# Study of Flight Departure Delay and Causal Factor Using Spatial Analysis

Authors: Shaowu Cheng; Yaping Zhang; Siqi Hao; Ruiwei Liu; Xiao Luo; Qian Luo
Journal: Journal of Advanced Transportation
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

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. [1] studied several determining causes of flight delay at the Newark International Airport (EWR) using a comprehensive approach. The results show that adverse weather conditions, low ceilings, and low visibility conditions strongly influence flight delays. Similarly, Asfe et al. [2] investigated the major causal factors of flight delays by ranking different factors using the analytical hierarchical process. They found technical failure and delayed entries as two of the most influential factors. Based on the identification of causal factors, further researches explored the quantitative effect of each factor on flight delay. By analyzing the characteristics of flight departure and arrival delays by constructing probability density functions, Mueller et al. [3] explored several causal factors of delays, such as traffic volume, aircraft type, aircraft maintenance, airline operations, weather conditions, change of procedures en route, capacity constraints, customer service issues, and late aircraft or crew arrival. The results show that weather contributed to 69% of the delays. Different results can be achieved by different method and variables; research results of Kwan and Hansen [4] show that airport congestion contributed to approximately 32% of the average delays, in which a series of econometric models was established to identify the key causal factors of flight delays, including airport congestion, total traffic, and en route weather. In addition to identifying the causal factors and their quantitative effect on flight delay, more studies focus on the development of models to determine the probability of aircraft delay. Wesonga et al. [5] proposed and evaluated a multiple parametric approach, which includes the apparently significant meteorological and aviation parameters, to predict the probability of aircraft delay. Recent research and development effort in delay probability prediction are seeking to develop asymmetric Bayesian logit model to take the asymmetric distribution pattern of the dependent variable into consideration (see Perez-Rodriguez et al. [6]). By using data from BTS and IATA, this article corroborates the necessity and superiority of the proposed asymmetric Bayesian logit model, as well as identifying new significant factors affecting the probability of arrival delay.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 [7, 8]. Artificial neural network was also utilized to examine the relationship between departure delay and different causal factors comparing to linear and nonlinear regressions [9]. Deep learning models have also been investigated for air traffic delay prediction tasks [10]. Moreover, a number of studies attempted to determine the major causal factors of flight delays by detecting the time series data trend. Abdel-Aty et al. [11] applied the “two-stage approach” to detect periods of regularly repeating patterns in their data and to identify the factors correlated with them. Tu et al. [12] employed a smoothing spline model to identify the relationship between seasonal trends, random effects, and daily delay propagation pattern. Delay propagation has also been deeply investigated by many researches to help to understand the air congestion [13–15] and alleviate fight delay [16, 17]. The effects of day and time were assumed to be additive, and the residuals were assumed to be identically and independently distributed in the study.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 [18] initially introduced the approach of spatial analysis for delay prediction, which is able to take the spatial dependence in every direction into account. In the study, delay was considered as a spatially distributed variable in a space coordinated by day and time. A spatial error model (SEM) was built to consider spatial dependence in error.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 [12], as the operation resources required by the current flight, such as the crew, aircraft, and passenger gates, might have been utilized by previously delayed flights. This resources correlation may lead to delay daily propagation. The spatial dependence exists in every direction since the aggregation is observed in both day of week and hour of day, which probably lead to error correlation and variable lag effect of causal factors on delay [19].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. Section2 introduces the spatial analysis methodology. Section 3 describes the data sources, defines the variables, and describes the data-processing methodology. Section 4.1 shows the exploration analysis of flight departure delay with a distribution map and trend analysis. Section 4.2 demonstrates identification of delay pattern. Section 4.3 maps the semivariogram to quantify the spatial dependence of flight departure delay. Based on the results of Section 4.3, Section 4.4 illustrates spatial prediction considering spatial autocorrelation by employing ordinary kriging method. Section 4.5 discusses the establishment of the classical regression model, SEM, and SLM, as well as the comparative analysis of the three models to explore the main causal factors of flight departure delays. Finally, this paper is concluded with a summary.


## 2. Methods

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 [20].


