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Development of network restructuring models for improved air traffic forecasts
Transportation Research Part C 18 (2010) 937–949
Contents lists available at ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Development of network restructuring models for improved air traffic forecasts Tatsuya Kotegawa *, Daniel A. DeLaurentis, Aaron Sengstacken Purdue University School of Aeronautics and Astronautics, 701 W. Stadium Avenue, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
Article history: Received 16 August 2008 Received in revised form 1 July 2009 Accepted 3 March 2010
Keywords: Forecast Network theory Air traffics
a b s t r a c t Current air traffic forecast methods employed by the United States Federal Aviation Administration function under the assumption that the structure of the network of routes operated by airlines will not change; that is, no new routes will be added nor existing ones removed. However, in reality the competitive nature of the airline industry is such that new routes are routinely added between cities possessing significant passenger demand; city-pairs are also removed. Such phenomena generates a gap between the forecasted and actual state of the US Air Transportation System in the long term, providing insufficient situational awareness to major stakeholders and decision-makers in their consideration of major policy and technology changes. To address this gap, we have developed and compared three algorithms that forecast the likelihood of un-connected city-pairs being connected by service in the future, primarily based on the nodal characteristics of airports in the US network. Validation is performed by feeding historical data to each algorithm and then comparing the accuracy and precision of new city-pairs forecasted using knowledge of actual new city-pairs that developed. While an Artificial Neural Network produces superior precision, fitness function and logistic regression algorithms provide good representation of the distribution of new route types as well as greater flexibility for modeling future scenarios. However, these latter two algorithms face difficulty in resolving differences among the large number of ‘spoke’ airports in the network – additional parameters that may be able to differentiate them are currently under review. These insights gained are valuable stepping stones for exploiting knowledge of restructuring in the service route network to improve overall forecasts that drive policy and technology decision-making. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In order to synthesize long term plans for new technology, infrastructure improvements, policy enhancements, and regulations for the Air Transportation System (ATS), an understanding of air traffic dynamics—how, when, and where air traffic arises or shifts in the future—is needed. To meet this need, the US Federal Aviation Administration (FAA) Air Traffic Organization (ATO) Office of Performance Analysis and Strategy (PAS) produces air traffic forecasts which are used to project future performance, identify operational shortfalls, determine workforce requirements, and estimate the benefits of future investments (Post et al., 2008). In the current forecast algorithm, the projected schedules are based upon the assumption that the future service route network structure will remain the same as the current network structure (Schaufele, 2007). This assumption precludes the establishment of new routes and limits the ability to forecast hub formation and deletion. * Corresponding author. Tel.: +1 11 1 847 903 3055; fax: +1 11 1 765 494 0307. E-mail addresses: [email protected] (T. Kotegawa), [email protected] (D.A. DeLaurentis), [email protected] (A. Sengstacken). 0968-090X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2010.03.004
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However, the service route network structure changes over time (Kotegawa and DeLaurentis, 2007). The competitive nature of the airline industry is such that new direct routes are routinely added between cities with significant passenger demand and routes are also removed when demand diminishes. In addition, the location and number of airline hubs are not fixed; within the past several years, two major hubs have been eliminated (St. Louis and Pittsburgh), one airline hub opened and subsequently closed (Washington Dulles International Airport), and several other hubs were substantially restructured (Compart, 2009; Michels, 2007). In order to enhance the ATS forecast precision, a better understanding of restructuring dynamics is required. Motivated by this goal, the objective of the research reported in this paper is to construct algorithms that capture the mechanism of new route formation in the US ATS. Three forecast algorithms have been developed: the logistic regression, fitness function and Artificial Neural Network (ANN). Each forecast algorithm identifies un-connected city-pairs that are strong candidates for service in the future, utilizing parameters describing the ATS network topology as predictor variables. The remainder of the paper is organized as follows. Section 2 provides a brief introduction to network theory and associated network measures used within all three algorithms while Section 3 describes the data source and assumptions for all analysis. The logic and formulation of each forecast algorithm are presented in Section 4. Section 5 summarizes the results from experimentation and validation with these forecast algorithms, along with some key implications. 2. Network theory as an enabling tool Network Theory has produced powerful insights from multiple domains (e.g. physics, information, social science, biology) in recent years concerning how real world networks evolve. Some researchers have explored application for analyzing air transportation networks. Guimera et al. (2005) analyzed the worldwide air transportation network topology and computed measures which characterized the relative importance of cities/airports. Bonnefoy and Hansman (2007) used a plot of the weighted degree distribution for light jet operations to understand the capability of airports to attract the use of Very Light Jets (VLJs). A significant body of works exists in the related domain of operations research on the design of optimal networks for particular instances and applications such as scheduling for an airline (Lederer and Nambimadom, 1998; Siddappa, 2006; Taaffe et al., 1996). However, these approaches generally do not pursue insight into the underlying structure of networks, the role this structure plays in future designs, nor the interplay between networks from multiple domains. Examination of the ATS using network theory at the national level and assessment of associated analysis models and techniques as a framework to provide insight on ATS structure and useful systems analysis has been a topic for our work. The forecast of service route restructuring presented in this paper is one example application. For this particular study in which the long term formation patterns of service routes is of interest, the dynamics of the transport network, defined as the network composed of airports as the nodes interconnected by service routes (links), is analyzed in the timescale of years. Table 1 summarizes key network theory parameters that will be discussed and utilized for the remainder of this paper. The manner in which some of these parameters translate into real world performance and operations metrics is also a topic of on-going research. For example, recent work includes the estimation of delayed operations at airports as a function of the network characteristic for the corresponding airports, which showed a very strong correlation (DeLaurentis et al., 2008). The objective of work reported in this paper is to determine if network theory parameters, embedded within the three algorithms, can be utilized to identify un-connected city-pairs that are most likely to connect in the future. More broadly, the study of network topologies enables focus on the high-level characteristics of the ATS that can generate deeper understanding on the nature of the ATS without being overwhelmed by its complexity (DeLaurentis et al., 2008; DeLaurentis and Kotegawa, 2006). 3. Data source and assumptions The data used for this study is obtained from the Air Carrier Statistics database family maintained by the US Bureau of Transportation Statistics (2007), BTS. In particular, the Form 41 T-100 Domestic segment (All US Carriers) database was used to construct the network studied. The BTS monitors 2627 total airports; however, our network only contains the 887 airport nodes that had at least one annual commercial flight between 1990 and 2005. The weight on a link between nodes can be defined by several different measures such as the number of passengers, available seats, flights scheduled, or flights actually performed. For this particular study, the transport network is explored; therefore, the weight on a link is constituted by performed passenger flights per year between airports. To reduce the impact of data irregularities (mostly originating from irregularities in operations themselves), the definition of link existence is set at 24 annual flights between two nodes; links that have a weight of less than 24 annual flights are considered to be un-connected. To further simplify the analysis, the network is assumed to be undirected; traffic from node i to j and j to i are averaged to compute a single value for the weight between the two nodes. Analysis of BTS data supports the validity of this assumption since the amount of traffic flow between almost all node pairs is near equal in both flow directions; assuming an undirected network reduces computational time required for analysis significantly. By analyzing the BTS data, ATS network evolution mechanisms can be organized in four categories – addition of new service routes due to network expansion or reconfiguration growth, and removal of service routes due to network contraction or reconfiguration decay. Reconfiguration growth and decay refers to the act of re-wiring links within a fixed set of nodes; no
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Table 1 Definitions for selected network measures. Measure
Symbol and equation
General implications
Node
N/A
Represents an airport within the US ATS
Non-weighted adjacency matrix
A
Mathematical expression for a network. The matrix size depends on how many nodes compose the network – if there are n nodes in the network, the matrix size will be n n. All entries are binary, indicating whether a link between two nodes are present (‘‘1”) or not (‘‘0”). For example, if a link between node 1 and 5 exist, the entry in A(1,5) = 1
Weighted adjacency matrix
Aw
Similar to non-weighted adjacency matrix but instead of a binary value, each link has a corresponding scalar weight that signifies some distinguishing trait such as distance, and number of operations, etc.
Node degree
ki ¼
X
Aij
In the transport network, degree refers to the total number of connection node i has with other nodes
Aw ij
In the transport network, node weight represents the amount of traffic (operations) associated with the node
j
Node weight, or strength, is the total load on all its links
si ¼
X
ð1Þ
j
Link weight Eigenvector centrality, introduced by Newman (2004), used for weighted networks, is proportional to the linear combination of its neighbors’ degree and strength of links
Aw ij
xi ¼ k
The link weight refers to the amount of traffic that occurs between node i and j
X 1
Aw ij xj
ð2Þ
j
In a compact form, Eq. (2) becomes Awx = kx, an eigenvector problem where x is an eigenvector of the adjacency matrix Clustering coefficient for a given node is the number of triangles centered on that node divided by the number of triples centered on that node
Population
X 1 ci ¼ Aij Aik Ajk ki ðki 1Þ i;j
popi
ð3Þ
In the transport network, the importance of one airport is determined not only by its own number of routes supported, but also the number of routes and traffic level of airports with which it directly connects (i.e., an airport with high eigenvector centrality is likely to be very busy itself and also connected to other busy airports)
Measure of local cohesiveness for a collection of nodes; a node with higher Ci > Cavg is ‘‘more interconnected’’ than the average. This measure has implications on local robustness (or global for the average value). Higher Ci, indicates greater robustness since alternate connection paths may exist when a neighboring node fails Population within 50 mile radius of node i. Although not a network parameter, this value is used in conjunction with ki, and si, to represent nodal strength
new nodes are added and no pre-existing nodes are eliminated to create or remove a link. On the other hand, link addition and removal via network expansion and contraction occur via creating a link to nodes that are newly introduced to the network or by removing a node from the network (thereby eliminating all associated links). Fig. 1 illustrates the morphing of the ATS network categorized in these four evolution mechanism sets. All forecast models and results described in the following section only examine the mechanism of service route addition due to reconfiguration. Explicating the mechanism for the other three evolution categories described above would require another iteration of the algorithms discussed using different input datasets.
