| I. Introductionhis paper is motivated by the need for selecting days with particular air traffic flow patterns and operational characteristics, as encapsulated in the performance metrics, for validating simulation models and evaluating next generation air traffic system concepts.Evaluation of system-wide impacts in terms of cost and benefits with one or two days of data, or with several days of data with similar traffic conditions, is of limited utility.Such evaluations therefore, have to be conducted with a set containing days with distinct characteristics.In order to balance the effort required against the quality of results achieved for these types of simulations and evaluations, a small set of days that covers all the possible traffic conditions is desirable.The multiple-metric classification method proposed in this paper makes it possible to create such a set of days. | |
| TPrior effort on the selection of days for validating simulation models is described in Refs. 1 and 2. Reference 1 contains a detailed description of the data and the procedure used, while Ref. 2 is a summary of the same.The approach consisted of using the K-Means algorithm, first proposed in Ref. 3, to partition the set of days into six significant groups, each with at least 2% of the days, and one outlier group for days that could not be assigned to the six significant groups.Each group was separated from others in terms of a single Euclidean distance metric composed of the eight chosen metrics.Based on analysis of the metrics associated with each of the six significant groups, they concluded that Ground Delay Program (GDP) minutes and total operations count, a measure of trafficvolume, were the primary determinants of group membership.Threshold values were computed for these two metrics and used within a decision-tree for labeling a given day as a typical day characterized by one of the six groups.The main limitation of the method is that the Euclidean distance metric, constructed by adding quadratic terms corresponding to metrics with different scales and units, partitions days in the transformed domain of the combined metrics.This obscures the relation of the groups to the individual metrics.Thus, grouping with a finer level of granularity cannot be achieved with this method.The method proposed in this paper overcomes the limitations of the previous approach by creating groups based on each metric individually using the K-Means algorithm.Each day is then tagged with a composite ID consisting of IDs of the groups it belongs to based on different metrics.For example, if a day is a member of Group 1 based on Metric 1, a member of Group 1 based on Metric 2, and a member of Group 3 based on Metric 3, it is tagged with the composite group ID of (1,1,3), where the first index indicates grouping based on Metric 1, the second based on Metric 2 and the third based on Metric 3.All days with the same tag are then placed in one group.A salient feature of the proposed algorithm is that the linguistic description of the group labels based on each metric is retained in the composite label of the final grouping.For example, if groups 1, 2 and 3 mean "low," "medium" and "high," respectively, the composite label (1,1,3) means "low" based on the first metric, "low" based on the second metric and "high" based on the third metric.Thus, the fidelity of partitions of individual metrics is retained in the final grouping.The rest of the paper is organized as follows.Major trends observed in the 517 days of NAS delay data are described in Section II.Total time delay in minutes is used as a distance metric within the K-Means algorithm to partition the set of 517 days into ten groups in Section III.Convergence characteristics of the K-Means algorithm and summary statistics of the ten groups are provided in this section.A multiple-metric classification technique that builds on the single-metric classification technique is then developed in Section IV.Two examples of grouping of days are provided in Section IV to highlight the salient features of the algorithm.Conclusions are discussed in Section V. | |
| II. National Airspace System Delay and Traffic-Volume CharacteristicsTo keep track of operational efficiency of the air traffic system, the Federal Aviation Administration (FAA) and the Bureau of Transportation Statistics (BTS) keep records of a multitude of metrics including delay, number of operations, conditions at airports, and traffic management initiatives in databases.Several of the frequently used databases are: Aviation System Performance Metrics (ASPM), Air Traffic Control System Command Center (ATCSCC) Logs, BTS data, Enhanced Traffic Management System (ETMS) and OPSNET.Detailed descriptions of the contents of these databases are available in Refs. 1 and4.All the analysis and the results in this paper are based on OPSNET data, which are available via http://www.apo.data.faa.gov.OPSNET data only include delays of fifteen minutes or more experienced by Instrument Flight Rule (IFR) flights that are reported by the FAA facilities.These data do not include delays caused by mechanical or other aircraft operator problems.Speed reductions and pilot initiated deviations around weather are also not reported.Taxi times spent under non-FAA facilities, for example under company/airport ramp towers, are not included in delay reports. 5ASPM also provides delay data that are computed based on the Out-Off-On-In (OOOI) data provided by nine commercial and cargo carriers, which can also be utilized for analyzing days via the methods discussed in this paper.Although the trends in ASPM and OPSNET data are similar, the two databases contain very different types of data that make comparisons between them difficult.They are both useful depending on the analysis desired.OPSNET delay data are provided in a tabular form; numbers of aircraft delayed are reported by category, by class and by cause.Delays by category consist of numbers of aircraft that were subject to departure delays, arrival delays, enroute delays and traffic management system (TMS) delays.The distinction between the enroute and TMS delays is discussed later in this section.Delays by class consist of numbers of air carrier, air taxi, general aviation and military aircraft that were delayed.Delays by cause consist of numbers of aircraft that were delayed due to weather, terminal volume, center volume, equipment limitations, runway issues and "other" issues.International delays are included in the "other" category.In addition to these, average time delay in minutes and total time delay in minutes are included in the table.Seventeen variables of OPSNET national