Fuzzy Logic and Ecological Indices

1. Fuzzy Logic and Ecological Indices This presentation was developed by

2. Words of Warning! The Fine Print: I�m not an expert on zooplankton. This presentation is based on my work developing indices of benthic conditions under fish farms.

3. Outline of Presentation Part 1 � About Indices Part 2 � Brief introduction to Fuzzy Logic Part 3 � Relevance to Zooplankton Indices Part 4 � The process of Defuzzification Final Summary Worked Example

4. Part 1 � Indices To develop a clear understanding of how best to develop indices of environmental conditions, whether for predicting the survival of fish larvae or the risk of cancer from industrial sites, we need to think about just what it is that indices tell us.

5. What are Indices? The basic idea behind indices is pretty simple. We start with a mess of environmental data, process it mathematically, and end up with a simplified representation that is supposedly informative about matters ecological.

6. Cooking the Data Start with a mess of ingredients (data) Process the ingredients (cook the data) Serve and digest the results

7. Getting Our Priorities Straight Data are far and away most expensive. We have to conserve resources, but we shouldn�t scrimp on the cheap stuff. What does it cost to create an index?

8. The Data are  out of Control! Although data collection can, and should, be driven by how data will be used, in practice there is not always much feedback from analysis to the design of field programs. Part of the reason for this is that in most cases the data are collected for wider purposes than the production of an index. For example, there is more to physical oceanography than larval fish survival!

9. Focus on Analysis Since data analysis is so much cheaper than data collection, we can afford to do a really good job of processing the data. This is especially true when the data are not ideally suited for our task.

10. The Myth of Numbers Numbers alone are misleading and don�t really tell us much. Is 10.000 t of fish a lot or a little? That depends: are they are salmon or sardines? Numerical indices can thus be misleading, especially if for use by non-specialists. Qualitative indices may not look scientific, but in reality they are most informative.

11. Zadeh�s Principle of Inconsistency Lotfi Zadeh made the interesting observation that as systems become more complex, it is increasingly hard to maintain both precision of measurement and meaningful results.

12. Take the Weather ... We can usually identify whether the weather is good or bad pretty easily. But try this ...

13. � Including Variability Once you have done that, introduce the next level of complexity, which is variability.

14. Too Much Precision! The lesson in this simple example is clear, too many precise variables are unmanageable.

15. Part 2 � Fuzzy Logic Fuzzy Logic offers a powerful mathematical language for the development of indices.

16. Crisp Classification Taxonomy is an example of crisp classification. We classify copepods as Calanus or Acartia or Euchaeta with no consideration of the possibility that some bug might be a mixture of Calanus and Acartia. (Woodger�s Paradox notwithstanding.)

17. Fuzzy Classification Not everything fits into �crisp� categories.

18. Fuzzy Set Theory in Early Oceanography �You have seen him spout; then declare what the spout is; can you not tell water from air? My dear sir, in this world it is not so easy to settle these plain things.� Herman Melville Moby Dick

19. Classifying  the Environment The boundaries between these categories are fuzzy. For example, as temperature increases, the suitability of the environment changes in a continuous, not discontinuous, fashion.

20. Environmental Categories

21. Fuzzy Classification is to �fuzzify� the boundaries between discrete sets by letting the system belong to more than one classification set. We can, for example, describe the state of the system as a mixture of �Good� and �Poor�, say 60% Good and 40% Poor. These fractions are called the �memberships� in the two sets.

22. Is that all there is? Fuzzy Logic is just a common-sense approach to using mathematics for real-world problems that don�t fit into neat categories. A traditional (�crisp�) set is just one in which the set �memberships� are only 0 or 100%. So a crisp set is just a special kind of fuzzy set.

23. So why bother? Fuzzy Logic offers the mathematical tools to use common sense in a quantitative way to deal with complicated systems. Fuzzy Logic lets us fit the mathematics to the biology. It is usually the other way around. To be specific ...

24. Advantages of Fuzzy We can use discrete categories for classification without introducing artificial discontinuities into our descriptions. We can reconcile contradictory evidence. We can deal with incomplete sets of data.

25. Part 3 � Zooplankton Indices based on plankton data are good candidates for the use of Fuzzy Logic. Many of the problems that arise from data quality and quantity are difficult to resolve in the context of traditional mathematics, but can easily be resolved with a fuzzy approach.

26. Problems Developing Plankton Indices Many different variables, leading to possibly inconsistent pictures of conditions. Incomplete data, reflecting the difficulty of consistent sampling at sea. Continuously varying quantities which cannot easily be put into sharp categories.