4. Network growth forecast algorithms The authors of this paper investigated various types of forecast models constructed upon different assumptions, logic and network measures (e.g. using aircraft type to extract out different networks to be fed into the forecast algorithms). Though several prototype forecast algorithms have been created and evaluated, in this paper, three of the most promising
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Fig. 1. Variation in source of ATS network topology evolution.
approaches are examined: the logistic regression model, fitness function model and the Artificial Neural Network model. A brief summary for each approach is provided next. The logistic regression is a statistical method that develops a probability curve for event occurrence based on historical data input. Here, the event for which the occurrence probability will be calculated refers to the addition of new service routes between un-connected city-pairs and the inputs consist of the network parameters of the node pair. The iteratively-reweighed least squares (IRLS) method (Guitton, 2000) is utilized as the algorithm to train the regression model with historical data. The fitness function model is based upon a network growth logic which operates under fundamentals of the ‘‘B–A model” (Albert and Barabási, 2002) where nodes with higher importance, or fitness value, are granted a higher probability to form new links with other nodes. The initial composition of the function projects growth that favors highly connected nodes (a hub-and-spoke type growth), typical in the ATS today. However, the fitness function can be modified to allow various types of network growth mechanisms corresponding to a different mixture of business models and operational constraints. Artificial Neural Network (ANN) concepts are similar to logistic regression in that an (optimal) mapping from input data to output is sought. An ANN usually has higher precision, but the relationship between input and output remains opaque – it cannot be clearly expressed as in the regression analysis. Also, due to the higher computational requirements of the ANN algorithm, the network to be analyzed via ANN must be kept relatively small. The technical approach for each of the algorithms is described in the following sub-sections, after a definition of the accuracy metrics that will be used to compare algorithmic performance. 4.1. Accuracy metrics Three accuracy metrics were constructed and employed. Accuracy 1, shown in (4), measures how many new routes the forecast algorithm is predicting in order to obtain the correct new route. If the algorithm is predicting thousands of new routes to acquire only few correct ones, accuracy 1 will be very low. On the other hand, accuracy 2, shown in (5), describes how many of the predicted new routes were correct with respect to the number of actual new routes.
Accuracy 1 ¼ number of correctly predicted routes=total number of predicted routes
ð4Þ
Accuracy 2 ¼ number of correctly predicted routes=number of actual new routes
ð5Þ
In addition to the numeric evaluation of the forecast algorithms accomplished via accuracy 1 and 2, comparison between the characteristic trends of new links in the data and forecast results is important. This aspect is addressed via accuracy 3. Characteristics of links are described by analyzing the network properties of the node pair that form the connection in two ways. The first way, termed parameter deviation, simply compares the network properties of the nodes that form the link. For example, if data shows that the difference in degree is large between most node pairs that make a connection, it can be concluded that most new connections are formed between airports with heterogeneous degree. However, only inspecting the pair-wise nodal properties provides limited insight on the characteristics of individual nodes that are being favored to form a new link both in historical data and by the forecast algorithm. Distribution of network properties for these individual nodes, referred to as parameter log, is thus also employed. By observing traits from the parameter deviation and parameter log of airport pairs that construct a new connection, the patterns in situations that facilitate new service routes can be extracted. To demonstrate the use of accuracy 3, Fig. 2 shows the actual distribution of nodal weight for airports and Fig. 3
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Fig. 2. Node weight list distribution for new service routes.
Fig. 3. Node weight list deviation for new service routes.
shows the actual distribution of nodal weight difference between node pairs that formed new service routes between 1990 and 2005. Fig. 2 shows the amount of operations for airports that constructed a new link are relatively small (between 1 and 100,000 annual operations). On the other hand, Fig. 3 shows that the difference in the amount of operations between the node pairs is also small. This implies that many of the new links occurred between small airports in terms of their amount of operations. Currently, there is no equation to determine a numerical value for accuracy 3; instead, a simple visual comparison in the distribution patterns of the parameter deviation and parameter log histograms is performed. In summary, accuracy 3 is utilized to check whether the forecast algorithms are on the right track for predicting the nature of new service routes. Even if the forecast algorithm cannot determine the exact moment and location of a new service route (low accuracy 1 and 2), high accuracy 3 traits show promising future potential for that algorithm. 4.2. Logistic regression As mentioned previously, logistic regression is a statistical technique that trains a probability curve based on historical events. For application in route forecasting, the historical data used to train the probability curve are the network parameters of nodes that were involved in a new route. This data is fed into the logistic regression model via design matrix B which ultimately gives node pairs that trend toward higher likelihood of connection. Design matrix B is structured as shown in Eq. (6) for which all network theory variables in Table 1 are included in a similar manner, along with the distance information
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between nodes i and j. The value of abs(ki kj) characterizes the degree homogeneity (low, or zero value indicates homogeneity while high value indicates heterogeneity). The particular set of field values in the design matrix that maximizes forecast accuracy is a topic of on-going research. In addition to the design matrix B, a corresponding target matrix Y is constructed in the form of Eq. (7); ri,j signifies the occurrence of a new service route construction for node i and j between observation years. If a new route is established between i and j a ‘1’ is placed in ri,j and a ‘0’ is placed otherwise. Once the design and target matrices are constructed, the IRLS method will ‘train’ the regression coefficient vector X in attempt to best map historical occurrences to the regression equation. The structure of X is shown in Eq. (8).
"
# ki kj absðki kj Þ wi B¼ . . where i ¼ 1 : n and j ¼ i þ 1 : n .. .. .. .. .. . . . " # r i;j where i ¼ 1 : n and j ¼ i þ 1 : n Y¼ . .. 0 X ¼ aki akj aabsðki kj Þ awi 1 Pconnect;ij ¼ 1 þ expðB2 XÞ
ð6Þ
ð7Þ ð8Þ ð9Þ
To forecast new city-pairs in a given network, un-connected city pair data will be structured in the format of the design matrix shown in Eq. (6). The design matrix constructed from the network to be forecasted for future connectivity is differentiated as B2 for clarification. An array for Pconnect,ij values, the probability of link formation between un-connected node pair i and j, is computed via Eq. (9). After Pconnect,ij is calculated, this value is compared to a uniformly distributed random number (rand) between 0 and 1 and the algorithm predicts a new service route construction between node i and j if Pconnect,ij > rand. 4.3. Artificial Neural Network The Artificial Neural Network (ANN) is composed of a set of interconnected neurons that mimic human brain activity. Through supervised back-propagation training techniques, an ANN is able to achieve desired input–output mapping by adjusting the weights associated with each neuronal connection in the network (Tsoukalas and Uhrig, 1997). The basic concept underlying ANN is similar to that of the logistic regression, but ANN usually has higher precision due to its higher degrees of freedom. However, the relationship between input and output for a trained ANN remains difficult to describe, unlike the logistic regression model. Also, due to the computational intensity of the ANN training algorithm, the amount of data that can be fed to the ANN (for this study using a single server) was restricted to a network size of approximately 250 nodes. In order to reduce the input data’s network size from 887 to approximately 250, nodes that had less than 40,000 cumulative arrival and departure operations or with less than 30 cumulative links between the years 1990–2005 were filtered out. This procedure removed the relatively inactive nodes in the US ATS, which ultimately reduced the network size down to 244 nodes. The MATLAB ANN toolbox was used to compute the dynamic ATS forecasting via a feed-forward, fully connected network. After training the ANN with historical BTS data, the trained ANN was used to predict connections between two airport nodes. To encapsulate the ATS dynamics, the airport metrics for the previous three years were used as the input neuron layer. The current ANN has an input layer of 63 neurons, a hidden layer consisting of 126 neurons, and a single output neuron. The input neurons represent two airport nodes and the hidden layer neurons used a tan-sig activation function, while the output neuron used a log-sig activation function. The single output neuron indicated the connectivity between the two airport nodes, 1 for connected, 0 un-connected. The training data consisted of 50% of the historical data, while 25% was used for testing and 25% for validation, based on guidance for setting up ANN procedures provided by Tsoukalas and Uhrig (1997). Clearly, our implementation of the ANN can be improved with further research, and here was used only for a subset of data of the ATS. Historical data from the American Airlines (composed of routes provided by American Airlines, American Eagle and Executive Airlines) and Southwest Airlines transport networks were employed to investigate the performance of the ANN algorithm. 4.4. Fitness function algorithm The fitness function model is an artificial network growth logic which operates under the fundamentals of preferential attachment in which nodes with higher importance, or fitness value, are granted a higher probability to construct a new link. Fig. 4 summarizes the main steps of the fitness function logic. The procedure is initialized with the previous year’s network topology. For each node in the network, a fitness value is calculated through a function of nodal metrics listed in Table 1. The initial composition of the function that computes nodal fitness is simply a ratio of the individual nodal parameter of airports and the network sum of that parameter. For example, if a particular node i has ki = 10 and the total k for the entire network is 100, its fitness function will be 10/100 = 0.1. This type of fitness function projects growth that favors highly connected and important nodes (a hub-and-spoke type growth), which is typical in the ATS today. However, the fitness function can be modified to allow various types of network growth mechanisms corresponding to a different mix of business models that
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Fig. 4. Fitness function logic.
