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Originally presented at ICES WGZE, Reykjavik, April 1999 Fuzzy Logic and Ecological Indices This presentation was developed by William Silvert William Silvert to show how Fuzzy Logic can be used to develop and apply ecological indices based on complex environmental data.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Words of Warning! The Fine Print: Im not an expert on zooplankton. This presentation is based on my work developing indices of benthic conditions under fish farms.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Outline of Presentation Part 1 About IndicesIndices Part 2 Brief introduction to Fuzzy LogicFuzzy Logic Part 3 Relevance to Zooplankton IndicesZooplankton Indices Part 4 The process of DefuzzificationDefuzzification Final SummarySummary Worked Example The talk will consist of six parts, as follows:

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Originally presented at ICES WGZE, Reykjavik, April 1999 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.

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. +We have to make sure that an index tells us something that we want to know! To answer different questions we need different indices.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Cooking the Data 4Start with a mess of ingredients (data) 4Process the ingredients (cook the data) 4Serve and digest the results Creating an index is a lot like cooking.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Getting Our Priorities Straight 4Data are far and away most expensive. 0% We have to conserve resources, but we shouldnt scrimp on the cheap stuff. What does it cost to create an index? 0% 25% 50% 75% 100% 25% 50% 75% 100% Data Analysis 4Analysis is relatively cheap, do it well. Presentation 4Presentation is not a major cost.

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Originally presented at ICES WGZE, Reykjavik, April 1999 The Data are out of Control! 4Although 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. 4Part of the reason for this is that in most cases the data are collected for wider purposes than the production of an index. KFor example, there is more to physical oceanography than larval fish survival!

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. It makes no sense to spend 100.000 on a survey cruise, and then analyse the data with linear regressions to save a day or two of computational time.

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Originally presented at ICES WGZE, Reykjavik, April 1999 The Myth of Numbers 4Numbers alone are misleading and dont really tell us much. Is 10.000 t of fish a lot or a little? That depends: are they are salmon or sardines? 4Numerical 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. To make the transition to Fuzzy Logic, lets take a look at the use of numerical index values.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Zadehs 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. He calls this tradeoff between quantitative values and significance the Principle of Inconsistency.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Take the Weather... We can usually identify whether the weather is good or bad pretty easily. But try this... Construct a scheme for classifying the weather based on precise measurements of variables, including (but not limited to): ^Temperature ^Wind Speed ^Precipitation ^Humidity ^Cloud Cover

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Originally presented at ICES WGZE, Reykjavik, April 1999 … Including Variability Once you have done that, introduce the next level of complexity, which is variability. For weather variables that means that we also have to include in our classification: ^Temperature Range ^Wind Gusts and Direction ^Precipitation Type (more Fuzzy Classification!) ^Is Precipitation Constant or Episodic? ^Cloud Type, e.g. Cirrus vs. Cumulo-Nimbus

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Originally presented at ICES WGZE, Reykjavik, April 1999 Too Much Precision! The lesson in this simple example is clear, too many precise variables are unmanageable. 4As we add more and more variables, the numbers tell us less and less. 4More information should enable us to gain a clearer picture of the system we are studying. 4Instead the opposite happens, and more data lead to a more confusing picture. 4Human beings do not assimilate information as tables of numbers, but as qualitative images.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Part 2 Fuzzy Logic Fuzzy Logic offers a powerful mathematical language for the development of indices. +It is not the great mathematical discovery that some of its proponents claim, but it makes it easy to develop indices that work with real data.

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. (Woodgers Paradox notwithstanding.) We normally classify everything into crisp categories.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Fuzzy Classification Not everything fits into crisp categories. 4Can any organism go from female to male in an instant? No, at some stage it is part female and part male. 4Even sex is sometimes best described by fuzzy classification. Although most organisms are male or female, hermaphrodites belong to both sexes. And some organisms can change sex.

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Originally presented at ICES WGZE, Reykjavik, April 1999 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

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Originally presented at ICES WGZE, Reykjavik, April 1999 Classifying the Environment 4The boundaries between these categories are fuzzy. 4For example, as temperature increases, the suitability of the environment changes in a continuous, not discontinuous, fashion. We can describe environmental conditions in terms of discrete categories, such as good, bad, moderately …, etc.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Environmental Categories Here is a typical classification scheme. How do we convert continuous data into discrete categories?

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Originally presented at ICES WGZE, Reykjavik, April 1999 Fuzzy Classification is to fuzzify the boundaries between discrete sets by letting the system belong to more than one classification set. 4We 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. 4The solution

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Originally presented at ICES WGZE, Reykjavik, April 1999 Is that all there is? 4Fuzzy Logic is just a common-sense approach to using mathematics for real-world problems that dont fit into neat categories. 4A 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. 4Fuzzy Logic is not a form of high-powered obscure mathematics (although some people like to pretend that it is).

