A Logical Characterization of Individual-Based Models James F . - PowerPoint PPT Presentation

A Logical Characterization of Individual-Based Models James F . Lynch Department of Computer Science Clarkson University June 26, 2008 James F. Lynch A Logical Characterization of Individual-Based Models

What Is an Individual-Based Model? (v. 1) An IBM consists of populations of individuals. It evolves via interactions among the individuals. James F. Lynch A Logical Characterization of Individual-Based Models

An Example Two species: Predator and Prey. Individuals move freely and rapidly in an enclosed space. James F. Lynch A Logical Characterization of Individual-Based Models

Interactions Death of Predator James F. Lynch A Logical Characterization of Individual-Based Models

Interactions Death of Predator ⇒ James F. Lynch A Logical Characterization of Individual-Based Models

Interactions Death of Predator ⇒ Birth of Prey James F. Lynch A Logical Characterization of Individual-Based Models

Interactions Death of Predator ⇒ Birth of Prey ⇒ James F. Lynch A Logical Characterization of Individual-Based Models

Interactions Death of Predator ⇒ Birth of Prey ⇒ Predation James F. Lynch A Logical Characterization of Individual-Based Models

Interactions Death of Predator ⇒ Birth of Prey ⇒ Predation ⇒ James F. Lynch A Logical Characterization of Individual-Based Models

Typical Behavior James F. Lynch A Logical Characterization of Individual-Based Models

Extinction Is Inevitable Extinction of Predator Extinction of Both James F. Lynch A Logical Characterization of Individual-Based Models

Continuous Approximation of Large Populations (A State-Variable Model) Lotka-Volterra Model dx dt = ax − bxy dy dt = − cy + bxy James F. Lynch A Logical Characterization of Individual-Based Models

Some Areas That Use Individual-Based Models Biology Molecular Biology Ecology Chemistry Computer Science Internet Graphs Economics Physics Statistical Mechanics Galaxy Formation James F. Lynch A Logical Characterization of Individual-Based Models

The Need for a Formal Approach Model complexity Many species of individuals Many types of interactions High cost of simulation and analysis Design issues Discrete vs. continuous Probabilistic vs. deterministic Individuals vs. aggregations Reasons for aggregation: More efficient More “realistic" model for large populations Rigorous justifications for using SVMs James F. Lynch A Logical Characterization of Individual-Based Models

Goals A unifying conceptual framework for IBMs Classification of IBMs Relationship between IBMs and SVMs Methodologies for dealing with complexity Determining appropriate level of abstraction Stepwise refinement Efficient algorithms for simulation and analysis James F. Lynch A Logical Characterization of Individual-Based Models

Our Results A formal language for IBMs Classification of IBMs, including SVMs Definition of abstraction Characterization of IBMs that can be abstracted to SVMs Examples of IBMs that can not be abstracted to SVMs James F. Lynch A Logical Characterization of Individual-Based Models

What Is an Individual-Based Model? (v. 2) An IBM is a dynamical system whose states are metafinite models metafinite models = finite models + weight functions + numeric functions + multiset operations State transitions are probabilistic rules defined on metafinite models James F. Lynch A Logical Characterization of Individual-Based Models

Metafinite Models Definition A weight function of arity k on a set A is a partial function 1 w : A k → R A numeric function of arity k is a function 2 f : R k → R A multiset over a set S is an unordered collection of 3 elements from S with repetitions allowed. Ex: { | 2 , 5 , 3 , 2 | } . A multiset operation on S is a function 4 Γ: { finite multisets over S } → R Example S = R , Γ( M ) = � r ∈ M r . James F. Lynch A Logical Characterization of Individual-Based Models

Metafinite Models Continued Definition A vocabulary is a triple ( W , F , G ) where 1 W is a set of weight function symbols F is a set of numeric function symbols G is a set of multiset operation symbols A metafinite model A over ( W , F , G ) is a structure 2 ( A , W A , F A , G A ) where A is the universe—a finite set (of individuals) W A is a set of interpretations on A of the weight function symbols in W F A is a set of interpretations of the numeric function symbols in F G A is a set of interpretations on R of the multiset operation symbols in G James F. Lynch A Logical Characterization of Individual-Based Models

A Logic of Metafinite Models The logic is a pure term calculus over the vocabulary. Two types of variables: Individual variables: values range over the universe A 1 Numeric variables: values range over R 2 Two kinds of atomic terms: Numeric variables. 1 w ( x 1 , . . . , x k ) where w is a k -ary weight function symbol 2 and x 1 , . . . , x k are free individual variables. James F. Lynch A Logical Characterization of Individual-Based Models