27. Many Different Variables (Conflicts) Rather than simply averaging conflicting variables, Fuzzy Logic lets us identify conflicting evidence by allowing simultaneous membership in different index sets.

28. Incomplete Data This creates problems for indices based on the average of specific measurements. With Fuzzy Logic it is possible to assign membership categories for each available measurement and combine these to provide indices based on as much data as is available.

29. Continuous Variables It makes little sense to define a precise level at which conditions change from Good to Poor, for example. It makes more sense to describe a gradual transition from 100% Good through (50% Good AND 50% Poor) to 100% Poor.

30. ReClassifying  the Environment The boundaries between these categories are fuzzy. For example, as the temperature rises, the suitability of the environment changes in a continuous, not discontinuous, fashion.

31. Classification Along an Environment Gradient

32. Classification Along an Environment Gradient

33. Part 4 - Defuzzification Consequently there exist techniques for converting fuzzy memberships into numerical indices that can be understood without going into Fuzzy Logic. These are called �Defuzzification�.

34. How to Defuzzify Defuzzification is usually straightforward. Suppose that we assign value 1 to Poor conditions, 2 to Good conditions, and 3 to Excellent conditions. Then if the Fuzzy classification is 40% Poor and 60% Good, the defuzzified score would be: 0.4*1 + 0.6*2 = 1.6

35. So why not Defuzzify? The catch to defuzzification is that there is information in the fuzzy representation that can be useful.

36. For example ...

37. Summary Fuzzy Logic is a flexible tool for developing indices under difficult conditions. It can be used to deal with incomplete multi-variate data sets. Fuzzy Logic offers ways to reconcile continuous measurements with discrete indices. Fuzzy indices contain more information than simple numerical indices, but can be expressed as single numbers if desired.

38. A Worked Example To illustrate some of the issues that we discussed at the WGZE session on 20 April 1999, here is an example of how one might use Fuzzy Logic to combine data on phytoplankton, physical factors, and zooplankton in an index of larval fish condition. The indices used are loosely based on a report by Harrison and Sameoto.

39. Consistent Data If all variables produce a consistent picture, there is no difficulty or ambiguity in combining them, as the following table shows: Variable Value Memberships Poor Fair Good Excellent Bloom Duration Good 0.0 0.0 1.0 0.0 Stratification Good 0.0 0.0 1.0 0.0 Zooplankton Biomass Good 0.0 0.0 1.0 0.0 Combined Index Good 0.0 0.0 1.0 0.0

40. Conflicting Evidence In general life is not this simple, and we are more likely to see a situation like the one shown below. Note that the categorisations of both Stratification and Zooplankton Biomass are themselves fuzzy. If we identify the categories Poor to Excellent with the numerical values 1 to 4, we can associate these with index values of 3.0 for Bloom Duration, 3.5 for Stratification, and 1.5 for Zooplankton Biomass. Variable Value Memberships Poor Fair Good Excel. Bloom Duration Good 0.0 0.0 1.0 0.0 Stratification Good/Excellent 0.0 0.0 0.5 0.5 Zooplankton Biomass Poor/Fair 0.5 0.5 0.0 0.0

41. Pessimistic Rules of Combination The table below is based on the idea that the worst conditions are the ones that are limiting, which is one way of combining fuzzy sets � you can think of this as the Minimum operator. Variable Value Memberships Poor Fair Good Excel. Bloom Duration Good 0.0 0.0 1.0 0.0 Stratification Good/Excellent 0.0 0.0 0.5 0.5 Zooplankton Biomass Poor/Fair 0.5 0.5 0.0 0.0 Combined Index 0.5 0.5 0.0 0.0

42. Averaging Approach The same values can be combined in a different way, reflecting more an averaging process, so that the Good and Excellent levels of Bloom Duration and Stratification compensate for the Poor to Fair levels of Zooplankton Biomass. This calls for a fuzzy operator more like an Averaging operator. Variable Value Memberships Poor Fair Good Excel. Bloom Duration Good 0.0 0.0 1.0 0.0 Stratification Good/Excellent 0.0 0.0 0.5 0.5 Zooplankton Biomass Poor/Fair 0.5 0.5 0.0 0.0 Combined Index 0.2 0.2 0.5 0.1

43. The Combined Indices In both cases the Combined Index involves membership in more than one fuzzy set, but note that this cannot really be represented adequately by a mean and variance, since the distributions are far from normal.

44. Further Reading Papers by Bill Silvert on Fuzzy Classification