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Analysis of flight delay and causal factors is crucial in maintaining airspace efficiency and safety. However, delay samples are not independent since they always show a certain aggregation pattern. Therefore, this study develops a novel spatial analysis approach to explore the delay and causal factors which is able to take dependence and the possible problem involved including error correlation and variable lag effect of causal factors on delay into account. The study first explores the delay aggregation pattern by measuring and quantifying the spatial dependence of delay. The spatial error model (SEM) and spatial lag model (SLM) are then established to solve the error correlation and the variable lag effect, respectively. Results show that the SEM and SLM achieve better fit than ordinary least square (OLS) regression, which indicates the effectiveness of considering dependence by employing spatial analysis. Moreover, the outcomes suggest that, aside from the well-known weather and flow control factors, delay-reduction strategies also need to pay more attention to reducing the impact of delay at the previous airport.

With the rapid development of the civil aviation industry, airspace has become increasingly crowded. This crowdedness causes increasingly frequent delays in most major airports worldwide. This situation seriously affects airports, airlines, and passengers. From 2007 to 2017, the annual flights in China consistently increased from 3.65 million to 10.83 million, with an average increasing rate of approximately 12.2% in the past five years. Meanwhile, the rate of flights arriving on time decreased from 83.19% in 2007 to 71.67% in 2017. The annual cost of flight delays in China was estimated to be more than $7.4 billion. Such high economic costs of delay necessitate delay causal factor analysis and delay-reduction strategies.

Several approaches have been taken to analyze the factors that affect flight arrival and departure delay. Allan et al. [

In addition to traditional statistical methods, machine learning algorithms were used by several studies. Bayesian network was a commonly used approach to establish delay model to explore the delay propagation mode and estimate delay [

However, delays show a certain aggregation pattern in the temporal dimension; high delays are normally clustered; and low delays tend to be surrounded by low delays. In other words, the delay value of samples with shorter distance between them is normally similar compared to the delay value of delays with longer distance between them. The correlation between two delay values depends on their spatial attribute such as spatial location and spatial distance. Without doubt, there is high degree of spatial dependence among delays in a space organized by hour and by day. Given that most of the aforementioned methods were based on certain assumptions which either ignore or simplify the correlation of samples in the dataset, Diana [

Actually, flight departure delay is a complex problem with substantial direct causal factors and many concealed indirect causal factors. Flight departure delay is caused by the abovementioned factors, as well as by the flight delays that occur earlier [

Motivated by the exploration of the main causal factors of flight departure delays in consideration of correlation between delay samples, our study analyzes departure delay as a geographic problem instead of a statistical problem by assuming delay as a spatially distributed variable organized by hour and by day. Causal factor analysis using spatial analysis enables the existence of spatial dependence in variables, which solves the problem of sample correlations among hours and days simultaneously. Specifically, spatial regression models were built to absorb the delay spatial dependence by adding a spatial independent variable. The spatial lag model (SLM) and spatial error model (SEM) are established in our study to solve the variable lagged effect and the error correlation, respectively. Comparisons between the SLM, the SEM, and the OLS estimation are also conducted.

This paper is structured as follows. Section

This study employs the spatial analysis method to explore the delay distribution pattern and causal factors of flight departure delays while considering delay spatial dependence. Delay is assumed to be a spatially distributed variable. Spatial analysis is a quantifying technique used in the study of spatial variables [

Moran’s

Most of the spatial weight matrices are built based on spatial connectivity and spatial distance. The weight matrix in this study is generated based on distance measured by the inverse Euclidean distance between two hour units. The value of Moran’s

When the

The cluster type in the flight departure delay is then identified, and the hot and cold spots of flight departure delay are explored.

After the identification of delay dependence, causal factor analysis is performed using spatial analysis, which enables the existence of spatial dependence in variables. To explore the causal factors of flight departure delay, spatial econometric models were built to absorb the delay spatial dependence by adding a spatial independent variable, and the outcomes of the SLM, SEM, and classical regression model are compared.

