Outline 1 The topic 2 Decision support systems 3 Modeling 3.3 Advanced Modeling 3.3.2 Qualitative Modeling Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 93 Group of the Technical University of Munich
Ecological Modeling and Decision Support Systems Motivation - Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 94 Group of the Technical University of Munich
The Algal Bloom – A „Numerical Model“ d 1 I = a - + > fall s ( T 20 ) 0 P P e (ln( ) 1 ), I I e max, 20 0 s 24 H I s d 1 I = a - falls ( T 20 ) 0 P P e , I I e max, 20 0 s 24 H I s Numerical model: only an approximation Extinction of light: – Not linear – Not a function Daylight: – Not a fraction (dawn and dusk) – Varying (clouds) Temperature dependence: … Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 95 Group of the Technical University of Munich
Intraspecific Competition Net rate equals r for small population K: maximal capacity Assumption: linear decrease of the rate Why linear decrease? Why not … 1/N* dN/dt Not a function, anyway .. r 0 N K Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 96 Group of the Technical University of Munich
Qualitative Models - Motivation Models capturing partial knowledge and information Why? What do we know? What can be observed? What needs to be distinguished? 1/N* dN/dt N trout r ? X X X X X X X X X X N t K Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 97 Group of the Technical University of Munich
Qualitative Modeling • Modeling systems with partial knowledge/information: • Only rough understanding • imprecise, or missing data • Qualitative results required • Treating classes of systems and conditions Tasks • Calculi for qualitative domains • Formal analysis of relationships among models of different granularity Expected benefit: • Finite representation • Efficiency • Intuitive representation Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 98 Group of the Technical University of Munich
Ecological Modeling and Decision Support Systems Interval-based Qualitative Modeling Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 99 Group of the Technical University of Munich
A (Very General) Representation of Behavior Models For instance, intraspecific competition dN/dt = N*r = N*r 0 *[1 – (N/K)] r = 1/N* dN/dt = r 0 *[1 – (N/K)] • What does it mean? • Not simply computation of dN/dt 1/N* dN/dt • Constrains the possible tuples of values • For instance, if r 0 = 2 and K = 1000 r 0 - (r, N) = (1, 500) is possible R r,N - (r, N) = (1, 100) is not - (r, N) = (-1/2, *) is not representation: a relation R r,N • N K Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 100 Group of the Technical University of Munich
Representation of Qualitative Behavior Models For instance, intraspecific competition dN/dt = N*r = N*r 0 *[1 – (N/K)] • Express qualitative knowledge: • N is never greater than K (and not negative) 1/N* dN/dt • r lies between 0 and r 0 • relation R q r,N = r 0 {[r 0 , r 0 ] [0, 0]} {[0, 0 ] [K, K]} (0 , r 0 ) (0 , K) (r, N) = (1, 500) R q r,N : i.e. consistent (r, N) = (1, 100) R q r,N : consistent! N K (r, N) = (-1/2, *) R q r,N : not consistent Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 101 Group of the Technical University of Munich
Extended Qualitative Model For instance, intraspecific competition dN/dt = N*r = N*r 0 *[1 – (N/K)] • Express qualitative knowledge: • N is never greater than K (and not negative) 1/N* dN/dt • r lies between 0 and r 0 • r decreases with increasing N r 0 r,N,dr DOM(r, N, dr/dN): • relation R q {[r 0 , r 0 ] [0, 0] [0, 0] } {[0, 0 ] [K, K] [0, 0] } (0 , r 0 ) (0 , K) (- , 0) N K Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 102 Group of the Technical University of Munich
Refined Qualitative Model Still not perfect For instance, intraspecific competition dN/dt = N*r = N*r 0 *[1 – (N/K)] Why? • “If N is close to 0, r is close to r 0 “ • “If N is close to K, r is close to 0” • “If N is in between, r is in between” 1/N* dN/dt r,N,dr DOM’(r, N, dr/dN): • R q’ {[r e , r 0 ] [0, K e ] [dr e , 0] } r 0 {[0, r ] [ K , K] [dr , 0] } r e (r e , r ) (K e , K ) (- , 0) R q’ r,N,dr = r { (small, small, neg e ) (large, large, neg ) N K e K K (medium, medium, neg)} Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 103 Group of the Technical University of Munich
