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Geography 625

Geography 625. Intermediate Geographic Information Science. Week3: Fundamentals: Maps as outcomes of process. Instructor : Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006. Outline. Introduction Processes and the patterns

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Geography 625

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  1. Geography 625 Intermediate Geographic Information Science Week3: Fundamentals: Maps as outcomes of process Instructor: Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006

  2. Outline • Introduction • Processes and the patterns • Predicting the pattern generated by a process • More definitions • Stochastic processes in lines, areas, and fields • Conclusion

  3. 1. Introduction • Maps as outcomes of process • Maps have the ability to suggest patterns in the phenomena they represent. • Patterns provide clues to a possible causal process. • Maps can be understood as outcomes of processes. Map Processes Patterns

  4. 2. Process and the Patterns A spatial process is a description of how a spatial pattern might be generated. Deterministic: it always produce the same outcome at each location. Z = 2x + 3y Where x and y are two spatial coordinates z is the numerical value for a variable y 2 x 2

  5. 2. Process and the Patterns y Deterministic Z = 2x + 3y x

  6. 2. Process and the Patterns Stochastic • More often, geographic data appear to be the result of a chance process, whose outcome is subject to variation that cannot be given precisely by a mathematical function. • This chance element seems inherent in processes involving the individual or collective results of human decisions. • Some spatial patterns are the results of deterministic physical laws, but they appear as if they are the results of chance process. z= 2x + 3y+ d Where d is a randomly chosen value at each location, -1 or +1. y 2 2 x

  7. 2. Process and the Patterns Stochastic: two realizations of z= 2x + 3y± 1 y y x x

  8. 2. Process and the Patterns Dot map with randomly distributed points Created Random numbers from Excel Int(10 * Rand()) Use these numbers as x and y coordinates Repeat this process

  9. 3. Predicting the Pattern Generated By a Process What would be the outcome if there were absolutely no geography to a process (completely random)? Independent random process (IRP) Complete spatial randomness (CSR) • Equal probability: any point has equal probability of being in any position or, equivalently, each small sub-area of the map has an equal chance of receiving a point. • Independence: the positioning of any point is independent of the positioning of any other point.

  10. A B 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) Event: a point in the map, representing an incident. Quadrats: a set of equal-sized and nonoverlapping areas Pattern Process (Complete spatial randomness)

  11. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) • Equal probability • Independence P (event A in Yellow quadrat) = 1/8 P (event A not in Yellow quadrat) = 7/8 A P (event A only in the Yellow quadrat) = P (event A in Yellow quadrat and other events not in the Yellow quadrat) B A B C D E F G H I J

  12. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) P (one event only) = P (event A only) + P (event B only) + … + P (event J only) = 10 × P (event A only) A B

  13. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) P (event A & B in Yellow quadrat) = 1/8 ×1/8 P (event A & B in Yellow quadrat only) = P ((event A & B in Yellow quadrat) and (other events not in Yellow quadrat)) A B A B C D E F G H I J

  14. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) P ( two events in Yellow quadrat) = P(A&B only) + P(A&C only) + … + P(I&J only) =(no. possible combinations of two events) × A B How many possible combinations?

  15. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) The formula for number of possible combinations of k events from a set of n events is given by A B In our case, n = 10, and k = 2

  16. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) P (k events) = A B p = quadrat area / area of study region

  17. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) Binomial distribution A B x is the number of quadrats used n is the number of events k is the number of events in a quadrat

  18. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR)

  19. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) The binomial expression derived above is often not very practical for serious work because of computation burden, the Poisson distribution is a good approximation to the binomial distribution. e is a constant, equal to 2.7182818

  20. 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) Comparison between binomial and Poisson distribution

  21. 4. More Definitions • The independent random process is mathematically elegant and forms a useful starting point for spatial analysis, but its use is often exceedingly naive and unrealistic. • If real-world spatial patterns were indeed generated by unconstrained randomness, geography would have little meaning or interest, and most GIS operations would be pointless.

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