are anticipated to emerge in the future. Subsequent to the fitness calculation, a pair-wise fitness is calculated for each node pair, and this is used to determine a probability of linking for all un-connected node pairs in the network. Links are then added to the topology based on those pairs with high link probability, though still under a randomized selection.
5. Algorithm application and performance comparison 5.1. Logistic regression results Results for an iteration of the logistic regression model are shown in Table 2; output for each year is an average over 10 runs. The model inputs consist of all the network theory parameters and deviation values for variables listed on Table 1, as well as the distance between airport pairs. Forecasts are done in a single prior-year span; that is, parameters from only the previous year are utilized for the forecast. The ‘correctly predicted’ column and ‘total prediction’ column in the table pertains to the total number of correctly predicted routes and total number of predicted routes from the forecast algorithm, respectively. The logistic regression model has relatively high accuracy 2 but low accuracy 1; the algorithm is able to correctly forecast a fair amount of new routes but forecasts too many in the process. Inspection of Table 3 for consideration of accuracy 3 indi-
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Table 2 Summary of logistic regression result. Year
Correctly predicted
Total predictions
Accuracy 1
Accuracy 2
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
N/A 83.7 88.7 68.8 80.0 146.7 82.6 78.2 62.7 71.3 113.6 52.0 388.8 128.2 120.5 104.4
N/A 1217.6 1427.6 1065.1 1365.8 1395.8 1409.8 1489.0 1177.6 1488.4 1980.2 1286.3 2328.9 1123.5 1043.1 1088.1
N/A 0.0700 0.0624 0.0658 0.0594 0.1077 0.0596 0.0535 0.0552 0.0488 0.0580 0.0410 0.1676 0.1159 0.1157 0.0970
N/A 0.4314 0.4264 0.4145 0.4645 0.3976 0.4325 0.3476 0.3968 0.4006 0.3381 0.3824 0.2878 0.2728 0.2953 0.2806
Average
111.3
1392.5
0.0785
0.3713
Table 3 Summary of logistic regression results for accuracy 3.
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cates that the parameter log and deviation distributions have some correspondence but also diverge in such a way that the historical trends are ‘over-exaggerated’ (largest value is made larger, smaller values are made smaller). The outcome of accuracy 1 and 3 was actually expected. Various historical network parameter log and deviation distributions (for nodes with new connections) of the ATS network indicated that new routes were being established between small airports (in terms of degree and centrality). Based on such input, the logistic regression model distributes higher probability to establish connections between small airports. The current ATS network is mainly a hub-and-spoke style structure with many small, spoke airports and very few large, hub airports (Conway, 2004). Since there are ultimately more small nodes in the network, many small to small airport pairs arise as candidates for service route construction. Abundant small to small airport connection candidates coupled with the forecast model favoring small to small airport connections from historical trends results in a significant number of over-predictions for small to small airport connections. This phenomenon is exposed in the accuracy metric 1 and 3 results. 5.2. Artificial Neural Network Results The trained ANN had an extremely high accuracy in predicting new service routes, with a minimum precision rate of 70%, for both Southwest and American Airlines Transport Network. Again, the ATS network used for the ANN forecast algorithm was reduced to 244 nodes due to current computational limits, where the logistic regression and fitness function considers 887 nodes. The 244 nodes included in the ANN training were the most active nodes in the ATS, excluding smaller, inactive airport nodes. Result tables, which are the averages over 10 runs for the two airline network forecasts along with translation to accuracy 1 and 2, are shown in Tables 4 and 5. The results tables are separated into four cells. The sum of rows in the table describes the forecast results by the ANN, and the sum of columns describes the actual status of the un-connected node pairs. For example in the American Airlines results table, the ANN forecasted a total of (7291 + 2788=) 10,079 new service routes. Out of the 10,079 in actuality 7291 formed connections and 2788 remained disconnected. Similarly, the ANN forecasted that (2962 + 372,357=) 375,319 node pairs will remain disconnected but historically 2962 out of the 375,319 forecasted node pairs made a connection. The overall precision results of the ANN are staggering; however it is difficult to extract any insights into the ATS evolution mechanism itself, since the trained ANN algorithm remains a black box-little insight is available as to why these trends are observed. In addition, the ANN shows very high accuracy in forecasting the false-positive node pairs (2962/375,319 = 99% accuracy). Using ANN to filter out strong false-positive candidates in conjunction with other forecast methods may produce interesting results and is a topic of current investigation. 5.3. Fitness function results Unlike the logistic regression model and the ANN approach where historical trends were directly projected to forecasting, the fitness function algorithm is a completely artificial algorithm based on insights of growth models developed from the network science community. All 325 possible combinations of the five variables (some combinations have less than five of variables) being investigated were tested and the fitness function that uses distance, degree, eigenvector centrality and nodal weight produced the forecast with highest accuracy. In this analysis, the coefficients for each variable are unweighted. Results for an iteration of this fitness function model are shown in Table 6; again, output for each year is an average over 10 runs. In comparison with the logistic regression model, the fitness function model produces poor results as measured by accuracy 1 and 2. The problem of ‘over-forecasting’ is not resolved either. Surprisingly however, the fitness function has imTable 4 Southwest Airlines Network forecast results (1990–2005). Historical data
Network Simulation
Connect Disconnect
Connect
Disconnect
2850 738
1104 380,706
Accuracy 1 = 72.08% Accuracy 2 = 79.43%
Table 5 American Airlines Network forecast results (1990–2005). Historical data
Network Simulation Accuracy 1 = 72.33% Accuracy 2 = 71.11%
Connect Disconnect
Connect
Disconnect
7291 2962
2788 372,357
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Table 6 Summary of fitness function results for accuracy 1 and 2. Year
Correctly predicted
Total predictions
Accuracy 1
Accuracy 2
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
N/A 36.9 45.3 28.5 31.1 75.3 33.6 46.7 30.9 29.8 51.6 23.3 139.5 54.7 45.4 48.1
N/A 4153.3 4328.5 4200.3 3934.8 3890.5 4183.2 4363.8 3842.3 3697.7 3887.7 4027.1 4001.1 4116.1 3960.6 3883.5
N/A 0.0091 0.0104 0.0069 0.0082 0.0201 0.0082 0.0109 0.0083 0.0081 0.0137 0.0060 0.0353 0.0138 0.0117 0.0133
N/A 0.1902 0.2178 0.1717 0.1808 0.2041 0.1759 0.2076 0.1956 0.1674 0.1536 0.1713 0.1033 0.1164 0.1113 0.1293
48.0.
4031.4
0.0123
0.1664
Average
Table 7 Summary of fitness function results for accuracy 3.
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proved accuracy 3 results especially in the parameter log histograms as seen in Table 7. The fitness function better captures the correct trend for choosing the appropriate nodes to add new routes, but the precision is relatively low, perhaps due to the large pool of new connection candidates (there are approximately 4 million un-connected node pairs to choose from). Even though accuracy 2 and 3 may be lower than the ANN or logistic regression, for an algorithm that is not dependent on historical trends, the results seem rather reasonable; in this context, it is difficult to compare the fitness function model performance with the other two algorithms that purely utilize historical trends as input.
5.4. Algorithm flexibility for future scenario network construction From the experimentation just summarized, the best forecast precision (accuracy 1 and 2) was found with the ANN, followed by the logistic regression and then fitness function algorithm. However, in terms of flexibility to modify growth rules for constructing sets of possible future scenarios (e.g. strict environmental restriction, shift in demand patterns, etc.), this ranking is completely the opposite. It is very difficult to tailor the set of equations the ANN and logistic regression model logic executes in order to simulate anticipated changes of network growth laws for such complex, non-linear forecast models that go through sophisticated training via historical data. The current logistic regression and ANN algorithms will be able to forecast the network restructuring at higher precision if and only if the ATS continues to evolve in the direction it has been evolving. However, if new policy, technology, or operation perturbations are assumed, the logistic regression and ANN will require some sort of additional training with fictional datasets that represents the ATS phase transition. With algorithms like the fitness function that is based on simpler logic, change in ATS characteristics can be more readily introduced without artificial training data by appropriately adjusting the fitness calculation to represent the phase transition in ATS.
