emds 3 3 2 qualitative modeling printmqm.in.tum.de/teaching/emds/ws1112/slides/emds_3_3_2...

Model-Based Systems & Qualitative ReasoningGroup of the Technical University of Munich WS 11/12 EMDS 3 93

1 The topic2 Decision support systems3 Modeling

3.3 Advanced Modeling3.3.2 Qualitative Modeling

Outline

Model-Based Systems & Qualitative ReasoningGroup of the Technical University of Munich WS 11/12 EMDS 3 94 94

Ecological Modeling and Decision Support Systems

Motivation


The Algal Bloom – A „Numerical Model“

Pd

P eH

I

II I

Pd

P eH

I

II I

T

s

s

T

s

s

× × ×

× × ×

24

11

24

1

20

20 00

20

20 00

max,

( )

max,

( )

(ln( ) ),

,

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: …

Numerical model: only an approximation Extinction of light:

– Not linear– Not a function

Daylight:– Not a fraction (dawn and dusk)– Varying (clouds)

Temperature dependence: …


Intraspecific Competition

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

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 ..

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


Qualitative Models - Motivation

Models capturing partial knowledge and information Models capturing partial knowledge and information

Why? What do we know? What can be observed? What needs to be distinguished?

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

XX ?


Qualitative Modeling

• Modeling systems with partial knowledge/information:• Only rough understanding• imprecise, or missing data• Qualitative results required• Treating classes of systems and conditions

• 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

Tasks• Calculi for qualitative domains• Formal analysis of relationships among models of

different granularity

Expected benefit:• Finite representation• Efficiency• Intuitive representation

Expected benefit:• Finite representation• Efficiency• Intuitive representation



Interval-based Qualitative Modeling


A (Very General) Representation of Behavior Models

For instance, intreaspecific competitiondN/dt = N*r = N*r0*[1 – (N/K)]r = 1/N* dN/dt = r0*[1 – (N/K)]

For instance, intreaspecific competitiondN/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

• 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


Representation of Qualitative Behavior Models

For instance, intreaspecific competitiondN/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) Rqr,N : i.e. consistent(r, N) = (1, 100) Rqr,N : consistent!(r, N) = (-1/2, *) Rqr,N : not consistent


(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) Rqr,N : i.e. consistent(r, N) = (1, 100) Rqr,N : consistent!(r, N) = (-1/2, *) Rqr,N : not consistent


Extended Qualitative Model



N

1/N* dN/dt

r0

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)


(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)


Refined Qualitative Model



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): {[r, r0 ] [0, K ] [dr , 0] } {[0, r ] [K, K] [dr ,0] } (r , r) (K , K) (- , 0)

Rq’r,N,dr =

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

• “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): {[r, r0 ] [0, K ] [dr , 0] } {[0, r ] [K, K] [dr ,0] } (r , r) (K , K) (- , 0)

Rq’r,N,dr =

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

r

r

K K

Still not perfect Why?

Still not perfect Why?


Generalization: Relational Behavior Models

• Representational space: (v, DOM(v))• v: Vector of local variables and

parameters• local

w.r.t Model fragment or aggregat• Dom(v): Domain of v

• Behavior description: Relation• R DOM(v)

• Composition: join of relations

• Representational space: (v, DOM(v))• v: Vector of local variables and

parameters• local

w.r.t Model fragment or aggregat• Dom(v): Domain of v

• Behavior description: Relation• R DOM(v)

• Composition: join of relations


Valid Behavior Models

• 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• sSIT Val(vS , vS,0, s) vS,0 RS

Real behavior

RS


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

of behaviors• e.g. between “landmarks”: intervals


Domain Abstraction - Formally

0 ... Ncrit ... K

small crit normal

General:• i: DOM0(vi) DOM1(vi)

Aggregation of values:• i: DOM0(vi) DOM1(vi) P(DOM0(vi))• P(X): power set of X

(Generalized) Intervals:• i: IR DOM1(vi) I(IR)

Real landmarks and intervals between them:• L IR• i: IR DOM1(vi) IL(IR)


Model Abstraction Induced by Domain Abstraction

• Domain abstraction • : DOM0(vS) DOM1(vS)• induces model abstraction RS DOM(vS) (RS) DOM1(vS)

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

Real behavior(RS)


Arithmetic on Signs

0 0 0

0 0 0 0 0 0 0


• Addition of intervals(1, 1) (2, 2) = (1+ 2, 1+ 2)

• Subtraction(1, 1) (2, 2) = (1 - 2, 1 - 2)

• Multiplication(1, 1) (2, 2) = ( min(1* 2 , 1* 2, 2 * 1 , 2 * 1),

max (1* 2 , 1* 2, 2 * 1 , 2 * 1))

Interval Arithmetic

0 ... Ncrit ... K

small crit normal

• Division(1, 1) (2, 2) = ( min(1/ 2 , 1/ 2, 1 / 2, 1 / 2),

max (1/ 2 , 1/ 2, 1 / 2, 1 / 2)) • for 0(2, 2) !• Because … ?


