Outline 1 The topic 2 Decision support systems 3 Modeling 3.3 - - PowerPoint PPT Presentation

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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1 The topic 2 Decision support systems 3 Modeling 3.3 Advanced Modeling 3.3.2 Qualitative Modeling

Outline

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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• Ecological Modeling and Decision Support Systems

Motivation

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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The Algal Bloom – A „Numerical Model“

P d P e H I I I I P d P e H I I I I

T s s T s s

=    + > =    

• 24

1 1 24 1

20 20 20 20 max, ( ) max, ( )

(ln( ) ), ,

a a

e e fall s falls

 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 Group of the Technical University of Munich

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Intraspecific Competition

 Net rate equals r for small population  K: maximal capacity  Assumption: linear decrease of the rate

N

1/N* dN/dt

r0 K

 Why linear decrease?  Why not …  Not a function, anyway ..

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Qualitative Models - Motivation

Models capturing partial knowledge and information Why?  What do we know?  What can be observed?  What needs to be distinguished?

N

1/N* dN/dt

r K

Ntrout

t

X X X X X X X X X X

?

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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 Group of the Technical University of Munich

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Ecological Modeling and Decision Support Systems Interval-based Qualitative Modeling

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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A (Very General) Representation of Behavior Models

For instance, intraspecific competition  dN/dt = N*r = N*r0*[1 – (N/K)]  r = 1/N* dN/dt = r0*[1 – (N/K)]

N

1/N* dN/dt

r0 K

• What does it mean?
• Not simply computation of dN/dt
• Constrains the possible tuples of values
• For instance, if r0 = 2 and K = 1000
• (r, N) = (1, 500) is possible
• (r, N) = (1, 100) is not
• (r, N) = (-1/2, *) is not
•  representation: a relation Rr,N    

Rr,N

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Representation of Qualitative Behavior Models

For instance, intraspecific competition  dN/dt = N*r = N*r0*[1 – (N/K)]

N

1/N* dN/dt

r0 K

• Express qualitative knowledge:
• N is never greater than K

(and not negative)

• r lies between 0 and r0
•  relation Rq

r,N =

{[r0, r0 ]  [0, 0]}  {[0, 0 ]  [K, K]}  (0 , r0)  (0 , K)  (r, N) = (1, 500)  Rq

r,N : i.e. consistent

 (r, N) = (1, 100)  Rq

r,N : consistent!

 (r, N) = (-1/2, *)  Rq

r,N : not consistent

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Extended Qualitative Model

For instance, intraspecific competition  dN/dt = N*r = N*r0*[1 – (N/K)]

N

1/N* dN/dt

r0 K

• Express qualitative knowledge:
• N is never greater than K

(and not negative)

• r lies between 0 and r0
• r decreases with increasing N
•  relation Rq

r,N,dr  DOM(r, N, dr/dN):

{[r0, r0 ]  [0, 0]  [0, 0] }  {[0, 0 ]  [K, K]  [0, 0] }  (0 , r0)  (0 , K)  (- , 0)

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

Refined Qualitative Model

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For instance, intraspecific competition  dN/dt = N*r = N*r0*[1 – (N/K)]

N

1/N* dN/dt

r0 K

• “If N is close to 0, r is close to r0“
• “If N is close to K, r is close to 0”
• “If N is in between, r is in between”
•  Rq’

r,N,dr  DOM’(r, N, dr/dN):

{[re, r0 ]  [0, Ke ]  [dre , 0] }  {[0, r ]  [K, K]  [dr ,0] }  (re , r)  (Ke , K)  (- , 0)  Rq’

r,N,dr =

{ (small, small, nege) (large, large, neg) (medium, medium, neg)}

re r  Ke K

 Still not perfect  Why?

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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 Group of the Technical University of Munich

Valid Behavior Models

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• Independently of the syntactical form:
• What set of states is allowed by the model?

