MODELING MOVING OBJECTS AND THEIR MOVEMENTS USING FUZZY LOGIC APPROACH

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1 MODELING MOVING OBJECTS AND THEIR MOVEMENTS

USING FUZZY LOGIC APPROACH

H. Mustafa Palancioglu and Kate Beard

NCGIA and Department of Spatial Information Science and Engineering University of Maine

Orono, ME 04469 hpalanci @spatial.maine.edu

beard@spatial.maine.edu

ABSTRACT

Despite the many application areas of spatio temporal (s-t) systems, there are only a few approaches dealing with the integration of spatial and temporal components to handle moving objects and their movement information. The features and requirements of such systems vary and depend on the characteristics of moving objects, their movement patterns, and the environment. Available systems try either to handle objects with continuously changing positions and properties or handle objects with positions and properties that change at discrete steps. Modeling moving objects and their movement patterns are key to designing such systems. Emerging technologies such as Global Positioning Systems (GPS), Remote Sensing, and Video Imagery are available to capture moving objects and their movement patterns in an accurate way close to a nearly continuous basis.

In this paper, we introduce fuzzy logic as a method for s-t reasoning about moving objects based on the properties and the characteristics that they and their environment may have during the course of movement. We specify the basics for building a fuzzy model for the representation of moving objects. Because it is not possible to store exact knowledge on the movement of objects (i.e., all their possible states) and their environment (i.e., weather condition and media conditions), we propose to use available knowledge and approximate conditions to cope with estimating the travel time, positions, and most appropriate paths for moving objects. We then discuss certain aspects of the implementation of the fuzzy model in a Spatio-Temporal System.

INTRODUCTION

Modeling the dynamics of the world requires at least the integration of space, time, and movement (Galton, 1993). Today’s geographic information system (GIS) data models are incapable of fulfilling the requirements of integrating space and time. Until now, space has been the focus of models of geographic phenomena. The GIS data models can only handle modeling and reasoning about space. Although time in GIS has been a research topic over the last ten years (Langran, 1989; Egenhofer and Golledge, 1998; Peuquet, 1994), models of GISs still treat space and time separately and support only a world that exists in the present or as a collection of temporal snapshots. Their ability to incorporate dynamics of geographic phenomena within geographic information systems (GISs) is only available in the most rudimentary fashion.

GISs should be able to represent and reason about dynamic geographic phenomena in both space and time. Integration of space and time for spatio-temporal analysis and reasoning has gained significant attention recently (Langran, 1988; Langran, 1993; Peuquet, 1994; Worboys, 1994, Erwig et al., 1999). There has been considerable research in modeling, representing, and reasoning about space and time, resulting in different forms of temporal (Allen, 1984) and spatial logic (Egenhofer, 1991; Randell, 1992). And recently, Galton (1993) discusses an integrated logic of space, time, and motion.

As space and time dimensions interact with each other, usually changes occur on geographic phenomena. Explanations of geographic phenomena often require the description of changes. Without including changes especially movements about geographic phenomena, it is impossible to model and reason about the real world, however, movement has not been considered a significant component of GISs traditionally. Currently, data stored in a GIS can be updated, but no records of changes are maintained. Therefore, new methods to incorporate change into GISs to model, represent, and reason about the dynamic world should be considered when examining a variety of problems ranging from urban growth to global change.

Understanding how geographic objects change over time is fundamental to the development of appropriate models of the real world. Differences between the states of dynamic geographic phenomena over time are referred to

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as spatio-temporal changes such as changes in the boundary of a country, in the position of a car, and in the trajectory of an wild fire. People are good at detecting change through visual observations from such documents as satellite images, photographs, and videos. Collectively these heterogeneous observations form a rich source for identifying change. People observe various types of changes, such as movements, in their daily lives, and have no difficulty in understanding and reasoning about them; however, how to model spatio-temporal aspects of the world and how to incorporate them into an information system has not been fully addressed as yet.

The increasing demand for information systems that are capable of dealing with scenarios of change emphasizes the need for research in this particular area. The concept of change must be formalized in order to create efficient spatio-temporal models for the dynamic world. Some spatio-temporal models have been developed (Langran, 1989; Langran, 1992; Peuquet, 1994; Güting, 2000; Erwig, 1997; Sistla; 1997), however, none of them can deal with dynamic geographic phenomena effectively. In order to develop a meaningful model of change it is paramount to analyze its ontological foundations.