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)



Lotka-Volterra - Qualitative


Lotka-Volterra Predator-Prey Model – A Qualitative Analysis

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


Time

P

N

Isoclines P = r/a N = q/(f*a)

Isoclines P = r/a N = q/(f*a)

P

N

ra

N

P P

Nqf*a

Model-Based Systems & Qualitative ReasoningGroup of the Technical University of Munich


WS 11/12 EMDS 3 114

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’]


Qualitative Lotka-Volterra - Relational Model

RLVPP DOM( P’, N’, N’, P’) :• {(-,-), (0,0), (+,+) } X { (-,+), (0,0), (+,-)}• Constraint Satisfaction ( Ch. 2.4)

• N’ [P’] = 0• P’ = [N’]


Qualitative Lotka-Volterra – Qualitative States

[P’] = -[N’] = P’ = 0N’ = +

[P’] = -[N’] = P’ = +N’ = +

[P’] = -[N’] = P’ = -N’ = +

[P’] = 0[N’] = P’ = 0N’ = 0

[P’] = 0[N’] = P’ = -N’ = 0

[P’] = +[N’] = P’ = + N’ = -

[P’] = +[N’] = P’ = -N’ = -

[P’] = +[N’] = P’ = 0N’ = -

[P’] = 0[N’] = P’ = +N’ = 0

• N’ [P’] = 0• P’ = [N’]


Qualitative Lotka-Volterra – Transitions between States

• N’ [P’] = 0• P’ = [N’]

[P’] = -[N’] = P’ = 0N’ = +

[P’] = -[N’] = P’ = +N’ = +

[P’] = -[N’] = P’ = -N’ = +

[P’] = 0[N’] = P’ = 0N’ = 0

[P’] = 0[N’] = P’ = -N’ = 0

[P’] = +[N’] = P’ = + N’ = -

[P’] = +[N’] = P’ = -N’ = -

[P’] = +[N’] = P’ = 0N’ = -

[P’] = 0[N’] = P’ = +N’ = 0

• Constraints on pairs of states• Constraint Satisfaction ( Ch. 2.4)


Qualitative Lotka-Volterra – Possible Terminal States

• N’ [P’] = 0• P’ = [N’]

[P’] = -[N’] = P’ = 0N’ = +

[P’] = -[N’] = P’ = +N’ = +

[P’] = -[N’] = P’ = -N’ = +

[P’] = 0[N’] = P’ = 0N’ = 0

[P’] = 0[N’] = P’ = -N’ = 0

[P’] = +[N’] = P’ = + N’ = -

[P’] = +[N’] = P’ = -N’ = -

[P’] = +[N’] = P’ = 0N’ = -

[P’] = 0[N’] = P’ = +N’ = 0


Qualitative Lotka-Volterra – InterpretationP’

N’

[P’] = -[N’] = P’ = 0N’ = +

[P’] = -[N’] = P’ = +N’ = +

[P’] = -[N’] = P’ = -N’ = +

[P’] = 0[N’] = P’ = 0N’ = 0

[P’] = 0[N’] = P’ = -N’ = 0

[P’] = +[N’] = P’ = + N’ = -

[P’] = +[N’] = P’ = -N’ = -

[P’] = +[N’] = P’ = 0N’ = -

[P’] = 0[N’] = P’ = +N’ = 0

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



Different forms and limitations of qualitative modeling


Qualitative Modeling with Deviations

Deviationsx := xact - xref Model Fragments

[Q1] [Q2] = [0]EquationsQ1 + Q2 = 0

x + y) = x + y x - y) = x - y

x * y) = xact * y + yact * x - x * y x / y) = (yact * x - xact * y) / (yact * ( yact * y)) y = f(x) monotonic x = y Reference can be unspecified!


Spurious Solutions in Interval-based Qualitative Modeling

y

• x+y = y+z xy yz• 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)


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)


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)

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