delay data for two days are summarized in Table 1.This table shows that the numbers of aircraft delayed by category (departure + arrival + enroute + TMS) add up to the total number of aircraft delayed.Similarly, the numbers of aircraft delayed by class and by cause also add up to the total number of aircraft delayed during the day of operation.Observe that the average delay is obtained by dividing the total time delay in minutes by the total number of delayed aircraft.There are three significant trends that are easily seen in Table 1.First, the sum of departure and TMS delayed flights account for most of the delayed flights.Second, most aircraft are delayed due to weather.Third, as expected, total time delays are proportional to total numbers of aircraft delayed.To understand NAS delay characteristics, OPSNET delay data covering a period of 517 days (17 months) spanning the period from January 1, 2003 through May 31, 2004 were analyzed.Figure 1 shows a scatter plot of the percentages of aircraft delayed due to weather as a function of days.The mean percentage of aircraft delayed due to weather was found to be 66% and the standard deviation was found to be 21% for this dataset.Additional statistical characteristics are summarized in Table 2.These results show that the number of aircraft delayed due to weather represents a large fraction of the number of aircraft that experience delay in the NAS, a finding consistent with Ref. 6, which states that weather is responsible for approximately 70% of NAS delays.The data shown earlier in Fig. 1 was reorganized as a function of total number of aircraft that experienced delay.These transformed data are shown in Fig. 2. Figure 2 shows that on days when a large number of aircraft are delayed, weather is the dominant cause of delays.Percentages of aircraft delayed due to weather are widely scattered when fewer aircraft are delayed, which indicates that factors other than weather are also responsible for delays on those days.Figure 3 shows the number of aircraft delayed due to weather versus the total number of aircraft delayed.Viewing the sample points with respect to the diagonal line across the figure, it is clear that a high degree of correlation exists between the number of aircraft delayed due to weather and the total number of aircraft delayed in the NAS.Assuming both the number of aircraft delayed due to weather and the total number of aircraft delayed are random variables, the correlation coefficient was computationally determined to be 0.95.Correlation between the number of aircraft delayed due to weather and the total time delay in minutes due to all reportable causes (see the last row of Table 1 for an example) was found to be 0.94.The causes of delay other than weather were also studied.Their statistics are summarized in Table 3 along with those of weather delays.Correlation coefficients 1 ρ in Table 3 are defined with respect to the total number of aircraft delayed, and correlation coefficients 2 ρ are defined with respect to the total accrued time delay in minutes.As evident from the correlation coefficient value of 0.21 in this table, the number of aircraft delayed due to volume has a weak linear correlation with the total number of aircraft delayed in the NAS.Correlation is even lower, 0.11, with respect to the total time delay in minutes.Similarly, the value of the correlation coefficient between the number of aircraft delayed due to equipment, runway and other non-US facilities, and the total number To determine relative contributions of delays attributed to departure, enroute and arrival phases of flights, and to TMS restrictions, percentages of aircraft delays by category were calculated for the 517-day dataset.It was determined that on an average, on any given day, 47% of the aircraft that are delayed in the NAS, are delayed during departure, 1% during enroute and 14% during arrival phases of flight.The average percentage of aircraft delayed due to TMS was 39%.Analysis of the data showed that, on average, departure delayed flights and TMS delayed flights roughly account for 86% of the flights that are delayed.Additional statistics that characterize these delays are summarized in Table 4.Note that the values of the correlation coefficients 1 ρ and 2 ρ listed in the table are with respect to the total number of aircraft delayed and the total accrued time delay in minutes, respectively.Since traffic management initiatives such as GDP and GS are applied to aircraft while they are on the ground, and rerouting and holding while they are airborne, TMS delays include both ground and airborne delay components.To separate TMS delay into ground delay and airborne delay components, analysis of GDP and GS data, that are also available via OPSNET, is needed.Like data in Table 1, these data are also provided in a tabular format with the following items: 1) date, 2) number of aircraft delayed, 3) total delay in minutes, and 4) average time delay in minutes.For example, GDP and GS data for two days, 10 April 2004 1.Ground delay and airborne delay components can be computed using NAS delay data (see delays by category in Table 1) and the GDP and GS delays (see Table 5) as follows.Let, ,, and be the numbers of aircraft delayed during departure, due to GDP, and due to GS, respectively.The total number of aircraft delayed on the ground is thend n GDP n GS n GS GDP d G n n n n + + = (1)Number of aircraft delayed during the airborne phase can be obtained after subtracting GDP and GS components from TMS delays as, ) ( 6 show that on average 74% of the aircraft that experience delay are delayed on the ground, compared to an average of 26% that are delayed while airborne.The last row of Table 6 shows that on some days NAS conditions are unusual in that a large percentage of delayed aircraft experience airborne In addition to the NAS delay metrics discussed in this section, past studies such as Ref. 1, 2 and 7 have used metrics of traffic volume to select days for analysis.There is consensus in the literature that the "traffic volume" and the delay taken together characterize NAS operations, therefore traffic volume metrics are discussed next.OPSNET database includes Towers: Summary Report, Instrument Operations: Summary Report and Centers: Summary of Domestic Operations Report that contain traffic volume data.These three reports count traffic from different perspectives.One is unable to separate departure counts from arrival counts in the Towers: Summary Report and in the Instrument Operations: Summary Report.A departure at one facility is counted as a departure at that facility and as an arrival at a different facility.Since