### 2.1. Delay Pattern Analysis

(1) Exploring Delay Distribution. The first step to analyze delay pattern is to explore the distribution. By defining a space with x coordinate of day of week and y coordinate of hour, the delay is added to each hour unit as an attribute. The delay distribution can be plotted with different colors as the delay minutes. The time when an intense delay occurred is recognized in the distribution map. 3D trend analysis can be used to visualize the departure delay distribution and trend in the temporal dimension.(2) Identifying the Pattern of Delay. The pattern of delay is then identified by calculating Moran’s I and general G to measure the degree of delay spatial dependence among observations. Positive autocorrelation suggests that the values of the one hour unit and its neighbors are similar. Negative autocorrelation suggests that the values of the one hour unit and its neighbors are different. No autocorrelation suggests that the values are randomly distributed over the space.Moran’sI is calculated as (1)I=∑in∑jnwijxi-x-xj-x-S2∑in∑jnwij(2)S2=1n∑inxi-x-2x-=1n∑i=1nxi(3)Z=1-EIVARIwhere I is the value of Moran’sI, xi is the total minutes of departure delay during the hour uniti, and wij is the spatial weight matrix.Z value is generally used to test Moran’s I value. A test result against the null hypothesis indicates that no spatial autocorrelation exists.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’sI ranges from −1 to 1. Moran’s I identifies the similarity between units with delay and the spatial distribution pattern. However, it cannot distinguish high- from low-value clusters. General G identifies the two different patterns of spatial cluster; it is computed as (4)Gd=∑in∑jnwijdxixj∑in∑jnxixjZ=G-EGVARGWhen theZ value is significant, a general G value that is greater than its average indicates a high-value cluster, as opposed to a general G value that is less than its average. A general G value that is equal to its average indicates no autocorrelation.The cluster type in the flight departure delay is then identified, and the hot and cold spots of flight departure delay are explored.(3) Quantifying Delay Spatial Dependence. After measuring the degree of spatial dependence, the semivariogram is modeled to quantify the departure delay spatial dependence and to analyze its random and structural properties. Departure delay is considered as a regionalized variable because it is correlated with the hour and the day. Structural property indicates the existence of the autocorrelation between the departure delay at location x and at location x+h (h is the distance from x). Semivariance calculates the average difference on departure delays between pairs of hour units at a given interval [9]; it is computed as (5)γh=12Nh∑i=1NhZxi-Zxi+h2,where Zxi is the total minutes of departure delay of location xi; Zxi+h is the total minutes of departure delay of the locations with h distance from xi; and Nh is the number of locations with h distance from xi.(4) Delay Prediction. After the spatial dependence structure of a variable is determined, the measured data can be used to estimate the variable at unmeasured locations. This interpolation method is known as kriging interpolation. Based on the unbiased estimation and the minimum variance principle, the kriging interpolation method can quantify the spatial dependence between the known sample and the estimated point according to the statistical characteristics and spatial variation of the sample.


### 2.2. Causal Factor Analysis

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.(1) Classical Regression Model. A classical regression model can be written as (6)Y=βX+ε,where Y represents the total minutes of departure delay at the target airport and X represents the factor variables. β represents the effect of the independent variables on the dependent variable, and ε is the random error term vector subjected to normal distribution.(2) SEM. SEM is able to consider the spatial dependence in error terms by adding spatial error term as an explanatory variable. The SEM takes the following form:(7)Y=βX+εε=λWε+μ,where Y is the total minutes of departure delay and X is the factor variables. β represents the effect of the independent variables on the dependent variable, ε is the random error term vector, λ is the spatial error coefficient, W is the spatial weight matrix of error term generated based on distance measured by the inverse Euclidean distance between two hour units, and μ is the random error term vector subjected to normal distribution.(3) SLM. SLM is able to consider the spatial autocorrelation in delay variable by adding a spatial lag variable as an explanatory variable. The SLM takes the following form:(8)Y=ρWY+βX+ε,where Y is the total minutes of departure delay and X is the factor variables. β represents the effect of the independent variables on the dependent variable; W is a spatial weight matrix of the dependent variables generated based on distance measured by the inverse Euclidean distance between two hour units; ρ is the spatial regression coefficient, which reflects the effects of the delay in the neighbor hours WY on the delay in one hourY; and ε is the random error term vector subjected to normal distribution.


