Gaussian Mixture Modelling for Grouping Traffic Patterns Using Automatic Traffic Recorder Data

Transcription

1 0 0 Gaussian Mixture Modelling for Grouping Traffic Patterns Using Automatic Traffic Recorder Data Sunil K. Madanu Graduate Research Assistant Department of Civil Engineering, the University of Texas at Arlington Nedderman Hall B, Yates Street, Arlington, TX 0 sunil.madanu@mavs.uta.edu Stephen P. Mattingly, Corresponding Author Associate Professor Department of Civil Engineering, the University of Texas at Arlington Nedderman Hall, Yates Street, Arlington, TX 0 Tel: --; Fax: --0; mattingly@uta.edu Christina A. McDaniel-Wilson, P.E. Oregon Department of Transportation Planning Analysis Unit th St. NE, Ste, Salem, OR 0 Tel: Christina.A.MCDANIEL-WILSON@odot.state.or.us Word count:,00 words for text + tables/figures x 0 words (each) =,00 words

2 Madanu, Mattingly, and McDaniel-Wilson 0 0 ABSTRACT The grouping of similar traffic patterns plays an important role while estimating Annual Average Daily Traffic (AADT) values from Short Term Traffic Counts (STTCs). Incorrect grouping often becomes a significant source of AADT estimation errors. For instance, grouping a commuter traffic trend pattern into a recreational traffic trend may produce an erroneous AADT value. The traditional knowledge based methods, often aided with visual interpretation, introduce subjective bias while grouping traffic patterns. In addition, the grouping requires additional time to process large amounts of data, and remains inefficient with unapparent traffic patterns. Under limited resources and constraints, better methods and techniques may group sites with similar characteristics. The paper presents a refined Gaussian Mixture Modelling (GMM) for clustering continuous count data. The number of mixture components identified using Bayesian Information Criteria (BIC) from mixture modelling leads to overestimation if the number of components is interpreted as the number of clusters. Hence, the authors adopt a hierarchical combination of the initial mixture components using an entropy criterion to identify the number of clusters. The researchers compare this modified clustering solution with model based agglomerative hierarchical clustering, volume factor grouping and a traditional approach. The study uses Automatic Traffic Recorder (ATR) data from the Oregon Department of Transportation (ODOT) as a comparative case study. Overall, the GMM clustering solution shows a lower average monthly and average daily Mean Absolute Percent Error (MAPE) than other methods; however, for summer months, the ODOT method shows less MAPE than the GMM solution. Keywords: Clustering Analysis, Gaussian Mixture Model, AADT, Grouping, Model-based Clustering, Short Term Traffic Count, Seasonal Traffic Trend Grouping, ATR, Automatic Traffic Recorder