departures and arrivals are counted together twice in these reports, the total number of operations excluding the overflight operations have to be halved to estimate the number of departures or the number of arrivals.The Centers: Summary of Domestic Operations Report directly provides a count of the number of departures from airports within the ARTCCs.Since departure count eventually drives the overflight count and the arrival count, it represents the traffic demand.Due to this reason, departure count from the Centers: Summary of Domestic Count Report has been used in this paper.Table 7 lists the departure counts and the overflight counts for the two days.Departure counts excluding military flights for the two days obtained by summing the air carrier, air taxi and general aviation departures are 31,959 and 42,062.GS GDP TMS e a A n n n n n n - - + + = (2)A histogram of the total domestic departure counts for the 517 days of data is shown in Fig. 5.The minimum and the maximum numbers of departures were found to be 25,677 on 11/27/2003 (Thanksgiving holiday) and 51,399 on 5/27/2004 (Thursday).Observe that the histogram is bimodal which indicates that days can be classified into two categories -low departure count day and high departure count day.Reference 1 noted similar observations and offered evidence that the bimodal distribution is primarily due to the weekday versus weekend traffic levels.This section described several delay and traffic volume metrics that are available in OPSNET data.Summary statistics described in the tables and the patterns observed in the figures suggest that these metrics can be used for distinguishing one day of NAS operations from another day of NAS operations.To illustrate the use of a metric for classifying days of operations, total time delay in minutes, in Eq. ( 3), is used as a distance metric within the K-Means method in the next section.T n | |
| III. Single-Metric ClassificationThe motivation for assigning or labeling days into groups with associated properties, such as mean delay values, is to aid selection of prototype days for analysis.For example, a few days from a group of days with large delays and from a group of days with small delays can be selected for system-wide studies using the National Aeronautics and Space Administration's air traffic simulation, concept evaluation, and decision support tools such as the Airspace Concept Evaluation System (ACES), the Center TRACON Automation System (CTAS) and the Future ATM Concepts Evaluation Tool (FACET). 8-11 All classification processes use metrics, or features, of the data to partition it into groups.A popular classification method, known as the K-Means method, partitions data such that the means associated with the groups are as widely separated as possible. 3The method labels the data elements based on their closeness to the group American Institute of Aeronautics and Astronautics means for reducing the group variance.The K-Means algorithm consists of two steps: 1) the initialization step, and 2) the iterative step.Data elements that are far apart from each other are chosen as the initial means of the groups during the initialization step.Each element is then assigned to the group that it is closest to, based on its distance with respect to the group's initial mean value.Group means are then recomputed based on the elements assigned.Each element is then reassigned to its closest group based on its distance with respect to the recomputed mean values.This process of computation of the means and reassignment of elements to groups is continued in subsequent iterative steps.Convergence is achieved when the numerical values of the group means do not change with reassignment of the elements.Iterations are halted once convergence is achieved.To further clarify the initialization and the iterative steps of the K-Means algorithm, consider a vector with the following elements [0, 0.5, 0.8, 1.2, 5, 7,12,15,20,25].If two groups are desired, the elements with values closer to 0 are assigned to the first group and the elements with values closer to 25 are assigned to the second group.Thus, elements one through seven are assigned to Group One and elements eight through ten are assigned to the Group Two in the initialization step.With this assignment of the elements to the groups, average values of the first group and the second group are 3.79 and 20 and the standard deviation values are 4.47 and 5. Reassignment of the elements based on the recomputed means results in the first six elements being assigned to Group One and the last four elements being assigned to Group Two in the first iterative step.Group means are recomputed in the next iterative step.These means are 2.42 and 18 and the standard deviations are 2.87 and 5.72.The next iterative step results in the same means and the standard deviations as those in the prior step; final grouping is therefore achieved in the previous step.For this example, the K-Means algorithm partitions the data into two groups in three steps.If three groups are desired for the above example, a value from the vector that is far away from both 0 and 25 needs to be selected as the initial value for the third group.Observe that this value is 12 since its minimum distance to 0 (12 units) and 25 (13 units) is a maximum compared to the minimum distances of other elements to 0 and 25.Other values in the vector are less than 12 units with respect to either 0 or 25.Once these initial group means are chosen, the subsequent iterative steps are the same as those described in the previous paragraph.It should be noted that good initial conditions are needed because the K-Means algorithm is sensitive to initial conditions.The K-Means algorithm was used for classifying 517 days into ten groups using total time delay in minutes as the discriminating metric.The choice of ten groups was arbitrary.The algorithm converged in thirteen iterations.Its convergence characteristics are shown in Figure 6.Properties of the ten groups, based on total time delay statistics of the elements assigned to the groups, are summarized in Table 8.The second column of the table shows the number of days in the group, with the group ID given in the first column.Columns three and four show the average delay and the standard deviation of the delays in minutes.Columns five and six show the minimum and maximum delays in minutes of the days belonging to the particular groups.The data in this table 7. Note that the extent of the abscissa is limited to the range of the delay data.Sixteen days (seven days in Group 8, seven days in Group 9 and two days in Group 10) that experienced large delays are listed in Table 9.Of the 517 days grouped by the K-Means algorithm, the least delay of 1,686 minutes occurred on 1/11/2003 (a Saturday) and most delay of 186,313 minutes occurred on 5/13/2004 (a Thursday). | |
