originally presented at ices wgze, reykjavik, april 1999 fuzzy logic and ecological indices this...

Originally presented at ICES WGZE, Reykjavik, April 1999

Fuzzy Logic and Ecological Indices

This presentation was developed by

William SilvertWilliam Silvertto show how Fuzzy Logic can be used to develop and apply ecological indices based on complex environmental data.


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.


Outline of Presentation

Part 1 — About IndicesPart 2 — Brief introduction to Fuzzy LogicPart 3 — Relevance to

Zooplankton IndicesPart 4 — The process of DefuzzificationFinal SummaryWorked Example

The talk will consist of six parts, as follows:


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.


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.


Cooking the Data

Start with a mess of ingredients (data)

Process the ingredients (cook the data)

Serve and digest the results

Creating an index is a lot like cooking.


Getting Our Priorities Straight

Data are far and away most expensive.

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We have to conserve resources, but we shouldn’t scrimp on the cheap stuff. What does it cost to create an index?

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Data

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Analysis is relatively cheap, do it well.

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Presentation is not a major cost.


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!


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.


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.

To make the transition to Fuzzy Logic, let’s take a look at the use of numerical index values.


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. He calls this tradeoff between quantitative values and significance the “Principle of Inconsistency”.


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):TemperatureWind SpeedPrecipitation HumidityCloud Cover


… 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 RangeWind Gusts and DirectionPrecipitation Type (more Fuzzy Classification!)Is Precipitation Constant or Episodic?Cloud Type, e.g. Cirrus vs. Cumulo-Nimbus


Too Much Precision!

The lesson in this simple example is clear, too many precise variables are unmanageable.

As we add more and more variables, the numbers tell us less and less.

More information should enable us to gain a clearer picture of the system we are studying.

Instead the opposite happens, and more data lead to a more confusing picture.

Human beings do not assimilate information as tables of numbers, but as qualitative images.


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.


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.)

We normally classify everything into “crisp” categories.


Fuzzy Classification

Not everything fits into “crisp” categories.

Can any organism go from female to male in an instant? No, at some stage it is part female and part male.

Even 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.


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 MelvilleMoby Dick


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.

We can describe environmental conditions in terms of discrete categories, such as “good”, “bad”, “moderately …”, etc.


Environmental Categories

D iscrete C lassification Schem e

Exce llen tC ond itions

G oodC onditions

PoorC ond itions

BadC onditions

C ontinuous D ata

Here is a typical classification scheme. How do we convert continuous data into discrete categories?


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.

The solution


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.

Fuzzy Logic is not a form of high-powered obscure mathematics (although some people like to pretend that it is).


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 ...

If Fuzzy Logic is just applied common sense, why not skip the mathematics altogether? What does Fuzzy Logic do for us?


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.


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.


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.

There are several problems in the development of indices based on plankton data which can be alleviated by the use of Fuzzy Logic:


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.


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.

It is not always possible to follow the same exact protocol every season when sampling at sea.


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.

With Fuzzy Logic we can use continuous variables to define discrete categories.


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.

As pointed our earlier, we can describe environmental conditions with discrete categories, such as “good”, “bad”, “moderately …”, etc.


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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Classification Along an Environment Gradient

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2 0 %

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But if we use Fuzzy Classification we can get a continuous transition like this:


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).


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


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.


For example ...

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Poor Good Excellent

If we defuzzify, we do not have any way to distinguish between this classification

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Poor Good Excellent

and this one.


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.


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.


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 MembershipsPoor Fair Good Excellent

Bloom Duration Good 0.0 0.0 1.0 0.0Stratification Good 0.0 0.0 1.0 0.0Zooplankton Biomass Good 0.0 0.0 1.0 0.0

Combined Index Good 0.0 0.0 1.0 0.0


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 0.0Stratification Good/Excellent 0.0 0.0 0.5 0.5Zooplankton Biomass Poor/Fair 0.5 0.5 0.0 0.0


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 MembershipsPoor Fair Good Excel.

Bloom Duration Good 0.0 0.0 1.00.0Stratification Good/Excellent 0.0 0.0 0.50.5Zooplankton Biomass Poor/Fair 0.5 0.5 0.0 0.0Combined Index 0.5 0.5 0.0 0.0


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 MembershipsPoor Fair Good Excel.

Bloom Duration Good 0.0 0.0 1.0 0.0Stratification Good/Excellent0.0 0.0 0.5 0.5Zooplankton Biomass Poor/Fair 0.5 0.5 0.0 0.0Combined Index 0.2 0.2 0.5 0.1


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.


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).