3 Madanu, Mattingly, and McDaniel-Wilson 0 0 INTRODUCTION The state Departments of Transportation (DOTs) allocate significant resources on collecting historic traffic data throughout the state-wide networks. The estimation of Annual Average Daily Traffic (AADT) has many practical applications in planning, operations, maintenance, and decision making. The literature proposes a wide range of techniques like regression analysis, geographically weighted regression, artificial neural networks, time-series analysis, genetic algorithms, and kriging-based methods (see references () and () for review). According to the Federal Highway Administration (FHWA) Traffic Monitoring Guide (TMG), monitoring traffic volume trends represents a key task for continuous traffic count program (). The deployment of Automatic Traffic Recorders (ATR) on a state-wide network helps state DOTs to collect and monitor traffic patterns. State agencies face a tough decision on how many ATRs to deploy, where to deploy, and how frequently to collect the data with limited resources. In addition, decisions on the type of traffic patterns to monitor, either by vehicle type, monthly, day of week, or hourly distribution complicates the monitoring process. The FHWA and state agencies have developed some guidelines for the Traffic Monitoring Analysis System (TMAS) to evaluate volume trends over a specified time period. Moreover, monitoring the AADT and its trends requires continuous data from only at a limited number of locations. In order to cover a specific location of interest, Short Term Traffic Counts (STTC) are taken and seasonally adjusted using factor groups (). Factor groups have reasonably homogeneous patterns, usually but not necessarily, calculated based on monthly traffic patterns. In addition, the TMG also suggests day of week, hourly patterns, and patterns by vehicle type (passenger or trucks) or geographic region. Many state DOTs group traffic patterns from ATR sites and calculate AADT values using seasonal factors (). In fact, the TMG outlines three types of analysis for grouping: Traditional Approach, Cluster Analysis, and Volume Factor Grouping. The traditional approach uses general knowledge of the road system with visual interpretation to identify groups. Cluster analysis is a procedure to group the patterns, often, using monthly factors (ratio of AADT to MADT-Monthly Average Daily Traffic) at continuous count stations. Volume factor grouping maintains separate volume factor groups by highway functional category. Finding groups through knowledge and functional class seems neither practical nor likely to produce better results because of large amount of continuous count data, and dynamic changes in travel activity patterns (irrespective of highway functional class). In addition, bias due to subjectivity, difficulty in analyzing these large datasets, and significant time resource requirements elevate the problems of the conventional methods (). A few alternative methods, generally labeled as clustering techniques, have evolved for the automatic grouping of traffic patterns. The paper aims to provide a modified clustering process using Gaussian Mixture Modelling (GMM). The study adopts an entropy difference criterion to systematically select clustering solutions using a fitted piecewise linear regression model. A hierarchical cluster combination method is introduced to select the number of clusters. In addition, the GMM solution is compared with model based agglomerative hierarchical clustering, traditional approach and volume factor grouping to assess its relative performance. After the background information, the paper outlines Gaussian mixture modelling and the cluster selection process. The following section describes the classification of test data according to the

4 Madanu, Mattingly, and McDaniel-Wilson 0 0 cluster solutions. Finally, the paper presents a comparative case study of clustering solutions for estimating AADT values. BACKGROUND Clustering techniques try to determine the structure of data when no information is available except observational data. Cluster analysis partitions the data into meaningful subgroups without knowing its components and structure. Cluster analysis is broadly divided into heuristic methods and statistical models, which follow either hierarchical or relocation strategies (). Hierarchical methods produce clusters in stages either using an agglomerative (merging) or divisive (splitting) method at each stage. The agglomerative hierarchical clustering treats each observation as a cluster by itself at the beginning and merges observations at each stage using a variety of methods, like single link (nearest neighbor), complete link (farthest neighbor), and Ward s distance. Model based hierarchical agglomerative clustering, on other hand, uses maximum-likelihood criteria for merging observations into groups (). A drawback of agglomerative methods is that they do not address the issue of determining the number of clusters as a result the criteria for merging and selecting clusters will produce different classifications from agglomerative clustering (). Relocation based methods assign the observations iteratively among the groups. The number of clusters or groups has to be specified in advance, and they do not change during the course of iterations. However, at each iteration, observations move from one group to other groups, usually, using some form of distance criteria. K-means clustering and finite mixture modelling represent two examples of relocation based methods. K-means clustering selects the number of clusters that minimize within-group variance (). Mixture modelling involves probability based cluster analysis, where observational data is assumed to come from a mixture of probability distributions. The Bayesian Information Criterion (BIC) is used to determine the number of components and model form for clustering. Often, mixture components are modelled to follow a Gaussian distribution where its maximum likelihood parameters are found using an Expected- Maximization (EM) algorithm (). Past research has categorized ATR data using a variety of techniques: agglomerative hierarchical grouping (,), k-means clustering (,), model-based clustering (), fuzzy C-means method (-), regression models (), Bayesian statistics (), mixture of regression models (), neural networks (), genetic algorithms (), quantum-frequency algorithm for automated identification of traffic patterns (), fuzzy logic (), Support Vector Machines (SVM) (), and wavelets (). These methods present a variety of drawbacks, which include inconsistent clustering results for different locations and time periods, clustering solution sensitivity to missing data, inability to provide sematic meaning (e.g. summer, commuter, recreational, etc...), theoretical nature of methods, and ill-suitability for wider implementation (). The groups formed from cluster analysis, unlike groupings based on expert judgment, avoid biasness because they are chosen by their data-driven similarity measure (). However, cluster analysis lacks guidelines on establishing the optimal number of clusters for a given data set. Priori information on the number of clusters, cluster initializations, and cluster evaluation criteria play an important role in cluster analysis outputs. Unfortunately, no standard procedure exists for selecting a priori