## 2.1. Delay Pattern Analysis

(1) Exploring Delay Distribution. The first step to analyze delay pattern is to explore the distribution. By defining a space with x coordinate of day of week and y coordinate of hour, the delay is added to each hour unit as an attribute. The delay distribution can be plotted with different colors as the delay minutes. The time when an intense delay occurred is recognized in the distribution map. 3D trend analysis can be used to visualize the departure delay distribution and trend in the temporal dimension.(2) Identifying the Pattern of Delay. The pattern of delay is then identified by calculating Moran’s I and general G to measure the degree of delay spatial dependence among observations. Positive autocorrelation suggests that the values of the one hour unit and its neighbors are similar. Negative autocorrelation suggests that the values of the one hour unit and its neighbors are different. No autocorrelation suggests that the values are randomly distributed over the space.Moran’sI is calculated as (1)I=∑in∑jnwijxi-x-xj-x-S2∑in∑jnwij(2)S2=1n∑inxi-x-2x-=1n∑i=1nxi(3)Z=1-EIVARIwhere I is the value of Moran’sI, xi is the total minutes of departure delay during the hour uniti, and wij is the spatial weight matrix.Z value is generally used to test Moran’s I value. A test result against the null hypothesis indicates that no spatial autocorrelation exists.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’sI ranges from −1 to 1. Moran’s I identifies the similarity between units with delay and the spatial distribution pattern. However, it cannot distinguish high- from low-value clusters. General G identifies the two different patterns of spatial cluster; it is computed as (4)Gd=∑in∑jnwijdxixj∑in∑jnxixjZ=G-EGVARGWhen theZ value is significant, a general G value that is greater than its average indicates a high-value cluster, as opposed to a general G value that is less than its average. A general G value that is equal to its average indicates no autocorrelation.The cluster type in the flight departure delay is then identified, and the hot and cold spots of flight departure delay are explored.(3) Quantifying Delay Spatial Dependence. After measuring the degree of spatial dependence, the semivariogram is modeled to quantify the departure delay spatial dependence and to analyze its random and structural properties. Departure delay is considered as a regionalized variable because it is correlated with the hour and the day. Structural property indicates the existence of the autocorrelation between the departure delay at location x and at location x+h (h is the distance from x). Semivariance calculates the average difference on departure delays between pairs of hour units at a given interval [9]; it is computed as (5)γh=12Nh∑i=1NhZxi-Zxi+h2,where Zxi is the total minutes of departure delay of location xi; Zxi+h is the total minutes of departure delay of the locations with h distance from xi; and Nh is the number of locations with h distance from xi.(4) Delay Prediction. After the spatial dependence structure of a variable is determined, the measured data can be used to estimate the variable at unmeasured locations. This interpolation method is known as kriging interpolation. Based on the unbiased estimation and the minimum variance principle, the kriging interpolation method can quantify the spatial dependence between the known sample and the estimated point according to the statistical characteristics and spatial variation of the sample.


## 2.2. Causal Factor Analysis

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.(1) Classical Regression Model. A classical regression model can be written as (6)Y=βX+ε,where Y represents the total minutes of departure delay at the target airport and X represents the factor variables. β represents the effect of the independent variables on the dependent variable, and ε is the random error term vector subjected to normal distribution.(2) SEM. SEM is able to consider the spatial dependence in error terms by adding spatial error term as an explanatory variable. The SEM takes the following form:(7)Y=βX+εε=λWε+μ,where Y is the total minutes of departure delay and X is the factor variables. β represents the effect of the independent variables on the dependent variable, ε is the random error term vector, λ is the spatial error coefficient, W is the spatial weight matrix of error term generated based on distance measured by the inverse Euclidean distance between two hour units, and μ is the random error term vector subjected to normal distribution.(3) SLM. SLM is able to consider the spatial autocorrelation in delay variable by adding a spatial lag variable as an explanatory variable. The SLM takes the following form:(8)Y=ρWY+βX+ε,where Y is the total minutes of departure delay and X is the factor variables. β represents the effect of the independent variables on the dependent variable; W is a spatial weight matrix of the dependent variables generated based on distance measured by the inverse Euclidean distance between two hour units; ρ is the spatial regression coefficient, which reflects the effects of the delay in the neighbor hours WY on the delay in one hourY; and ε is the random error term vector subjected to normal distribution.