 RS  DOM(vS) A valid model of a behavior:

• RS covers all states of the behavior
• "sSIT Val(vS , vS,0, s)  vS,0  RS

Real behavior

RS

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Types of Qualitative Abstraction

0 ... Ncrit ... K small crit normal  “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
• f behaviors
• e.g. between “landmarks”: intervals

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• Addition of intervals

(a1, 1)  (a2, 2) = (a1+ a2, 1+ 2)

• Subtraction

(a1, 1)  (a2, 2) = (a1 - 2, 1 - a2)

• Multiplication

(a1, 1)  (a2, 2) = ( min(a1* a2 , a1* 2, a2 * 1 , 2 * 1), max (a1* a2 , a1* 2, a2 * 1 , 2 * 1))

Interval Arithmetic

• Division

(a1, 1)  (a2, 2) = ( min(a1/ a2 , a1/ 2, 1 / a2, 1 / 2), max (a1/ a2 , a1/ 2, 1 / a2, 1 / 2))

• for 0(a2, 2) !
• Because … ?

0 ... Ncrit ... K small crit normal

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Properties of Interval Arithmetic

• Associative
• Commutative
• Sub-distributive:
• i1(i2 i3)  (i1 i2)  (i1 i3)
•  intervals may include spurious real-valued solutions

Solutions of interval equations

• x1=i1, x2=i2 , …
• satisfies
• fl(x1, x2, …, xn )  fr(x1, x2, …, xn)
• iff
• fl(i1, i2, …, in)  fr(i1, i2, …, in)  

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Special Case: Arithmetic on Signs



• +
• +

+ + + 

• +
• +
• +
• +

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Domain Abstraction - Formally

0 ... Ncrit ... K small crit normal General:

• ti: DOM0(vi)  DOM1(vi)

Aggregation of values:

• ti: DOM0(vi)  DOM1(vi)  P(DOM0(vi))
• P(X): power set of X

(Generalized) Intervals:

• ti: IR  DOM1(vi)  I(IR)

Real landmarks and intervals between them:

• L  IR
• ti: IR  DOM1(vi)  IL(IR)
• IL: intervals with boundaries in L

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Model Abstraction Induced by Domain Abstraction

• Domain abstraction
• t: DOM0(vS)  DOM1(vS)
• induces model abstraction

 RS  DOM(vS)  t(RS) DOM1(vS) Theorem:

• If the base relation is a valid model of a behavior
• then so is its abstraction
• Important for consistency check

t(RS)

Real behavior

RS

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Ecological Modeling and Decision Support Systems Lotka-Volterra - Qualitative

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Lotka-Volterra Predator-Prey Model – A Qualitative Analysis

 dN/dt = (r – a*P)*N  dP/dt = (f*a*N – q)*P Time P N r a N P P N q f*a P N

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

 dN/dt = (r – a*P)*N  dP/dt = (f*a*N – q)*P

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Qualitative Lotka-Volterra Predator-Prey Model

Transformation:

• N‘ = N – q/(f*a)
• P‘ = P – r/a

 dN’/dt = -a*P’*(N’-q/(f*a)) dP’/dt = f*a*N’*(P’-r/a) Qualitative Abstraction:

• [x] := sign (x)
• x := [dx/dt]

 N’  [P’]  [N’-q/(f*a)] = 0 P’ = [N’]  [P’-r/a]

• N, P > 0
• N’  [P’] = 0
• P’ = [N’]

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Qualitative Lotka-Volterra - Relational Model

RLVPP  DOM(P’, N’, N’, P’) :

• {(-,-), (0,0), (+,+) } X { (-,+), (0,0), (+,-)}
• Constraint Satisfaction ( Ch. 3.4)
• N’  [P’] = 0
• P’ = [N’]

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Qualitative Lotka-Volterra – Qualitative States

[P’] = - [N’] = P’ = 0 N’ = + [P’] = - [N’] = P’ = + N’ = + [P’] = - [N’] = P’ = - N’ = + [P’] = 0 [N’] = P’ = 0 N’ = 0 [P’] = 0 [N’] = P’ = - N’ = 0 [P’] = + [N’] = P’ = + N’ = - [P’] = + [N’] = P’ = - N’ = - [P’] = + [N’] = P’ = 0 N’ = - [P’] = 0 [N’] = P’ = + N’ = 0