6. Impact of network size on forecast precision In the previous sections, the abundant possibilities of small to small airport connections were described as a possible factor in lowering accuracy 1 and 2 for the fitness function and logistic regression algorithm. The ANN method, on the other hand, maintains high accuracy 1 and 2 when utilizing a network of significantly smaller size (244 nodes compared to the 887 nodes that are used for the fitness function and logistic regression). The ANN network was further downsized from the fitness function/logistic regression network by omitting nodes that have had less than 30 cumulative links or 40,000 cumulative arrival and departure traffic between 1990 and 2005. This downsized network was applied to the fitness function and logistic regression algorithm to examine whether or not there was a change of performance depending on the input data’s network size. Tables 8 and 9 show the comparative results between the non-reduced, baseline network and the downsized network. Two different approaches were taken to downsize the network. The first method (downsize method 1) omitted the network parameter data of nodes that did not satisfy the aforementioned criteria; the network structure itself remained the same as the baseline network. Downsize method 2 completely eliminates nodes that did not meet the criteria when constructing the ATS network from the BTS data. For example, in the downsize method 1 case, even if there was a link between node A and node B, but node B failed to satisfy the downsize criteria, node B will be eliminated but the degree of node A will also include the link with B. On the other hand in the down-
Table 8 Comparison of logistic regression results (downsized net). Year
Logistic regression ATS network Baseline (887 nodes) Accuracy 1 (%)
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Average
6.25 6.01 4.11 4.06 6.56 4.71 5.05 3.75 4.64 3.86 3.00 17.00 9.31 10.45 6.34
Accuracy 2 (%) 0.4191 0.3851 0.3295 0.3446 0.3149 0.367 0.3249 0.3032 0.3657 0.2387 0.3066 0.3087 0.3768 0.4265 34.37
Downsize method 1 (244 nodes)
Downsize method 2 (244 nodes)
Accuracy 1 (%)
Accuracy 2 (%)
Accuracy 1 (%)
Accuracy 2 (%)
0.00 0.14 0.22 0.06 0.81 0.29 0.15 0.28 0.92 1.52 0.53 0.00 0.40 0.49
0.00 0.91 1.43 0.54 5.00 3.64 1.85 2.11 6.89 7.71 3.55 0.00 2.86 3.60
2.18 2.93 1.91 1.83 1.14 1.99 1.53 2.15 2.72 1.60 1.22 1.42 3.90 2.35
6.93 13.24 5.32 8.32 5.98 8.65 8.47 12.02 6.71 6.88 6.39 4.19 18.44 10.59
0.42
2.86
2.06
8.72
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Table 9 Comparison of fitness function results (downsized net). Year
Fitness function ATS network Baseline (887 nodes)
Downsize method 1 (244 nodes)
Downsize method 2 (244 nodes)
Accuracy 1 (%)
Accuracy 2 (%)
Accuracy 1 (%)
Accuracy 2 (%)
Accuracy 1 (%)
Accuracy 2 (%)
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
0.91 1.04 0.69 0.82 2.01 0.82 1.09 0.83 0.81 1.37 0.60 3.53 1.38 1.17
19.02 21.78 17.17 18.08 20.41 17.59 20.76 19.56 16.74 15.36 17.14 10.33 11.64 11.13
4.05 3.22 1.92 3.01 5.12 3.66 2.97 1.99 3.26 3.63 1.61 8.87 4.31 4.68
20.43 19.86 17.30 16.51 15.00 15.74 18.68 16.47 19.52 21.25 14.44 21.81 14.30 15.89
3.34 4.19 0.98 1.48 0.65 0.18 0.34 1.43 1.46 1.75 0.75 0.13 0.59 1.18
17.86 8.06 5.28 3.45 6.73 1.48 4.71 7.03 5.71 8.59 5.81 1.05 3.73 5.26
Average
1.22
16.91
3.74
17.66
1.32
6.05
Fig. 5. Clustering coefficient distribution for networks.
size method 2 case, the degree of node A will not include the link with B, once node B is eliminated. Therefore, there will be some deviation between the network structure of the baseline and downsized network. Surprisingly, the forecast algorithms actually tended to perform worse when fed a downsized network. The question naturally arises whether this degradation of performance was due to the nature of the forecast algorithms or due to the misrepresentation of the ATS network through downsizing. Fig. 5 displays the clustering coefficient distribution of nodes between 1990 and 2004 for the two downsized and the baseline network. Besides the magnitude of the histograms, there is a significant deviation in the distribution pattern among the three networks. However, with the current understanding of networks, it is still very difficult to describe how similar or dissimilar a network is from another, especially for networks of different size and when the distribution variation exists in multiple parameters. Techniques that can further characterize the uniqueness of each network are a subject for future research. In addition, more effective scaling methods that can change the network size yet maintain its generic properties are of urgent need, not just for the study presented here, but for any future research that requires simplifying the network to reduce computational cost.
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7. Conclusion and future work Research described provided an initial examination and effort to understand the US Air Transportation System network dynamics. In particular, this research is aimed to expand the capabilities of existing air traffic forecast methods developed by the Federal Aviation Administration, which would ultimately lead to improved decision-support to maintain and enhance the Air Transportation System. Employing network theory variables and concept as a baseline to measure the air transportation network characteristics, several network growth algorithms were created in attempt to mimic the its dynamics. Logistic regression, fitness function method and the Artificial Neural Network algorithm are strong candidates although each method contains its unique strengths and drawbacks that need to be tackled. There is much room for expansion in the current network forecast capabilities portrayed in this report. First is to increase all accuracy fields for each forecast algorithm. Some proposed methods to meet this goal include the implementation of more accurate and precise air traffic data, parsing the Air Transportation System network into sub-networks such as specific aircraft class or service provider and remove variables that are insignificant to the model or cause high multi-collinearity. Forecasting not only based on the single but multiple previous years for the logistic regression and fitness function model may also increase its precision. Eventually, multiple forecast methods may be merged to go beyond the limit of individual methods. The level of economic activity both at the local and national scale also play a central role in forming demand patterns for air travel (Bhadra, 2002) which will need to be implemented to the forecast models in the near future. Second, enhanced ability to implement future scenarios will greatly enhance the value of this research. The current state of all forecast methods are essentially based under an assumption in which the future Air Transportation network will grow in the way it has in the past. However, this is not necessarily true. New type of airline services, innovative technologies as well as new regulations and policies will emerge in the future; each of these factors has the potential to drastically change the fundamental principles of the Air Transportation System. We envision future work to use forecast algorithms that implement ‘alternative scenarios’ through constructing parameter list and deviation histograms that represents the scenario of interest. These curves will be inputted to the forecast algorithm and the probability to construct new service routes will be adjusted based on those probability curves. Acknowledgments This research was sponsored by the FAA Air Traffic Organization Office of Performance Analysis and Strategy via the PARTNER Center of Excellence (awards 07-C-NE-PU-001 and 07-C-NE-PU-011). The contributions of Donald Fry, En-pei Han, Kyle Noth and Sricharan Ayyalasomayajula at Purdue University are also acknowledged and appreciated. References Albert, R., Barabási, A., 2002. Statistical mechanics of complex networks. Reviews of Modern Physics 74, 47–97. Bhadra, D., 2002. Demand for air travel in the United States: bottom-up econometric estimation and implications for forecasts by origin–destination pairs.f AIAA Aircraft Technology, Integration and Operations. Los Angeles, CA, 1–3 October 2002. Bonnefoy, P., Hansman, R., 2007. Potential impacts of very light jets in the national airspace system. Journal of Aircraft 44 (4), 1318–1326. Compart, M., 2009. American Leaving Love Field. Aviation Week. (accessed 24.06.09). Conway, S., 2004. Scale-free networks and commercial air carrier transportation in the United States. In: Congress of the International Council of the Aeronautical Sciences. Yokohama, Japan, 29 August–3 September 2004. DeLaurentis, D., Kotegawa, T., 2006. Establishment of a Network-based Simulation of Future Air Transportation Concepts. AIAA Aircraft Technology, Integration and Operations. Witchita, KS, 25–27 September 2006. DeLaurentis, D., Han, E-P., Kotegawa, T., 2008. Network-theoretic approach for analyzing connectivity in air transportation capacity networks. American Institute of Aeronautics and Astronautics Journal of Aircraft 45 (5), 1669–1679. Guimera, R., Mossa, S., Turtschi, A., Amaral, L., 2005. The worldwide air transportation network: anomalous centrality, community structure, and cities’ global roles. Proceedings of the National Academy of Sciences of the United States of America 102 (22), 7794–7799. Guitton, A., 2000. Huber Solver versus IRLS Algorithm for Quasi L1 Inversion. (accessed 18.06.09). Kotegawa, T., DeLaurentis, D., 2007. Evolution of service provider behaviors via network-based analysis. AIAA Aircraft Technology, Integration and Operations. Belfast, Northern Ireland, 18–20 September 2007. Lederer, P., Nambimadom, R., 1998. Airline network design. Operations Research 46 (6), 785–804. Michels, J., 2007. Pennsylvania Delegation Grills Parker on Philly Slots. Aviation Week. (accessed 24.06.09). Newman, M., 2004. Analysis of weighted networks. Physical Review E 70 (5), 056131. Post, P., Gulding, J., Noonan, K., Murphy, D., Bonn, J., Graham, M., 2008. The modernized national airspace system performance analysis capability. In: International Congress of the Aeronautical Sciences, Anchorage, AK, 14–19 September 2008. Schaufele, R., 2007, Terminal Area Forecast Summary. Federal Aviation Administration. (accessed 24.06.09). Siddappa, S., 2006. Statistical Modeling Approach to Airline Revenue Management with Overbooking (Doctoral dissertation, University of Texas at Arlington, 2006). Retrieved from . Taaffe, E.J., Gauthier, H.L., O’Kelly, M.E., 1996. Geography of Transportation, second ed. Prentice-Hall, Inc., Upper Saddle River, NJ. Tsoukalas, L.H., Uhrig, R.E., 1997. Fuzzy and Neural Approaches in Engineering. John Wiley and Sons, Inc., New York, NY. US Bureau of Transportation Statistics, Air Carrier Statistics (Form 41 Traffic) – US Carriers. (accessed 10.01.07).