Spatio-temporal systems must be designed to deal with changes of spatial and temporal properties. Among the three components of change—boundary redefinition, thematic state change, and movement (Beard et al., 2000)—the main focus of this research is on the latter. Movements with overwhelmingly detailed information represent complex types of spatio-temporal changes. Although movements occur continuously, people often perceive them discretely. Improving the understanding of the concept of movement for geographic phenomena is challenging.

FUZZY LOGIC SYSTEMS

Fuzzy logic is used in this paper as a convenient and useful mathematical tool for representing and manipulating spatio-temporal information.

Fuzzy logic starts with and builds on a set of user-supplied human language rules. The fuzzy systems convert these rules to mathematical equivalents. This simplifies the job of the system designer and the computer, and results in much more accurate representations of the way systems behave in the real world. Additional benefits of fuzzy logic include its simplicity and flexibility. Fuzzy logic can handle problems with imprecise and incomplete data, and can model nonlinear functions of arbitrary complexity.

Fuzzy logic models, called fuzzy inference systems, consist of a number of conditional "if-then" rules. For the designer who understands the system, these rules are easy to write, and as many rules as necessary can be supplied to describe the system adequately.

In fuzzy logic, unlike standard conditional logic, the truth of any statement is a matter of degree. (How cold is it? How high should we set the heat?) We are familiar with inference rules of the form p -> q (p implies q). With fuzzy logic, it's possible to say (.5* p ) -> (.5 * q). For example, for the rule if (weather is cold) then (heat is on), both variables, cold and on, map to ranges of values. Fuzzy inference systems rely on membership functions to explain to the computer how to calculate the correct value between 0 and 1. The degree to which any fuzzy statement is true is denoted by a value between 0 and 1.

Not only do the rule-based approach and flexible membership function scheme make fuzzy systems straightforward to create, but they also simplify the design of systems and ensure that one can easily update and maintain the system over time.

Fuzzy Inference Systems (FISs)

The FIS (Jang et al., 1995; Wang, 1994) is a popular computing framework based on the concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. Basically, a FIS is composed of four functional blocks as shown in Figure 1 :

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Fuzzification Fuzzy Rule Base Fuzzy Inference Engine Non fuzzy input(s) Defuzzification Non fuzzy output(s)

Figure 1. Basic fuzzy inference system.

Fuzzification maps the crisp inputs into fuzzy sets, which are subsequently used as inputs to the inference engine. A fuzzy set U is characterized by a membership function (MF) µ: Uà {0,1}. The membership functions are labeled by a linguistic term such as “small”, “medium” or “large” (Mendel et al., 1997). In the following, several classes of parameterized functions commonly used to define membership functions are given

i ) Gaussian MFs

÷ ÷ ø ö ç ç è æ ÷ ø ö ç è æ − − = b c a x ;a,b,c) gaussian(x 2 exp (1)

ii) Generalized bell MFs

b c a x ,c) bell(x;a,b 2 1 1 − + = (2)

iii) Trapezoidal MFs

ï ï ï ï ï î ïï ï ï ï í ì ≤ ÷ ø ö ç è æ + − > ÷ ø ö ç è æ − + − > < − − ≤ + ≥ = c x a b c+ x a c x x a b c+ a b c r x o b c+ x a b c or x a b c+ x x;a,b,c) Trapezoid( , 1 2 , 1 2 2 2 , 1 1 2 1 2 , 0 (3)

where x is an input variable, and a, b, and c are membership function parameters.

Fuzzy rule base is a set of fuzzy rules in the form of if-then clauses. For a multi input single output case, the ith rule can be expressed by

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where x and y are the input variables, z is the output variable, and Ai and Bi are the labels of membership functions associated to the input variables and Ci is the label of the membership function associated with the output variable, y, in the rule i.

Fuzzy inference engine is a decision-making logic which performs the inference operations on the rules and a given condition to derive a reasonable output or conclusion. Three types of FISs (Jang et al., 1995), Mamdani fuzzy model, Sugeno fuzzy model, and Tsukamoto fuzzy model, have been widely employed in various applications. The differences between these three FISs lie in the consequents of their fuzzy rules, and thus their aggregation and defuzzification procedures differ accordingly.

The Mamdani fuzzy model (Mamdani et al., 1975) was proposed as a first attempt to control a steam engine and boiler combination using a set of linguistic control rules obtained from experienced human operators. Figure 2 shows how a two-rule fuzzy inference system of the Mamdani type works to calculate the overall output z when subjected to two crisp inputs x and y.

