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

2. University of Wisconsin-Milwaukee Geographic Information Science Outline 1.Introduction 2.Processes and the patterns 3.Predicting the pattern generated by a process 4.More definitions 5.Stochastic processes in lines, areas, and fields 6.Conclusion

3. University of Wisconsin-Milwaukee Geographic Information Science 1. Introduction Maps as outcomes of process 1.Maps have the ability to suggest patterns in the phenomena they represent. 2.Patterns provide clues to a possible causal process. 3.Maps can be understood as outcomes of processes. ProcessesPatterns Map

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

5. University of Wisconsin-Milwaukee Geographic Information Science 2. Process and the Patterns Z = 2x + 3y Deterministic x y

6. University of Wisconsin-Milwaukee Geographic Information Science  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. 2 2 x y Stochastic 2. Process and the Patterns

7. University of Wisconsin-Milwaukee Geographic Information Science y Stochastic: two realizations of z= 2x + 3y ± 1 2. Process and the Patterns xx y

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

9. University of Wisconsin-Milwaukee Geographic Information Science 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) 1.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. 2.Independence: the positioning of any point is independent of the positioning of any other point.

10. University of Wisconsin-Milwaukee Geographic Information Science 3. Predicting the Pattern Generated By a Process A B 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. University of Wisconsin-Milwaukee Geographic Information Science A B 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) 1)Equal probability 2)Independence P (event A in Yellow quadrat) = 1/8 P (event A not in Yellow quadrat) = 7/8 P (event A only in the Yellow quadrat) = P (event A in Yellow quadrat and other events not in the Yellow quadrat) A B C D E F G H I J

12. University of Wisconsin-Milwaukee Geographic Information Science 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) A B P (one event only) = P (event A only) + P (event B only) + … + P (event J only) = 10 × P (event A only)

13. University of Wisconsin-Milwaukee Geographic Information Science 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) A B 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 C D E F G H I J

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

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

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

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

18. University of Wisconsin-Milwaukee Geographic Information Science 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR)

19. University of Wisconsin-Milwaukee Geographic Information Science 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. University of Wisconsin-Milwaukee Geographic Information Science 3. Predicting the Pattern Generated By a Process Complete spatial randomness (CSR) Comparison between binomial and Poisson distribution

21. University of Wisconsin-Milwaukee Geographic Information Science 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.