5 Madanu, Mattingly, and McDaniel-Wilson 0 information. Cluster analysis output groups, often, appear relatively unidentifiable on a given state-wide network due to their pure mathematical nature (). Bad clustering or grouping is the chief source of error while estimating AADT using a seasonal factoring approach (). In addition, day to day traffic variation, missing or bad data due to ATR malfunctions, and incorrect assignment of STTCs to seasonal factor groups make significant contributions to AADT estimation errors (). Cluster analysis may compute a different number of clusters across years and ATR sites may change cluster year by year (). These drawbacks highlight the challenges of cluster analysis and the difficulty of identifying clusters for practical situations. Despite the previous criticism, clustering analysis with evolution of machine learning algorithms provides an opportunity to improve the clustering analysis outputs. The objective of this paper is to provide a modified clustering process using Gaussian Mixture Modelling. The study introduces a hierarchical cluster combination method to systematically select a clustering solution using an entropy difference criterion. GAUSSIAN MIXTURE MODELLING In mixture modelling, a distribution f is a mixture of K component distributions like f, f f K if f(x) = K k= P k f k (x) () Where P k is called mixing weight or mixing proportion with P k > 0 and k P k =. The component distributions can be completely arbitrary and may come with different distributions. Most cases select parametric mixture models where component distributions come from the same family (all Gaussian, all Gamma, etc ) (). In general, Gaussian mixture models use different parameters (mean, variances, and mixing weights). Let x = {x, x x n } in R n d is a d-dimensional data patterns with a sample size of n. The likelihood or probability density f (x i K,θ K ) of a data point x i belongs to a sample from a finite K mixture density equal to k= P k f k (x i a k ), where θ K are the parameters (P P K-, a a K- ). The component densities are Gaussian densities with parameter a k = (µ k, Σ k ), µ k being the mean and Σ k the variance matrix of component k. The group or class, z = (z z n ), to which a data point belongs is unknown or to be determined. The group of a data point x i is expressed as z i = (z i z ik ) where z ik takes a value of one if x i belong to group k. n Assuming x to be independent samples, then the likelihood is expressed as f(x i K, θ K ) and Log-likelihood is n n K l(k, θ K ) = i= log f(x i K, θ K ) = log( P k f k (x i a k ) i=, i= k= ) () The parameters of mixture models are found using an Expected-Maximization (EM) algorithm (). Data generated from mixture models are characterized by either groups, clusters or classes. Cluster geometrical features (shape, volume, and orientation) are determined by the covariance matrices (Σ k ). The Eigenvalue decomposition of Σ k provides a general framework for clustering analysis, which is often called model based clustering (). The covariance matrix ( Σ k ) is decomposed into: Σ k = λ k D k A k D k T ()