A Logic of Metafinite Models Continued Recursively, If f is a k -ary numeric function symbol 1 and τ 1 , . . . , τ k are terms, then f ( τ 1 , . . . , τ k ) is a term. If Γ is a multiset operation symbol 2 and τ is a term with free individual variables x 1 , . . . , x k , y , then (Γ y τ ) is a term with free variables x 1 , . . . , x k . For a 1 , . . . , a k ∈ A , (Γ y τ ) A ( a 1 , . . . , a k ) = Γ( { | τ ( a 1 , . . . , a k , b ) | b ∈ A | } ) James F. Lynch A Logical Characterization of Individual-Based Models

Example A weighted graph G = ( V , { w G 1 , w G 2 } , { + , × , − , / } , {| . . . | , � } ) where V = vertices for a ∈ V , w G 1 ( a ) = 1. � weight of edge ( a , b ) if it exists for a , b ∈ V , w G 2 ( a , b ) = undef otherwise | . . . | is the cardinality operator on multisets: |{ | 2 , 5 , 3 , 2 | }| = 4. Expressing the number of vertices: |{ | w 1 ( v ) | v ∈ V | }| Expressing the outdegree of vertex v : |{ | w 2 ( v , u ) | u ∈ V | }| The average outdegree of G : � { ||{ | w 2 ( v , u ) | u ∈ V | }| | v ∈ V | } / |{ | w 1 ( v ) | v ∈ V | }| James F. Lynch A Logical Characterization of Individual-Based Models

Transition Rules A = ( A , W A , F , G ) and A ′ = ( A ′ , W A ′ , F , G ) denote the states of the IBM before and after a transition. Probability of a transition from A to A ′ is defined by a term in the vocabulary ( { A , A ′ } ∪ W ∪ W ′ , F , G ) . Example Graph growth model with preferential attachment. Probability of transition A → A ′ is � ∈ A ∧ A ′ = A ∪ { v } ∧ outdeg ( v ) = 1 � v / v �� �� �� � E ′ ( v , u ) × indeg ( u ) × / u ∈ A × indeg ( u ) u u James F. Lynch A Logical Characterization of Individual-Based Models

What Is an Individual-Based Model? (v. 3) Definition An IBM over vocabulary ( W , F , G ) is a pair ( S , δ ) where S is a set of metafinite models over ( W , F , G ) δ is a term over ( { A , A ′ } ∪ W ∪ W ′ , F , G ) that defines a Markov process on S . James F. Lynch A Logical Characterization of Individual-Based Models

Abstractions Definition Let ( S , δ ) be an IBM over vocabulary ( W , F , G ) , ( S α , δ α ) be an IBM over vocabulary ( W α , F , G ) . ( S α , δ α ) is an abstraction of ( S , δ ) if For every w ∈ W α there is a term τ w in the logic of ( W , F , G ) of the same arity as w . There is a map α : S → S α such that for every A ∈ S , if A α = α ( A ) then A α ⊆ A for all a 1 , . . . , a i ∈ A α and w ∈ W α of arity i , w A α ( a 1 , . . . , a i ) = τ A w ( a 1 , . . . , a i ) James F. Lynch A Logical Characterization of Individual-Based Models

Predator-Prey IBM Abstracted to Lotka-Volterra SVM Example States in S are metafinite models of the form ( A , { P A , X A , Y A , Z A } , { + , × , − , / } , {| . . . | , � } ) where A is the set of all predators and prey P A ( a ) = 1 if a is a predator; 0 otherwise X A ( a ) = x -coordinate of a similarly for Y A and Z A States in S α are of the form ( ∅ , { w A α 0 () , w A α 1 () } , { + , × , − , / } , {| . . . | , � } ) where � w 0 () ≡ | A | − P ( a ) (number of prey) a ∈ A � w 1 () ≡ P ( a ) (number of predators) a ∈ A James F. Lynch A Logical Characterization of Individual-Based Models

Accuracy of Abstraction Definitions For any time t , let A t (resp. A α t ) be the state of ( S , δ ) (resp. ( S α , δ α ) ) at time t . Let [ t , t + ∆ t ] be a time interval. For A ∈ S and r ∈ R , let Q ( A , r ) = Pr ( τ A t +∆ t ≤ r | A t = A ) w and for A α ∈ S α , let Q α ( A α , r ) = Pr ( w A α t +∆ t ≤ r | A α t = A α ) (The conditional cumulative distribution function of τ A t +∆ t w (resp. w A α t +∆ t ).) James F. Lynch A Logical Characterization of Individual-Based Models