| Table 9. Days with large delays.Results presented in this section demonstrate the use of the K-Means algorithm for partitioning a set of days into groups organized in order of a single metric like total time delays.The next section describes a labeling technique that enables use of the single-metric K-Means classification technique for achieving classification based on multiple metrics. | |
| IV. Multiple-Metric ClassificationMotivation for multiple-metric classification stems from the desire for finer levels of partitioning.For example, a group with large mean delay contains days when aircraft were delayed due to weather and also days when aircraft were delayed due to runway conditions.In order to discern which ones were affected by weather and which ones were affected by runway conditions, one would need metrics such as numbers of aircraft delayed due to weather and due to runway conditions, in addition to total delays.Fidality Classification based on multiple metrics has been traditionally accomplished by weighing and combining several metrics into a single metric, and then using it in a K-Means algorithm.For example, if day ' ' was characterized by metrics:, , …, and if day 'q f 1 , q u 2 , q u f q u ,r ' was similarly characterized by , , …, , a weighted quadratic function of the form1 , r u 2 , r u f r u , 2 , , 1 , ) ( l r l q f l l r q u u w d - = ∑ ≤ ≤ (4)can be defined as the distance metric between days and q r .Note that through are the weights corresponding to the different metrics.Interpretation of the distance metric, Eq. ( 4), for grouping days with the K-Means algorithm is straightforward with as the mean of measure l in groupm k n j p u w d l j l k f l l j k ≤ ≤ ≤ ≤ - = ∑ ≤ ≤ 1 ; 1 ; ) ( 2 , , 1, where is the number of metrics, m is the number of days and is the number of groups.f nAlthough the distance metric, Eq. ( 5), enables transformation of a multiple-metric classification problem into a single-metric classification problem, its deficiencies are noteworthy.Limitations stem from the fact that metrics have different scales and units, and that only their combined contribution is available in the distance metric; classification is insensitive to individual contribution.For example, consider the two metrics in Table 1: 1) number of aircraft affected by departure delays and 2) total time delay.Units of the two metrics are quite different, number of aircraft and minutes.The scales are also different by an order of magnitude; 257 aircraft impacted by departure delays versus 12,616 minutes of total time delay on 10 April 2004.To compensate for these differences, the associated weights have to be scaled correctly, and their units have to be chosen appropriately to enable summation of quantities with disparate units.References 1 and 2 suggest that the inverse of the statistical variance of the metric should be used to weigh its contribution.Even with this scaling, a meaning cannot be ascribed to the grouping in the native domain of the metrics.In order to overcome the limitations of the weighted quadratic distance function used in the prior approach of Refs. 1 and 2, a multiple metric classification technique that treats each metric independently of others in an dimensional metric space is proposed.Since each metric is treated independently, the single metric K-Means algorithm described earlier in Section III can be used for assigning days to groups based solely on each metric.IDs of these groups are then coordinates in the -dimensional metric space.For the sake of discussion, consider the problem of classifying days into four groups using two metrics.Using the K-Means algorithm twice, days are first assigned to four groups based on Metric One, and then to four groups based on Metric Two.The resulting sixteen possible groups are labeled using two indices as follows: (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3,4), (4, 1), (4, 2), (4, 3) and (4,4).The first index denotes group ID based on Metric One and the second index denotes group ID based on Metric Two.Thus, a day which is assigned to the second group based on Metric One and to the first group based on Metric Two is a member of group (2, 1).Since a unique group is labeled using two indices in this two metric classification problem, the combined group IDs are coordinates in a twodimensional metric space.f f Generalization of the technique to metrics such that days are classified into groups using metric results inf l n l ∏ ≤ ≤ = f l l g n n 1 (6)number of possible groups.Equation (6) shows that if the same number of groups is desired for each metric, the number of possible groups is given in terms of the power of .For example if groups are desired with each metric, the number of possible groups is .Thus, it is seen that the growth in the number of groups is explosive with increasing number of metrics.Should one conclude that the growth is unbounded based on this observation, or is there an upper bound on the number of groups?The answer is provided by the following.If each day is classified into its own group, one would have the same number of groups as the number of days; hence, number of days is the upper bound.This fact also implies that if is the number of days and , there are at least number of empty groups, groups without any members.Removing these empty groups, the number of possible groups, , is given asf n f n m m n g > m n g - g n ∏ ≤ ≤ = f l l g n m n 1 ), min( (7) where each .m n l ≤ Is it possible that several of the groups counted in Eq. ( 7) are empty?One can demonstrate this to be true by constructing the following examples.Consider the problem of classifying ten days into two sets of five groups using American Institute of Aeronautics and Astronautics two metrics.Following the nomenclature of Eq. ( 7), 2 = f for the two metrics, 5 1 = n using Metric One, 5 2 = n using Metric Two, and for the ten days.Substituting these numerical values in Eq. ( 7) it is seen that .Assume that the first two days are assigned to Group 1, the third and the fourth to Group 2, the fifth and the sixth to Group 3, the seventh and the eighth to Group 4, and the final two to Group 5 based on Metric One, and also based on Metric Two.In this scenario, groups with members are (1, 1), (2, 2), (3,3), (4,4) and (5, 5).All other groups labeled with two different indices, such as (1, 2), (1, 3) and (5, 1), are empty.7).If the first five days are assigned to Group 1 and the next five to Group 