GeoComputation

1. GeoComputation Gordon Green Hunter College Department of Geography 11/2006

2. What is GeoComputation? • “…the eclectic application of computational methods and techniques to portray spatial properties, to explain geographic phenomena, and to solve geographical problems” -Couclelis, 1998 • “The application of a computational science paradigm to study a wide range of problems in geographical and earth systems.” –Openshaw, 2000 • "The universe of computational techniques applicable to spatial problems" -Reed and Tuton, 1998

3. What is GeoComputation? • Its champions consider it a new discipline. • Others consider it just a toolbox of computationally intensive techniques.

4. What is GeoComputation? • Geo - emphasizes geographical and spatial aspects. Some other approaches entail spatial additions to methods that arose in non-spatial fields. GeoComputational methods tend to be inherently spatial. • Computation - equally important as the spatial aspect. The approaches tend to come out of computer science, rather than geography. • Traditional statistics tends to reduce and summarize information; GeoComputation tends to retain or generate more complexity in its operations.

5. What is GeoComputation? • Fotheringham differentiates between Weak GC, where the computer is used to augment existing quantitative methods; and Strong GC, where the computer "drives" the form of analysis. • Confirmatory versus exploratory techniques -- GC tends to fall in the latter category. • “The challenge of GC is to develop the ideas, the methods, the models, and the paradigms able to use the increasing computer speeds to do useful, worthwhile, innovative ad new science in a variety of geo-contexts.” (Openshaw)

6. What is GeoComputation? Aspects of GeoComputation: • Multi-Agent Modeling • Cellular Automata • Fuzzy Modeling • Genetic Programming • Neural Networks

7. Agent-Based Models • An agent-based model is one in which the basic unit of activity is a software agent. • Usually a model will contain many agents (at least tens, occasionally many thousands). • Its outcomes are determined by the interactions of the agents. • A agent is an identifiable unit of computer program code which is autonomous and goal-directed. (from Schelhorn, O’Sullivan, Haklay, Thurstain-Goodwin, 1999)

8. Agent-Based Models • An early example of agent-based modeling is Boids (Reynolds, 1987), a model that describes flocking behavior. • Explaining flocking using functional models is complex. • By approaching it by modeling autonomous agents with simple rules, it was easier to create an explanatory model. • Agent models typically work by maintaining a collection of agents, and a timer that advances the state of each agent with each tick, each agent updating its state based on characteristics and location of the other agents and other entities in the landscape. • Agent models are best implemented in an object-oriented fashion

9. Agent-Based Models Each agent has a simple set of rules defining its behavior. Creating multiple instances of the model and adjusting the rules made it possible to model flocking behavior concisely. Boids animation More elaborate example

10. Agent-Based Models • These animations exhibit the phenomenon of emergence, whereby complex behavior emerges from the behavior of simple components. • Another example of agent-based modeling is pedestrian modeling: • Pedestrian model – note formation of lanes. • Other sample pedestrian models

11. Agent-Based Models Pedestrian model implementation (Batty, 1999)

12. Agent-Based Models Pedestrian model implementation (Batty, 1999)

13. Agent-Based Models Pedestrian model implementation (Batty, 1999)

14. Agent-Based Models Pedestrian model implementation (Batty, 1999)

15. Pedestrian model design (Schelhorn, O’Sullivan, Haklay, Thurstain-Goodwin, 1999)

17. Software for agent-based modeling (from Najlis, Janssen, and Parker, 2001)

18. Agent-Based Models When to use agents? Axtell (1999) describes three basic cases when agent-based models might be used: 1. Agent models as simulation of mathematical models – agent models can be used to implement monte-carlo simulations that can also be solved using other numerical methods. Even though the simulation can be solved in a functional manner, an agent-based implementation can act as an alternative implementation that can validate the results. Similarly, an agent-based model can be useful as an illustration of a complex numerical model. Even though the agent-based model may not be necessary to come up with a solution, it may be very helpful in illustrating the results to a general audience.

