Introduction

1. Introduction • Errors result in uncertainty. However, uncertainty can also be caused by vagueness in the conceptual model. • This can be modelled using fuzzy set theory. • Today we will look at two topics: • Modelling conceptual vagueness using fuzzy sets • Decision support, and how to allow for uncertainty

2. Conceptual Vagueness • Aristotelian logic has three laws: • The law of identity • The law of non-contradiction • The principle of the excluded middle • Eastern civilisations place emphasis on the balance between opposites (e.g. yin and yang)

3. Crisp And Fuzzy Sets • Aristotelian logic is binary, and cannot cope with situations where there may be conceptual vagueness. • Boolean or crisp sets permit only two states; membership function can have only two values (0 or 1). • Fuzzy sets can accommodate non-binary situations; membership function can have any value between 0 and 1. • Membership functions are not probabilities; sometimes called possibilities. • Two approaches to assigning membership values (other than arbitrarily): expert knowledge; taxonomy.

4. Semantic Import Approach • The semantic import (SI) approach is useful in situations where you have a good qualitative idea of how the data should be grouped or classified, but where it is difficult to define exact boundaries. • The approach basically involves selecting a suitable membership function and appropriate parameters.

6. Fuzzy k-Means • The semantic import approach is ‘theory’-driven; the fuzzy k-means uses numerical taxonomy i.e. it is empirically-determined. • For example, you may wish to classify EDs within a city into similar areas in multidimensional attribute space. • Procedure: • Arbitrarily assign each ED to one of k categories. • Calculate the ‘centre of gravity’ in each category. • Reassign each ED to the nearest ‘centre of gravity’. • Repeat until no EDs are reassigned. • Assign membership values for each ED for each category.

7. Operations On Crisp Sets • Crisp sets can be compared using operations such as intersect (logical AND), union (OR) and complement (NOT). 0 1 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 AND OR NOT

8. Operations On Fuzzy Sets • Analogous operations can be carried out on fuzzy sets: • Intersect: take the lowest value in the two sets • Union: take the highest value in the two sets • Complement: 1 minus the membership function • Crisp sets are just a special case of fuzzy sets • Gives a more nuanced result in sieve mapping

9. Fuzzy Polygon Boundaries • A similar logic can be applied to locational data. • For example, inside of identifying a forest as a polygon with a sharp boundary (forest, non-forest), could have a gradation at the edges where membership function changed gradually from 1 (forest) to 0 (non-forest).

10. Decision Support • Decision support refers to the application of GIS technology to aid the decision making process. • Types of decision: • Resource allocation decisions • Policy decisions • The process is usually informal and intuitive, but a more formal approach may yield insights.

11. Concepts • Decision • Decision frame (e.g. alternative landuses) • Candidate set (e.g. land parcels) • Decision set (e.g. a particular allocation of landuses to land parcels) • Criteria: • Factors • Constraints

12. Decision Process • Decision rule – often involves calculation of a composite index along with rules for deciding between alternatives: • Choice function – i.e. formula • Choice heuristic – i.e. algorithm • Types of decision: • Classification (including hypothesis testing) • Selection (identification of sites most suitable foe a given purpose) • Objectives are shaped by social, cultural political and other factors. Objectives should be reflected by decision rules. • Evaluation: the application of decision rules

13. Evaluations • Evaluations may be: • Single criterion • Multiple criteria (MCE) • Boolean overlay (– e.g. sieve mapping) • Weighted linear combinations • Evaluations may also be: • Single objective • Multiple objective • Complementary • Conflicting

14. Problem Types • Criteria and objectives give four possibilities, but few examples of non-trivial examples of multi-objective / single criterion problems • De facto three types • single objective / single criteria; • single objective / multi-criteria; and • multi-objective / multi-criteria

15. Uncertainty • All decisions involve a certain degree of uncertainty. • Sources of uncertainty: • Database uncertainty – i.e. measurement errors (can be modelled using probability theory) • Decision rule uncertainty – • Conceptual vagueness (can be modelled using fuzzy sets) • Suboptimal data • Specification errors – vagueness in the decision rule itself.

16. Risk Assessment • Need to weigh up the implications of making the wrong decision. • Need to weigh up: • Decision risk – likelihood of making the wrong decision • Decision cost – cost of a wrong decision

17. Bayesian Statistics • Classical statistics use the Neyman-Pearson paradigm. This restricts itself to empirical data. • It may be possible to take account of additional information using a Bayesian approach. • Bayesian approach permits new information to be used to update older information.