2 based on both the metrics, groups (1, 1) and (2, 2) are non-empty while groups (1, 2) and (2, 1) are empty.These two examples clearly show that it is always possible to have empty groups.An aspect of multiple-metric classification that has not been discussed so far is the semantics associated with the group IDs.Without a linguistic meaning, it is difficult to interpret what do group IDs such as (1, 1) and (1, 2) mean.One of the ways of attributing a meaning to the indices is to order them according to the increasing values of the group means.For example if total time delay in minutes was the metric being considered, the index with the least value would correspond to the group with the minimum mean total time delay while the index with the highest value would correspond to the group with the maximum mean total time delay.From an implementation perspective, once classification into specified number of groups is accomplished with the K-Means algorithm using a single metric, and group means are computed based on the metric values of the assigned members, the group means are sorted in an increasing order.Indices of the groups are then re-labeled to reflect the sorted order.This procedure is repeated for each metric to obtain the complete set of indices required for labeling the groups.Three metrics-1) total domestic departure counts, 2) number of aircraft delayed on the ground, and 3) number of aircraft delayed in the air were used as the basis for classification in the multiple-metric K-Means algorithm described in this paper.Recollect that the total domestic departure counts were obtained from the Centers: Summary of Domestic Counts Report discussed in Section II.Numbers of aircraft delayed on the ground and delayed in the air were computed using Eqs.( 1) and (2).Days were initially organized into three groups using the single-metric K-Means algorithm.When number of aircraft delayed in the air was used as a metric, 393 days were assigned to Group 1, 128 to Group 2 and one to Group 3. The mean and the standard deviation values derived from the number of aircraft delayed in the airborne phase metric of the assigned members for these groups are listed in Table 10.The sole member of Group 3, 7/15/2005, had excessive amount of airborne delay of 1891 minutes.1661 aircraft were delayed on the ground on this day.Since this day is an outlier, it has the effect of increasing the standard deviation of the other groups.Due to this reason, it was removed from the dataset.American Institute of Aeronautics and Astronautics The analysis was repeated with the remaining 521 days.The resulting grouping showed that 6/5/2005 became the sole member of Group 3 with 970 aircraft delayed in the airborne phase and 753 aircraft delayed on the ground.Table 11 summarizes these results.Comparing Table 10 to Table 11, it is seen that the removal of 7/15/2005 data lowers the standard deviation values of the groups.Since 6/5/2005 is an outlier day, it was also removed from the dataset.Classification based on number of aircraft delayed in the airborne phase for the remaining 520 days are summarized in Table 12.Observe that the standard deviation values decrease further for groups 1 and 2. It increases for Group 3. The mean values decrease for all three groups.Days belonging to Group 1 can be thought of as days with low number of aircraft delayed while airborne; in Group 2 as days with medium number of aircraft delayed and the ones in Group 3 as days with large number of aircraft delayed.Similar categorization based on number of aircraft delayed on the ground partitions the days in Groups 1 through 3, whose statistics are summarized in Table 13.Results obtained using total domestic departure counts are provided in Table 14.Note that the days were classified into two groups based on the bimodal distribution seen in Fig. 5.Comparing Tables 12 and13, it is seen that the trends are similar with the largest number of days assigned to groups with lower mean and lower standard deviation values.The trends are different in Table 14.More days are assigned to Group 2 with higher mean departure counts.Given that three groups were created using two metrics and two groups using one metric, the total number of possible composite groups, determined using Eq. ( 7), is 18.The range of IDs for these groups is (1, 1, 1) to (2,3,3).With the first index being the group number associated with the total domestic departure counts metric, the second index being the group number associated with the number of aircraft delayed on the ground metric, and the third index being the group number associated with the number of aircraft delayed during the airborne phase metric, each day in the set of 520 days has a three index group ID associated with it.Many of the salient features of multiple-metric classification algorithm are apparent from data in Table 15.A majority of days, 94 and 89, are assigned to Group (1, 1, 1) and Group (2, 1, 1).These groups represent days on which few aircraft were delayed.The difference between them is that Group (1, 1, 1) represents low-volume days while Group (2, 1, 1) represents high-volume days.Table 15 shows that a large number of days were assigned to groups (2, 2, 1) and (2, 2, 2) that represent high-volume days on which many aircraft were delayed on the ground.There are several groups with few days assigned to them; five groups had three or less than three days assigned to them.Table 16 lists the corresponding dates.Member days belonging to small groups can be considered to be special.Days in Group 3 with Group ID (1, 1, 3) experienced relatively low total departure counts, fewer aircraft affected by ground delay, and higher number of aircraft affected by airborne delay.Two members of Group 6 had more aircraft delayed on the ground compared to members of Group 3. Three member days of Group 7 had many more aircraft delayed on the ground and few during the airborne phase.The sole member of Group 9 had many aircraft delayed while on ground and while in the air.The member days of Group 12 experienced high departure counts, relatively few aircraft delayed on the ground, and a large number of aircraft delayed during the airborne phase.Another example of multiple-metric classification using total domestic departure counts and delays by cause: 1) weather, 2) volume, 3) equipment, runway and other, as metrics for partitioning 522 days into groups is summarized in Table 17.Numbers of aircraft delayed due to terminal volume and due to center volume (see "Delays by Cause" in Table 1) were combined to obtain the number of aircraft delayed due to volume.Similarly, numbers of aircraft delayed due equipment, runway and other issues were added together for obtaining the number of aircraft impacted due to these causes.As in the previous example, days were categorized into groups with the K-Means algorithm using each of these four metrics one at a time.Observe that in this example, days are partitioned into 37 groups out of 54 possible groups. | |
