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Uncertainty and sensitivity analysis methods are introduced, concerning the quality of spatial data as well as that of output information from Global Positioning System (GPS) and Geographic Information System (GIS) integrated applications for transportation. In the methods, an error model and an error propagation method form a basis for formulating characterization and propagation of uncertainties. They are developed in two distinct approaches: analytical and simulation. Thus, an initial evaluation is performed to compare and examine uncertainty estimations from the analytical and simulation approaches. The evaluation results show that estimated ranges of output information from the analytical and simulation approaches are compatible, but the simulation approach rather than the analytical approach is preferred for uncertainty and sensitivity analyses, due to its flexibility and capability to realize positional errors in both input data. Therefore, in a case study, uncertainty and sensitivity analyses based upon the simulation approach is conducted on a winter maintenance application. The sensitivity analysis is used to determine optimum input data qualities, and the uncertainty analysis is then applied to estimate overall qualities of output information from the application. The analysis results show that output information from the nondistancebased computation model is not sensitive to positional uncertainties in input data. However, for the distancebased computational model, output information has a different magnitude of uncertainties, depending on position uncertainties in input data.
As the Geographic Information System (GIS) has been used for a wide range of transportation applications, positional errors inherent in spatial data become critical for ensuring spatial problemsolving and decisionmaking. However, GIS involves spatial data from multiple sources and different types. People are used to making decisions without knowledge of either positional errors in the data or their impact on output information. In GIS for transportation, various datacollection methods or devices have been used to maintain and update a spatial database, of which the Global Positioning System (GPS) provides a cost effective and efficient means of collecting spatial and nonspatial data along roadways. One emerging GPSbased method is to equip vehicles with Differential Global Positioning System (DGPS) receivers and numerous sensors [
However, positional uncertainties inevitably exist in GPS data points and roadway centerline maps. Although numerous mapmatching algorithms have been proposed to correctly integrate GPS data points with a roadway centerline map [
Characterization and propagation of positional uncertainties are not well formulated to determine a positional accuracy requirement for input data and a quality requirement for output information.
Relative importance of multiple input data for output information is not well assessed and addressed.
Overall uncertainties in output information are not well assessed and addressed.
Numerous approaches have been proposed to deal with problems concerning the quality of input data as well as that of output information from GIS applications such as hydrology, environment, and soil science [
Modeling of positional errors and their propagation is necessary to understand error and its impact on GPS and GIS integrated applications for transportation. Generally, there are two approaches: analytical and simulation. The analytical approach estimates uncertainties in output information by applying the law of error propagation, assuming uncertainty properties of spatial data are known [
The analytical approach involves the analytical GPS error model (or error curve) and the error propagation model, assuming that a test roadway centerline map is representative of roadway centerline maps with the same nominal scale [
Also, it implies a probability of proximity with other measurements of the same position when the same measurement technique is applied under the same conditions. The error model becomes circular when two variances are equal with no correlation. However, when there is high correlation, the error curve becomes highly curvilinear.
When GPS points are integrated with a cartographic roadway network in GIS, the potential bounds of linearlyreferenced errors are estimated by projecting the GPS error model onto roadway centerlines. For example, in
The simulation approach involves error models that generate a population of errorcorrupted versions of GPS data points and roadway centerline map [
The general procedure for the simulation approach is described in
For comparison and evaluation of the analytical and simulationbased approaches for modeling positional errors and their propagation in GPS and GIS integrated applications, the spatial data employed in this paper are DGPS data points from probe vehicles, and roadway centerline maps (
Two roadway spatial databases with different nominal scales were obtained. The reference map, directly compiled from stereomodels by a mapping firm, depicts roadway centerlines with a 1:4,800 nominal scale. In an error modeling, the reference map is treated as representing the actual locations of roadways. The test map, obtained from a commercial vendor, was developed for vehicle navigation purposes. However, metadata for the test map were not available. The test map is assumed to have multiple sources, including maps of varying scales and DGPSderived roadway centerline components from vehicles driven over roadways. Also, it is expected that different magnitudes of positional uncertainties exist depending upon the complexity and curvilinearity of roadways. Thus, test areas are divided into three groups: (1) straightaway roadway; (2) curvilinear roadway; and (3) ramp (
For each test area shown in
For assessing the impact of positional uncertainties, a reference route measure from reference input data is compared to route measures estimated by the analytical and simulation approaches. However, different from the simulation approach, the analytical approach assumes that positional uncertainties in input data are independent. Their offsets and confidence intervals discretely reflect sensitivities to positional uncertainties in a roadway centerline map and DGPS data points, respectively. For example, in the straightaway roadway (
Also, fundamental properties of the analytical and simulation approaches are identified. The analytical approach requires a complex mathematical process to model positional uncertainties in input data and their propagation through spatial operations. However, when a computational model for outputs is nonlinear, an error propagation model is simplified by applying a firstorder Taylor series. Further simplification can be made by assuming that errors in input data are independent of one another. Thus, uncertainty estimation by the analytical approach is oftentimes unrealistic. Also, due to its inflexibility and impracticality, the analytical approach is limited to specific applications, even if positional errors in input data and utilized spatial operations are known. Different from the analytical approach, the computational cost of the simulation approach is heavy due to Monte Carlo simulation. But, as the spatial correlation of positional errors can be realized, the simulation approach is suitable for many situations to predict uncertainties in output information. Also, it can be applied to various applications due to its flexibility and simplicity.
