# An EMD–SARIMA-Based Modeling Approach for Air Traffic Forecasting

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Objective of Paper

#### 1.3. Paper Organization

## 2. Literature Review

#### 2.1. On Single Theory-Based Forecasting Models

#### 2.2. On Hybrid Theory-Based Forecasting Models

#### 2.3. On Specific Models for Air Traffic Forecasting

## 3. The Hybrid EMD–SARIMA Forecasting Framework

#### 3.1. Stage 1: EMD Modeling

- (I)
- Identify all the local extrema (i.e., local maxima and minima) in the original time series x(t).
- (II)
- Interpolate all the local maxima (minima) by a cubic spline from the upper envelope e
_{max}(t) and lower envelope e_{min}(t). - (III)
- Calculate the mean of the envelope m(t) from the upper and lower envelope.$$m(t)=\frac{{e}_{\mathrm{max}}(t)+{e}_{\mathrm{min}}(t)}{2},$$
- (IV)
- Extract the mean of the envelope from the original signal to obtain a new signal h(t).$$h(t)=x(t)-m(t),$$
- (V)
- Check whether h(t) is an IMF: (1) If h(t) is an IMF then set d(t) = h(t) and replace x(t) with the residual r(t) = x(t) − d(t); (2) if h(t) is not an IMF, replace x(t) with z(t) then repeat the steps from II to IV until the following stopping criterion is met in the iterative process. As has been suggested by Huang et al. in literature [39], a typical value of δ is between 0.2 and 0.3.$$SD={\displaystyle \sum _{t=1}^{m}\frac{{[{h}_{i-1}(t)-{h}_{i}(t)]}^{2}}{{[{h}_{i-1}(t)]}^{2}}}\le \delta \text{\hspace{1em}}(i=1,2,\cdots ,m)$$

^{i}(t) is the IMF obtained from EMD:

#### 3.2. Stage 2: SARIMA Modeling

_{t}is a stationary process. The seasonal component is s

_{t}= s

_{t−h}where h is the length of the series and

- (I)
- Test the stationarity of the time series through a unit root test and examine its power spectrum for trend and seasonality.
- (II)
- Do necessary differencing. If there is seasonality and no trend, take a difference of lag h; if there is both a trend and seasonality, do a seasonal difference to the time series and evaluate the trend. If a trend still exists, take the first difference.
- (III)
- Examine the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the differenced time series.
- (IV)
- Estimate the model and examine the residuals, and compare the Akaike information criterion (AIC) and Bayesian information criterion (BIC) to identify a best model if multiple models are tried.

## 4. Experimental Setup

#### 4.1. Data Description

#### 4.2. EMD–SARIMA Modeling

_{t}

_{+1}= x

_{t}is set to serve as a basic benchmarking model to evaluate the overall performance of different models.

## 5. Result Analysis

#### 5.1. Domestic Cargo

- (1)
- The performance of the EMD–SARIMA model is superiorto that of the SARIMA model, the Holt–Winters model, and the naive model, especially for the local maximum and minimum values;
- (2)
- The performance of the Holt–Winters model is the second best, and the performance is close to the SARIMA model; this is consistent with previous research, which claimed that the Holt–Winters model and the SARIMA model are comparable based on situations [41];
- (3)
- The hybrid model improves the forecasting accuracy of the traditional SARIMA model; the MAPEs of the EMD–SARIMA model for the six-month and twelve-month forecasting horizons are 1.42% and 4.64%, respectively, compared with the MAPEs of the SARIMA model for the six-month and twelve-month forecasting horizons, which are 4.62% and 9.94%, respectively;
- (4)
- By comparing with other models, the EMD–SARIMA model forecasts with better accuracy consistently regardless of whether it is single- or multi-step forecasting.

#### 5.2. Domestic Passenger

- (1)
- The EMD–SARIMA model forecasts with the best accuracy, followed by the SARIMA model, the Holt–Winters model, and the naive model, especially for the local maximum and minimum values;
- (2)
- Due to greater seasonality of the domestic passenger data, the performance of the SARIMA model is the second best, superior to the Holt–Winters model. This is again consistent with previous research claiming that the Holt–Winters model and the SARIMA model are comparable based on different time series [41];
- (3)
- Again, the EMD–SARIMA model forecasts with better accuracy consistently regardless of whether it is single- or multi-step forecasting.