| American Institute of Aeronautics and AstronauticsAmerican Institute of Aeronautics and Astronautics The results of the two examples considered here show that 1) days can be classified into the specified number of groups based on each individual metric, 2) the individual metric group labels can be used for creating multiplemetric group labels, and 3) linguistic description of the individual metric grouping is retained in the composite group label.These examples also illustrate that the multiple-metric classification method does not require that the number of groups be the same based on each metric for creating composite group IDs.In the first example, days were organized in two groups using the total number of departure counts metric and in three groups using the number of aircraft delayed on the ground and the number of aircraft delayed in the air.This technique of maintaining different numbers of groups along different axes of the metric space is in contrast with the method described in Ref. 1 and 2 that only partitions data along the single distance metric.Results demonstrate that the multiple-metric classification method generates groups with several members and also groups with few members; thus, identifying both nominal and off-nominal days.By selecting a typical day from each group, and then using traffic data corresponding to those days, enough data diversity can be assured for validation of simulations and for Monte Carlo type of benefits analysis of novel air traffic management concepts.Resulting benefits metrics can be weighed with number of members in the group that each day is associated with for estimating overall benefits. | |
| V. ConclusionsConsistent with other studies, analysis of 517 days of National Airspace System (NAS) delay data, which were obtained from the Federal Aviation Administration's Air Traffic Operations Network (OPSNET) database, showed that weather is the predominant causal factor for delays; equipment and runway conditions, and traffic-volume are the other major causal factors.It was also determined the departure and traffic management system delays account for about 86% of the aircraft that are delayed.Ground Delay Program and Ground Stop delay data, also obtained from OPSNET, were combined with the NAS delay data to obtain the number of aircraft delayed on the ground and in the air.The results obtained indicate that on an average 74% of the delayed aircraft are delayed on the ground while only 24% are delayed in the air. | |
| American Institute of Aeronautics and AstronauticsThe daily total time delay in minutes was used as a discriminating metric within the K-Means algorithm for partitioning 517 days into ten groups.Mean time delay values associated with the resulting groups, computed using time delay values of the member days, arranged in increasing order were found to be approximately equidistant from the preceding and succeeding mean values.Differences between the standard deviation values associated the groups were also found to be small.Most of the days were assigned to groups with smaller mean time delays.Days with large delays were also identified by the algorithm.A multiple-metric algorithm was synthesized with the single-metric K-Means algorithm at its core.The technique consists of creating groups using each metric individually as a distance metric within the single-metric K-Means algorithm.Member days are labeled with the group numbers associated with the metrics.Final grouping is achieved by assigning days with a common label to the same group, such that groups are labeled by the same number of indices as the number of metrics.The multiple-metric algorithm was applied to the problem of organizing the 522 days into groups using a) total domestic departure counts, b) number of aircraft delayed on the ground and c) number of aircraft delayed in the air as the three metrics.Two days that were found to be outliers were removed and the remaining 520 days were classified into 18 groups.Six of the 18 groups had six or fewer days as members.Although these groups represent unusual days, their inclusion in a set of days that represents diverse air traffic conditions is essential for evaluating concepts and validating simulations.The other 12 groups had 14 or more days as members.Another example of multiple-metric classification of 522 days into groups with a) total domestic departure counts, b) weather delays, c) volume delays and d) equipment, runway and other delays as the chosen metrics was presented.In this instance, 37 groups out of 54 possible groups had member days.Of the 37 groups, 20 groups had five or fewer days as members.Comparing the results obtained via the two examples, it was seen that different sets of days can be created and certain unusual days can be identified based on the choice of metrics.The two examples serve as illustrations of the ability of the multiple-metric algorithm to create datasets, consisting of days classified into groups, with enough data diversity for concept evaluation and simulation validation.Figure 2 .2Figure 2. Percentage of aircraft delayed due to weather as a function of total number of aircraft delayed. | |
| Figure 3 .3Figure 3. Proportion of number of aircraft delayed due to weather to the total number of aircraft delayed in the NAS. | |
| the numbers of aircraft delayed in arrival phase, in enroute phase, and due to TMS, number of aircraft delayed in the NAS.Results obtained using Eqs.(1) through (3) with 517 days of OPSNET data are shown in Fig. 4; n and values are plotted against values.This figure shows that when more aircraft are delayed, a significantly higher number of them are delayed on the ground compared to in the air.Statistical trends summarized in Table | |
| Figure 4 .4Figure 4. Proportion of number of aircraft delayed on the ground and in the air to the total number of aircraft delayed in the NAS. | |
| delay.Of the 665 aircraft that were delayed on 2/22/2003, 513 aircraft (77%) were delayed during their airborne phase of flight.The airborne and ground delay values for this day are marked with a large 'X' and a large 'O' in Fig.4. | |
| Figure 5 .5Figure 5. Histogram of 517 days of total domestic departure counts. | |
| Figure 6 .6Figure 6.Convergence characteristics of the K-Means algorithm as it partitions the 517 days into ten groups. | |