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Originally presented at ICES WGZE, Reykjavik, April 1999 So why bother? 4Fuzzy Logic offers the mathematical tools to use common sense in a quantitative way to deal with complicated systems. 4Fuzzy Logic lets us fit the mathematics to the biology. It is usually the other way around. To be specific... If Fuzzy Logic is just applied common sense, why not skip the mathematics altogether? What does Fuzzy Logic do for us?

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Originally presented at ICES WGZE, Reykjavik, April 1999 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.

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Originally presented at ICES WGZE, Reykjavik, April 1999 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.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Problems Developing Plankton Indices Many different variables, leading to possibly inconsistent pictures of conditions. Many different variables Incomplete data, reflecting the difficulty of consistent sampling at sea. Incomplete data Continuously varying quantities which cannot easily be put into sharp categories. Continuously varying quantities There are several problems in the development of indices based on plankton data which can be alleviated by the use of Fuzzy Logic:

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. With so many different planktonic variables involved biomass, diversity, chlorophyll, size structure as well as environmental variables like temperature and stratification, it is unlikely that a consistent picture will emerge.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Incomplete Data 4This creates problems for indices based on the average of specific measurements. 4With 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. It is not always possible to follow the same exact protocol every season when sampling at sea.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Continuous Variables 4It makes little sense to define a precise level at which conditions change from Good to Poor, for example. 4It makes more sense to describe a gradual transition from 100% Good through (50% Good AND 50% Poor) to 100% Poor. With Fuzzy Logic we can use continuous variables to define discrete categories.

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Originally presented at ICES WGZE, Reykjavik, April 1999 ReClassifying the Environment 4The boundaries between these categories are fuzzy. For example, 4as the temperature rises, the suitability of the environment changes in a continuous, not discontinuous, fashion. As pointed our earlier, we can describe environmental conditions with discrete categories, such as good, bad, moderately …, etc.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Classification Along an Environment Gradient When we look at classification (e.g., Good vs. Poor) along an environmental gradient with crisp classification we get discontinuities:

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Originally presented at ICES WGZE, Reykjavik, April 1999 Classification Along an Environment Gradient But if we use Fuzzy Classification we can get a continuous transition like this:

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. Although Fuzzy Logic has many advantages in the development of indices, it is a novel approach that may not be well understood or appreciated by many potential users (clients in the current jargon of science managers).

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Originally presented at ICES WGZE, Reykjavik, April 1999 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

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Originally presented at ICES WGZE, Reykjavik, April 1999 So why not Defuzzify? The catch to defuzzification is that there is information in the fuzzy representation that can be useful. If all variables indicate Good conditions, then the defuzzified score is 2. But if the variables do not give a consistent picture, with as many Poor indications as Excellent ones, the defuzzified score is also 2. By defuzzification we lose information about the consistency of the indices.

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Originally presented at ICES WGZE, Reykjavik, April 1999 For example... If we defuzzify, we do not have any way to distinguish between this classification and this one.

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Originally presented at ICES WGZE, Reykjavik, April 1999 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.

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Originally presented at ICES WGZE, Reykjavik, April 1999 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.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Consistent Data If all variables produce a consistent picture, there is no difficulty or ambiguity in combining them, as the following table shows: VariableValueMemberships PoorFairGoodExcellent Bloom DurationGood0.00.01.00.0 StratificationGood0.00.01.00.0 Zooplankton BiomassGood0.00.01.00.0 Combined IndexGood0.00.01.00.0

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. VariableValueMemberships PoorFairGoodExcel. Bloom DurationGood0.00.01.00.0 StratificationGood/Excellent0.00.00.50.5 Zooplankton BiomassPoor/Fair0.50.50.00.0

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. VariableValueMemberships PoorFairGoodExcel. Bloom DurationGood0.00.01.00.0 Stratification Good/Excellent0.00.00.50.5 Zooplankton BiomassPoor/Fair0.50.50.00.0 Combined Index0.50.50.00.0

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Originally presented at ICES WGZE, Reykjavik, April 1999 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. VariableValueMemberships PoorFairGoodExcel. Bloom DurationGood0.00.01.00.0 StratificationGood/Excellent0.00.00.50.5 Zooplankton BiomassPoor/Fair0.50.50.00.0 Combined Index0.20.20.50.1

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Originally presented at ICES WGZE, Reykjavik, April 1999 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.

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Originally presented at ICES WGZE, Reykjavik, April 1999 Further Reading Papers by Bill Silvert on Fuzzy Classification William Silvert, 1979. Symmetric summation: a class of operations on fuzzy sets. IEEE Trans. Syst., Man, Cyber. SMC-9: 657-659. William Silvert, 1997. Ecological impact classification with fuzzy sets. Ecological Modelling 96:1-10. Dror Angel, Peter Krost and William Silvert. 1998. Describing benthic impacts of fish farming with fuzzy sets: theoretical background and analytical methods. J. Appl. Ichthyology 14: 1-8. William Silvert, 1999? Fuzzy Indices of Environmental Conditions. Ecological Modelling (in press).

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