## 3. Data Collection

### 3.1. Data Sample

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.


### 3.2. Definitions of Variables

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 [21, 22]. Chinese aviation determined the following factors of flight delay. Technical failure includes technical failure at the target airport (T) and technical failure at the previous airport (TL). Weather refers to weather conditions at the target airport (W), at the previous airport (WL), at the destination airport (WD), and en route (WR). Control factors include flow control (CF) and route restriction (CR). Other factors include the airline (A), airport facility (F), passenger (P), and capacity allocation (D).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 Table1 is measured by average delay minutes caused by each factor. As shown in Table 1, the frequency of the flow control factor is significantly higher than the others, but the average delay minutes caused by the flow control factor is lower. Conversely, the frequency of the weather condition at the target airport and the technical failure at the previous airport are significantly lower, with high average delay minutes.Table 1
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 9All 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 asαin=5% and a significant level of excluded independent variable as αout=10%. Seven factors are excluded, and the remaining 8 explanatory variables comprise the regression model (Table 2). All variables are calculated in an hour unit.Table 2
Descriptive statistics and definitions of the variables used in the model.
Variables Definition Mean s.d. Min Max Delay Total minutes of departure delay 587.824 428.433 0 2341 T Technical failure at the target airport (equals 1 if technical failure occurs) 0.039 0.194 0 1 TL Technical failure at the previous airport (equals 1 if technical failure occurs) 0.065 0.246 0 1 W Weather condition at the target airport (equals 1 if weather is adverse) 0.018 0.132 0 1 WL Weather condition at the previous airport (equals 1 if weather is adverse) 0.251 0.434 0 1 WR Weather condition en route (equals 1 if weather is adverse) 0.016 0.124 0 1 CF Flow control (equals 1 if conduct flow control) 0.571 0.496 0 1 CR Route restriction (equals 1 if conduct route restriction) 0.122 0.327 0 1 NF Scheduled departure traffic 16.943 4.414 4 27


### 3.3. Data Processing

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.


## 3.1. Data Sample

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.


## 3.2. Definitions of Variables

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 [21, 22]. Chinese aviation determined the following factors of flight delay. Technical failure includes technical failure at the target airport (T) and technical failure at the previous airport (TL). Weather refers to weather conditions at the target airport (W), at the previous airport (WL), at the destination airport (WD), and en route (WR). Control factors include flow control (CF) and route restriction (CR). Other factors include the airline (A), airport facility (F), passenger (P), and capacity allocation (D).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 Table1 is measured by average delay minutes caused by each factor. As shown in Table 1, the frequency of the flow control factor is significantly higher than the others, but the average delay minutes caused by the flow control factor is lower. Conversely, the frequency of the weather condition at the target airport and the technical failure at the previous airport are significantly lower, with high average delay minutes.Table 1
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 9All 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 asαin=5% and a significant level of excluded independent variable as αout=10%. Seven factors are excluded, and the remaining 8 explanatory variables comprise the regression model (Table 2). All variables are calculated in an hour unit.Table 2
Descriptive statistics and definitions of the variables used in the model.
Variables Definition Mean s.d. Min Max Delay Total minutes of departure delay 587.824 428.433 0 2341 T Technical failure at the target airport (equals 1 if technical failure occurs) 0.039 0.194 0 1 TL Technical failure at the previous airport (equals 1 if technical failure occurs) 0.065 0.246 0 1 W Weather condition at the target airport (equals 1 if weather is adverse) 0.018 0.132 0 1 WL Weather condition at the previous airport (equals 1 if weather is adverse) 0.251 0.434 0 1 WR Weather condition en route (equals 1 if weather is adverse) 0.016 0.124 0 1 CF Flow control (equals 1 if conduct flow control) 0.571 0.496 0 1 CR Route restriction (equals 1 if conduct route restriction) 0.122 0.327 0 1 NF Scheduled departure traffic 16.943 4.414 4 27


## 3.3. Data Processing

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.