6 Madanu, Mattingly, and McDaniel-Wilson 0 0 Where D k is an orthogonal matrix of Eigenvectors (gives cluster orientation), A k is a diagonal matrix whose elements are proportional to the eigenvalues (gives cluster shape), and λ k governs the volume of the cluster. Restricting the cluster orientation, shape and volume yields different models for clustering (). These models are broadly categorized into three main model families: spherical, diagonal, and general. A three-letter code describes the volume, shape, and orientation of the cluster groups. Each letter in the code may take a value of E for equal, V for variable, or I for identity (). For instance, model VEV stands for a cluster model with variable volume, equal shape and variable orientation. Selection of mixture models is usually done using Bayesian Information Criteria (BIC). In model-based clustering, a decisive first local maximum of BIC indicates strong evidence for a good model (). Usually, the number of mixture components is taken as the number of clusters (). However, this can lead to an overestimation, because BIC selects the number of mixture components only to provide a good approximation to the density rather than the number of clusters (). Also, a mixture of Gaussian components may better represent a cluster than a single Gaussian component. Thus, a best approximating Gaussian mixture model may not represent the correct number of clusters in the data (). Biernacki et al. () proposed the Integrated Complete Likelihood (ICL) criteria to assess the number of clusters. Often, the number of clusters identified by ICL criteria is smaller than the number of components selected by BIC because ICL is approximately equal to BIC with a penalty term of mean entropy (). The ICL usage for estimating mixture components may give an underestimated number of components (). In order solve the dilemma of cluster selection, Baudry et al. () proposed a method to achieve best clustering using entropy criterion. The mixture components produced by a BIC solution are hierarchically combined using entropy criterion. The best number of clusters are selected based on a piecewise linear regression fitted to the rescaled entropy plot. The entropy of K mixture components is defined as (): K n k= i= () Ent(K) = t ik (θ K ) log t ik (θ K ) 0 Where t ik is the conditional probability of x i from the kth mixture component, which is given by t ik (θ K ) = p k f k (x i a k ) K j= p j f j (x i a j ) STUDY METHODOLOGY The study methodology has three stages. In the first stage, the GMM clustering technique is applied to Oregon Department of Transportation (ODOT) ATR data. The initial clustering solution provides the best model form and number of components using BIC criteria. The initial clustering solution (based on BIC criteria) is hierarchically combined until obtaining a single cluster. At any step, during the combination process, only two components are added based on the minimization of entropy criterion. Once the clustering solution is found, each data point is assigned to a group number or class. The next stage designs a data classifier using the group labels assigned during the first stage. In the final stage, the designed classifier helps in assigning a group number to the test data. The relative merits of clustering solutions are also assessed by comparing estimated and actual AADT values. In particular, the GMM solution is compared ()

7 Madanu, Mattingly, and McDaniel-Wilson 0 with model based agglomerative hierarchical clustering, and ODOT s seasonal trend grouping method and functional class grouping method. CASE STUDY Data The data sets are developed from ATR counters throughout the State of Oregon for the years and. For the purpose of analysis, a Short Term Traffic Count (STTC) is defined as a complete hour count on a given day. At each ATR station, STTCs are sampled and used for clustering analysis. Table provides a summary of the ATR data characteristics. Regions, and share more ATRs and also have higher travel activity. The researchers try to generalize the grouping methodology by considering all possible variations in the traffic trends. Thus, the analysis does not subset the data by vehicle type, weekdays, weekends, and seasonal patterns when establishing groupings. The data is used only for clustering and design of a classifier. The clustering solution and its relative performance is tested with the data. The next section presents the clustering analysis outputs and evaluates the solution. Year Table Study Data Characteristics ATRs per Region (#) Mean ATR AADT per Region (veh/year) Total ATRs Annual ADT Number of STTCs per year 0,,,,,, 0,,0,0,,,,, First Stage - Clustering The clustering analysis uses the data with a sample size of 0,. Each sample has continuous hourly traffic volume (veh/hour), daily traffic volume (DT in veh/day), and the ratio of AADT to DT. At the beginning of the first stage, GMM clustering is fit to the data and refined to obtain a final cluster solution. The researchers, then, present model based Gaussian agglomerative hierarchical clustering analysis, and other clustering solutions for comparative analysis. The study uses the mclust package available in the R programming language to analyze clustering (). The following sections present these steps. GMM - Initial Clustering Solution Sample hourly traffic data (with dimensionality d = ) represents the input data for the GMM clustering. The model based clustering chooses mixture components using both BIC and ICL criteria. An unconstrained model VVV (variable shape, volume and orientation of covariance matrix) is chosen in both cases. The mix proportions of the fifteen components are not uniform; five components have a proportion between and percent, three components between and percent, four components between and percent, and the proportion varies for the other three components. GMM Solution Refinement Starting with the initial BIC solution (having components), hierarchically, components are added until obtaining a single cluster (clusters,, to ). At each step, two mixture