Contents lists available at ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Development of network restructuring models for improved air traffic forecasts Tatsuya Kotegawa *, Daniel A. DeLaurentis, Aaron Sengstacken Purdue University School of Aeronautics and Astronautics, 701 W. Stadium Avenue, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
Article history: Received 16 August 2008 Received in revised form 1 July 2009 Accepted 3 March 2010
Keywords: Forecast Network theory Air traffics
a b s t r a c t Current air traffic forecast methods employed by the United States Federal Aviation Administration function under the assumption that the structure of the network of routes operated by airlines will not change; that is, no new routes will be added nor existing ones removed. However, in reality the competitive nature of the airline industry is such that new routes are routinely added between cities possessing significant passenger demand; city-pairs are also removed. Such phenomena generates a gap between the forecasted and actual state of the US Air Transportation System in the long term, providing insufficient situational awareness to major stakeholders and decision-makers in their consideration of major policy and technology changes. To address this gap, we have developed and compared three algorithms that forecast the likelihood of un-connected city-pairs being connected by service in the future, primarily based on the nodal characteristics of airports in the US network. Validation is performed by feeding historical data to each algorithm and then comparing the accuracy and precision of new city-pairs forecasted using knowledge of actual new city-pairs that developed. While an Artificial Neural Network produces superior precision, fitness function and logistic regression algorithms provide good representation of the distribution of new route types as well as greater flexibility for modeling future scenarios. However, these latter two algorithms face difficulty in resolving differences among the large number of ‘spoke’ airports in the network – additional parameters that may be able to differentiate them are currently under review. These insights gained are valuable stepping stones for exploiting knowledge of restructuring in the service route network to improve overall forecasts that drive policy and technology decision-making. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In order to synthesize long term plans for new technology, infrastructure improvements, policy enhancements, and regulations for the Air Transportation System (ATS), an understanding of air traffic dynamics—how, when, and where air traffic arises or shifts in the future—is needed. To meet this need, the US Federal Aviation Administration (FAA) Air Traffic Organization (ATO) Office of Performance Analysis and Strategy (PAS) produces air traffic forecasts which are used to project future performance, identify operational shortfalls, determine workforce requirements, and estimate the benefits of future investments (Post et al., 2008). In the current forecast algorithm, the projected schedules are based upon the assumption that the future service route network structure will remain the same as the current network structure (Schaufele, 2007). This assumption precludes the establishment of new routes and limits the ability to forecast hub formation and deletion. * Corresponding author. Tel.: +1 11 1 847 903 3055; fax: +1 11 1 765 494 0307. E-mail addresses: [email protected] (T. Kotegawa), [email protected] (D.A. DeLaurentis), [email protected] (A. Sengstacken). 0968-090X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2010.03.004
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However, the service route network structure changes over time (Kotegawa and DeLaurentis, 2007). The competitive nature of the airline industry is such that new direct routes are routinely added between cities with significant passenger demand and routes are also removed when demand diminishes. In addition, the location and number of airline hubs are not fixed; within the past several years, two major hubs have been eliminated (St. Louis and Pittsburgh), one airline hub opened and subsequently closed (Washington Dulles International Airport), and several other hubs were substantially restructured (Compart, 2009; Michels, 2007). In order to enhance the ATS forecast precision, a better understanding of restructuring dynamics is required. Motivated by this goal, the objective of the research reported in this paper is to construct algorithms that capture the mechanism of new route formation in the US ATS. Three forecast algorithms have been developed: the logistic regression, fitness function and Artificial Neural Network (ANN). Each forecast algorithm identifies un-connected city-pairs that are strong candidates for service in the future, utilizing parameters describing the ATS network topology as predictor variables. The remainder of the paper is organized as follows. Section 2 provides a brief introduction to network theory and associated network measures used within all three algorithms while Section 3 describes the data source and assumptions for all analysis. The logic and formulation of each forecast algorithm are presented in Section 4. Section 5 summarizes the results from experimentation and validation with these forecast algorithms, along with some key implications. 2. Network theory as an enabling tool Network Theory has produced powerful insights from multiple domains (e.g. physics, information, social science, biology) in recent years concerning how real world networks evolve. Some researchers have explored application for analyzing air transportation networks. Guimera et al. (2005) analyzed the worldwide air transportation network topology and computed measures which characterized the relative importance of cities/airports. Bonnefoy and Hansman (2007) used a plot of the weighted degree distribution for light jet operations to understand the capability of airports to attract the use of Very Light Jets (VLJs). A significant body of works exists in the related domain of operations research on the design of optimal networks for particular instances and applications such as scheduling for an airline (Lederer and Nambimadom, 1998; Siddappa, 2006; Taaffe et al., 1996). However, these approaches generally do not pursue insight into the underlying structure of networks, the role this structure plays in future designs, nor the interplay between networks from multiple domains. Examination of the ATS using network theory at the national level and assessment of associated analysis models and techniques as a framework to provide insight on ATS structure and useful systems analysis has been a topic for our work. The forecast of service route restructuring presented in this paper is one example application. For this particular study in which the long term formation patterns of service routes is of interest, the dynamics of the transport network, defined as the network composed of airports as the nodes interconnected by service routes (links), is analyzed in the timescale of years. Table 1 summarizes key network theory parameters that will be discussed and utilized for the remainder of this paper. The manner in which some of these parameters translate into real world performance and operations metrics is also a topic of on-going research. For example, recent work includes the estimation of delayed operations at airports as a function of the network characteristic for the corresponding airports, which showed a very strong correlation (DeLaurentis et al., 2008). The objective of work reported in this paper is to determine if network theory parameters, embedded within the three algorithms, can be utilized to identify un-connected city-pairs that are most likely to connect in the future. More broadly, the study of network topologies enables focus on the high-level characteristics of the ATS that can generate deeper understanding on the nature of the ATS without being overwhelmed by its complexity (DeLaurentis et al., 2008; DeLaurentis and Kotegawa, 2006). 3. Data source and assumptions The data used for this study is obtained from the Air Carrier Statistics database family maintained by the US Bureau of Transportation Statistics (2007), BTS. In particular, the Form 41 T-100 Domestic segment (All US Carriers) database was used to construct the network studied. The BTS monitors 2627 total airports; however, our network only contains the 887 airport nodes that had at least one annual commercial flight between 1990 and 2005. The weight on a link between nodes can be defined by several different measures such as the number of passengers, available seats, flights scheduled, or flights actually performed. For this particular study, the transport network is explored; therefore, the weight on a link is constituted by performed passenger flights per year between airports. To reduce the impact of data irregularities (mostly originating from irregularities in operations themselves), the definition of link existence is set at 24 annual flights between two nodes; links that have a weight of less than 24 annual flights are considered to be un-connected. To further simplify the analysis, the network is assumed to be undirected; traffic from node i to j and j to i are averaged to compute a single value for the weight between the two nodes. Analysis of BTS data supports the validity of this assumption since the amount of traffic flow between almost all node pairs is near equal in both flow directions; assuming an undirected network reduces computational time required for analysis significantly. By analyzing the BTS data, ATS network evolution mechanisms can be organized in four categories – addition of new service routes due to network expansion or reconfiguration growth, and removal of service routes due to network contraction or reconfiguration decay. Reconfiguration growth and decay refers to the act of re-wiring links within a fixed set of nodes; no
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Table 1 Definitions for selected network measures. Measure
Symbol and equation
General implications
Node
N/A
Represents an airport within the US ATS
Non-weighted adjacency matrix
A
Mathematical expression for a network. The matrix size depends on how many nodes compose the network – if there are n nodes in the network, the matrix size will be n n. All entries are binary, indicating whether a link between two nodes are present (‘‘1”) or not (‘‘0”). For example, if a link between node 1 and 5 exist, the entry in A(1,5) = 1
Weighted adjacency matrix
Aw
Similar to non-weighted adjacency matrix but instead of a binary value, each link has a corresponding scalar weight that signifies some distinguishing trait such as distance, and number of operations, etc.