## 4. Results and Discussion

### 4.1. Exploring Delay Distribution

(1) Distribution Map. The distribution map is a commonly used spatial data visualization method. Each grid defined by day and hour is colored according to the departure delay in minutes that occurred in an hour unit. Red represents high departure delay, whereas dark blue indicates low departure delay. The distribution map highlights delay intensity, the day on which the delay occurred, and the duration of the delay. As demonstrated in Figure 1, the delay levels between neighborhoods are usually similar, which indicates obvious spatial cluster characteristics. The distribution map also shows that intense delay occurred mostly at 16:00, 18:00, and 21:00, especially from June 18 to June 22 in our dataset.Figure 1
Distribution map of departure delay.(2) 3D Trend Analysis. The trend analysis generated a 3D trend map of the departure delay. In Figure 2, thex-axis andy-axis represent the day and the hour of delay, respectively, and thez-axis represents the total minutes of departure delay. The green line in thex–z plane and the blue line in they–z plane indicate the trend of the delay. The figure shows that flight departure delay is intensively late in the month and exhibits a parabola with the peak at 18:00 in our dataset.Figure 2
3D trend map of departure delay.


### 4.2. Identifying the Pattern of Delay

As mentioned in Introduction, there exists a high degree of spatial dependence among delays. Moran’sI and general G are calculated to measure the degree of spatial autocorrelation and to identify the pattern of delay. The Moran’s I and general G values of all variables are calculated. Values with a significant autocorrelation are listed in Table 3.Table 3
Autocorrelation statistics for selected variables.
Variables Coefficient Observed Expected Stddev Z P Delay Moran’sI 0.580509 -0.001965 0.001016 18.275923 0.000000 GeneralG 0.010119 0.007496 0.000000 19.158604 0.000000 W Moran’sI 0.397390 -0.001965 0.000915 13.198725 0.000000 GeneralG 0.194444 0.007496 0.000201 13.177839 0.000000 WL Moran’sI 0.161547 -0.001965 0.001021 5.116695 0.000000 GeneralG 0.011565 0.007496 0.000001 5.576605 0.000000 WR Moran’sI 0.248777 -0.001965 0.000901 8.352514 0.000000 GeneralG 0.142857 0.007496 0.000260 8.398764 0.000000 CF Moran’sI 0.341630 -0.001965 0.001024 10.738782 0.000000 GeneralG 0.009717 0.007496 0.000000 11.624289 0.000000 NF Moran’sI 0.532762 -0.001965 0.001018 16.761879 0.000000 GeneralG 0.007859 0.007496 0.000000 14.762860 0.000000 NP Moran’sI 0.489845 -0.001965 0.001019 15.405064 0.000000 GeneralG 0.007888 0.007496 0.000000 12.831494 0.000000Table3 shows significant positive spatial autocorrelation in variables such as the total minutes of departure delay, weather conditions at the target airport, weather conditions at the previous airport, weather conditions en route, flow control, total departure traffic, and number of passengers. General G test showed that all of the abovementioned variables are high-value clusters, indicating that the hours of intense delay are clustered.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 Figure3. Among all the factors responsible for flight departure delay, this large-scale cluster is probably caused by an exogenous variable such as sudden adverse weather conditions. This conclusion corresponded to the actual weather report record of the target airport in June 2016. Between June 18 and 22, there were cloudy skies for 5 days and a thunderstorm for 3 days.Figure 3
Hot and cold spots of departure delay.


### 4.3. Quantifying Delay Spatial Dependence

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:( 1 ) Nugget effect C0 is estimated from the empirical variogram at h=0.This represents the measurement error or random property of the departure delay. The C0 value reflects the variation caused by the stochastic factor.( 2 ) Range A0 is the distance where the variogram reaches plateau. This represents the largest distance of autocorrelation. Data can be considered uncorrelated if their distance exceeds the range.( 3 ) Sill C+C0 is the plateau at which the variogram reaches the range. This represents the total variance of regionalized variables, which is equal to the sum of the autocorrelation and stochastic variances.The following are the other two parameters that can be calculated from the three parameters mentioned above:( 4 ) Structural variance C represents the structural property of delay. The value of C reflects the variance caused by the autocorrelation.( 5 ) Nugget–sill ratio C0/(C+C0) represents the percentage of variance caused by randomness. A low nugget–sill ratio reflects that the variation is mainly affected by the autocorrelation factors.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 Table4. The comparison shows that, when fitting the isotropic semivariogram, the exponential model is more effective than others, such as the spherical and Gaussian models.Table 4
Theoretical model fit comparison of isotropic semivariogram.
Model Nugget Sill Range RSS R2 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 0.947According 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 Figure4, the delays separated by short intervals are strongly correlated to one another. The correlation decreases as the intervals increase to a distance of 17.60.Figure 4
Isotropic semivariogram of departure delay.