8 Madanu, Mattingly, and McDaniel-Wilson 0 0 components are combined based on the minimization of entropy criterion calculated at each step. The same steps are repeated for the ICL criteria solution. As the model form and number of mixture components are the same for the BIC and ICL criteria, a hierarchical combination of the initial solution is done only once. The hierarchical combination does not change the likelihood and distribution of the observations from the initial cluster solution. Hence, no formal statistical inference is available to choose between different numbers of clusters produced at every consecutive combination step. However, the decrease in entropy at each step guides the selection of a cluster solution. Figure (a) shows the entropies of the combined solutions, and Figure (b) shows the difference in entropy. Reducing the clusters from to appears beneficial because the decrease in entropy is large. Similarly, reducing from to clusters provides a large decrease in entropy. Each of the five subsequent reduction steps (from down to clusters) provide an entropy reduction roughly half as large as the initial change from to. The next cluster reduction steps (from down to ) each decrease the entropy a little less than one third of the first change from to. The change in entropy alone does not provide sufficient guidance for the number of clusters. To obtain more insight, Figure (c) shows the decrease in entropy with respect to the number of observations merged at each step. On the abscissas, the difference between two successive points shows the number of observations used for merging at a given step in the hierarchical combination process. Using the difference in observations and its corresponding entropy difference, a normalized difference in entropy is computed (Figure (d)). The ratio of entropy difference (from K+ clusters to K clusters) to number of merged observations gives a normalized entropy difference. At the first merging, a large entropy difference over a small number of observations leads to a large normalized difference in entropy. Byers and Raftery () proposed fitting a piecewise linear regression model to the values in the entropy plot, and selecting the number of clusters corresponding to an estimated breakpoint (elbow) in the plot. The regression line is shown as the dotted lines for both the entropy and normalized entropy plots in Figure (a) and Figure (c). The fit of piecewise linear regression plot shows one elbow and chooses eight clusters. Model based Gaussian agglomerative Hierarchical Clustering Analysis (MHCA) The GMM clustering solution is compared with the Model based Gaussian agglomerative Hierarchical Clustering Analysis (MHCA). Since the MHCA is more efficient than conventional agglomerative hierarchical clustering and model based clustering shows improved estimates over K-means and other types of clustering while estimating AADT (), the study focuses on a comparison of the GMM and MHCA methods. The MHCA uses maximum-likelihood criteria for merging observations into groups (). Even, the MHCA requires a prioi information on the number of clusters. Unfortunately, such information is often not available before hand; instead the algorithm is tested for a number of clusters varying from to. The best number of groups is selected using one of commonly used cluster evaluation criteria. Calinski-Harabasz index (0), Davies-Bouldin index (), Dunn index () and Silhouette values () are some of the frequently used clustering evaluation criteria. The study selects six cluster solutions for MHCA as the optimum number of clusters using the Dunn Index.

9 Madanu, Mattingly, and McDaniel-Wilson (a) (b) (c) (d) FIGURE Entropy and normalized entropy plots for case study data. Other Clustering In addition, the clustering solutions are compared with the ODOT seasonal trend grouping method (TMG labeled traditional approach) and Functional Class (FC) grouping method (TMG labeled volume factor grouping method). The ODOT seasonal trend grouping method uses knowledge of the road system with aid of visual interpretation. The ODOT, historically, identifies ten seasonal trends: Summer < 00 AADT, Recreational Summer, Interstate Non- Urbanized, Interstate Urbanized, Agricultural, Commuter, Recreational Summer /Winter, Coastal Destination, Summer, and Coastal Destination Route. Functional class grouping bundles the data patterns based on ODOT highway functional class: Rural Interstate, Urban Interstate,