The data in this study are obtained from the database of an international hub airport in China in June 2016. To maintain the privacy of the institution, the name of the airport is not revealed. In June 2016, 8788 flights departed from the target airport, among which 18 flights returned, 51 flights were canceled, and 5357 flights (60.96%) were delayed for more than 15 minutes; 3180 flights (36.19%) were delayed for more than half an hour; 1528 flights (17.39%) were delayed for more than one hour; and 489 flights (5.56%) were delayed for more than two hours. The most severe delay lasted for 888 minutes. Approximately 70% of the delays were within 60 minutes. The data are organized by day of week and hour of day. To demonstrate the spatial dependence of delay distribution intuitively, the study assumed delay as a spatially distributed variable. The space is defined with day of week as the x coordinate and hour of day as the y coordinate. Compared with the total number of flights (8788), there were few flights (72) from 0:00 to 7:00, and hour units with less than five flights are not considered since it could bias the average. The study area covers 7:00 to 24:00, including a total of 510 hour units with departure delays.

First step of variable construction is to find out factors affecting flight departure delay. The flight delay determinants considered in previous studies include weather, delay propagation, flight schedule, airplane shortage, air route, airplane type, flight order, air traffic flow, hub airport, ability of the airline to pay debt, ability of the airline to profit, load factors of the airline, load rate of the airline, and other factors [

Then, nominal factors are selected by calculating the frequency and the effect of each factor in our dataset. Effect of each factor of flight delay in Table

Frequency and effect of each factor of flight departure delay.

Factors of departure delay | Frequency | Rank of frequency | Average delay minutes | Rank of average delay minutes |
---|---|---|---|---|

T | 21 | 7 | 206.5 | 4 |

TL | 34 | 5 | 235.8 | 2 |

W | 30 | 6 | 238.6 | 1 |

WL | 161 | 2 | 154.8 | 6 |

WD | 9 | 9 | 176.1 | 5 |

WR | 13 | 8 | 212.2 | 3 |

CF | 650 | 1 | 99.2 | 11 |

CR | 71 | 4 | 126.9 | 8 |

A | 115 | 3 | 134.8 | 7 |

F | 3 | 11 | 110.7 | 10 |

P | 2 | 12 | 98.0 | 12 |

D | 7 | 10 | 118.0 | 9 |

All factors are classified into three categories: high frequency and low effect, low frequency and high effect, and low frequency and low effect. Flow control, airline factor, route restriction, and weather condition at the previous airport caused most of the departure delays; however, these factors can be usually controlled well, and the delay can be eliminated in a short time. The effects of weather conditions at the target airport and en route and the technical failure at the target, previous, and destination airports, although they did not happen often, have dramatic impacts with long departure delays. Airport facility, passenger, and capacity allocation are the minor reasons for flight departure delay, and we will not focus on these factors in the following discussion.

In addition, delay can be related to time period (morning, afternoon, night, and weekday or weekend). The total traffic and passengers are also important factors. Aviation industry experts are interviewed about the limitations of the data collection, and the final list included 15 factors that affected flight departure delays.

We then conducted a stepwise-backwards regression in variable construction and determined a significant level of introduced independent variable as

Descriptive statistics and definitions of the variables used in the model.

Variables | Definition | Mean | s.d. | Min | Max |
---|---|---|---|---|---|

| Total minutes of departure delay | 587.824 | 428.433 | 0 | 2341 |

| Technical failure at the target airport (equals 1 if technical failure occurs) | 0.039 | 0.194 | 0 | 1 |

| Technical failure at the previous airport (equals 1 if technical failure occurs) | 0.065 | 0.246 | 0 | 1 |

| Weather condition at the target airport (equals 1 if weather is adverse) | 0.018 | 0.132 | 0 | 1 |

| Weather condition at the previous airport (equals 1 if weather is adverse) | 0.251 | 0.434 | 0 | 1 |

| Weather condition en route (equals 1 if weather is adverse) | 0.016 | 0.124 | 0 | 1 |

| Flow control (equals 1 if conduct flow control) | 0.571 | 0.496 | 0 | 1 |

| Route restriction (equals 1 if conduct route restriction) | 0.122 | 0.327 | 0 | 1 |

| Scheduled departure traffic | 16.943 | 4.414 | 4 | 27 |

The data are processed with various software. The exploration analysis module in ArcGIS 10.2 is used for the distribution mapping and the 3D trend analysis. The geostatistic module in ArcGIS 10.2 software is adopted to generate the theoretical and empirical semivariograms, as well as the kriging interpolation. The Geoda software is used to develop the spatial econometric models.

Distribution map of departure delay.

3D trend map of departure delay.