19. Agent-Based Models An example of this first use can be found in the classic problem of modeling a bank-teller line: • The queuing process is commonly simulated using a the Monte Carlo method to arrive at a distribution of waiting times. • This is can be equivalently modeled using agents that each have different arrival times and other parameters, and running the model with many agents to similarly build up the resulting distribution function. (Axtell, 1999)

20. Agent-Based Models 2. Agent models as complementary to mathematical models – a mathematical model may adequately model some aspects of a problem but not others, or may be awkward or incapable of a comprehensive solution. Or a mathematical model may be known, but so complex as to be practically insoluble. An artificial stock market is an example of such a model. Trading agents have the choice of investments with varying stochastic levels of risk, adapt their behavior based on the results of prior trading. Even though the basic features of such a model may be functionally describable, an agent-based model can evolve over time into a system that actually replicates some of the complexity of a real stock market. These complexities are impractical to model using a mathematical model. (Axtell, 1999)

21. Agent-Based Models 3. Agent models as substitutes for mathematical models – some problems are not amenable to mathematical modeling. Examples include the behavior of individuals in animal population, or groups of pedestrians as described earlier.

22. Agent-Based Models Pros: • An argument from economics, which could also be applied to geography: “The reason why large scale computable general equilibrium problems are difficult for economists to solve is that they are using the wrong hardware and software. Economists should design their computations to mimic the real economy, using massively parallel computers and decentralized algorithms that allow competitive equilibria to arise as ‘emergent computations’...[T]he most promising way for economists to avoid the computational burdens associated with solving realistic large scale general equilibrium models is to adopt an “agent-based” modeling strategy where equilibrium prices and quantities emerge endogenously from the decentralized interactions of agents.” (Rust, 1996, in Axtell) • The quality of emergent behavior doesn’t correspond to any part of a traditional statistical analysis. Agent-based models uniquely provide the opportunity to model a process, and see what happens when it runs (Axtell).

23. Agent-Based Models Cons: • When dealing with agent models, we are quickly involved in a world where everything - both the agents and their environment - are designed, and the reliable scientific analysis of the real world may be compromised by the complexity of the models (paraphrased from Couclelis, Why I No Longer Work with Agents, discussing human/environment agent models). • Agent models don’t have a convenient way of measuring their accuracy, unlike statistical models. This can only be overcome by running the agent model many times, varying the parameters to discover the robustness of the results. There is a limit as to how many such variations can be executed (although this is increasing with increases in computer power) (Axtell).

24. Cellular Automata • Cellular automata are closely related to agent-based models. • Instead of an autonomous agents being given simple behavioral rules, the states of cells in a surface are dictated by similarly simple rules. • Those rules use the state of surrounding cells in a grid to determine the state of any given cell in the next iteration of the model.

25. Cellular Automata • CAs develop in space and time. • Space and time are defined in discrete steps. • Cells are lined up in a string for one-dimensional automata, or arranged in a two or higher dimensional lattice for two- or higher- dimensional automata. • The number of states of each cell is finite. • The states of each cell are discrete. • All cells are identical. • The future state of each cell depends only of the current state of the cell and the states of the cells in the neighborhood, determined by rules (from Alexander Schatten).

26. Cellular Automata • The simplest CAs are one-dimensional. For example, here are a set of rules and the results of a few iterations (from David R. Green):

27. Cellular Automata For geographical applications, CAs are usually 2-dimensional, often using one of these cell neighborhoods:

28. Cellular Automata Conway’s Game of life rules Mathematician John Conway invented CAs in his Game of Life, first described in a 1970 Scientific American article. • A cell that is dead at the time step t, becomes alive at time t+1 if exactly three of the eight neighboring cells at time t were alive. • A cell that is alive at time t dies at time t+1 if at time t less than two (loneliness) or more than three cells (overcrowding) are alive. (Alexander Schatten) From these very simple rules, very complex behaviors can be modeled.