### 4.4. Delay Prediction

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 Figure5, in which thex-axis represents the day of week,y-axis represents the hour of day, and thez-axis represents the value of delay. In this way the value of unmeasured locations can be interfered according to the value of measure locations and the spatial relation between them.Figure 5
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 (Table5).Table 5
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


### 4.5. Spatial Econometric Analysis

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 (6)). The estimation results of each variable are demonstrated as β^ in Table 6. After the model parameters are estimated, it is necessary to perform the statistical test of the model, which includes the goodness of fit test, the significance test of the equation, and the significance test of the variables.Table 6
Estimation results of delay and causal factors for each model.
Variable OLS estimation Spatial lag model Spatial error model β ^ p β ^ p β ^ p Constant -189.2450∗∗∗ 0.0008 -275.3832∗∗∗ 0.0000 81.4065 0.2570 T 192.5201∗∗∗ 0.0066 163.5734∗∗∗ 0.0029 197.2891∗∗∗ 0.0003 TL 179.6183∗∗∗ 0.0013 150.6237∗∗∗ 0.0005 140.6904∗∗∗ 0.0006 W 1059.5640∗∗∗ 0.0000 699.4476∗∗∗ 0.0000 769.7684∗∗∗ 0.0000 WL 247.8793∗∗∗ 0.0000 181.4880∗∗∗ 0.0000 161.2410∗∗∗ 0.0000 WR 275.8793∗∗ 0.0187 90.3014 0.3228 147.4748 0.1265 CF 292.8883∗∗∗ 0.0000 139.7609∗∗∗ 0.0000 128.7156∗∗∗ 0.0000 CR 205.2254∗∗∗ 0.0000 141.3503∗∗∗ 0.0002 120.0013∗∗∗ 0.0002 NF 28.3640∗∗∗ 0.0000 19.8473 0.0000 28.7060∗∗∗ 0.0000 ρ   0.5806∗∗∗ 0.0000 λ     0.6926∗∗∗ 0.0000  R2 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:∗∗∗, ∗∗, and ∗ represent significance in 1%, 5%, and 10% levels, respectively.The goodness of fit test can be reflected by R2. R2 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 (H0) 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.Thep-value measures the probability of correctly rejecting the null hypothesis when testing the significance of a single variable. A largerp-value indicates greater probability of erroneously rejecting the null hypothesis.In Table6, the OLS estimation shows an F value of 62.0591 at a 1% level of significance and a goodness of fit R2 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.p-value indicates significant variables such as the technical fault at the target airport (T), the technical fault at the previous airport (TL), the weather condition at the previous airport (WL), the weather condition at the airport of departure (W), the flow control (CF), the route restriction (CR), and the number of scheduled departure flights (NF).However, the Moran’sI test shows a significant spatial autocorrelation in the residuals of the OLS estimators. The spatial dependence test results of the error and lag are positive, as shown in Table 7.Table 7
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 0.0000Classical 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 [23].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 (R2). 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 Table6, the goodness of fit R2 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 H0 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 orp-value. In this way, we can judge whether the difference between models is significant. The results of the likelihood ratio test of SLM-OLS and SEM-OLS show that the likelihood ratio values are greater than the chi-squared distribution with the degree of freedom of 1 at 1% significance level, and the null hypotheses are rejected, which means that the SLM and SEM provide significantly better fit compared with the OLS estimation, and the explanatory capacity is enhanced by adding the spatial effect to the model.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 Table6 show that the WR variable, which is significant in the OLS estimation, is insignificant in the SLM and SEM. Delay-reduction strategies may focus on the weather prediction on route according to the OLS estimation but would not achieve a significant effect in delay reduction according to SLM and SEM. The SLM shows that the spatial lag variable is at the 1% significance level, which indicates a strong spillover effect of the departure delay in the temporal dimension.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.