Uncertainty and sensitivity analysis methods are developed based on the error modeling approaches. However, as evaluated in the Section 3, uncertainty estimations by the analytical approach are oftentimes unrealistic due to the independence assumption among positional errors. Even though heavy computational time is required, the simulation approach is utilized for uncertainty and sensitivity analyses due to its capability to realize positional errors in each input data type.
The uncertainty analysis method is designed to estimate the overall quality of output information based on spatial varieties of spatial data. A conceptual view of uncertainty analysis is illustrated in
In the development of GIS applications, data collection is the most important and expensive component. Thus, to determine optimal input data qualities for applications, a sensitivity analysis is designed to explore contributions of positional uncertainties in input data to variations in output information. The conceptual procedure for sensitivity analysis is illustrated in
The first sensitivity analysis is designed to estimate the impact of input data on output information with varying combinations of input data quality (e.g., 5 m accuracy GPS data and 1:12,000 scale roadway map; 2 m accuracy GPS data and 1:24,000 scale roadway map). Reference information from reference input data is compared to estimates of output information. The offset between estimated and reference information indicates impacts of positional uncertainties in input data on output information. On the other hand, the second sensitivity analysis is designed to estimate relative contributions of positional uncertainties in input data to uncertainties in output information. In this analysis, the confidence interval for output information is a degree of uncertainty that indicates a consistency (or repeatability) of output information from a given set of input data qualities. To measure the relative sensitivity, positional errors in one input data type are simulated while another input data type remains unperturbed.
The sensitivity and uncertainty analysis methods, proposed in this paper, provide means to formulate characterization and propagation of positional uncertainties in input data so the impact of the uncertainties on applications can be analyzed. For verification and demonstration purposes, the methods are applied to a winter highway maintenance application. The overall procedure for uncertainty and sensitivity analyses is illustrated in
The winter maintenance application, referred to as WiscPlow, was developed as a GPS and GIS application for winter storm reporting and resource management [
All data coming from the vehicles are represented as twodimensional DGPS point features along roadway centerlines and have a temporal resolution equal to 2 s. WiscPlow calculates winter maintenance performance measures and develops analytical decision tools using spatial and nonspatial data. The performance measures, presented as reports and charts, provide a basis for decisionmaking on allocation of resources and enhancement of overall performance of winter operations [
Computational models dealing with average salt application rate and total quantity of salt are selected to generate performance measures. The performance measure “Average salt application rate for each operator and event” in WiscPlow is computed by a nondistancebased computational model (the unit for the application rate is pounds of salt per lane mile):
The performance measure “Quantity of salt used for all events and each patrol section” is computed by a distancebased computational model (the unit for the quantity of salt is kilograms):
Spatial data covers a county roadway and an interstate highway in Columbia County, WI, USA (
The sensitivity analysis aims to characterize sensitivity of output information to positional uncertainties in input data, considering complexity and curvilinearity of roadways. The sensitivity analysis method was applied to roadway centerline maps and DGPS data points in Patrol Section 1, Patrol Section 3, and Patrol Section 4, which represent the straightaway roadway, the roadway including ramps, and the curvilinear roadway, respectively (
The first sensitivity analysis is designed to estimate the impact of errors in the roadway centerline map on performance measures, for which performance measures are computed from DGPS data points and roadway centerline maps altering from the commercial map to the public map. The reference performance measure from the reference input data is the basis for estimating the sensitivity of performance measures from test maps. However, “Average salt application rate for each operator and event,” computed from the nondistancebased computational model, is not sensitive to positional uncertainties in input data. Thus, the sensitivity analysis is conducted for “Quantity of salt used for all events and each patrol section” from the distancebased computational model.
For further analysis, the second sensitivity analysis method simulates route measures by the following combinations of error models: DGPS Data, roadway map, and DGPS Data and roadway map. Effects of input data on route measures are examined independently and in combination (
When a combination of quality levels for each input data type is determined, uncertainty analysis was conducted on two datasets from winter maintenance vehicles driven by different operators in
The quality of spatial data becomes one of critical factors to be considered before utilizing GPS and GIS applications for spatial problemsolving and decisionmaking. For this reason, this paper introduced the sensitivity and uncertainty analysis methods that provide means for formulating characterization and propagation of uncertainties in the applications. The proposed methods analyze two aspects of uncertainty propagation: sensitivity analysis aims to estimate contributions of positional uncertainties in input data to variations in output information, and uncertainty analysis aims to predict an overall quality of output information from given qualities of input data.