Figure 2. Mamdani type Fuzzy Inference System

The Sugeno fuzzy model was proposed in an effort to develop a systematic approach to generating fuzzy rules from a given input-output data set. Figure 3 shows how a two-rule fuzzy inference system of the Sugeno type works to calculate the overall output z when subjected to two crisp inputs x and y.

Figure 3. Sugeno type Fuzzy Inference System A typical fuzzy rule in a Sugeno fuzzy model has the form

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where Ai and Bi are fuzzy sets and z = fi (x; y) is a crisp function. Usually f(x; y) is a polynomial in the input variables x and y and the fuzzy inference system that uses a first-order polynomial f(x; y) is called a first-order Sugeno fuzzy model (Takagi et al., 1985; Sugeno et al., 1988). If f is a constant, it is called a zero-order Sugeno fuzzy model that is functionally equivalent to a radial basis function network under certain minor constraints (Jang et al., 1993).

In the Tsukamoto fuzzy models (Tsukamoto, 1979), as shown in Figure 4, the result of each fuzzy if-then rule is represented by a fuzzy set with a monotonical MF. The overall output is taken as the weighted average of each rule's output. The whole reasoning procedure for a two-input two-rule system can be seen in Figure 4.

Figure 4. Tsukomato type fuzzy Inference System

Since each rule infers a crisp output, the Tsukamoto fuzzy model aggregates each rule's output by the method of weighted average and thus also avoids the time-consuming process of defuzzification.

Defuzzification transforms the fuzzy results of the inference into a crisp output. The most commonly used defuzzification strategy is the centroid of area, which is defined as

( )

ò

= Z

C Z

dz z

dz z z output

µ µ

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where µC(z) is the aggregated output MF. Other defuzzification strategies arise for specific applications, which

includes bisector of area, mean of maximum, largest of maximum, and smallest of maximum, and so on. These defuzzification strategies are shown in Figure 5. These strategies are computationally intensive and there is no rigorous way to analyze them except through experimental-based work.

Smallest of max. Largest of max. Mean Bisector Centroid

of max. of area of area

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PROBLEM DEFINITION

The need for a more detailed examination and better understanding of dynamic world at all scales has become an increasing priority (Mark, 1999). People’s perceptions of geographic space have received attention through consideration of the spatial cognitive aspects of GIS. Among the important aspects of how people represent geographic phenomena are storing and retrieving spatial information about past states. What has not been addressed so far is how to reason and model about dynamic properties of geographic phenomena in a GIS setting.

To represent and reason about the dynamic world, we need to model movements of moving objects such as cars, animals, airplanes, and oil spills. Examples of movements are cars traveling on the interstate, moose roaming through the forest, emergency dispatch vehicles responding to an accident, a wild fire spreading, an oil spill expanding, a glacier shifting, and a dessert eroding. In all these applications there is a need to answer questions about moving objects. Such questions may include where objects have been in the past, where they are now, and where they will be at a specific time, and what are their expected travel times and paths.

A traditional GIS approach to estimate and calculate a list of trajectories of moving objects between two locations has been done using only spatial information. Adding the time and semantic information into the system has started to be considered recently (Beard et al., 2000; Moreira et al.,, 1999; Pfoser et al., 1999). Along with the obtained trajectories, travel times and distances can also be calculated in this system. In addition to these, there are constraints such as traffic volume, road conditions, weather conditions, special events (Christmas, holidays, weekends, etc.) that have not been considered in the calculations. In the real world, one needs to take these constraints into account to get accurate estimations. The fuzzy logic approach has been considered in this work to include these constraints in reasoning about moving objects, estimating their travel times and paths.

In cognitive science, it is known that humans often use fuzzy logic methods to reason about events in their daily life. For humans, navigation in space can be handled by using approximate values that can also be semantic values, fuzzy relations, and by taking the constraints (i.e., weather and road conditions) into account. People draw on previous experience to decide on the most appropriate trajectory to choose among different options to travel between two locations. For example, a person who travels between home and work can predict certain information related with the trajectory such as how long this trajectory might take according to the time of day, weather and other constraints. In this manner, fuzzy logic offers tools to include semantic and fuzzy information in reasoning about moving objects over time.