## 4.1. Exploring Delay Distribution

(1) Distribution Map. The distribution map is a commonly used spatial data visualization method. Each grid defined by day and hour is colored according to the departure delay in minutes that occurred in an hour unit. Red represents high departure delay, whereas dark blue indicates low departure delay. The distribution map highlights delay intensity, the day on which the delay occurred, and the duration of the delay. As demonstrated in Figure 1, the delay levels between neighborhoods are usually similar, which indicates obvious spatial cluster characteristics. The distribution map also shows that intense delay occurred mostly at 16:00, 18:00, and 21:00, especially from June 18 to June 22 in our dataset.Figure 1
Distribution map of departure delay.(2) 3D Trend Analysis. The trend analysis generated a 3D trend map of the departure delay. In Figure 2, thex-axis andy-axis represent the day and the hour of delay, respectively, and thez-axis represents the total minutes of departure delay. The green line in thex–z plane and the blue line in they–z plane indicate the trend of the delay. The figure shows that flight departure delay is intensively late in the month and exhibits a parabola with the peak at 18:00 in our dataset.Figure 2
3D trend map of departure delay.


## 4.2. Identifying the Pattern of Delay

As mentioned in Introduction, there exists a high degree of spatial dependence among delays. Moran’sI and general G are calculated to measure the degree of spatial autocorrelation and to identify the pattern of delay. The Moran’s I and general G values of all variables are calculated. Values with a significant autocorrelation are listed in Table 3.Table 3
Autocorrelation statistics for selected variables.
Variables Coefficient Observed Expected Stddev Z P Delay Moran’sI 0.580509 -0.001965 0.001016 18.275923 0.000000 GeneralG 0.010119 0.007496 0.000000 19.158604 0.000000 W Moran’sI 0.397390 -0.001965 0.000915 13.198725 0.000000 GeneralG 0.194444 0.007496 0.000201 13.177839 0.000000 WL Moran’sI 0.161547 -0.001965 0.001021 5.116695 0.000000 GeneralG 0.011565 0.007496 0.000001 5.576605 0.000000 WR Moran’sI 0.248777 -0.001965 0.000901 8.352514 0.000000 GeneralG 0.142857 0.007496 0.000260 8.398764 0.000000 CF Moran’sI 0.341630 -0.001965 0.001024 10.738782 0.000000 GeneralG 0.009717 0.007496 0.000000 11.624289 0.000000 NF Moran’sI 0.532762 -0.001965 0.001018 16.761879 0.000000 GeneralG 0.007859 0.007496 0.000000 14.762860 0.000000 NP Moran’sI 0.489845 -0.001965 0.001019 15.405064 0.000000 GeneralG 0.007888 0.007496 0.000000 12.831494 0.000000Table3 shows significant positive spatial autocorrelation in variables such as the total minutes of departure delay, weather conditions at the target airport, weather conditions at the previous airport, weather conditions en route, flow control, total departure traffic, and number of passengers. General G test showed that all of the abovementioned variables are high-value clusters, indicating that the hours of intense delay are clustered.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 Figure3. Among all the factors responsible for flight departure delay, this large-scale cluster is probably caused by an exogenous variable such as sudden adverse weather conditions. This conclusion corresponded to the actual weather report record of the target airport in June 2016. Between June 18 and 22, there were cloudy skies for 5 days and a thunderstorm for 3 days.Figure 3
Hot and cold spots of departure delay.


## 4.3. Quantifying Delay Spatial Dependence

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:( 1 ) Nugget effect C0 is estimated from the empirical variogram at h=0.This represents the measurement error or random property of the departure delay. The C0 value reflects the variation caused by the stochastic factor.( 2 ) Range A0 is the distance where the variogram reaches plateau. This represents the largest distance of autocorrelation. Data can be considered uncorrelated if their distance exceeds the range.( 3 ) Sill C+C0 is the plateau at which the variogram reaches the range. This represents the total variance of regionalized variables, which is equal to the sum of the autocorrelation and stochastic variances.The following are the other two parameters that can be calculated from the three parameters mentioned above:( 4 ) Structural variance C represents the structural property of delay. The value of C reflects the variance caused by the autocorrelation.( 5 ) Nugget–sill ratio C0/(C+C0) represents the percentage of variance caused by randomness. A low nugget–sill ratio reflects that the variation is mainly affected by the autocorrelation factors.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 Table4. The comparison shows that, when fitting the isotropic semivariogram, the exponential model is more effective than others, such as the spherical and Gaussian models.Table 4
Theoretical model fit comparison of isotropic semivariogram.
Model Nugget Sill Range RSS R2 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 0.947According 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 Figure4, the delays separated by short intervals are strongly correlated to one another. The correlation decreases as the intervals increase to a distance of 17.60.Figure 4
Isotropic semivariogram of departure delay.


