Bayesian Methods Weights of Evidence

1. Bayesian Methods Weights of Evidence RESM 575 Spring 2011 Lecture 5

2. Today • Quick review of methods we discussed to this point • Weights of evidence • Spatial data modeler (Arc-SDM)

3. Last time For ranking or prioritizing known alternatives • Spatial Compromise Programming • CP ranking tool

6. Building suitability models, steps • Define the problem or goal • Decide on evaluation criteria • Normalize and create utility scales • Boolean method • Index overlay with utility scales • Fuzzy membership functions • Define weights for criteria • Point allocation • Ranking methods (rank sum, exponent, reciprocal) • Pairwise comparison (AHP) • Calculate a ranking model result • Linear weighted summation model • Compromise programming • Evaluate result • Sensitivity analysis

7. Combining maps together • Boolean operations • Simplest and best known, popular • Index overlay • Popular and very customizable for weighted combinations • Fuzzy logic • Allows for more flexibility for combining GIS data

8. Summary of those methods • Based on subjective modeling approaches • Rules, weights, or fuzzy membership values assigned subjectively • Used a knowledge of the process to estimate the relative importance on the input maps

9. Now, however… • We will start to include a probabilistic framework in our modeling process not just suitability! • The weighting of the individual map layers will be based on a Bayesian probability model • In particular, a weights of evidence model will be presented in a map context http://yudkowsky.net/rational/bayes

11. Background • In general, Bayesian methods can be applied to maps whether they are categorical or use higher levels of measurement • Weights of evidence is a log-linear version of the general Bayesian model normally applied where the evidence is binary

12. Bayesian methods • The Bayesian method to combining datasets uses a probability framework • One of the main concepts is the idea of prior and posterior probability

13. Example

14. Goal: predict the likelihood of rain tomorrow • Say it rains on average 80 days a year • The prior probability of rain the next day may be the average proportion P{rain} = 80/365 This initial or prior estimate can be modified by other sources of information

15. Other info • Month of year • Use month to modify our probability by multiplying by a factor that varies with the time of the year P{rain|Time of year} = P{rain} * Time of year factor Prior probability Posterior probability

16. Even more info • Has rain occurred the previous day? • Rain tomorrow is more likely if it rains today P{rain|Evidence} = P{rain} * Time of year factor * Rain today factor Evidence NOTE: the Evidence is determined from historical data

17. Note P{rain|Evidence} = P{rain} * Time of year factor * Rain today factor Prior probability Evidence Posterior probability So, several sources of data can be used to provide evidence about tomorrow’s weather and can be combined using such a model. Some evidence will cause the posterior probability to increase as compared to the prior (the likelihood of rain is more than average). In such cases, the evidence will have a multiplying factor greater than 1.

18. Note P{rain|Evidence} = P{rain} * Time of year factor * Rain today factor Prior probability Evidence Posterior probability But where the multiplying factor is less than 1 (but always must be positive), the evidence causes the posterior probability to be less than the prior. The prior probability can be successively updated with the addition of new evidence, so that the posterior probability from adding one piece of evidence can be treated as the prior for adding a new piece of evidence.

19. Weights of evidence • The Bayesian model in a log-linear form is know as weights of evidence • It has been applied where sufficient data is available to estimate the relative importance of evidence by statistical means • It is data driven!!!!!

20. Applications • Ecological GIS application Aspinall, R.J., 1992, An inductive modeling procedure based on Bayes’ theorem for analysis of pattern in spatial data: International Journal of Geographical Information Systems, V. 6(2), p. 105-121. G70.2 I59 on 2nd of downtown lib • Others on the class web site

21. Weights of evidence method • Originally developed as a medical diagnostic system • Relationships between symptoms and disease evaluated from a large patient database • Each symptom either presence/absent • Weight for present/weight for absent (W+/W-) • Apply weighting scheme to new patient • Add the weights together to get results

23. Weights of evidence modeling • A quantitative spatial statistical method of overlaying several different maps to identify areas where the interested phenomenon may be present. • The larger the number and magnitude of appropriate overlapping high values in data maps the greater the qualitative indication that the interested phenomenon may be present.

24. Weights of evidence modeling • When several maps are combined, the areas with the greatest coincidence of weights produce the greatest probability of occurrence of undiscovered points related to the interested phenomenon.

25. Weights of evidence general overview • Data driven technique • Requires training sites • Statistical calculations are used to derive the weights based upon training sites • Evidence (maps) are generally reclassified (as multiclass or binary patterns)

26. Mapping example • Acid mine drainage potential mapping • Goal: to predict the presence of acidic mined deposits • The acidic deposits are binary (either present or absent)

27. Mapping example • Find acidic deposits in a region that covers 10,000 km2 • Suppose that 200 acidic deposits are known within the total region, T, occupying 1 cell or 1 km2 N{T} = 10,000 unit cells Where N{} is the notation used to denote the count of unit cells

28. The average density of known acidic deposits in the region is N{D} / N{T} or 200 / 10,000 = 0.02 Where N{D} is the total number of deposits This is the probability that a 1 km2 cell, chosen at random contains a known acidic deposit Where no other info is available, this ratio can be can be use as the prior probability of a deposit, P{D}

29. Now suppose that that a binary indicator map, like the freeport coal seam covers the same area and that 180 out of the 200 deposits occur where the freeport coal seam also occurs

30. Clearly the probability of finding an acidic deposit “x” is much greater than the 0.02 if the coal seam is known to be present • Conversely, the probability is less than 0.02 if the coal seam is absent Study region, T X X X X X X X X Binary pattern, B = present X X X X X X X X Binary pattern, B = absent

31. The chance of finding a deposit given the presence of the evidence can be expressed by the conditional probability P{D∩B} EQ #1 P{D|B} = P{B} P{D|B} is the conditional probability of a deposit given the presence of a binary pattern, B. P{D∩B} is equal to the proportion of total area occupied by D and B together or P{d|B} = N{D ∩B}/N{T}

32. And similarily P{D|B} = N{B} /N{T}, where P{B} and N{B} are respectively the probability and area where pattern B is present. Substituting these expressions into the previous probability gives us N{D∩B} P{D|B} = EQ #2 N{B}

33. Venn diagram Areas in unit cells N{T} = 10,000 (total area) N{B} = 3,600 (area of pattern) N{D} = 200 Area of deposits N{B∩D} = 180 (area of deposits on pattern) B T B∩D B∩D B∩D B∩D D

34. In the present case, P{D|B} = .02 * 0.15625 = 0.003125 • So, a map based on this single source of evidence effectively reduces the area of search from 10,000 km2 to 3,600 km2 because the chances of finding a deposit where the coal seam is absent is significantly smaller (about 50 times) then where it is present.

35. Summary of weights of evidence • Method is objective, avoids subjective choice of weighting factors • Multiple map patterns can be combined • Process gives insight into spatial data relationships • Output is probability

36. Summary of weights of evidence • Assumes conditional independence when combining input maps • Only really applicable when the response variable is pretty well known for an area (there are good evidence maps)

37. Reference • Bonham-Carter, G.F. 1994. Geographic Information Systems for Geoscientists: Modeling with GIS, Pergannon Press, Oxford, 398p.

38. Software (free) http://arcscripts.esri.com/details.asp?dbid=15341