Spatio-Temporal Movement

The phenomenon of movement arises whenever the same object occupies different positions in space at different times (Galton, 1995). Erwig et al. (1999) refers to all geometric change as movement, including changes in shape (growing or shrinking). We define movement as comprising a change in the position (e.g., a translation) or in the orientation (e.g., a rotation) of a geographic phenomenon over time. We categorize change to an object’s boundary as shape change and distinguish it from movement. This distinction allows us to capture important behavioral differences between real world objects, for example lakes and cars.

Unless we have observations that are continuous (e.g., video stream, GPS observations) we cannot account for every instant of a movement. Descriptions of movement are thus limited to the actual observations that we use to detect them and the interpolation methods that we have available.

Improvements in data collection technologies, such as high-resolution satellite imagery, videogrammetry, and GPS in combination with wireless communication, are rapidly increasing the feasibility of obtaining information on moving objects and supporting new research and development. Complete knowledge of any particular movement is impossible, but movements can be detected and predicted with some degree of accuracy.

In addition, development of new and more powerful techniques and tools in various fields has started to provide us with better modeling, representation, and reasoning about dynamic geographic phenomena. Fuzzy logic is one of the recent approaches that deal with modeling of dynamic systems.

Method

Several applications require support for representing and analyzing moving objects. Examples of such applications are wildlife tracking, environmental monitoring, emergency dispatch, vehicle navigation, fleet management, storm tracking, and military applications. Fuzzy logic can be easily applied to those applications to carry out some of the requests. Wildfire modeling, as an example, is one of the research areas that can utilize fuzzy logic. The factors such as wind speed and direction, tree types, forest density, temperature, and humidity, play important roles in wildfire modeling. Those factors, obtained from observations and stored in GIS, can be evaluated

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using a fuzzy logic approach to estimate the spread and possible effects of the fire. Also, the spread of an oil spill can be modeled in a similar manner.

Tracking, dispatching and courier services, that are main areas of research for this work, have a need to better estimate travel times and most appropriate trajectories between two known locations. In this work, we claim that adding temporal information to calculate the travel time and to find most appropriate trajectory between two locations may yield better results. In traditional approaches, linear interpolation methods have been used to find the most appropriate trajectories.

Fuzzy logic method provides additional tools to reason about moving objects, their travel times and trajectories adding temporal information such as time intervals in a day (morning, noon, 8 am to 11 am), weekdays, weekend days, and holidays and semantic information such as traffic density (low, average, high) and weather condition (cold, average, hot).

Using a fuzzy logic approach, temporal and semantic information are included in the calculations of travel times and most appropriate trajectories. In our approach a road might have multiple sections with each section represented by different temporal characteristics such as traffic volume, accidents, construction activity and other constraints that may change the travel time. The data about each section is stored in GIS and evaluated employing fuzzy logic approach. Section characteristics such as traffic flow direction, number of open lanes, and temporal information such as time intervals in a day, are entered as inputs into the fuzzy systems. As can be seen from Figure 6 such section information as traffic density that is given as heavy, moderate, and light can be continuously obtained from monitoring systems. Using these inputs and the rules defined by an expert, the fuzzy system calculates the optimum travel time for each section in the trajectory. The fuzzy rules are defined using temporal characteristics of the roads from the observations taken the experiences of the expert into consideration. The output of the fuzzy system, travel times for each section, are used by GIS to calculate most appropriate trajectories between two locations.

Figure 6. Traffic Report of Boston, MA

In this work, we determined five input and one output variables given as shown Table 1. A Mamdani type fuzzy inference system is used for rule evaluation.

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Table 1. Input and output variables of the fuzzy logic system.

Input Variables Output Variable

Time of the Day

Special Occasions

Number of Lanes

Speed Limit

Traffic Volume

Weather Travel time per mile

Early morning Morning Late morning Noon Afternoon Late afternoon Night

Midnight

Weekdays Weekend Holidays

Single Two Three Four

25 35 45 55 65

Low Average Dense

Bad Mind Average Good

Very low Moderate low Low

Average High

Moderate high Very high

The implementation of this application has been carried out using the Fuzzy Toolbox of Matlab software. The input and output variables are defined in the FIS editor as shown in Figure 7. Among those, only membership functions for time and traffic volume input variables are given in Figure 8 as an example. Although, there are several membership function types that can be used such as gaussian, generalized bell, and trapezoid, the gaussian is implemented in this work.