## 4.4. Delay Prediction

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 Figure5, in which thex-axis represents the day of week,y-axis represents the hour of day, and thez-axis represents the value of delay. In this way the value of unmeasured locations can be interfered according to the value of measure locations and the spatial relation between them.Figure 5
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 (Table5).Table 5
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


## 4.5. Spatial Econometric Analysis

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 (6)). The estimation results of each variable are demonstrated as β^ in Table 6. After the model parameters are estimated, it is necessary to perform the statistical test of the model, which includes the goodness of fit test, the significance test of the equation, and the significance test of the variables.Table 6
Estimation results of delay and causal factors for each model.
Variable OLS estimation Spatial lag model Spatial error model β ^ p β ^ p β ^ p Constant -189.2450∗∗∗ 0.0008 -275.3832∗∗∗ 0.0000 81.4065 0.2570 T 192.5201∗∗∗ 0.0066 163.5734∗∗∗ 0.0029 197.2891∗∗∗ 0.0003 TL 179.6183∗∗∗ 0.0013 150.6237∗∗∗ 0.0005 140.6904∗∗∗ 0.0006 W 1059.5640∗∗∗ 0.0000 699.4476∗∗∗ 0.0000 769.7684∗∗∗ 0.0000 WL 247.8793∗∗∗ 0.0000 181.4880∗∗∗ 0.0000 161.2410∗∗∗ 0.0000 WR 275.8793∗∗ 0.0187 90.3014 0.3228 147.4748 0.1265 CF 292.8883∗∗∗ 0.0000 139.7609∗∗∗ 0.0000 128.7156∗∗∗ 0.0000 CR 205.2254∗∗∗ 0.0000 141.3503∗∗∗ 0.0002 120.0013∗∗∗ 0.0002 NF 28.3640∗∗∗ 0.0000 19.8473 0.0000 28.7060∗∗∗ 0.0000 ρ   0.5806∗∗∗ 0.0000 λ     0.6926∗∗∗ 0.0000  R2 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:∗∗∗, ∗∗, and ∗ represent significance in 1%, 5%, and 10% levels, respectively.The goodness of fit test can be reflected by R2. R2 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 (H0) 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.Thep-value measures the probability of correctly rejecting the null hypothesis when testing the significance of a single variable. A largerp-value indicates greater probability of erroneously rejecting the null hypothesis.In Table6, the OLS estimation shows an F value of 62.0591 at a 1% level of significance and a goodness of fit R2 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.p-value indicates significant variables such as the technical fault at the target airport (T), the technical fault at the previous airport (TL), the weather condition at the previous airport (WL), the weather condition at the airport of departure (W), the flow control (CF), the route restriction (CR), and the number of scheduled departure flights (NF).However, the Moran’sI test shows a significant spatial autocorrelation in the residuals of the OLS estimators. The spatial dependence test results of the error and lag are positive, as shown in Table 7.Table 7
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 0.0000Classical 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 [23].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 (R2). 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 Table6, the goodness of fit R2 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 H0 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 orp-value. In this way, we can judge whether the difference between models is significant. The results of the likelihood ratio test of SLM-OLS and SEM-OLS show that the likelihood ratio values are greater than the chi-squared distribution with the degree of freedom of 1 at 1% significance level, and the null hypotheses are rejected, which means that the SLM and SEM provide significantly better fit compared with the OLS estimation, and the explanatory capacity is enhanced by adding the spatial effect to the model.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 Table6 show that the WR variable, which is significant in the OLS estimation, is insignificant in the SLM and SEM. Delay-reduction strategies may focus on the weather prediction on route according to the OLS estimation but would not achieve a significant effect in delay reduction according to SLM and SEM. The SLM shows that the spatial lag variable is at the 1% significance level, which indicates a strong spillover effect of the departure delay in the temporal dimension.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.


## 5. Conclusions

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.



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*Source: 3525912-2019-06-04.xml*