10 Madanu, Mattingly, and McDaniel-Wilson 0 0 Freeways and Expressways, Principal Arterial, Principal Arterial Urban, Minor Arterial, and Major Collector. Next, the research team tests the relative merit of the cluster solutions for estimating the AADT. The study adopts the GMM eight cluster solution and MHCA six cluster solution for estimation. First, the cluster solution classifies each data pattern into one of the groups (either one of eight or one of six depending on the solution). However, the test data needs a classifier that assigns data patterns into one of the groups. In the past, researchers have developed numerous methods for assigning traffic data to factor groups (see reference ()). Both tasks, reasonably grouping the traffic data (current study focus) and assigning traffic data to correct groups (classification) play an important role when computing AADT values. Second Stage - Classification The study uses the commonly adopted Quadratic Discriminant Analysis (QDA) technique for classifying test data patterns. The QDA statistically assigns a given data pattern to the groups of known characteristics. The error rate or misclassification rate (percent of data patterns whose groups are misclassified) shows the predictability of the QDA. The data with known group labels from GMM clustering solution is divided into two parts, namely, training data and test data using 0 and 0 proportion rule. Next, the QDA is performed to classify the training data patterns. Trained QDA classifier has a -fold cross validation error rate of. percent for the GMM clustering solution. When tested for QDA predictability, a test error rate of. percent is observed. A separate QDA is performed for the MHCA solution using the same training and testing data. Final Stage - AADT Estimation After the clustering step, each cluster group is again sub grouped by month to compute the average ratio of AADT to DT. For instance, the GMM eight cluster solution produces average ratios of AADT to DT ( groups months) after sub-grouping. Subgrouping helps to address monthly and seasonal variation of traffic data. Depending upon the analysis type, subgrouping continues only on a part of the test data. For instance, while studying weekday patterns, subgrouping is performed only for the weekdays of the test data. In total, three subgrouping tables namely weekday, weekend, and daily or total are developed. The subgrouping of each cluster by month continues for GMM, MHCA, ODOT seasonal trend grouping (ODOT), and highway functional class (FC) grouping methods. The ATR data set with a sample of, data patterns is used for testing. The dataset has hourly traffic data showing time of day variation and the groupings according to the ODOT seasonal trend grouping method and highway functional class. The QDA classify the test data and assign the group number according to the GMM and MHCA solution. The AADT is calculated by matching group number and month of the test patterns (using data). The product of the matched average ratio of AADT to DT (corresponding to a matched group number and month) and sum of -hour traffic volume (daily traffic or DT) estimates the AADT. The computed AADT value is compared with the actual AADT value to obtain an error. The Mean Absolute Percent Error (MPAE) given in Equation () is used to compare the estimates from clustering methods (, ):

11 Madanu, Mattingly, and McDaniel-Wilson MAPE = N ( AADT Estimated AADT Actual n= 0) () AADT Actual n RESULTS First, the study examines MAPEs among the clustering solutions by day of week patterns (see Table ). The research team follows the ODOT definition of weekdays (Monday to Thursday) and weekends (Friday to Sunday). The daily pattern type refers to any day of week that combines both weekday and weekend patterns. The GMM clustering solution gives less percent error for the weekday pattern type compared to weekends. The percent error for daily patterns lies between the weekday and weekend types. Compared to the MHCA solution, the GMM solution yields a lower error rate for any pattern type. Grouping by functional class shows a higher error rate than any other method. The ODOT method shows almost two percent less error than the next best solution by the GMM method for the weekday pattern. The ODOT method considers seasonal trends based on a dedicated weekday and weekend traffic patterns. This allows the ODOT method to show less error than the GMM method, which is based on daily traffic patterns. For other patterns, the GMM and ODOT solutions appear similar, while the GMM holds a little advantage when comparing daily patterns. Table does not give a clear picture on whether the GMM solution performs better than the conventional ODOT method. The team develops Figure, which shows monthly variation of MAPEs for different clustering solutions. Table Mean Absolute Percent Error (MPAE) among Clustering Solutions Mean Absolute Percent Error (MPAE) in % Pattern ODOT Seasonal Trend Grouping Functional Class Grouping GMM MHCA (ODOT) (FC) Weekday....0 Weekend.... Daily Both GMM and ODOT methods have more MAPEs for winter months compared to summer months. Overall, the monthly variation of the MAPE is lower for the GMM method compared to the ODOT method except for summer months. The summer month trends typically represent recreational activities that are mostly based on weekend patterns. The daily traffic pattern based GMM solution does not capture the AADT variation compared to the ODOT method for those months. However, application of the monthly subgrouping from the GMM cluster solution seems to produce less AADT estimation error. The monthly error variation remains high for the MHCA and the functional class grouping methods.