#### 5.3. International Cargo

- (1)
- Consistent with the domestic cargo cases, the EMD–SARIMA model forecasts with the best accuracy, followed by the Holt–Winters model, the SARIMA model, and the naive model, especially for the local maximum and minimum values;
- (2)
- Due to greater variability of the international cargo data, the performances of the forecasting models generate larger prediction errors compared with the domestic cargo series;
- (3)
- Again, the EMD–SARIMA model forecasts with better accuracy consistently regardless of whether it is single- or multi-step forecasting.

#### 5.4. International Passenger

- (1)
- Consistent with the domestic passenger cases, the EMD–SARIMA model forecasts with the best accuracy, followed by the SARIMA model, the Holt–Winters model, and the naive model, especially for the local maximum and minimum values;
- (2)
- Due to greater variability of the international passenger data, the performances of the forecasting models generate larger prediction errors compared with the domestic passenger series;
- (3)
- Again, the EMD–SARIMA model forecasts with better accuracy consistently regardless of whether it is single- or multi-step forecasting.

## 6. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Summary of research on air traffic forecasting. SVM: support vector machine; ANN: artificial neural network; EEMD: ensemble empirical mode decomposition.

**Figure 2.**The hybrid EMD–SARIMA modeling framework. IMF: intrinsic mode function; SARIMA: seasonal autoregressive integrated moving average.

**Table 1.**Comparison of different model performances on six- and twelve-month horizons for domestic cargo.

Six-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 5.02 | 15.80 | 14.30 | 21.33 |

MAPE(%) | 1.42 | 4.62 | 4.20 | 6.34 |

Twelve-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 15.94 | 31.90 | 26.02 | 44.02 |

MAPE(%) | 4.64 | 9.94 | 8.33 | 13.5 |

**Table 2.**Comparison of different model performances on six- and twelve-month horizons for domestic passengers.

Six-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 0.89 | 2.22 | 2.46 | 2.47 |

MAPE(%) | 2.84 | 6.82 | 7.75 | 7.56 |

Twelve-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 0.91 | 1.83 | 1.81 | 1.89 |

MAPE(%) | 3.06 | 6.02 | 5.97 | 6.07 |

**Table 3.**Comparison of different model performances on six and twelve-month horizon for international cargo.

Six-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 12.06 | 18.19 | 13.56 | 19.7 |

MAPE(%) | 9.97 | 15.24 | 12.19 | 15.83 |

Twelve-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 8.59 | 11.71 | 12.22 | 12.86 |

MAPE(%) | 7.33 | 9.46 | 9.34 | 9.83 |

**Table 4.**Comparison of different model performances on six- and twelve-month horizons for international passengers.

Six-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 0.15 | 0.25 | 0.31 | 0.23 |

MAPE(%) | 5.45 | 9.10 | 11.33 | 8.09 |

Twelve-Month Horizon Forecast | ||||

EMD-SARIMA | SARIMA | Holt–Winters | Naive | |

MAE | 0.15 | 0.19 | 0.21 | 0.18 |

MAPE(%) | 6.01 | 7.46 | 8.06 | 6.96 |

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**MDPI and ACS Style**

Nai, W.; Liu, L.; Wang, S.; Dong, D.
An EMD–SARIMA-Based Modeling Approach for Air Traffic Forecasting. *Algorithms* **2017**, *10*, 139.
https://doi.org/10.3390/a10040139

**AMA Style**

Nai W, Liu L, Wang S, Dong D.
An EMD–SARIMA-Based Modeling Approach for Air Traffic Forecasting. *Algorithms*. 2017; 10(4):139.
https://doi.org/10.3390/a10040139

**Chicago/Turabian Style**

Nai, Wei, Lu Liu, Shaoyin Wang, and Decun Dong.
2017. "An EMD–SARIMA-Based Modeling Approach for Air Traffic Forecasting" *Algorithms* 10, no. 4: 139.
https://doi.org/10.3390/a10040139