As mentioned in Introduction, there exists a high degree of spatial dependence among delays. Moran’s

Autocorrelation statistics for selected variables.

Variables | Coefficient | Observed | Expected | Stddev | Z | P |
---|---|---|---|---|---|---|

| Moran’s | 0.580509 | -0.001965 | 0.001016 | 18.275923 | 0.000000 |

General | 0.010119 | 0.007496 | 0.000000 | 19.158604 | 0.000000 | |

| Moran’s | 0.397390 | -0.001965 | 0.000915 | 13.198725 | 0.000000 |

General | 0.194444 | 0.007496 | 0.000201 | 13.177839 | 0.000000 | |

| Moran’s | 0.161547 | -0.001965 | 0.001021 | 5.116695 | 0.000000 |

General | 0.011565 | 0.007496 | 0.000001 | 5.576605 | 0.000000 | |

| Moran’s | 0.248777 | -0.001965 | 0.000901 | 8.352514 | 0.000000 |

General | 0.142857 | 0.007496 | 0.000260 | 8.398764 | 0.000000 | |

| Moran’s | 0.341630 | -0.001965 | 0.001024 | 10.738782 | 0.000000 |

General | 0.009717 | 0.007496 | 0.000000 | 11.624289 | 0.000000 | |

| Moran’s | 0.532762 | -0.001965 | 0.001018 | 16.761879 | 0.000000 |

General | 0.007859 | 0.007496 | 0.000000 | 14.762860 | 0.000000 | |

| Moran’s | 0.489845 | -0.001965 | 0.001019 | 15.405064 | 0.000000 |

General | 0.007888 | 0.007496 | 0.000000 | 12.831494 | 0.000000 |

Table

The hot and cold spots of delay are explored after the degree of autocorrelation is measured and the hour units of high-value clusters in flight departure delay are identified. A high degree of delay from June 18 to 22 that lasted for 8 hours from 14:00 to 22:00 is noted, as shown by the red area in Figure

Hot and cold spots of departure delay.

After measuring the degree of delay spatial dependence between observations, the variogram is utilized to quantify the spatial dependence based on the theory of regionalized variables. The experimental semivariogram is mapped to quantify the spatial dependence of delays and to provide the spatial structure for the subsequent kriging interpolation.

The following are the key parameters in the semivariogram:

The following are the other two parameters that can be calculated from the three parameters mentioned above:

Theoretical semivariogram is necessary to obtain the spatial structure of delay. The experimental semivariogram generated from limited samples is used to estimate the correlation in the whole area by fitting a theoretical semivariogram to an empirical semivariogram. Different theoretical models are compared in Table

Theoretical model fit comparison of isotropic semivariogram.

Model | Nugget | Sill | Range | RSS | R^{2} |
---|---|---|---|---|---|

Exponential | 66979.4200 | 219070.4200 | 17.6000 | 5.48E+08 | 0.977 |

Spherical | 90023.7000 | 206937.5788 | 14.0439 | 1.12E+09 | 0.955 |

Gaussian | 105523.0000 | 203931.7816 | 11.4025 | 1.35E+09 | 0.947 |

According to results of semivariogram, the low nugget–sill ratio (30.6%) suggests that the variation of delay is mainly caused by autocorrelation (69.4%). In Figure

Isotropic semivariogram of departure delay.

Spatial interpolation allows us to further comprehend the overall situation of the entire study area from a limited number of spatial sample points. We randomly select 10% of the sample dataset as the test set and the remaining 90% as the training set. Spatial autocorrelation undermines the accuracy and effectiveness of some commonly used interpolation methods such as trend surface method or inverse distance weighting (IDW) method. We use the ordinary kriging method to interpolate delays since it can take spatial dependence into account by considering spatial structure obtained by semivariogram. Similar to the IDW method, the ordinary kriging method predicts the value on unmeasured position by generating weights of the surrounding points. IDW generates weights according to the distance between unmeasured position and surrounding points. Different from IDW, kriging method generates weight from the semivariogram, which is developed by considering spatial properties and spatial structure of the data. The interpolation results of the prediction surface are shown in Figure

Prediction surface of delay.