Figure 7. FIS Editor for defining input and output variables. Sample set of rules, defined by an expert using past observations over time, are given below :

o If volume is dense then time-per-mile is moderate high.

o If weather is bad and lane is single then time-per-mile is too high.

o If time is early morning and lane is four and speed-limit is 65 and volume is empty and weather is

good then time-per-mile is too low.

o If time is morning and lane is single and speed-limit is 25 and volume is dense and weather is bad

then time-per-mile is too high.

The proposed technique assumes that the expert already obtained enough observations to make evaluations and come up with appropriate rules for the fuzzy system. The number of rules defined in the rule base can be increased or decreased by an expert depending on the experience and the system need. There is a tradeoff between number of rules, accuracy of the output and evaluation time of the system.

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Figure 8. Membership functions for (a) time and (b) traffic volume input variables

Figure 9. Sample inputs and the output values and defined rules in the fuzzy system.

The following sample set of inputs is given to the fuzzy system. Those are : Time is 8:00am, number of lanes is 1, speed limit is 50 miles/hour, traffic volume is 0.7 (0.5 is average volume), weather is 0.8 (0.5 is average),

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occasion is 1 (weekdays). The output, time per mile, is obtained as 4.07 that is a numerical value that will be used in the travel time calculations in GIS. For different input values, different output values can be obtained. Accuracy of the output depends on the rules that should reflect the reality. The optimum rules and membership functions can be derived from observed data by training them in FIS. However, the training process is computationally expensive and time consuming. Therefore, rules and membership functions used in this work are defined by an expert.

CONCLUSION

In this work, we demonstrated how to use a fuzzy logic approach to carry out estimations using semantic information in spatio-temporal systems. Although, linguistic terms, fuzzy values and relationships that are often used in daily life to reason about real world phenomena may not be included in linear approaches fuzzy logic systems can handle them successfully. These systems offer new solutions and approaches for spatio-temporal systems especially for modeling moving objects.

The continuous change in the spatial and temporal behavior of the moving objects makes it difficult to estimate their movement patterns such as travel time, trajectory, and position. Fuzzy logic approach can be employed to carry out these types of estimations. Fuzzy logic approach has benefits when there is lack of observations and have experiences instead. However, the communication between available GISs and Fuzzy systems needs to be improved as do the methods of how to use fuzzy approach in spatio-temporal systems.

REFERENCES

Allen J.F. (1984). A General Model of Action and Time. Artificial Intelligence 23(2): 123-154.

Beard K., Palancioglu H. M. (2000). Estimating Positions and Paths of Moving Objects. In: Proceedings of Seventh International Workshop on Temporal Representation and Reasoning, The IEEE Computer Society Press, Cape Breton, Nova Scotia, CANADA, ISBN 0-7695-0756-5, pp. 155-162.

Egenhofer Max J., Franzosa Robert D. (1991). Point Set Topological Relations. International Journal of Geographical Information Systems, 5:161-174 .

Egenhofer Max J., Golledge Reginald G. (1998). Spatial and Temporal Reasoning in Geographic Information Systems. Oxford University Press, London, ISBN: 0-19-510342-4.

Erwig M., Güting R. H., Schneider M., Vazirgiannis M. (1999). Spatio-Temporal Data Types: An Approach to Modeling and Querying Moving Objects in Databases. GeoInformatica 3.

Galton A. (1993). Towards an Integrated Logic of Space, Time, and Motion. In: Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI'93), Chambery, France.

Galton A. (1995). Towards a Qualitative Theory of Movement, In: Spatial Information Theory: A Theoretical Basis for GIS (Proceedings of International Conference COSIT'95). Springer-Verlag, Semmering, Austria, pp. 377-396.

Guting R.H. (1994). An Introduction to Spatial Database Systems. VLDB Journal. 4:357-399.

Güting R.H., Böhlen M.H., Erwig M., Jensen C.S., Lorentzos N.A., Schneider M., Vazirgiannis M. (2000). A Foundation for Representing and Querying Moving Objects. TODS, 25(1): 1-42.

Jang J. S. Roger, Sun C. T. (1993). Functional equivalence between radial basis function networks and fuzzy inference systems. IEEE Trans. on Neural Networks, 4(1):156-159.

Jang J. S. Roger, Sun C. T. (1995). Neuro-Fuzzy Modeling and Control. Proceedings of the IEEE, 83:378-406 Langran G. (1988). Temporal Design Tradeoffs. In: Proceedings of GIS/LIS'88, Falls Church, VA, 2:890-899. Langran G. (1989). A review of temporal database research and its use in GIS applications. Int. Journal of GIS, 3(3):

215-232.