Node degree
ki ¼
X
Aij
In the transport network, degree refers to the total number of connection node i has with other nodes
Aw ij
In the transport network, node weight represents the amount of traffic (operations) associated with the node
j
Node weight, or strength, is the total load on all its links
si ¼
X
ð1Þ
j
Link weight Eigenvector centrality, introduced by Newman (2004), used for weighted networks, is proportional to the linear combination of its neighbors’ degree and strength of links
Aw ij
xi ¼ k
The link weight refers to the amount of traffic that occurs between node i and j
X 1
Aw ij xj
ð2Þ
j
In a compact form, Eq. (2) becomes Awx = kx, an eigenvector problem where x is an eigenvector of the adjacency matrix Clustering coefficient for a given node is the number of triangles centered on that node divided by the number of triples centered on that node
Population
X 1 ci ¼ Aij Aik Ajk ki ðki 1Þ i;j
popi
ð3Þ
In the transport network, the importance of one airport is determined not only by its own number of routes supported, but also the number of routes and traffic level of airports with which it directly connects (i.e., an airport with high eigenvector centrality is likely to be very busy itself and also connected to other busy airports)
Measure of local cohesiveness for a collection of nodes; a node with higher Ci > Cavg is ‘‘more interconnected’’ than the average. This measure has implications on local robustness (or global for the average value). Higher Ci, indicates greater robustness since alternate connection paths may exist when a neighboring node fails Population within 50 mile radius of node i. Although not a network parameter, this value is used in conjunction with ki, and si, to represent nodal strength
new nodes are added and no pre-existing nodes are eliminated to create or remove a link. On the other hand, link addition and removal via network expansion and contraction occur via creating a link to nodes that are newly introduced to the network or by removing a node from the network (thereby eliminating all associated links). Fig. 1 illustrates the morphing of the ATS network categorized in these four evolution mechanism sets. All forecast models and results described in the following section only examine the mechanism of service route addition due to reconfiguration. Explicating the mechanism for the other three evolution categories described above would require another iteration of the algorithms discussed using different input datasets.
4. Network growth forecast algorithms The authors of this paper investigated various types of forecast models constructed upon different assumptions, logic and network measures (e.g. using aircraft type to extract out different networks to be fed into the forecast algorithms). Though several prototype forecast algorithms have been created and evaluated, in this paper, three of the most promising
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Fig. 1. Variation in source of ATS network topology evolution.
approaches are examined: the logistic regression model, fitness function model and the Artificial Neural Network model. A brief summary for each approach is provided next. The logistic regression is a statistical method that develops a probability curve for event occurrence based on historical data input. Here, the event for which the occurrence probability will be calculated refers to the addition of new service routes between un-connected city-pairs and the inputs consist of the network parameters of the node pair. The iteratively-reweighed least squares (IRLS) method (Guitton, 2000) is utilized as the algorithm to train the regression model with historical data. The fitness function model is based upon a network growth logic which operates under fundamentals of the ‘‘B–A model” (Albert and Barabási, 2002) where nodes with higher importance, or fitness value, are granted a higher probability to form new links with other nodes. The initial composition of the function projects growth that favors highly connected nodes (a hub-and-spoke type growth), typical in the ATS today. However, the fitness function can be modified to allow various types of network growth mechanisms corresponding to a different mixture of business models and operational constraints. Artificial Neural Network (ANN) concepts are similar to logistic regression in that an (optimal) mapping from input data to output is sought. An ANN usually has higher precision, but the relationship between input and output remains opaque – it cannot be clearly expressed as in the regression analysis. Also, due to the higher computational requirements of the ANN algorithm, the network to be analyzed via ANN must be kept relatively small. The technical approach for each of the algorithms is described in the following sub-sections, after a definition of the accuracy metrics that will be used to compare algorithmic performance. 4.1. Accuracy metrics Three accuracy metrics were constructed and employed. Accuracy 1, shown in (4), measures how many new routes the forecast algorithm is predicting in order to obtain the correct new route. If the algorithm is predicting thousands of new routes to acquire only few correct ones, accuracy 1 will be very low. On the other hand, accuracy 2, shown in (5), describes how many of the predicted new routes were correct with respect to the number of actual new routes.
Accuracy 1 ¼ number of correctly predicted routes=total number of predicted routes
ð4Þ
Accuracy 2 ¼ number of correctly predicted routes=number of actual new routes
ð5Þ
In addition to the numeric evaluation of the forecast algorithms accomplished via accuracy 1 and 2, comparison between the characteristic trends of new links in the data and forecast results is important. This aspect is addressed via accuracy 3. Characteristics of links are described by analyzing the network properties of the node pair that form the connection in two ways. The first way, termed parameter deviation, simply compares the network properties of the nodes that form the link. For example, if data shows that the difference in degree is large between most node pairs that make a connection, it can be concluded that most new connections are formed between airports with heterogeneous degree. However, only inspecting the pair-wise nodal properties provides limited insight on the characteristics of individual nodes that are being favored to form a new link both in historical data and by the forecast algorithm. Distribution of network properties for these individual nodes, referred to as parameter log, is thus also employed. By observing traits from the parameter deviation and parameter log of airport pairs that construct a new connection, the patterns in situations that facilitate new service routes can be extracted. To demonstrate the use of accuracy 3, Fig. 2 shows the actual distribution of nodal weight for airports and Fig. 3
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Fig. 2. Node weight list distribution for new service routes.
Fig. 3. Node weight list deviation for new service routes.
shows the actual distribution of nodal weight difference between node pairs that formed new service routes between 1990 and 2005. Fig. 2 shows the amount of operations for airports that constructed a new link are relatively small (between 1 and 100,000 annual operations). On the other hand, Fig. 3 shows that the difference in the amount of operations between the node pairs is also small. This implies that many of the new links occurred between small airports in terms of their amount of operations. Currently, there is no equation to determine a numerical value for accuracy 3; instead, a simple visual comparison in the distribution patterns of the parameter deviation and parameter log histograms is performed. In summary, accuracy 3 is utilized to check whether the forecast algorithms are on the right track for predicting the nature of new service routes. Even if the forecast algorithm cannot determine the exact moment and location of a new service route (low accuracy 1 and 2), high accuracy 3 traits show promising future potential for that algorithm. 4.2. Logistic regression As mentioned previously, logistic regression is a statistical technique that trains a probability curve based on historical events. For application in route forecasting, the historical data used to train the probability curve are the network parameters of nodes that were involved in a new route. This data is fed into the logistic regression model via design matrix B which ultimately gives node pairs that trend toward higher likelihood of connection. Design matrix B is structured as shown in Eq. (6) for which all network theory variables in Table 1 are included in a similar manner, along with the distance information
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between nodes i and j. The value of abs(ki kj) characterizes the degree homogeneity (low, or zero value indicates homogeneity while high value indicates heterogeneity). The particular set of field values in the design matrix that maximizes forecast accuracy is a topic of on-going research. In addition to the design matrix B, a corresponding target matrix Y is constructed in the form of Eq. (7); ri,j signifies the occurrence of a new service route construction for node i and j between observation years. If a new route is established between i and j a ‘1’ is placed in ri,j and a ‘0’ is placed otherwise. Once the design and target matrices are constructed, the IRLS method will ‘train’ the regression coefficient vector X in attempt to best map historical occurrences to the regression equation. The structure of X is shown in Eq. (8).