After the generation of prediction surface, it is important to evaluate the interpolation precision, which is conducted by cross-validation. Cross-validation leaves one point out and uses the rest to predict a value at that location. The point is then changed to another in turn, and finally this process is performed for all samples in the dataset. Similar to another typical interpolation method, prediction performance can be evaluated by Mean Error (M) and Root Mean Square Error (RMS). The smaller the RMS, the better. Besides, ordinary kriging has other indicators to evaluate prediction performance, including Average Standard Error (A_Std), which measures the average of the prediction standard errors; Mean Standardized Error (Std_M), whose value should be close to 0; Root Mean Square Standardized Error (Std_RMS), which should be close to 1. A Std_RMS greater than 1 indicates underestimating the variability in the predictions. A Std_RMS less than 1 indicates overestimating the variability in the predictions.

Cross-validation can also be an effective selection approach between different interpolation methods. Comparing the cross-validation results, exponential semivariogram shows the minimum RMS and Std_RMS closest to 1. Therefore, the best result is obtained in this study using the exponential fitting semivariogram for ordinary kriging interpolation (Table

Results of the comparison of cross-validation between different interpolation methods.

Interpolation method | M | RMS | A_Std | Std_M | Std_RMS | |
---|---|---|---|---|---|---|

IDW | 8.0616 | 298.4739 | ||||

Ordinary kriging | Exponential | 1.7808 | 301.0585 | 312.6256 | 0.0040 | 0.9665 |

Spherical | 3.0301 | 307.9240 | 331.4682 | 0.0077 | 0.9312 | |

Gaussian | 2.9745 | 316.4641 | 336.4019 | 0.0075 | 0.9426 |

The regression model is commonly used to analyze the factors of departure delay. First, we perform an ordinary least square (OLS) estimation based on the classical regression model (as in (

Estimation results of delay and causal factors for each model.

Variable | OLS estimation | Spatial lag model | Spatial error model | |||
---|---|---|---|---|---|---|

| | | | | | |

| -189.2450 | 0.0008 | -275.3832 | 0.0000 | 81.4065 | 0.2570 |

| 192.5201 | 0.0066 | 163.5734 | 0.0029 | 197.2891 | 0.0003 |

| 179.6183 | 0.0013 | 150.6237 | 0.0005 | 140.6904 | 0.0006 |

| 1059.5640 | 0.0000 | 699.4476 | 0.0000 | 769.7684 | 0.0000 |

| 247.8793 | 0.0000 | 181.4880 | 0.0000 | 161.2410 | 0.0000 |

| 275.8793 | 0.0187 | 90.3014 | 0.3228 | 147.4748 | 0.1265 |

| 292.8883 | 0.0000 | 139.7609 | 0.0000 | 128.7156 | 0.0000 |

| 205.2254 | 0.0000 | 141.3503 | 0.0002 | 120.0013 | 0.0002 |

| 28.3640 | 0.0000 | 19.8473 | 0.0000 | 28.7060 | 0.0000 |

| 0.5806 | 0.0000 | ||||

| 0.6926 | 0.0000 | ||||

| ||||||

R^{2} | 0.5977 | 0.7928 | 0.7884 | |||

F | 62.0591 | |||||

Log likelihood | -3638.21 | -3539.13 | -3554.8640 | |||

SC | 7332.53 | 7140.61 | 7165.84 | |||

AIC | 7294.43 | 7098.26 | 7127.73 | |||

Likelihood Ratio | 198.1630 | 166.6973 |

Note:

The goodness of fit test can be reflected by R^{2}. R^{2} value is the ratio of the sum of the squares of the regression and the sum of the squares of the total deviations, and it indicates the degree of interpretation of all the explanatory variables to the variation of the dependent variables. The value is between 0 and 1; the closer to 1, the better the estimated regression model fits.

The F test is a joint significance test for multiple coefficients to infer whether the linear relationship between the dependent variable and explanatory variables is significant. The null hypothesis (H_{0}) of the F test is that all the parameters to be estimated are simultaneously zero. The larger the F value, the less likely the null hypothesis.

The

In Table ^{2} value of 0.5977, which indicates that the explanation variables and the dependent variable have relatively significant linear correlation, and the dependent variable can be effectively predicted by the explanation variables.

However, the Moran’s

Test results of OLS residuals’ spatial dependence.