| can be constructed to show that empty groups are possible even when . | |
| Table 1 . OPSNET NAS delay summary data.1Data Variable4/10/20044/13/2004Delays by CategoryTotal # of Aircraft3912,312Departure257651Arrival101391Enroute012TMS331,258Delays by ClassAir Carrier3381,769Air Taxi26474General Aviation2769Military00Delays by CauseWeather2352,049Terminal Volume5927Center Volume4113Equipment126Runway3024Other25173Time DelayAverage Time (min.)32.2753.51Total Time (min.)12,616123,709American Institute of Aeronautics and Astronautics | |
| Table 2 . Statistical characteristics of percentages of aircraft delayed due to weather.2CharacteristicAircraft delayed by weatherMean66%Standard deviation21%Minimum5%Maximum98%Median70% | |
| Figure 1. Percentage of aircraft delayed due to weather.American Institute of Aeronautics and Astronautics of aircraft delayed in the NAS was found to be 0.27.It was found to be 0.16 with respect to the total time delay in minutes.In the hierarchy of prime causal factors for delays, equipment and runway conditions follow weather.Results presented in this section suggest that delay metrics that encapsulate weather characteristics are likely to be useful in the classification of days.Delays attributed to weather, volume, andequipment, runway and other causes are realized viadeparture, arrival, enroute and TMS restrictions.Departure delays incur by holding aircraft at thegate, on the taxiway, short of the runway, and on therunway. Arrival delays accrue when aircraft aredelayed in the arrival Center's airspace or inTerminal Radar Approach Control airspace due torestrictions at arrival airports. Enroute delays occurwhen aircraft are held as a result of initiativesimposed by a facility for traffic managementreasons such as volume regulation, frequencyoutage and weather. The other major category ofdelays in the OPSNET data is TMS delays, whichresult from national or local Center (coordinatedwith Air Traffic Control System Command Center)traffic flow management initiatives such as GroundDelay Programs, local Ground Stops (GS),Departure Sequencing Programs, Enroute SpacingPrograms, Arrival Sequencing Programs, airborneholding, rerouting, Miles-in-Trail, Minutes-in-Trailand Fuel Advisory. | |
| Table 3 . Summary of weather, volume, and equipment, runway and other delay characteristics.3CharacteristicWeatherEquip.,VolumeRunway& OtherMean66%20%14%Standard21%16%11%deviation 1 ρ0.950.270.210.940.160.112 ρ | |
| Table 4 . Summary of departure, enroute, arrival and TMS delay characteristics.4Characteristic Departure TMS Arrival EnrouteMean47%39%14%1%Standard17%17%8%1%deviation 1 ρ0.820.860.550.450.730.860.510.492 ρAmerican Institute of Aeronautics and Astronautics and 13 April 2004, are shown in Table5.NAS delay data for the same two days were previously itemized in Table | |
| Table 6 . Summary of aircraft delayed on ground versus aircraft delayed in the air.6CharacteristicDelayed onDelayed in AirGroundMean74%26%Median76%24%Standard Deviation11%11%Minimum23%7%Maximum93%77% | |
| Table 5 . OPSNET GS and GDP delay data.5Data Variable4/10/20044/13/2004Ground Stops# of Aircraft Delayed398Minutes of Delay2927,079Average Delay97.3372.23Ground Delay Program# of Aircraft Delayed61,044Minutes of Delay24485,166Average Delay40.6681.57Total Delays Due to GS and GDP# of Aircraft Delayed91,142Minutes of Delay53692,245Average Delay59.5580.77American Institute of Aeronautics and Astronautics | |
| Table 7 . Centers: Summary of Domestic Operations Report.7Data Variable4/10/20044/13/2004DeparturesAir Carrier17,12220,231Air Taxi9,13212,831General Aviation5,7059,000OverflightsAir Carrier23,02123,753Air Taxi3,5564,318General Aviation2,7634,763 | |
| Table 8 . Summary of properties of the ten groups.8American Institute of Aeronautics and Astronautics show that there are fewer days in groups associated with large delays.For example, group number ten consists of only two days compared to group number one with 145 days.Observe that the mean values associated with the groups are approximately equally spaced and that the standard deviation values are fairly close to each other.Standard deviation values can be expected to increase with fewer groups.Probability density functions corresponding to Gaussian distributions with the group means and standard deviations listed in Table8are shown in FigureGroupNumberMeanStandardMinimumMaximumIDof DaysDelayDeviationDelayDelay(min.)(min.)(min.)(min.)114511,3024,3101,68618,834212626,6264,48919,10734,62839243,0265,09534,94752,11345761,2395,19852,56270,01153880,0905,42171,18690,464632 102,8205,94494,023112,031711 124,6523,612119,692131,17287 141,6335,854133,884148,34197 163,5015,887156,717172,211102 183,4264,083180,539186,313 | |
| Table 10 . Three groups based on number of aircraft delayed in the airborne phase.10To illustrate the utility of the multiple-metric classification technique, an example of classifying 522 days, which included the 517 days discussed previously and five special days used in earlier studies, into groups is presented next.These five special days are 5/17/2002, 4/17/2005, 4/21/2005, 6/5/2005 and 7/15/2005.5/17/2002 is the ACES baseline day.The other four days were used earlier in Ref. 7. They were categorized as a low-volume low-weather day, high-volume low-weather day, low-volume high-weather day and high-volume high-weather day, respectively in Ref. 7.GroupNumberMeanStandardMinimumMaximumNumberof DaysNumberDeviationNumberNumberDelayedDelayedDelayed13931244972182128314100222970311891018911891 | |
| Table 13 . Three groups based on number of aircraft delayed on the ground.13GroupNumberMeanStandardMinimumMaximumNumberof DaysNumberDeviationNumberNumberDelayedDelayedDelayed12493791604864621859131736511,2933861,6893281,3092,778 | |
| Table 12 . Groups based on number of aircraft delayed in the airborne phase (excluding 7/15/2005 and 6/5/2005).12GroupNumberMeanStandardMinimumMaximumNumberof DaysNumberDeviationNumberNumberDelayedDelayedDelayed129110336715821742143815929535538271304604 | |
| Table 11 . Three groups based on number of aircraft delayed in the airborne phase (excluding 7/15/2005).11GroupNumberMeanStandardMinimumMaximumNumberof DaysNumberDeviationNumberNumberDelayedDelayedDelayed1386123477213213430481214604319700970970 | |
| Table 14 . Two groups based on total domestic departure counts.14GroupNumberMeanStandardMinimumMaximumNumberof DaysDelayDeviationDelayDelay(min.)(min.)(min.)(min.)116834,7922,92025,67740,528235246,3352,24340,59651,759 | |