Generalization: Relational Behavior Models • Representational space : (v, DOM(v)) • v: Vector of local variables and parameters • local w.r.t Model fragment or aggregate • DOM(v): Domain of v • Behavior description : Relation • R DOM(v) • Composition: join of relations Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 104 Group of the Technical University of Munich
Valid Behavior Models • Independently of the syntactical form: • What set of states is allowed by the model? R S DOM(v S ) A valid model of a behavior: • R S covers all states of the behavior • " s SIT Val(v S , v S,0 , s) v S,0 R S Real behavior R S Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 105 Group of the Technical University of Munich
Types of Qualitative Abstraction “Increase of Diclofenac carcasses decreases vulture population size” “Variation in cloud coverage is not relevant to algae biomass in trout streams” “Population size is below a critical value” Domain Abstraction • Aggregate values leading to the same class 0 ... N crit ... K of behaviors • e.g. between “landmarks”: intervals small crit normal Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 106 Group of the Technical University of Munich
Interval Arithmetic • Addition of intervals ( a 1 , 1 ) ( a 2 , 2 ) = ( a 1 + a 2 , 1 + 2 ) • Subtraction ( a 1 , 1 ) ( a 2 , 2 ) = ( a 1 - 2 , 1 - a 2 ) • Multiplication ( a 1 , 1 ) ( a 2 , 2 ) = ( min( a 1 * a 2 , a 1 * 2 , a 2 * 1 , 2 * 1 ), max ( a 1 * a 2 , a 1 * 2 , a 2 * 1 , 2 * 1 )) 0 ... N crit ... K • Division ( a 1 , 1 ) ( a 2 , 2 ) = ( min( a 1 / a 2 , a 1 / 2 , 1 / a 2 , 1 / 2 ), max ( a 1 / a 2 , a 1 / 2 , 1 / a 2 , 1 / 2 )) for 0 ( a 2 , 2 ) ! • • Because … ? small crit normal Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 107 Group of the Technical University of Munich
Properties of Interval Arithmetic • Associative • Commutative • Sub-distributive: i 1 (i 2 i 3 ) (i 1 i 2 ) (i 1 i 3 ) • • intervals may include spurious real-valued solutions Solutions of interval equations • x 1 =i 1 , x 2 =i 2 , … • satisfies f l (x 1 , x 2 , …, x n ) f r (x 1 , x 2 , …, x n ) • • iff f l (i 1 , i 2 , …, i n ) f r (i 1 , i 2 , …, i n ) • Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 108 Group of the Technical University of Munich
Special Case: Arithmetic on Signs - + 0 - - - - + 0 0 + + + - + 0 - + - 0 0 0 0 0 + - + 0 Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 109 Group of the Technical University of Munich
Domain Abstraction - Formally General: t i : DOM 0 (v i ) DOM 1 (v i ) • Aggregation of values: t i : DOM 0 (v i ) DOM 1 (v i ) P(DOM 0 (v i )) • • P(X): power set of X 0 ... N crit ... K (Generalized) Intervals: t i : IR DOM 1 (v i ) I( IR ) • Real landmarks and intervals between them: L IR • t i : IR DOM 1 (v i ) I L ( IR ) • • I L : intervals with boundaries in L small crit normal Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 110 Group of the Technical University of Munich
Model Abstraction Induced by Domain Abstraction • Domain abstraction t : DOM 0 (v S ) DOM 1 (v S ) • • induces model abstraction R S DOM(v S ) t (R S ) DOM 1 (v S ) Theorem: • If the base relation is a valid model of a behavior • then so is its abstraction • Important for consistency check t (R S ) Real behavior R S Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 111 Group of the Technical University of Munich
Ecological Modeling and Decision Support Systems Lotka-Volterra - Qualitative Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 112 Group of the Technical University of Munich
Lotka-Volterra Predator-Prey Model – A Qualitative Analysis dN/dt = (r – a*P)*N dP/dt = (f*a*N – q)*P N P Time P P P r a N N N q f*a Model-Based Systems & Qualitative Reasoning WS 14/15 EMDS 3 - 113 Group of the Technical University of Munich
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