"
# ki kj absðki kj Þ wi B¼ . . where i ¼ 1 : n and j ¼ i þ 1 : n .. .. .. .. .. . . . " # r i;j where i ¼ 1 : n and j ¼ i þ 1 : n Y¼ . .. 0 X ¼ aki akj aabsðki kj Þ awi 1 Pconnect;ij ¼ 1 þ expðB2 XÞ
ð6Þ
ð7Þ ð8Þ ð9Þ
To forecast new city-pairs in a given network, un-connected city pair data will be structured in the format of the design matrix shown in Eq. (6). The design matrix constructed from the network to be forecasted for future connectivity is differentiated as B2 for clarification. An array for Pconnect,ij values, the probability of link formation between un-connected node pair i and j, is computed via Eq. (9). After Pconnect,ij is calculated, this value is compared to a uniformly distributed random number (rand) between 0 and 1 and the algorithm predicts a new service route construction between node i and j if Pconnect,ij > rand. 4.3. Artificial Neural Network The Artificial Neural Network (ANN) is composed of a set of interconnected neurons that mimic human brain activity. Through supervised back-propagation training techniques, an ANN is able to achieve desired input–output mapping by adjusting the weights associated with each neuronal connection in the network (Tsoukalas and Uhrig, 1997). The basic concept underlying ANN is similar to that of the logistic regression, but ANN usually has higher precision due to its higher degrees of freedom. However, the relationship between input and output for a trained ANN remains difficult to describe, unlike the logistic regression model. Also, due to the computational intensity of the ANN training algorithm, the amount of data that can be fed to the ANN (for this study using a single server) was restricted to a network size of approximately 250 nodes. In order to reduce the input data’s network size from 887 to approximately 250, nodes that had less than 40,000 cumulative arrival and departure operations or with less than 30 cumulative links between the years 1990–2005 were filtered out. This procedure removed the relatively inactive nodes in the US ATS, which ultimately reduced the network size down to 244 nodes. The MATLAB ANN toolbox was used to compute the dynamic ATS forecasting via a feed-forward, fully connected network. After training the ANN with historical BTS data, the trained ANN was used to predict connections between two airport nodes. To encapsulate the ATS dynamics, the airport metrics for the previous three years were used as the input neuron layer. The current ANN has an input layer of 63 neurons, a hidden layer consisting of 126 neurons, and a single output neuron. The input neurons represent two airport nodes and the hidden layer neurons used a tan-sig activation function, while the output neuron used a log-sig activation function. The single output neuron indicated the connectivity between the two airport nodes, 1 for connected, 0 un-connected. The training data consisted of 50% of the historical data, while 25% was used for testing and 25% for validation, based on guidance for setting up ANN procedures provided by Tsoukalas and Uhrig (1997). Clearly, our implementation of the ANN can be improved with further research, and here was used only for a subset of data of the ATS. Historical data from the American Airlines (composed of routes provided by American Airlines, American Eagle and Executive Airlines) and Southwest Airlines transport networks were employed to investigate the performance of the ANN algorithm. 4.4. Fitness function algorithm The fitness function model is an artificial network growth logic which operates under the fundamentals of preferential attachment in which nodes with higher importance, or fitness value, are granted a higher probability to construct a new link. Fig. 4 summarizes the main steps of the fitness function logic. The procedure is initialized with the previous year’s network topology. For each node in the network, a fitness value is calculated through a function of nodal metrics listed in Table 1. The initial composition of the function that computes nodal fitness is simply a ratio of the individual nodal parameter of airports and the network sum of that parameter. For example, if a particular node i has ki = 10 and the total k for the entire network is 100, its fitness function will be 10/100 = 0.1. This type of fitness function projects growth that favors highly connected and important nodes (a hub-and-spoke type growth), which is typical in the ATS today. However, the fitness function can be modified to allow various types of network growth mechanisms corresponding to a different mix of business models that
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Fig. 4. Fitness function logic.
are anticipated to emerge in the future. Subsequent to the fitness calculation, a pair-wise fitness is calculated for each node pair, and this is used to determine a probability of linking for all un-connected node pairs in the network. Links are then added to the topology based on those pairs with high link probability, though still under a randomized selection.
5. Algorithm application and performance comparison 5.1. Logistic regression results Results for an iteration of the logistic regression model are shown in Table 2; output for each year is an average over 10 runs. The model inputs consist of all the network theory parameters and deviation values for variables listed on Table 1, as well as the distance between airport pairs. Forecasts are done in a single prior-year span; that is, parameters from only the previous year are utilized for the forecast. The ‘correctly predicted’ column and ‘total prediction’ column in the table pertains to the total number of correctly predicted routes and total number of predicted routes from the forecast algorithm, respectively. The logistic regression model has relatively high accuracy 2 but low accuracy 1; the algorithm is able to correctly forecast a fair amount of new routes but forecasts too many in the process. Inspection of Table 3 for consideration of accuracy 3 indi-
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Table 2 Summary of logistic regression result. Year
Correctly predicted
Total predictions
Accuracy 1
Accuracy 2
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
N/A 83.7 88.7 68.8 80.0 146.7 82.6 78.2 62.7 71.3 113.6 52.0 388.8 128.2 120.5 104.4
N/A 1217.6 1427.6 1065.1 1365.8 1395.8 1409.8 1489.0 1177.6 1488.4 1980.2 1286.3 2328.9 1123.5 1043.1 1088.1
N/A 0.0700 0.0624 0.0658 0.0594 0.1077 0.0596 0.0535 0.0552 0.0488 0.0580 0.0410 0.1676 0.1159 0.1157 0.0970
N/A 0.4314 0.4264 0.4145 0.4645 0.3976 0.4325 0.3476 0.3968 0.4006 0.3381 0.3824 0.2878 0.2728 0.2953 0.2806
Average
111.3
1392.5
0.0785
0.3713
Table 3 Summary of logistic regression results for accuracy 3.
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cates that the parameter log and deviation distributions have some correspondence but also diverge in such a way that the historical trends are ‘over-exaggerated’ (largest value is made larger, smaller values are made smaller). The outcome of accuracy 1 and 3 was actually expected. Various historical network parameter log and deviation distributions (for nodes with new connections) of the ATS network indicated that new routes were being established between small airports (in terms of degree and centrality). Based on such input, the logistic regression model distributes higher probability to establish connections between small airports. The current ATS network is mainly a hub-and-spoke style structure with many small, spoke airports and very few large, hub airports (Conway, 2004). Since there are ultimately more small nodes in the network, many small to small airport pairs arise as candidates for service route construction. Abundant small to small airport connection candidates coupled with the forecast model favoring small to small airport connections from historical trends results in a significant number of over-predictions for small to small airport connections. This phenomenon is exposed in the accuracy metric 1 and 3 results. 5.2. Artificial Neural Network Results The trained ANN had an extremely high accuracy in predicting new service routes, with a minimum precision rate of 70%, for both Southwest and American Airlines Transport Network. Again, the ATS network used for the ANN forecast algorithm was reduced to 244 nodes due to current computational limits, where the logistic regression and fitness function considers 887 nodes. The 244 nodes included in the ANN training were the most active nodes in the ATS, excluding smaller, inactive airport nodes. Result tables, which are the averages over 10 runs for the two airline network forecasts along with translation to accuracy 1 and 2, are shown in Tables 4 and 5. The results tables are separated into four cells. The sum of rows in the table describes the forecast results by the ANN, and the sum of columns describes the actual status of the un-connected node pairs. For example in the American Airlines results table, the ANN forecasted a total of (7291 + 2788=) 10,079 new service routes. Out of the 10,079 in actuality 7291 formed connections and 2788 remained disconnected. Similarly, the ANN forecasted that (2962 + 372,357=) 375,319 node pairs will remain disconnected but historically 2962 out of the 375,319 forecasted node pairs made a connection. The overall precision results of the ANN are staggering; however it is difficult to extract any insights into the ATS evolution mechanism itself, since the trained ANN algorithm remains a black box-little insight is available as to why these trends are observed. In addition, the ANN shows very high accuracy in forecasting the false-positive node pairs (2962/375,319 = 99% accuracy). Using ANN to filter out strong false-positive candidates in conjunction with other forecast methods may produce interesting results and is a topic of current investigation. 5.3. Fitness function results Unlike the logistic regression model and the ANN approach where historical trends were directly projected to forecasting, the fitness function algorithm is a completely artificial algorithm based on insights of growth models developed from the network science community. All 325 possible combinations of the five variables (some combinations have less than five of variables) being investigated were tested and the fitness function that uses distance, degree, eigenvector centrality and nodal weight produced the forecast with highest accuracy. In this analysis, the coefficients for each variable are unweighted. Results for an iteration of this fitness function model are shown in Table 6; again, output for each year is an average over 10 runs. In comparison with the logistic regression model, the fitness function model produces poor results as measured by accuracy 1 and 2. The problem of ‘over-forecasting’ is not resolved either. Surprisingly however, the fitness function has imTable 4 Southwest Airlines Network forecast results (1990–2005). Historical data
Network Simulation
Connect Disconnect
Connect
Disconnect
2850 738
1104 380,706
Accuracy 1 = 72.08% Accuracy 2 = 79.43%
Table 5 American Airlines Network forecast results (1990–2005). Historical data
Network Simulation Accuracy 1 = 72.33% Accuracy 2 = 71.11%
Connect Disconnect
Connect
Disconnect
7291 2962
2788 372,357
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Table 6 Summary of fitness function results for accuracy 1 and 2. Year
Correctly predicted
Total predictions
Accuracy 1
Accuracy 2
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
N/A 36.9 45.3 28.5 31.1 75.3 33.6 46.7 30.9 29.8 51.6 23.3 139.5 54.7 45.4 48.1
N/A 4153.3 4328.5 4200.3 3934.8 3890.5 4183.2 4363.8 3842.3 3697.7 3887.7 4027.1 4001.1 4116.1 3960.6 3883.5
N/A 0.0091 0.0104 0.0069 0.0082 0.0201 0.0082 0.0109 0.0083 0.0081 0.0137 0.0060 0.0353 0.0138 0.0117 0.0133
N/A 0.1902 0.2178 0.1717 0.1808 0.2041 0.1759 0.2076 0.1956 0.1674 0.1536 0.1713 0.1033 0.1164 0.1113 0.1293
48.0.
4031.4
0.0123
0.1664
Average
Table 7 Summary of fitness function results for accuracy 3.
