# Missing Value Imputation of Time-Series Air-Quality Data via Deep Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Datasets

_{2.5}and PM

_{10}values for our experiments on both of the datasets. We split each dataset into two subsets according to a target variable, PM

_{2.5}or PM

_{10}; in total, the four datasets are used for the experiments in this study.

#### 2.2. Imputation Method

#### 2.3. Experimental Details

^{®}Xeon

^{®}Processor E5-2650 v4 machine equipped with 128GB RAM is used to conduct the experiments. The models are trained on a single NVIDIA Titan X GPU with a random seed 42 in an Ubuntu 16.04.6 LTS environment. All the experiments are implemented in the PyTorch 1.7.0 deep learning framework [13] using Python 3.6.10.

#### 2.4. Evaluation Metric

_{2.5}and PM

_{10}vary from zero to over a thousand, the scale-invariant metric can accurately measure the imputation performance of the model. Moreover, even when an observed value is zero, sMAPE can be utilized, in contrast to the mean absolute percentage error, $\mathrm{MAPE}=\frac{1}{B}{\sum}_{i=1}^{B}\frac{|\widehat{y}-y|}{y}\xb7100\%$, one of the commonly used scale-invariant metrics.

#### 2.5. Baseline Models

- Mean substitution (Mean): The missing values are substituted with the average value of the training dataset.
- Spatial average value substitution (SA): We replace the missing values with the average value of the data collected from different locations. The value is calculated as $\widehat{{y}_{i}}=\frac{1}{N}{\sum}_{j=1}^{N}{x}_{i}^{j}(1-{m}_{i}^{j})$, where ${x}_{i}^{j}$ indicates the input data at time step i that is collected at the j-th data collection location.
- Multivariate imputation by chained equations (MICE): We use MICE [3] to impute the missing values. MICE makes multiple imputations using chained equations. MICE is implemented using the FancyImpute library.

## 3. Results

_{10}(9.291 MAE and 31.408 sMAPE). The proposed model consistently outperforms the baselines by a large margin.

_{2.5}test data collected at Guro-gu. In real application, the missing values (gray background) are imputed with the final model prediction (red line).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MAE | Mean absolute error |

sMAPE | Symmetric mean absolute percentage error |

## References

- Wong, C.M.; Vichit-Vadakan, N.; Kan, H.; Qian, Z. Public Health and Air Pollution in Asia (PAPA): A multicity study of short-term effects of air pollution on mortality. Environ. Health Perspect.
**2008**, 116, 1195–1202. [Google Scholar] [CrossRef] [PubMed][Green Version] - Landrigan, P.J. Air pollution and health. Lancet Public Health
**2017**, 2, e4–e5. [Google Scholar] [CrossRef][Green Version] - Van Buuren, S.; Groothuis-Oudshoorn, K. Mice: Multivariate imputation by chained equations in R. J. Stat. Softw.
**2011**, 45, 1–67. [Google Scholar] [CrossRef][Green Version] - Honaker, J.; King, G.; Blackwell, M. Amelia II: A program for missing data. J. Stat. Softw.
**2011**, 45, 1–47. [Google Scholar] [CrossRef] - Che, Z.; Purushotham, S.; Cho, K.; Sontag, D.; Liu, Y. Recurrent neural networks for multivariate time series with missing values. Sci. Rep.
**2018**, 8, 1–12. [Google Scholar] - Luo, Y.; Cai, X.; Zhang, Y.; Xu, J.; Yuan, X. Multivariate time series imputation with generative adversarial networks. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, Montréal, QC, Canada, 3–8 December 2018; pp. 1603–1614. [Google Scholar]
- Luo, Y.; Zhang, Y.; Cai, X.; Yuan, X. E2gan: End-to-End Generative Adversarial Network for Multivariate Time Series Imputation; AAAI Press: Menlo Park, CA, USA, 2019; pp. 3094–3100. [Google Scholar]
- Oreshkin, B.N.; Carpov, D.; Chapados, N.; Bengio, Y. N-BEATS: Neural basis expansion analysis for interpretable time series forecasting. arXiv
**2019**, arXiv:1905.10437. [Google Scholar] - Park, J.; Jo, W.; Cho, M.; Lee, J.; Lee, H.; Seo, S.; Lee, C.; Yang, W. Spatial and Temporal Exposure Assessment to PM
_{2.5}in a Community Using Sensor-Based Air Monitoring Instruments and Dynamic Population Distributions. Atmosphere**2020**, 11, 1284. [Google Scholar] [CrossRef] - Maas, A.L.; Hannun, A.Y.; Ng, A.Y. Rectifier nonlinearities improve neural network acoustic models. In Proceedings of the ICML, Atlanta, GA, USA, 16–21 June 2013; Volume 30, p. 3. [Google Scholar]
- Kim, J.; Kim, T.; Choi, J.H.; Choo, J. End-to-end Multi-task Learning of Missing Value Imputation and Forecasting in Time-Series Data. In Proceedings of the 2020 25th International Conference on Pattern Recognition (ICPR), Milan, Italy, 10–15 January 2021; pp. 8849–8856. [Google Scholar]
- Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Paszke, A.; Gross, S.; Massa, F.; Lerer, A.; Bradbury, J.; Chanan, G.; Killeen, T.; Lin, Z.; Gimelshein, N.; Antiga, L.; et al. PyTorch: An Imperative Style, High-Performance Deep Learning Library. Adv. Neural Inf. Process. Syst. (NeurIPS)
**2019**, 32, 8026–8037. [Google Scholar] - Cao, W.; Wang, D.; Li, J.; Zhou, H.; Li, L.; Li, Y. Brits: Bidirectional recurrent imputation for time series. arXiv
**2018**, arXiv:1805.10572. [Google Scholar]