Langran G. (1992). Time in Geographic Information Systems. Taylor & Francis, Bristol, PA.

Langran G. (1993). Issues of Implementing a Spatiotemporal System, International Journal of Geographical Information Systems, 7(4):305-314.

Mamdani E. H., Assilian S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. In:

International Journal of Man-Machine Studies, 7(1):1-13.

Mark D. M. (1999). Geographic Information Science: Critical Issues In An Merging Cross-Disciplinary Research Domain.

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Mendel J.M., Mouzouris G.C. (1997). Designing fuzzy logic systems. IEEE Trans Circuit and Systems-II: Analog and Digital Signal Processing, 44:885-895.

Moreira J, Cristina R., Saglio J.-M. (1999). Representation and Manipulation of Moving Points: An Extended Data Model for Location Estimation. In: Cartography and Geographic Information Systems, 26(2):109-123.

Peuquet D. J. (1994). It's about time: a conceptual framework for the representation of temporal dynamics in geographic information systems. Annals of the Association of American Geographers, 84(3): 441-462.

Pfoser D, Jensen C. S. (1999). Capturing the Uncertainty of Moving Object Representations. In: Proceedings of the SSDBM Conference, pp. 111-132.

Randell David A., Cui Zhan, Cohn Anthony G. (1992). A Spatial Logic based on Regions and Connection. In:

Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR'92), Cambridge, MA, Morgan Kaufmann, ISBN 1-55860-262-3, pp. 165-176

Sistla P., Wolfson O., Chamberlain S., Dao S. (1997). Modeling and Querying Moving Objects. In: Proceedings of the 13th International Conference on Data Engineering. Birmingham, UK.

Sugeno M., Kang G. T. (1988). Structure identication of fuzzy model. Fuzzy Sets and Systems, 28:15-33.

Takagi T., Sugeno M. (1985). Fuzzy identication of systems and its applications to modeling and control. IEEE Trans. on Systems, Man, and Cybernetics, 15:116-132.

Tsukamoto Y. (1979). An approach to fuzzy reasoning method. In: Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, pp. 137-149.

Wang L.X. (1994). Adaptive fuzzy systems and control. NJ: Prentice-Hall, Englewood-Cliffs.

Worboys Michael F. (1994). A Unified Model for Spatial and Temporal Information. The Computer Journal, 37 (1): 36-34.

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PROBLEM DEFINITION

A traditional GIS approach to estimate and calculate a list of trajectories of moving objects between two locations has been done using only spatial information. Adding the time and semantic information into the system has started to be considered recently (Beard et al., 2000; Moreira et al.,, 1999; Pfoser et al., 1999). Along with the obtained trajectories, travel times and distances can also be calculated in this system. In addition to these, there are constraints such as traffic volume, road conditions, weather conditions, special events (Christmas, holidays, weekends, etc.) that have not been considered in the calculations. In the real world, one needs to take these constraints into account to get accurate estimations. The fuzzy logic approach has been considered in this work to include these constraints in reasoning about moving objects, estimating their travel times and paths.

Spatio-Temporal Movement

The phenomenon of movement arises whenever the same object occupies different positions in space at different times (Galton, 1995). Erwig et al. (1999) refers to all geometric change as movement, including changes in shape (growing or shrinking). We define movement as comprising a change in the position (e.g., a translation) or in the orientation (e.g., a rotation) of a geographic phenomenon over time. We categorize change to an object’s boundary as shape change and distinguish it from movement. This distinction allows us to capture important behavioral differences between real world objects, for example lakes and cars.

Method

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using a fuzzy logic approach to estimate the spread and possible effects of the fire. Also, the spread of an oil spill can be modeled in a similar manner.

Figure 6. Traffic Report of Boston, MA

In this work, we determined five input and one output variables given as shown Table 1. A Mamdani type fuzzy inference system is used for rule evaluation.

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Table 1. Input and output variables of the fuzzy logic system.