29. Cellular Automata Sample game of life pattern (“glider”):

30. Cellular Automata • There are many other examples available on the web, such as: http://www.ibiblio.org/lifepatterns/ • These simple rules can be expanded to generate many, many patterns of change: http://www.collidoscope.com/modernca/

31. Cellular Automata There are many enhancements involved in generating more complex evolutionary patterns: • Probabilistic CAs, where, instead of having binary rules (and binary states), the changes are described by probabilities, give an increased level of control over how a CA develops. The rules can express the chance of a cell changing state, and each step of the CA involves selecting the state of each cell based on a random number falling within its probability range. • CAs can also be self-modifying, that is, they can respond to changes in the states of the grid as it develops. • Irregular grid cell sizes • Action a distance, instead of just immediately adjacent cells. • Incorporating agents within CA landscapes

32. Cellular Automata These more sophisticated CAs have been used to model diverse geographic phenomena, such as: • Wildfire • Formation of a Megalopolis • West Nile Virus Infection Risk #2 from Paul Torrens, http://geosimulation.org/simulating-sprawl/, #3 from Sean Ahearn.

33. Cellular Automata Cons (Batty and Xie, 2005): • Most CA rules have little relationship to the actual causes of the phenomenon being studied. Even if the model happens to model the process successfully, it doesn’t really prove anything, and is difficult to use as the basis of any kind of policy decision. • They tend not to model spatial interaction well. • In practice, CA cells tend not to match units of the phenomenon under study.

34. Fuzzy Modeling • A frequent problem in geography is how to identify an geographic entity. • Geographic phenomena tend to be described by vague terms. • For example: a study concerns the major cities in Europe. Each of the terms “major”, “city”, “in”, and “Europe” are somewhat vague. • Fuzzy set theory is an attempt to deal with the problems posed to traditional set and logic theory by vagueness. (section paraphrased from Fisher, 2000)

35. Fuzzy Modeling • Sorites paradox: • If a grain of sand is placed on a surface, is there a heap? • If a second grain of sand is placed next to it, is there a heap? • If a third grain of sand is placed next to the second, is there a heap? • If a ten-millionth grain of sand is placed next to the 9,999,999th, is there a heap? • Heap is a vague concept. Other examples are “tall”, “near”, “far”, etc. • Many geographic values such as vegetation coverage, soil types, etc, are similarly “Sorites-susceptible”.

36. Fuzzy Modeling • Classical sets follow the logic described in familiar Venn diagram (examples drawn from behavior of boolean search terms):

37. Fuzzy Modeling These can also be expressed using linear graphs:

38. Fuzzy Modeling • Membership in a set is determined by some threshold value. This value can be subject to error, and can be assigned a probability, Those probabilities can then be used in set-based calculations:

39. Fuzzy Modeling • Boolean set theory is used throughout most conventional set and statistical analysis, For example: does the result of a test disprove the null hypothesis? We set a threshold value to the t test statistics and respond based on whether or not the results of the hypothesis are above or below that test. In many cases, there is actually a continuous probability of the null hypothesis being correct. • Zadeh 1965 put forward concept of fuzzy sets in response to shortcomings of boolean sets.

40. Fuzzy Modeling • Fuzzy set membership is at the core of fuzzy set theory. Boolean sets are encoded with 0 or 1, wherein an entity is or is not part of a set. Fuzzy set membership is defined by any value between (and including) 0 and 1.

41. Fuzzy Modeling • Fuzzy set membership can take varied forms:

42. Fuzzy Modeling • These then result in boolean calculations that take into account partial set membership:

43. Fuzzy Modeling • Boolean operations become functions rather than boolean values:

44. Fuzzy Modeling

45. Fuzzy Modeling Example of fuzzy versus crisp classifications

46. Fuzzy Modeling The concept of fuzziness can also be applied to polygon boundaries

47. Fuzzy Modeling Key question: How do you choose the membership function (MF)? Is it linear or curved? • The “semantic import” (SI) approach estimates based on domain expertise and iterative evaluation of the results. • Fuzzy K-means approach uses iterative mathematical estimates of the best function.

48. Fuzzy Modeling Fuzzy K-Means • Usually starts with random allocation of objects into k clusters. • The center of each cluster is calculated • Objects are re-allocated based on similarity of attributes using a similarity index, usually a distance measurement. • This process is repeated until a stable solution is reached. • Membership in each cluster is calculated as a range of from 0 to 1, instead of 1 as in “crisp” k-means (Burrough and McDonnell)

49. Fuzzy Modeling Fuzzy K-Means

50. Fuzzy Modeling • Fuzzy modeling is often used in conjunction with genetic algorithms, which provide a way of iteratively refining the results of automated models. • The following few slides will review this before concluding the section on fuzzy modeling.