**Figure 1.**Example of time-series data with missing values $\mathbf{X}$, its corresponding missing value mask $\mathbf{M}$, and time-series data without missing values $\mathbf{Y}$. The slash (/) denotes the missing values. ${x}_{i}\in {\mathbb{R}}^{N}$ is the i-th observation of the target variable collected from N different locations, where $N=3$ for this example.

**Figure 2.**Overview of proposed model. $\mathbf{X}$ and $\widehat{\mathbf{Y}}$ indicate the input and output time-series of the model, respectively.

**Figure 3.**Qualitative time-series imputation results of our method in comparison with baseline models. The results are obtained with PM

_{10}test data collected at Guro-gu. We mark the missing values in the input as gray background. For all the compared models, the non-missing values (white background) are the same as the label. Label denotes the original time-series without missing values. SA indicates the spatial average value imputation method.

**Figure 4.**Cumulative prediction results of the bias, slope, seasonality, and residual blocks. The results are obtained with PM

_{2.5}test data collected at Guro-gu. Label denotes the original time-series without missing values.

**Table 1.**Summarization of two air quality datasets. We report the statistics with the whole dataset including training data, validation data and test data. Stdev. denotes the standard deviation of the data. # locations indicates the number of data collection locations.

Dataset | Mean | Stdev. | # Observations | # Locations | Missing Rate (%) |
---|---|---|---|---|---|

Guro-gu (PM_{2.5}) | 21.931 | 30.593 | 827,051 | 24 | 26.014 |

Guro-gu (PM_{10}) | 34.275 | 47.650 | 827,049 | 24 | 26.027 |

Dangjin-si (PM_{2.5}) | 24.916 | 41.423 | 464,720 | 42 | 28.964 |

Dangjin-si (PM_{10}) | 43.914 | 190.288 | 464,720 | 42 | 28.963 |

**Table 2.**Imputation performances of the proposed method and of other imputation methods on the Guro-gu dataset. Our results show the performance of the proposed method, which achieves the best imputation accuracy.

Target Variable | Metric | Ours | Mean | SA | MICE |
---|---|---|---|---|---|

PM_{2.5} | MAE | 1.170 | 18.634 | 8.972 | 4.825 |

sMAPE | 7.155 | 75.236 | 36.771 | 28.865 | |

PM_{10} | MAE | 2.738 | 30.024 | 17.646 | 9.291 |

sMAPE | 9.385 | 73.259 | 43.464 | 31.408 |

**Table 3.**Imputation performances of proposed method and other imputation methods on the Dangjin-si dataset. The proposed method shows the best imputation accuracy.

Target Variable | Metric | Ours | Mean | SA | MICE |
---|---|---|---|---|---|

PM_{2.5} | MAE | 1.149 | 16.780 | 9.646 | 4.524 |

sMAPE | 9.710 | 81.604 | 52.389 | 34.859 | |

PM_{10} | MAE | 4.664 | 33.521 | 20.465 | 12.279 |

sMAPE | 13.702 | 86.151 | 56.168 | 44.624 |

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## Share and Cite

**MDPI and ACS Style**

Kim, T.; Kim, J.; Yang, W.; Lee, H.; Choo, J. Missing Value Imputation of Time-Series Air-Quality Data via Deep Neural Networks. *Int. J. Environ. Res. Public Health* **2021**, *18*, 12213.
https://doi.org/10.3390/ijerph182212213

**AMA Style**

Kim T, Kim J, Yang W, Lee H, Choo J. Missing Value Imputation of Time-Series Air-Quality Data via Deep Neural Networks. *International Journal of Environmental Research and Public Health*. 2021; 18(22):12213.
https://doi.org/10.3390/ijerph182212213

**Chicago/Turabian Style**

Kim, Taesung, Jinhee Kim, Wonho Yang, Hunjoo Lee, and Jaegul Choo. 2021. "Missing Value Imputation of Time-Series Air-Quality Data via Deep Neural Networks" *International Journal of Environmental Research and Public Health* 18, no. 22: 12213.
https://doi.org/10.3390/ijerph182212213