Input Variables Output Variable

Time of the Day Special Occasions Number of Lanes Speed Limit Traffic Volume

Weather Travel time per mile Early morning Morning Late morning Noon Afternoon Late afternoon Night Midnight Weekdays Weekend Holidays Single Two Three Four 25 35 45 55 65 Low Average Dense Bad Mind Average Good Very low Moderate low Low Average High Moderate high Very high

Figure 7. FIS Editor for defining input and output variables. Sample set of rules, defined by an expert using past observations over time, are given below :

o If volume is dense then time-per-mile is moderate high.

o If weather is bad and lane is single then time-per-mile is too high.

o If time is early morning and lane is four and speed-limit is 65 and volume is empty and weather is good then time-per-mile is too low.

o If time is morning and lane is single and speed-limit is 25 and volume is dense and weather is bad then time-per-mile is too high.

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(a)

(b)

Figure 8. Membership functions for (a) time and (b) traffic volume input variables

Figure 9. Sample inputs and the output values and defined rules in the fuzzy system.

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CONCLUSION

REFERENCES

Allen J.F. (1984). A General Model of Action and Time. Artificial Intelligence 23(2): 123-154.

Beard K., Palancioglu H. M. (2000). Estimating Positions and Paths of Moving Objects. In: Proceedings of Seventh International Workshop on Temporal Representation and Reasoning, The IEEE Computer Society Press, Cape Breton, Nova Scotia, CANADA, ISBN 0-7695-0756-5, pp. 155-162.

Egenhofer Max J., Franzosa Robert D. (1991). Point Set Topological Relations. International Journal of Geographical Information Systems, 5:161-174 .

Egenhofer Max J., Golledge Reginald G. (1998). Spatial and Temporal Reasoning in Geographic Information Systems. Oxford University Press, London, ISBN: 0-19-510342-4.

Erwig M., Güting R. H., Schneider M., Vazirgiannis M. (1999). Spatio-Temporal Data Types: An Approach to Modeling and Querying Moving Objects in Databases. GeoInformatica 3.

Galton A. (1993). Towards an Integrated Logic of Space, Time, and Motion. In: Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI'93), Chambery, France.

Galton A. (1995). Towards a Qualitative Theory of Movement, In: Spatial Information Theory: A Theoretical Basis for GIS (Proceedings of International Conference COSIT'95). Springer-Verlag, Semmering, Austria, pp. 377-396.

Guting R.H. (1994). An Introduction to Spatial Database Systems. VLDB Journal. 4:357-399.

Güting R.H., Böhlen M.H., Erwig M., Jensen C.S., Lorentzos N.A., Schneider M., Vazirgiannis M. (2000). A Foundation for Representing and Querying Moving Objects. TODS, 25(1): 1-42.

Jang J. S. Roger, Sun C. T. (1993). Functional equivalence between radial basis function networks and fuzzy inference systems. IEEE Trans. on Neural Networks, 4(1):156-159.

Jang J. S. Roger, Sun C. T. (1995). Neuro-Fuzzy Modeling and Control. Proceedings of the IEEE, 83:378-406 Langran G. (1988). Temporal Design Tradeoffs. In: Proceedings of GIS/LIS'88, Falls Church, VA, 2:890-899. Langran G. (1989). A review of temporal database research and its use in GIS applications. Int. Journal of GIS, 3(3):

215-232.

Langran G. (1992). Time in Geographic Information Systems. Taylor & Francis, Bristol, PA.

Langran G. (1993). Issues of Implementing a Spatiotemporal System, International Journal of Geographical Information Systems, 7(4):305-314.

Mamdani E. H., Assilian S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. In: International Journal of Man-Machine Studies, 7(1):1-13.

Mark D. M. (1999). Geographic Information Science: Critical Issues In An Merging Cross-Disciplinary Research Domain.

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Mendel J.M., Mouzouris G.C. (1997). Designing fuzzy logic systems. IEEE Trans Circuit and Systems-II: Analog and Digital Signal Processing, 44:885-895.

Moreira J, Cristina R., Saglio J.-M. (1999). Representation and Manipulation of Moving Points: An Extended Data Model for Location Estimation. In: Cartography and Geographic Information Systems, 26(2):109-123.

Peuquet D. J. (1994). It's about time: a conceptual framework for the representation of temporal dynamics in geographic information systems. Annals of the Association of American Geographers, 84(3): 441-462.

Pfoser D, Jensen C. S. (1999). Capturing the Uncertainty of Moving Object Representations. In: Proceedings of the SSDBM Conference, pp. 111-132.

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MODELING MOVING OBJECTS AND THEIR MOVEMENTS USING FUZZY LOGIC APPROACH

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