fuzzy aspects of systems science william silvert ipimar, portugal

Fuzzy Aspects of Systems Science

William Silvert

IPIMAR, Portugal

Fuzzy and Crisp Sets

A traditional (“crisp”) set

In

Out

Fuzzy and Crisp Sets

A fuzzy set

In

Out???

Does Fuzzy = Fantastic?

Some people think that fuzzy set theory is absolutely great,

Some don’t.

the best idea since the Arabs invented zero.

Literature on Fuzzy Sets

“Alles das ist nicht Politik ist Nichtpolitik.” Thomas Mann

“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

And Current Thinking?

“Liquid–vapour interfaces, particularly those involving water, are common in both natural and artificial environments. They were first described as regions of continuous variation of density, caused by density fluctuations within the bulk phases.” Fradin et al. In Nature, 24 Feb. 2000

Legality of Fuzzy Sets

Even Sex is Fuzzy

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

We usually say that an organism is male or female, but this doesn’t work for those (like corn) that possess both male and female organs, or that change sex.

Remote Sensing

Land

LandWater

Water

(Marsh)

What is Land?

100%

100%

0%

0%

40%

Ideal Food Chain

Plants

Herbivores

Carnivores

TL = 0

TL = 1

TL = 2

Realistic Food Web

Plants

Herbivores

Omnivores

TL = 0

TL = 1

TL = ?

+

The Ecological Niche

0%

100%

10o Temperature 20o

Factory Emissions

0

1

NOx SO2 H2S

Threshold Level

95% 95%

55% 55% 55%

45%

Ambiguous Data

0%

100%

Good Moderate Bad

H2S

Bacteria

Sea Grass

Temperature Control

We can design a thermal controller that adjusts the heater output by a formula such as

Heat = F (Temperature, Rate of Change)

We can also design a fuzzy controller that does not use a continuous function to control the heating.

Very Fuzzy Control

Hot Cold

Rising Open windows. Keep yourcoat on.

Falling Keep the firegoing.

Burn thefurniture!

Typical Fuzzy Control

Hot Cold

Rising Heat = 0AC = Full

Heat = LowAC = 0

Falling Heat = 0AC = Low

Heat = FullAC = 0

Forecasting

A forecast requires specification of a set.

If we predict that a pair of dice will come up with ten, then of the 36 (6 x 6) possibilities, 3 are in the prediction set -{6+4}, {5+5}, {4+6}

Fuzzy Forecasting

Most forecasts do not use crisp sets.

If we say that the dice are loaded and favour high numbers, how can we check that? (For this example we can ignore the obvious statistical test).

A Fuzzy Prediction Set

0%

100%

2 3 4 5 6 7 8 9 10 11 12

The prediction set of high values for a throw of two dice

Are Fuzzy Sets Unusual?

Alles das ist nicht Fuzzy ist Nichts.Most categories used in science (and in

everything else) are fuzzy, whether we say so or not.

Even astronomy is fuzzy:what about Red Giants?

and psychology - what about“borderline schizophrenics”?

Summary

Fuzzy set theory is a natural way to explain most natural categories.

Fuzzy set theory is not especially fancy from a mathematical point of view.

Objections to “fuzzy thinking” seem to be more psychological than scientific.

So give it a try!