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proved accuracy 3 results especially in the parameter log histograms as seen in Table 7. The fitness function better captures the correct trend for choosing the appropriate nodes to add new routes, but the precision is relatively low, perhaps due to the large pool of new connection candidates (there are approximately 4 million un-connected node pairs to choose from). Even though accuracy 2 and 3 may be lower than the ANN or logistic regression, for an algorithm that is not dependent on historical trends, the results seem rather reasonable; in this context, it is difficult to compare the fitness function model performance with the other two algorithms that purely utilize historical trends as input.
5.4. Algorithm flexibility for future scenario network construction From the experimentation just summarized, the best forecast precision (accuracy 1 and 2) was found with the ANN, followed by the logistic regression and then fitness function algorithm. However, in terms of flexibility to modify growth rules for constructing sets of possible future scenarios (e.g. strict environmental restriction, shift in demand patterns, etc.), this ranking is completely the opposite. It is very difficult to tailor the set of equations the ANN and logistic regression model logic executes in order to simulate anticipated changes of network growth laws for such complex, non-linear forecast models that go through sophisticated training via historical data. The current logistic regression and ANN algorithms will be able to forecast the network restructuring at higher precision if and only if the ATS continues to evolve in the direction it has been evolving. However, if new policy, technology, or operation perturbations are assumed, the logistic regression and ANN will require some sort of additional training with fictional datasets that represents the ATS phase transition. With algorithms like the fitness function that is based on simpler logic, change in ATS characteristics can be more readily introduced without artificial training data by appropriately adjusting the fitness calculation to represent the phase transition in ATS.
6. Impact of network size on forecast precision In the previous sections, the abundant possibilities of small to small airport connections were described as a possible factor in lowering accuracy 1 and 2 for the fitness function and logistic regression algorithm. The ANN method, on the other hand, maintains high accuracy 1 and 2 when utilizing a network of significantly smaller size (244 nodes compared to the 887 nodes that are used for the fitness function and logistic regression). The ANN network was further downsized from the fitness function/logistic regression network by omitting nodes that have had less than 30 cumulative links or 40,000 cumulative arrival and departure traffic between 1990 and 2005. This downsized network was applied to the fitness function and logistic regression algorithm to examine whether or not there was a change of performance depending on the input data’s network size. Tables 8 and 9 show the comparative results between the non-reduced, baseline network and the downsized network. Two different approaches were taken to downsize the network. The first method (downsize method 1) omitted the network parameter data of nodes that did not satisfy the aforementioned criteria; the network structure itself remained the same as the baseline network. Downsize method 2 completely eliminates nodes that did not meet the criteria when constructing the ATS network from the BTS data. For example, in the downsize method 1 case, even if there was a link between node A and node B, but node B failed to satisfy the downsize criteria, node B will be eliminated but the degree of node A will also include the link with B. On the other hand in the down-
Table 8 Comparison of logistic regression results (downsized net). Year
Logistic regression ATS network Baseline (887 nodes) Accuracy 1 (%)
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Average
6.25 6.01 4.11 4.06 6.56 4.71 5.05 3.75 4.64 3.86 3.00 17.00 9.31 10.45 6.34
Accuracy 2 (%) 0.4191 0.3851 0.3295 0.3446 0.3149 0.367 0.3249 0.3032 0.3657 0.2387 0.3066 0.3087 0.3768 0.4265 34.37
Downsize method 1 (244 nodes)
Downsize method 2 (244 nodes)
Accuracy 1 (%)
Accuracy 2 (%)
Accuracy 1 (%)
Accuracy 2 (%)
0.00 0.14 0.22 0.06 0.81 0.29 0.15 0.28 0.92 1.52 0.53 0.00 0.40 0.49
0.00 0.91 1.43 0.54 5.00 3.64 1.85 2.11 6.89 7.71 3.55 0.00 2.86 3.60
2.18 2.93 1.91 1.83 1.14 1.99 1.53 2.15 2.72 1.60 1.22 1.42 3.90 2.35
6.93 13.24 5.32 8.32 5.98 8.65 8.47 12.02 6.71 6.88 6.39 4.19 18.44 10.59
0.42
2.86
2.06
8.72
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Table 9 Comparison of fitness function results (downsized net). Year
Fitness function ATS network Baseline (887 nodes)
Downsize method 1 (244 nodes)
Downsize method 2 (244 nodes)
Accuracy 1 (%)
Accuracy 2 (%)
Accuracy 1 (%)
Accuracy 2 (%)
Accuracy 1 (%)
Accuracy 2 (%)
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
0.91 1.04 0.69 0.82 2.01 0.82 1.09 0.83 0.81 1.37 0.60 3.53 1.38 1.17
19.02 21.78 17.17 18.08 20.41 17.59 20.76 19.56 16.74 15.36 17.14 10.33 11.64 11.13
4.05 3.22 1.92 3.01 5.12 3.66 2.97 1.99 3.26 3.63 1.61 8.87 4.31 4.68
20.43 19.86 17.30 16.51 15.00 15.74 18.68 16.47 19.52 21.25 14.44 21.81 14.30 15.89
3.34 4.19 0.98 1.48 0.65 0.18 0.34 1.43 1.46 1.75 0.75 0.13 0.59 1.18
17.86 8.06 5.28 3.45 6.73 1.48 4.71 7.03 5.71 8.59 5.81 1.05 3.73 5.26
Average
1.22
16.91
3.74
17.66
1.32
6.05
Fig. 5. Clustering coefficient distribution for networks.
size method 2 case, the degree of node A will not include the link with B, once node B is eliminated. Therefore, there will be some deviation between the network structure of the baseline and downsized network. Surprisingly, the forecast algorithms actually tended to perform worse when fed a downsized network. The question naturally arises whether this degradation of performance was due to the nature of the forecast algorithms or due to the misrepresentation of the ATS network through downsizing. Fig. 5 displays the clustering coefficient distribution of nodes between 1990 and 2004 for the two downsized and the baseline network. Besides the magnitude of the histograms, there is a significant deviation in the distribution pattern among the three networks. However, with the current understanding of networks, it is still very difficult to describe how similar or dissimilar a network is from another, especially for networks of different size and when the distribution variation exists in multiple parameters. Techniques that can further characterize the uniqueness of each network are a subject for future research. In addition, more effective scaling methods that can change the network size yet maintain its generic properties are of urgent need, not just for the study presented here, but for any future research that requires simplifying the network to reduce computational cost.
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7. Conclusion and future work Research described provided an initial examination and effort to understand the US Air Transportation System network dynamics. In particular, this research is aimed to expand the capabilities of existing air traffic forecast methods developed by the Federal Aviation Administration, which would ultimately lead to improved decision-support to maintain and enhance the Air Transportation System. Employing network theory variables and concept as a baseline to measure the air transportation network characteristics, several network growth algorithms were created in attempt to mimic the its dynamics. Logistic regression, fitness function method and the Artificial Neural Network algorithm are strong candidates although each method contains its unique strengths and drawbacks that need to be tackled. There is much room for expansion in the current network forecast capabilities portrayed in this report. First is to increase all accuracy fields for each forecast algorithm. Some proposed methods to meet this goal include the implementation of more accurate and precise air traffic data, parsing the Air Transportation System network into sub-networks such as specific aircraft class or service provider and remove variables that are insignificant to the model or cause high multi-collinearity. Forecasting not only based on the single but multiple previous years for the logistic regression and fitness function model may also increase its precision. Eventually, multiple forecast methods may be merged to go beyond the limit of individual methods. The level of economic activity both at the local and national scale also play a central role in forming demand patterns for air travel (Bhadra, 2002) which will need to be implemented to the forecast models in the near future. Second, enhanced ability to implement future scenarios will greatly enhance the value of this research. The current state of all forecast methods are essentially based under an assumption in which the future Air Transportation network will grow in the way it has in the past. However, this is not necessarily true. New type of airline services, innovative technologies as well as new regulations and policies will emerge in the future; each of these factors has the potential to drastically change the fundamental principles of the Air Transportation System. We envision future work to use forecast algorithms that implement ‘alternative scenarios’ through constructing parameter list and deviation histograms that represents the scenario of interest. These curves will be inputted to the forecast algorithm and the probability to construct new service routes will be adjusted based on those probability curves. Acknowledgments This research was sponsored by the FAA Air Traffic Organization Office of Performance Analysis and Strategy via the PARTNER Center of Excellence (awards 07-C-NE-PU-001 and 07-C-NE-PU-011). The contributions of Donald Fry, En-pei Han, Kyle Noth and Sricharan Ayyalasomayajula at Purdue University are also acknowledged and appreciated. References Albert, R., Barabási, A., 2002. Statistical mechanics of complex networks. Reviews of Modern Physics 74, 47–97. Bhadra, D., 2002. Demand for air travel in the United States: bottom-up econometric estimation and implications for forecasts by origin–destination pairs.f AIAA Aircraft Technology, Integration and Operations. Los Angeles, CA, 1–3 October 2002. Bonnefoy, P., Hansman, R., 2007. Potential impacts of very light jets in the national airspace system. Journal of Aircraft 44 (4), 1318–1326. Compart, M., 2009. American Leaving Love Field. Aviation Week.