Gauging Bio-Diversity . . . Why These Measures? An Intuitive Meaning . . . Which Bio-Diversity Indices Fuzzy Logic Justifies . . . Bio-Diversity of . . . Are Most Adequate Localness Property Possibility of Scaling Olga Kosheleva 1 , Craig Tweedie 2 , Main Result and Vladik Kreinovich 3 Home Page Title Page 1 Department of Teacher Education 2 Environmental Science and Engineering Program ◭◭ ◮◮ 3 Department of Computer Science University of Texas at El Paso ◭ ◮ 500 W. University Page 1 of 21 El Paso, Texas 79968, USA olgak@utep.edu, ctweedie@utep.edu Go Back vladik@utep.edu Full Screen Close Quit
Gauging Bio-Diversity . . . 1. Gauging Bio-Diversity Is Important Why These Measures? An Intuitive Meaning . . . • One of the main objectives of ecology is to study and Fuzzy Logic Justifies . . . preserve bio-diversity. Bio-Diversity of . . . • Most existing measures of diversity are based on the Localness Property relative frequencies p i of different species. Possibility of Scaling • The most widely used measures are: Main Result Home Page n – the Shannon index H = − � p i · ln( p i ) , and Title Page i =1 n ◭◭ ◮◮ p 2 � – the Simpson index D = i . i =1 ◭ ◮ • Ecologists also use indices related to D : 1 D and 1 − D . Page 2 of 21 n Go Back p q • They also use indices related to the sum � i , such as Full Screen i =1 � n � 1 � p q Close R´ enyi entropy H q = 1 − q · ln . i i =1 Quit
Gauging Bio-Diversity . . . 2. Why These Measures? Why These Measures? An Intuitive Meaning . . . • The above measures of diversity are, empirically, in Fuzzy Logic Justifies . . . good accordance with the ecologists’ intuition. Bio-Diversity of . . . • However, from the theoretical viewpoint, the success Localness Property of these measures of diversity is somewhat puzzling. Possibility of Scaling Main Result • Why these expressions and not other possible expres- Home Page sions? Title Page • In this talk, we provide possible justification for the above empirically successful measures. ◭◭ ◮◮ ◭ ◮ • We provide two possible justification: Page 3 of 21 – we start with a simple fuzzy logic-based justifica- tion which explains Simpson index, and then Go Back – we provide a more elaborate justification that ex- Full Screen plains all the above diversity measures. Close Quit
Gauging Bio-Diversity . . . 3. An Intuitive Meaning of Bio-Diversity Why These Measures? An Intuitive Meaning . . . • An ecosystem is perfectly diverse if all its species are Fuzzy Logic Justifies . . . reasonably frequent but not dominant. Bio-Diversity of . . . • In other words, the ecosystem is healthy if: Localness Property Possibility of Scaling – the first species is reasonably frequent but not dom- Main Result inant, and Home Page – the second species is reasonably frequent but not Title Page dominant, – etc. ◭◭ ◮◮ ◭ ◮ • This statement uses an imprecise (“‘fuzzy”) natural- language terms like “reasonably frequent”. Page 4 of 21 • We need to translate this statement into precise terms. Go Back • We will use fuzzy logic, since fuzzy logic was invented Full Screen exactly for such a translation. Close Quit
Gauging Bio-Diversity . . . 4. Let Us Use Fuzzy Logic Why These Measures? An Intuitive Meaning . . . • For each p i , let µ ( p i ) be the degree to which the species Fuzzy Logic Justifies . . . is reasonably frequent and not dominant. Bio-Diversity of . . . • To compute bio-diversity, we need to combine use Localness Property “and”-operation to combine these degrees. Possibility of Scaling • The general strategy in applications of fuzzy techniques Main Result is to select the simplest possible “and”-operation. Home Page • The two simplest (and most frequently used) “and”- Title Page operations are the product and the minimum. ◭◭ ◮◮ • Our objective is to optimize bio-diversity, and most ◭ ◮ efficient optimization techniques use differentiation. Page 5 of 21 • From this viewpoint, it is desirable to come up with Go Back the differentiable measure of diversity. Full Screen • This eliminates min (since min( a, b ) is not differen- n Close � tiable when a = b ), so we use the product µ ( p i ) . i =1 Quit
