Download
1 / 17
170 likes | 292 Vues
This session delves into the role of fuzzy set theory in modeling conceptual vagueness and addressing uncertainty in decision support systems. We will cover how traditional Aristotelian logic limits understanding of vague concepts and how fuzzy sets provide a more nuanced approach. The discussion includes techniques such as the Semantic Import Approach and fuzzy k-Means clustering for classifying data without strict boundaries. We will also highlight decision-making processes, types of evaluations, and the impact of uncertainty on decisions, including database and decision rule uncertainties.
E N D
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
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)
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.
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.
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.
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
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
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).
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.
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
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
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
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
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.
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
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.