12 MAPE (%) Madanu, Mattingly, and McDaniel-Wilson Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec GMM Month MHCA ODOT Seasonal Trend Grouping Functional Class Grouping FIGURE Monthly variation of mean absolute percent error among clustering methods. CONCLUDING REMARKS Summary and Conclusions Cluster analysis partitions the data into meaningful subgroups without knowing its components and structure. However, choosing the number of clusters or patterns in the traffic data represents a challenging task. The study introduces a statistical clustering method, Gaussian Mixture Modelling, for grouping the ATR traffic data patterns. The study methodology has three stages: clustering, classification, and AADT estimation. In the clustering stage, first, the GMM is fit to the data and the number of components is selected using BIC criteria. Matching the number of components, from the BIC criteria, with the number of clusters overestimates the number of necessary clusters. Hence, the research team introduces a strategy to refine the initial GMM cluster solution using entropy criterion minimization. Next, the authors design a quadratic discriminant classifier to classify the data with known groups from the GMM clustering solution. Finally, The GMM solution is tested for estimating AADT values using a data set. The GMM clustering solution shows lower MAPEs for most months compared to other clustering solutions; however, the current study has the following limitations and possible improvements. Practical Implication The study uses a classifier designed based on a GMM clustering solution to obtain a group number for the field-collected STTCs. The AADT is computed by taking a product of the corresponding group s ratio of AADT to DT and daily traffic of that test pattern (see AADT Estimation section). Usually, STTCs span multiple days across multiple months. The average of AADT values estimated for all those days will become the AADT for the given study facility.

13 Madanu, Mattingly, and McDaniel-Wilson 0 0 Study Limitations and Future Directions The GMM clustering solution is constructed using daily traffic patterns. The clustering solution may be enhanced by further dividing the data set into subgroups by weekdays, weekends, season, and vehicle type. However, the current methodological framework needs only modified input for performing clustering analysis on these data sets. As the traffic patterns change by geographical location on a state-wide transportation network, performing a clustering analysis within each region may provide consistent results for different locations. Similarly, clustering with historic data, instead of a single year of data, may capture the traffic variations more effectively. The entropy criterion guides the choice of the number of clusters, but it does not have a statistical interpretation. An inferential means of identifying the number of clusters, for instance using a gap statistic, could provide a robust choice of clustering. The consideration of land-use and socio-economic characteristics while grouping the traffic patterns may provide more significant groups, too. Like GMM, the mixture of either linear or non-linear models with AADT as a dependent variable and land-use and socio-economic characteristics as independent variables may help reduce the AADT estimation errors. ACKNOWLEDGMENTS The authors would like to thank Mr. Brian Dunn, Mr. Doug Norval, Mr. Peter Schuytema, and Mr. Sam Ayash of the Oregon Department of Transportation (ODOT) for their valuable advice and help. The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the data, opinions, findings, and conclusions presented herein. The contents do not necessarily reflect the official view or policies of the ODOT, and the university that the authors represent. REFERENCES. Duddu, V. R., and S. S. Pulugurtha. Principle of Demographic Gravitation to Estimate Annual Average Daily Traffic: Comparison of Statistical and Neural Network Models. Journal of Transportation Engineering, Vol., No.,, pp... Gastaldi, M., G. Gecchele, and R. Rossi. Estimation of Annual Average Daily Traffic from One-week Traffic Counts - A Combined ANN-Fuzzy Approach. Transportation Research: Part C, vol.,, PP. -.. Traffic Monitoring Guide. Federal Highway Administration (FHWA). U.S. Department of Transportation,. Accessed July,.. Fraley, C., and A. E. Raftery. How Many Clusters? Answers via Model-Based Cluster Analysis. The Computer Journal, vol.,, pp... Sharma, S. C., and A. Werner. Improved Method of Grouping Province Wide Permanent Traffic Counters. In Transport Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp... Faghri, A., and J. Hua. Roadway Seasonal Classification using Neural Networks. Journal of Computing in Civil Engineering, Vol., No.,, pp... Flaherty, J. Cluster Analysis of Arizona Automatic Traffic Record Data. In Transport Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp..

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