TEST | MI/DF | VALUE | PROB |
---|---|---|---|

Moran’s I (error) | 0.3640 | 11.5242 | 0.0000 |

Lagrange Multiplier (lag) | 1 | 195.3046 | 0.0000 |

Robust LM (lag) | 1 | 68.0750 | 0.0000 |

Lagrange Multiplier (error) | 1 | 127.4078 | 0.0000 |

Robust LM (error) | 1 | 10.1783 | 0.0000 |

Classical regression model fails to reflect the spatial dependence between hour units and the influence of their interactions on the total minutes of departure delay. Therefore, spatial factors are introduced into the regression model, and spatial econometric analysis is necessary. SEM and SLM are built to measure the spatial dependence in error terms and the spatial dependence of delay between the hour units, respectively [

The spatial lag variable and spatial error terms are considered as the explanatory variables because of the spatial effects. The use of the OLS results in a biased and irregular estimation. Therefore, the maximum likelihood estimation method is used in this study. The model selection is based on the value of Log likelihood (Log L), the Akaike information criterion (AIC), and the Schwartz criterion (SC), which are fit statistic measures of the accuracy of the model, as well as the test for goodness of fit (R^{2}). A greater Log likelihood and goodness of fit value and smaller Akaike information and Schwartz criteria indicate a better model fit.

Comparing the estimation results between the OLS estimation, spatial lag model, and spatial error model in Table ^{2} is 0.7918 for the SLM, which is greater than 0.5978 in the OLS estimation and 0.7884 in the SEM. The AIC (7098.26) and SC values (7140.61) of the SLM are both less than the values of the OLS estimation (AIC 7294.43, SC 7332.53) and the SEM (AIC 7127.73, SC 7165.84). Moreover, the SLM and OLS estimations are nested, as with the SEM. Increasing the model parameters must result in high likelihood scores. Therefore, judging the fit of the model based on the log likelihood value is inaccurate. We conduct the likelihood ratio test for both models.

The likelihood ratio test uses a likelihood function to evaluate a simple model and a complex model with parameter constraints. The likelihood ratio is defined as the ratio of the maximum value of the likelihood function under constrained conditions to that under unconstrained conditions. A statistic that obeys the chi-square distribution can be constructed based on the likelihood ratio. The null hypothesis H_{0} is that there is no significant difference in the goodness of fit between model A and model B. The rejection or acceptance of the null hypothesis can be judged based on the constructed chi-square statistic value or

Among the explanatory variables, the effects of the weather condition at the previous airport, the weather condition at the airport of departure, flow control, and the number of scheduled departure flights are the most significant. Moreover, the technical failure at the target airport and the previous airport and the route restriction also significantly affect departure delay. Adverse weather is the primary cause of flight departure delays with harsher influence than flow control. Delay reduction primarily focuses on weather forecasts and dynamically adjusts to weather changes.

Besides, the comparison results in Table

Comparing with the results of causal factors obtained from the previous study, this study also indicates that the effect of weather condition at the target airport on flight delay is much greater than that of other factors. However, this study exhibits an interesting finding that the technical failure and weather condition at the previous airport have a larger effect on departure delays than flow control, which is one of the two most significant factors that affect delays aside from weather condition. This finding suggests that dealing with technical failure and weather prediction at the previous airport is crucial in delay reduction.

This study studied the flight departure delay and its causal factors by developing a novel spatial analysis method, which enables the correlation in data samples. The main conclusion can be presented as below.

First, spatial analysis is confirmed as a useful method in the delay and causal factor analysis in this study. Exploration analysis can intuitively demonstrate the distribution pattern of flight departure delay in the temporal dimension, semivariogram can quantify the spatial structure of the delay, and kriging interpolation allows delay estimation at unmeasured locations.

Besides, the results of the spatial econometrics models achieve better fit performance by taking the spatial dependence into consideration, since the fit of SLM and SEM is better than that of OLS estimation. Results achieved by this study reconfirm the significant effect of the weather condition and technical failure on flight departure delay.

This study also indicates that the weather condition and technical failure at the previous airport significantly affect departure delay. These effects are more significant than the flow control factor, which is regarded as one of the two most important factors that affect delay. This result suggests that delay-reduction strategies must also focus on reducing the impact of delay at the previous airport.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to acknowledge the financial support from the Research and Development Project of Scientific and Technological Cooperation between Sichuan Provincial Colleges and Universities (Grant No. 2019YFSY0024), Key Research and Development Projects of Sichuan Science and Technology Plan Project (Grant No. 2019YFG0050), and National Natural Science Foundation of China (Grant No. U1533203) (Grant No. 61179069).