• N’  [P’] = 0
• P’ = [N’]

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Qualitative Lotka-Volterra – Transitions between States

• N’  [P’] = 0
• P’ = [N’]

[P’] = - [N’] = P’ = 0 N’ = + [P’] = - [N’] = P’ = + N’ = + [P’] = - [N’] = P’ = - N’ = + [P’] = 0 [N’] = P’ = 0 N’ = 0 [P’] = 0 [N’] = P’ = - N’ = 0 [P’] = + [N’] = P’ = + N’ = - [P’] = + [N’] = P’ = - N’ = - [P’] = + [N’] = P’ = 0 N’ = - [P’] = 0 [N’] = P’ = + N’ = 0

• Constraints on pairs of states
• Constraint Satisfaction ( Ch. 3.4)

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Qualitative Lotka-Volterra – Possible Terminal States

• N’  [P’] = 0
• P’ = [N’]

[P’] = - [N’] = P’ = 0 N’ = + [P’] = - [N’] = P’ = + N’ = + [P’] = - [N’] = P’ = - N’ = + [P’] = 0 [N’] = P’ = 0 N’ = 0 [P’] = 0 [N’] = P’ = - N’ = 0 [P’] = + [N’] = P’ = + N’ = - [P’] = + [N’] = P’ = - N’ = - [P’] = + [N’] = P’ = 0 N’ = - [P’] = 0 [N’] = P’ = + N’ = 0

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Qualitative Lotka-Volterra – Interpretation

[P’] = - [N’] = P’ = 0 N’ = + [P’] = - [N’] = P’ = + N’ = + [P’] = - [N’] = P’ = - N’ = + [P’] = 0 [N’] = P’ = 0 N’ = 0 [P’] = 0 [N’] = P’ = - N’ = 0 [P’] = + [N’] = P’ = + N’ = - [P’] = + [N’] = P’ = - N’ = - [P’] = + [N’] = P’ = 0 N’ = - [P’] = 0 [N’] = P’ = + N’ = 0

• Oscillatory behavior as one possibility
• Other possible behaviors (terminal states)

P’ N’

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Ecological Modeling and Decision Support Systems Different forms and limitations of qualitative modeling

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

Qualitative Modeling with Deviations

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Deviations Dx := xact - xref Model Fragments [DQ1]  [DQ2] = [0] Equations Q1 + Q2 = 0  D(x + y) = Dx + Dy  D(x - y) = Dx - Dy  D(x * y) = xact * Dy + yact * Dx - Dx * Dy  D(x / y) = (yact * Dx - xact * Dy) / (yact * ( yact * Dy))  y = f(x) monotonic  Dx = Dy  Reference can be unspecified!

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Spurious Solutions in Interval-based Qualitative Modeling y

• x+y = y+z  xy  yz
• x=(1,2), y=(0,1), z=(0,1)
• satisfies all constraints
• BUT
• contains no real-valued solution:
• x+y = y+z  x = z

+

x (0,1)

+

z



(0,1) (1,2) (1,3) (0,2)

Solutions of interval equations

• x1=i1, x2=i2 , …
• satisfies
• fl(x1, x2, …, xn )  fr(x1, x2, …, xn)
• iff
• fl(i1, i2, …, in)  fr(i1, i2, …, in)  

Model-Based Systems & Qualitative Reasoning Group of the Technical University of Munich

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Qualitative Models - Implementation

• Usually:
• Finite set of variables
• Finite set of qualitative values

 Propositional logic  Finite constraint satisfaction ( ch. 3.4!)

y

+

x (0,1)

+

z



(0,1) (1,2) (1,3) (0,2)

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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”  Abstraction of functional dependencies  Orders of magnitude  Approximation vs. abstraction  Domain abstraction (this section)