In the uncertainty and sensitivity analyses, an error model and an error propagation method are basis for estimating the quality of output information from given qualities of input data. They are constructed in two distinct ways: an analytical approach and a simulation approach. Therefore, before implementation to the real application, the evaluation is preceded to compare and examine the analytical and simulation approaches. The evaluation results show that estimated ranges of output information from the analytical and simulation approaches are compatible. However, as the independence assumption among positional errors, uncertainty estimations by the analytical approach are oftentimes unrealistic. Even though heavy computational time is required, the simulation approach is more realistic due to error models for both types of input data. Therefore, in the development of the uncertainty and sensitivity analysis methods, the simulation approach rather than the analytical approach is preferred when considering the simplicity and flexibility for implementation.
For verification and demonstration purposes, the sensitivity and uncertainty analyses were conducted on the winter highway maintenance application. The sensitivity analysis method was used to determine input data qualities for the application. The uncertainty analysis method was then applied to estimate overall qualities of output information from the application. The results from the sensitivity and uncertainty analyses showed that consistent output information was calculated from the nondistancebased computation model, regardless of input data qualities. However, for the distancebased computational model, output information had a different magnitude of sensitivities depending on the input data quality and the spatial dependency in position errors along roadway centerlines.
In this paper, the uncertainty and sensitivity analysis methods were built upon positional error models for GPS data points and roadway centerline maps. However, GIST involves spatial data from various data collection devices and methods. Further research is necessary to develop error models for spatial measurements that are commonly applied to LRS (Linear Referencing System) such as distance measuring instruments, inertial measuring units, and dead reckoning systems. Also, the uncertainty and sensitivity analyses should be conducted under various environments affecting positional uncertainties in GPS data points and cartographic roadway network models. In addition, uncertainties in output information were most probable within acceptable bounds for decisionmaking, no matter the scale or quality of the available digital roadway centerline maps. Therefore, a rigorous approach needs to be designed to determine optimal levels of input data quality that produce acceptable levels of uncertainties in output information and, ultimately, to determine optimal quality of output information for decisionmaking with acceptable levels of risk.
The authors gratefully acknowledge the support of the Midwest Regional University Transportation Center at the University of WisconsinMadison and the Wisconsin Department of Transportation in USA.
The author declares no conflict of interest.
Analytical GPS error model (standard error curve).
Uncertainty propagation in GPS and GIS applications for transportation. (
Simulation error model. (
Simulation approach for modeling positional errors in GPS and GIS integrated applications for transportation.
Positional uncertainties in DGPS data.
DGPS data points and roadway centerline maps on orthophotos. (
Conceptual view of uncertainty analysis method.
Conceptual view of sensitivity analysis.
Overall procedure of uncertainty and sensitivity analyses.
Winter Maintenance Application (WiscPlow). (
Winter maintenance vehicle wired with sensors and DGPS receiver.
Patrol section map in Columbia County, WI, USA.
DGPS data points in patrol sections. (
DGPS data points in Columbia County, WI, USA. (
Uncertainty properties in DGPS data.
1.46 m^{2}  3.84 m^{2}  0.06  
Columbia County, Wisconsin Land Information Office 
1:4800 
Uncertainties in route measures
Straightaway Roadway  2456.56  2456.53 ± 2.74 (−)0.03  2456.62 ± 2.72 (0.06) 
Ramp 1  660.38 

645.86 ± 4.94 (−14.52) 
Ramp 2  704.27  694.68 ± 2.46 (−9.59)  694.40 ± 2.66 (−9.87) 
Curvilinear Roadway  3358.83  3368.24 ± 2.66 (9.41)  3365.64 ± 3.13 (6.81) 
* Values in parentheses are offsets between estimated and reference route measures.
Test maps in Columbia County, WI, USA.
Commercial Vendor  5 m but 15 m in certain areas  
USGS 7.5 min quadrangles  1:24,000 
The first sensitivity analysis: “Quantity of salt used for all event and each patrol section (kg).”
999.75  999.71 ± 0.11  999.96 ± 0.12  
877.64  878.10 ± 0.69  876.30 ± 0.73  
693.59  693.38 ± 0.24  696.03 ± 0.56 
The second sensitivity analysis: “Quantity of salt used for all events and each patrol section (kg).”
 

999.74 ± 0.11  999.71 ± 0.03  999.71 ± 0.11  
878.15 ± 0.47  878.10 ± 0.54  878.10 ± 0.69  
693.64 ± 0.23  693.38 ± 0.05  693.38 ± 0.24  
 
 
999.96 ± 0.11  999.96 ± 0.04  999.96 ± 0.12  
876.42 ± 0.48  876.30 ± 0.55  876.30 ± 0.73  
695.37 ± 0.24  696.03 ± 0.50  696.03 ± 0.56 
Uncertainty analysis: “Average salt application rate.”
402.58 ± 0.00  
17.09 ± 0.00 
“Uncertainty analysis: Quantify of salt.”
113.33 ± 0.38  3.46  
14.49 ± 0.36  0.13  
739.10 ± 0.99  5.14  
80.36 ± 0.23  0.60  
211.20 ± 0.80  2.43  
8.57 ± 0.43  0.42 