EXPLORATORY DATA ANALYSES - A Generalized Non-parametric Approach for Uncertainty Evaluation in

3.4.1 Empirical Data from San Antonio, Texas

It was necessary to explore the characteristics of the data—temporally and spatially—in order to choose the most appropriate estimating and prediction models, both for the mean as well as the confidence interval measures. It has been shown that point data such as that from loop detectors have a nonstationary distribution and exhibit a general periodicity (May 1990). Namely, the traffic flow on each day exhibits a repeating pattern such as a huge peak at 8:00 AM, a smaller peak at around 4:00 PM and an absolute minimum around 2:00 PM. Additionally, the data are also weakly dependent. That is, the dependence diminishes as the aggregation interval becomes large (i.e., traffic characteristics on Monday at 7:30 AM influence traffic characteristics on Monday at 7:35 AM, but do not influence traffic characteristics on the next Monday at 7:35AM). This section presents exploratory analyses conducted on the speed, volume and occupancy data (empirical and simulated). The characteristics of travel times estimated as a function

of the speed, volume and occupancy are also presented within this chapter. The travel times are estimated using a methodology that is presented in Chapter Four.

A periodic behavior (repeating pattern) can be observed by plotting each day‘s data side-by-side. Figures 3-5 to 3-8 show the data for the morning peak period (7 AM – 9 AM) as a function of time of day. A noticeable pattern can be observed in each of the plots. The volume, occupancy and travel times exhibit a distinct repeating pattern as seen in Figures 3-5, 3-7 and 3-8, respectively. However, Figure 3-6 shows that a defined pattern is not observable for the speed because they do not vary much during the 7-9 AM period. It should be noted that it was difficult to view the entire data (130 days) when plotted at once, therefore only data for five days (305 data points) were plotted. As an example, data from the main lane detectors located at milepost 160.504 are presented.

Also, the estimated travel times on the link are illustrated.

FIGURE 3-5: Peak hour traffic volume data at milepost 160.504

0 50 100 150 200 250 300 350

7:00 AM 7:16 AM 7:32 AM 7:48 AM 8:04 AM 8:20 AM 8:36 AM 8:52 AM 7:06 AM 7:22 AM 7:38 AM 7:54 AM 8:10 AM 8:26 AM 8:42 AM 8:58 AM 7:12 AM 7:28 AM 7:44 AM 8:00 AM 8:16 AM 8:32 AM 8:48 AM 7:02 AM 7:18 AM 7:34 AM 7:50 AM 8:06 AM 8:22 AM 8:38 AM 8:54 AM 7:08 AM 7:24 AM 7:40 AM 7:56 AM 8:12 AM 8:28 AM 8:44 AM 9:00 AM

Average Volume (veh/2-min)

Time of Day

Average Volumes (veh/2-min)

Detector at milepost 160.504

Day 1 Day 2 Day 3 Day 4 Day 5

FIGURE 3-6: Peak hour traffic speed data at milepost 160.504

FIGURE 3-7: Peak hour traffic occupancy data at milepost 160.504

FIGURE 3-8: Peak hour travel time between milepost 159.998 and 160.504

Autocorrelation functions (correlograms) were also plotted to further explore the characteristics of the data. A correlogram is a plot of the autocorrelation coefficients against the time lags. The autocorrelation coefficients measure the correlation between observations at different lags or distances apart (Chatfield 1996). The autocorrelation coefficient at time lag k is computed as:

where:

0 20 40 60 80 100 120

Average 2-min Travel Times (secs)

Time of Day

Average 2-min Travel Times (secs)

Link 2 (mileposts 159.998 to 160.504)

Day 1 Day 2 Day 3 Day 4 Day 5

= correlation coefficient at lag k;

= observation at current time;

= observation at one lag ahead and N = size of series.

Figures 3-9 to 3-12 illustrate the correlograms for the volume, speed, occupancy and travel time, respectively. The dashed horizontal lines are the confidence bounds:

1.96 (1.96 is the .975 quantile of the standard normal distribution). The 60-lags represent the 7-9 AM time period with each lag denoting a 2-minute interval.

The correlograms illustrate that more than 95% of the autocorrelation coefficients are outside the confidence bounds. This indicates that there is time dependence in the data (Brockwell and Davis 2003). That is, data from the current time step is dependent on the data from the previous time step. As an example, travel time at 7:12 AM is dependent on travel time at 7:10 AM. As well, the correlation coefficients tend to decay slowly as the lag increases. This is indicative of the data being nonstationary. That is, the statistical characteristics, the mean and variance, of the data change over time.

It can be concluded that point data (volume, speed and occupancy) and interval data (travel time) all exhibit a periodic behavior and are time dependent as defined by common statistical tests. Additionally, the data also have a nonstationary distribution.

