurtado, E.; Sotomayor-Olmedo, A.
Airborne Particulate Matter Modeling: A Comparison of Three Methods Using a Topology Performance Approach. *Appl. Sci.* **2022**, *12*, 256.
https://doi.org/10.3390/app12010256

**AMA Style**

Ramírez-Montañez JA, Aceves-Fernández MA, Pedraza-Ortega JC, Gorrostieta-Hurtado E, Sotomayor-Olmedo A.
Airborne Particulate Matter Modeling: A Comparison of Three Methods Using a Topology Performance Approach. *Applied Sciences*. 2022; 12(1):256.
https://doi.org/10.3390/app12010256

**Chicago/Turabian Style**

Ramírez-Montañez, Julio Alberto, Marco Antonio Aceves-Fernández, Jesús Carlos Pedraza-Ortega, Efrén Gorrostieta-Hurtado, and Artemio Sotomayor-Olmedo.
2022. "Airborne Particulate Matter Modeling: A Comparison of Three Methods Using a Topology Performance Approach" *Applied Sciences* 12, no. 1: 256.
https://doi.org/10.3390/app12010256