| Table 15 . Final grouping with three-metric classification.15Organizing the resultingtriple index group IDsusing a "dictionary sort" algorithm, each unique group and its members areGroup NumberGroup ID Number of Days1 μμ2μ3σ1σ2σ3determined. The values of1 (1, 1, 1)94 33,97031083 2,718 15837the metrics of the members2 (1, 1, 2)14 33,698444 224 3,179 14941are determination of minimum, then used for3 (1, 1, 3) 4 (1, 2, 1)2 34,650 27 36,060373 348 6,588 120 880 110 2,565 15241 33maximum,meanand5 (1, 2, 2)19 36,816951 221 1,916 15937standard deviation values associated with the groups.6 (1, 2, 3) 7 (1, 3, 1)2 36,583 3 35,219 1,713 131 4,418 301 951 390 4,185 30595 8Results of this process for8 (1, 3, 2)6 37,067 1,586 213 2,296 20430thethree-metric9 (1, 3, 3)1 36,485 1,462 355000classification being discussed here, are problem, outlined in Table 15. Group means for the three metrics are listed in the columns labeled 1 μ , 2 μ and 3 μ ; standard deviation values are listed in the columns labeled as 1 σ , 2 σ and 3 σ .10 (2, 1, 1) 11 (2, 1, 2) 12 (2, 1, 3) 13 (2, 2, 1) 14 (2, 2, 2) 15 (2, 2, 3) 16 (2, 3, 1) 17 (2, 3, 2) 18 (2, 3, 3)89 45,495 47 46,195 3 43,944 60 47,330 50 46,393 27 45,934 18 46,686 1,614 116 2,190 326 383 110 2,200 151 483 204 2,008 117 505 447 2,494 148 898 113 1,961 188 911 205 2,383 156 953 359 2,201 196 38 47,027 1,728 231 1,947 380 20 46,539 1,721 408 2,537 26131 33 52 31 36 58 31 41 81 | |
| Table 16 . Days in small groups.16GroupGroup ID DateNumber3(1, 1, 3)2/22/20033(1, 1, 3)5/16/20046(1, 2, 3)6/14/20036(1, 2, 3)3/28/20047(1, 3, 1)9/14/20037(1, 3, 1)5/23/20047(1, 3, 1)5/30/20049(1, 3, 3)1/4/200412(2, 1, 3)5/2/200312(2, 1, 3)5/5/200312(2, 1, 3)9/30/2003 | |
| Table 17 . Final grouping obtained using departure counts and delays by cause metrics.17Group NumberGroup IDNumber of Days1 μμ2μ3μ4σ1σ2σ3σ41 (1, 1, 1, 1)86 33,6482684972 2,932 16036442 (1, 1, 1, 2)22 35,24335760 252 2,172 17439643 (1, 1, 1, 3)2 33,69115479 498175 1685844 (1, 1, 2, 1)7 35,796355 21289 1,339 15961345 (1, 1, 2, 2)3 37,448402 198 259794 25128616 (1, 1, 2, 3)2 36,120352 229 4958894020 1447 (1, 1, 3, 1)1 35,484296 401 17300008 (1, 1, 3, 2)1 39,454394 541 23800009 (1, 2, 1, 1)25 35,7709635090 2,920 209354610 (1, 2, 1, 2)7 36,85496080 238 2,653 302386811 (1, 2, 1, 3)2 37,02196273 587378 38530 17012 (1, 2, 2, 1)2 37,118960 291 105 3,430 35312813 (1, 2, 2, 2)2 38,540741 306 24965941 100 16414 (1, 2, 3, 2)1 36,485898 710 209000015 (1, 3, 1, 1)4 37,143 1,56740 148 3,049 132322516 (1, 3, 2, 1)2 37,814 1,867 177 1295802711117 (2, 1, 1, 1)78 45,50832580 104 1,774 171263618 (2, 1, 1, 2)21 45,74033085 234 2,349 173214719 (2, 1, 1, 3)8 45,69037091 458 1,609 210248320 (2, 1, 2, 1)29 46,356337 19498 2,204 181463621 (2, 1, 2, 2)21 47,496319 215 274 2,293 129516522 (2, 1, 2, 3)7 47,610314 195 491 1,867 148365823 (2, 1, 3, 1)2 45,004411 412 147 6,233 270331324 (2, 1, 3, 2)2 50,358192 382 265 1,98185181125 (2, 1, 3, 3)1 48,983407 551 540000026 (2, 2, 1, 1)64 46,0909408096 2,318 217303927 (2, 2, 1, 2)27 46,39297796 250 2,481 220284828 (2, 2, 1, 3)3 47,117 1,172 106 555 2,916 33124 29329 (2, 2, 2, 1)19 46,817898 20596 1,910 216513830 (2, 2, 2, 2)17 47,587 1,021 205 262 2,533 241535831 (2, 2, 2, 3)3 47,360765 424 407 4,315 142 337 11332 (2, 2, 3, 3)2 45,890744 429 434 1,87064207133 (2, 3, 1, 1)26 46,221 1,9406680 1,646 341304234 (2, 3, 1, 2)13 47,142 1,89788 252 2,020 279266135 (2, 3, 1, 3)2 45,527 1,943 109 410 3,341 5868336 (2, 3, 2, 1)5 47,467 1,735 209 120482 227343737 (2, 3, 2, 2)3 49,992 2.055 160 280 1,271 9151770 | |
| Table 1818lists the group membership of holidays and special days -ACES baseline day, Joint Planning and Development Office (JPDO) baseline day and days studied in Ref. 7. Group IDs fromTable 15 are listed under Group ID 1 heading and from Table 17 under Group ID 2 heading.Results in this table show that the domestic departure counts are generally lower on holidays.Group ID 1 (2, 2, 2) indicates that the ACES baseline day has high departure counts, moderate number of aircraft delayed on the ground, and moderate number of aircraft delayed in the air; Group ID 2 (2, 2, 1, 1) indicates high departure counts, moderate number of aircraft delayed due to weather, low number of aircraft delayed due to volume and low number of aircraft delayed due to equipment, runway and other issues.The two group IDs for the JPDO baseline day indicate high departure counts, moderate number of aircraft delayed on ground, low number of aircraft delayed in the air, low number of aircraft delayed due to weather, moderate number of aircraft delayed due to volume and high number of aircraft delayed due to equipment, runway and other conditions.Ref. 7 considered 4/17/2005 to be a low departure count, low-delay due to weather day.The group IDs in Table 18 label this day as a high departure count, low-delay due to weather day.The departure count of 40,653 on this day is at the lower end of the high departure count group can be inferred from the statistics given in | |
| Table 18 . Classification of holidays and special days.18Number DateSignificanceDay of WeekGroup ID 1Group ID 215/17/2002 ACES Baseline DayFriday(2, 2, 2)(2, 2, 1, 1)21/1/2003 New Year's DayWednesday(1, 1, 1)(1, 1, 1, 1)31/20/2003 Martin Luther King DayMonday(1, 1, 1)(1, 1, 1, 1)42/17/2003 President's DayMonday(1, 1, 1)(1, 1, 1, 1)55/26/2003 Memorial DayMonday(1, 1, 1)(1, 1, 1, 1)67/4/2003 Independence DayFriday(1, 1, 1)(1, 1, 1, 1)79/1/2003 Labor DayMonday(1, 2, 1)(1, 2, 1, 1)8 10/13/2003 Columbus DayMonday(2, 1, 1)(2, 1, 2, 1)9 11/11/2003 Veterans DayTuesday(2, 2, 1)(2, 1, 2, 1)10 11/25/2003 Two Days Before ThanksgivingTuesday(2, 3, 1)(2, 1, 3, 3)11 11/27/2003 Thanksgiving DayThursday(1, 1, 2)(1, 1, 1, 1)12 12/25/2003 Christmas DayThursday(1, 1, 1)(1, 1, 1, 1)131/1/2004 New Year's DayThursday(1, 1, 1)(1, 1, 1, 1)141/19/2004 Martin Luther King DayMonday(2, 1, 1)(2, 1, 2, 2)152/16/2004 President's DayMonday(2, 2, 2)(2, 2, 2, 2)162/19/2004 JPDO Baseline DayThursday(2, 2, 1)(2, 1, 2, 3)175/31/2004 Memorial DayMonday(1, 3, 2)(1, 3, 2, 1)184/17/2005 Ref. 7 L/L DaySunday(2, 1, 1)(2, 1, 1, 2)194/21/2005 Ref. 7 H/L DayThursday(2, 1, 2)(2, 1, 3, 2)206/5/2005 Ref. 7 L/H DaySundayNot included (1, 3, 1, 1)217/15/2005 Ref. 7 H/H DayFridayNot included (2, 3, 2, 2) | |
| Table 14 .14Classification of 4/21/2005 as high departure count, low-delay due to weather day is in agreement with Ref. 7 except that many aircraft were delayed due to volume on this day.The results in Table 18 for 6/5/2005 and 7/15/2005 are in agreement with Ref. 7. Both these days experienced an inordinate amount of airborne and ground delays due to weather.A number of aircraft were also delayed due to volume, and equipment, runway and other issues on 7/15/2005. | |
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