Gauging Bio-Diversity . . . 5. Let Us Use Fuzzy Logic (cont-d) Why These Measures? An Intuitive Meaning . . . n � • Maximizing µ ( p i ) is equivalent to maximizing its Fuzzy Logic Justifies . . . i =1 Bio-Diversity of . . . n def logarithm L = � f ( p i ) , where f ( p i ) = ln( µ ( p i )). Localness Property i =1 Possibility of Scaling • In a diverse ecosystem all the frequencies p i are rather Main Result small. Home Page • Indeed, if one of the values is large, this means that we Title Page have a dominant species, not a diversity. ◭◭ ◮◮ • For small p i , we can replace each value f ( p i ) with the ◭ ◮ sum of the few first terms in its Taylor expansion. Page 6 of 21 • In the first approximation, f ( p i ) = a 0 + a 1 · p i , so Go Back L = a 0 · n + a 1 . Full Screen • This expression does not depend on the frequencies p i Close and thus, cannot serve as a measure of diversity. Quit
Gauging Bio-Diversity . . . 6. Fuzzy Logic Justifies the Simpson Index Why These Measures? An Intuitive Meaning . . . n • We want to maximize L = � f ( p i ). Fuzzy Logic Justifies . . . i =1 Bio-Diversity of . . . • We have shown that linear terms in f ( p i ) are not suf- Localness Property ficient. Possibility of Scaling • So, to adequately describe diversity, we need to take Main Result into account quadratic terms Home Page f ( p i ) = a 0 + a 1 · p i + a 2 · p 2 Title Page i . ◭◭ ◮◮ • In this approximation, ◭ ◮ n � p 2 L = a 0 · n + a 1 + a 2 · i . Page 7 of 21 i =1 Go Back • Maximizing this expression is equivalent to maximizing Full Screen n � p 2 the Simpson index D = i . Close i =1 Quit
Gauging Bio-Diversity . . . 7. The Ultimate Purpose of Diversity Estimation Why These Measures? Is Decision Making An Intuitive Meaning . . . Fuzzy Logic Justifies . . . • We want to describe which combinations of frequencies Bio-Diversity of . . . p = ( p 1 , . . . , p n ) are preferred and which are not. Localness Property • Most plans succeed only with a certain probability. Possibility of Scaling • So, we need to consider “lotteries”, in which different Main Result Home Page combinations A i appear with different probabilities P i . Title Page • The main result of utility theory states that: – if we have a consistent ordering relation L � L ′ (“ L ◭◭ ◮◮ is preferable to L ′ ”) between such lotteries, ◭ ◮ – then there exists a function u (called utility ) s.t. Page 8 of 21 L � L ′ if and only if u ( L ) ≥ u ( L ′ ) , where Go Back Full Screen u ( L ) = P 1 · u ( A 1 ) + . . . + P n · u ( A n ) . Close • In our case, we need a utility function u ( p 1 , . . . , p n ). Quit
Gauging Bio-Diversity . . . 8. Bio-Diversity of Subsystems Why These Measures? An Intuitive Meaning . . . • An important intuitive feature of bio-diversity is the Fuzzy Logic Justifies . . . localness property: Bio-Diversity of . . . – that, in addition to the bio-diversity of the whole Localness Property ecosystem, Possibility of Scaling – we may be interested in the bio-diversity of its sub- Main Result systems. Home Page • For the whole ecosystem, the sum of frequencies is 1. Title Page • When we analyze a subsystem, we only take into ac- ◭◭ ◮◮ count some of the species. ◭ ◮ • So the sum of the frequencies can be smaller than 1. Page 9 of 21 • Thus, we need to consider the values u ( p ) for tuples Go Back for which � p i < 1. Full Screen i • It makes sense to compare possible arrangements Close within a subsystem. Quit
Gauging Bio-Diversity . . . 9. Localness Property Why These Measures? An Intuitive Meaning . . . • We only compare tuples p = ( p 1 , . . . , p n ) and Fuzzy Logic Justifies . . . n n p ′ = ( p ′ 1 , . . . , p ′ p ′ � � n ) for which p i = i . Bio-Diversity of . . . i =1 i =1 Localness Property • Let us assume that for all species i from some set I , Possibility of Scaling the frequencies are the same: p i = p ′ i . Main Result • Suppose also that, from the point of bio-diversity, the Home Page tuple p is preferable to tuple p ′ : p � p ′ ; so: Title Page – while in the two tuples, the level of diversity is the ◭◭ ◮◮ same for species from the set I , ◭ ◮ – species from the complement set − I have a higher Page 10 of 21 degree of bio-diversity. Go Back • Thus, if we replace the values p i = p ′ i for i ∈ I with some other values q i = q ′ i , we will still have q � q ′ . Full Screen Close Quit
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