FIGURE 3-9: Autocorrelation plot for traffic volume data

FIGURE 3-10: Autocorrelation plot for traffic speed data

0 10 20 30 40 50 60

-0.5 0 0.5 1

Lag

Sample Autocorrelation

Sample Autocorrelation Function (ACF) - Average Volume (veh/2-min) Detector at milepost 160.504

95% Confidence Bounds

0 10 20 30 40 50 60

-0.5 0 0.5 1

Lag

Sample Autocorrelation

Sample Autocorrelation Function (ACF) - 2minute Average Speed (mph) Detector at milepost 160.504

95% Confidence Bounds

FIGURE 3-11: Autocorrelation plot for traffic occupancy data

FIGURE 3-12: Autocorrelation plot for travel time

0 10 20 30 40 50 60

-0.5 0 0.5 1

Lag

Sample Autocorrelation

Sample Autocorrelation Function (ACF) - 2minute Average Occupancy (%) Detector at milepost 160.504

95% Confidence Bounds

0 10 20 30 40 50 60

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Lag

Sample Autocorrelation

Sample Autocorrelation Function (ACF) - Average Travel Times (secs) Link between mileposts 159.998 and 160.504

95% Confidence Bounds

3.4.2 Simulated Data From Omaha, Nebraska

The data set from the I-80 simulation corridor was also explored to investigate for any characteristics of periodicity, non-stationarity and time dependence in its structure.

Figures 3-13 to 3-16 illustrate plots for each data type as a function of the time of day. A clearly visible repeating pattern is not present in the plots. Instead, it can be observed that the data fluctuate in a random pattern around an expected mean value. The mean value around which the data fluctuate is 58veh/2-min, 53mph, 10% and 35 secs for the volume, speed occupancy and travel time, respectively. It can be concluded that the data are not periodic and have a nonstationary distribution.

FIGURE 3-13: Peak hour traffic volume data

0 10 20 30 40 50 60 70 80 90

Average Volumes (veh/2-min)

Time of Day

Average Volume (veh/2min)

Day 1 Day 2 Day 3 Day 4 Day 5

Mean 58 veh/2-min.

FIGURE 3-14: Peak hour traffic speed data

FIGURE 3-15: Peak hour traffic occupancy data

FIGURE 3-16: Peak hour estimated travel time data

Figures 3-17 to 3-20 illustrate the correlograms for the volume, speed, occupancy and travel time data, respectively. In all the correlograms, it can be seen that the autocorrelation coefficients are inside the 95% confidence bounds. This suggests that there is no time dependence in the dataset. As such, data from the current time step is not dependent on the data from the time step ahead.

0 5 10 15 20 25 30 35 40 45 50

Average Travel Time (secs)

Time of Day

Average Travel Time (secs)

Day 1 Day 2 Day 3 Day 4 Day 5

Mean 35 secs

FIGURE 3-17: Autocorrelation plot for 2-minute average traffic volume data

FIGURE 3-18: Autocorrelation plot for 2-minute average traffic speed data

0 10 20 30 40 50 60

Sample Autocorrelation Function (ACF) - Average Volume (veh/2-min)

95% Confidence Bounds

Sample Autocorrelation Function (ACF) - 2minute Average Speed (mph)

95% Confidence Bounds

FIGURE 3-19: Autocorrelation plot for 2-minute average traffic occupancy data

FIGURE 3-20: Autocorrelation plot for 2-minute average estimated travel time data

0 10 20 30 40 50 60

Sample Autocorrelation Function (ACF) - 2minute Average Occupancy (%)

95% Confidence Bounds

Sample Autocorrelation Function (ACF) - Average Travel Time (secs)

95% Confidence Bounds

3.5 CONCLUDING REMARKS

This chapter has described the data collection and study corridors used in this study. It described the empirical dataset obtained from a freeway corridor in San Antonio, Texas.

Archived inductive loop detector data were extracted for a section of Interstate 35 for a period starting April 1, 2007 and ending September 30, 2007. The chapter also describes the simulated dataset obtained from a calibrated and validated traffic micro-simulation model of Interstate 80 in Nebraska.

This chapter also presented the steps taken to ensure that the data was ‗cleaned‖

and of quality. The techniques employed to perform the quality control on the empirical dataset were based upon proven methods adopted for cleaning similar data sets in previous research. Finally, the chapter also described some exploratory data analyses conducted to identify certain characteristics in the empirical and simulated datasets.

Specifically, the datasets were investigated for characteristics of periodicity (exhibiting a periodic behavior), being dependent and having a nonstationary distribution. It was found that the empirical data (point and interval) exhibits a periodic behavior, are time dependent and also nonstationary. A clearly identifiable repeating pattern could not be observed for the simulated dataset. The simulated data were also found to have a nonstationary distribution and no time dependence exists.

CHAPTER 4 ESTIMATION AND PREDICTION OF TRAVEL TIME

4.1 INTRODUCTION

The ability to predict future link travel time is critical for many Intelligent Transportation Systems‘ applications such as route guidance systems and traffic management systems.

As presented in Chapter Two, a variety of modeling approaches to compute future link travel time are available in the literature. Of these modeling approaches, neural networks have proven to perform better than competing approaches especially for multiple period forecasting. Moreover, neural networks can perform nonlinear mappings and are nonparametric. The latter part of this chapter discusses a neural network model that was developed to obtain predicted values of link travel time for this dissertation.

An essential step in developing a neural network is the training process. During training, the model weights (or parameters) are adjusted iteratively such that the difference between the network output (predicted travel time) and its corresponding target travel time is minimized. In general, the target travel times are directly measured, such as with AVI data. However, for this dissertation, the target travel times were not available.

In the absence of directly measured travel times, an indirect method developed by Vanajakshi was used. This approach has been shown to provide accurate estimates of travel time (Vanajakshi 2004; Vanajakshi and Rilett 2006; Vanajakshi et al. 2009). First,

a brief background into the Vanajakshi estimation method is presented within this chapter. Next, the Vanajakshi method, its application to an empirical dataset in this dissertation and the results are presented.

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