Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Different Monotonicity Definitions in stochastic modelling Im` ene KADI Nihal PEKERGIN Jean-Marc VINCENT ASMTA 2009 Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Plan Introduction 1 Models ?? 2 Stochastic monotonicity 3 Realizable monotonicity 4 Relations between monotonicity concepts 5 Realizable monotonicity and Partial Orders 6 Conclusion 7 Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Introduction Concept of monotonicity Lower and Upper bounding Coupling of trajectories ( perfect Sampling) − → Reduce the complexity. Different notions of monotonicity Order on trajectories( Event monotonicity). Order on distribution (Stochastic monotonicity). Monotonicity concepts depends on the relation order considerd on the state space Partial order and total order Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Main results Relations between monotonicity concepts in Total and Partial Orders Event System Transition Matrix Strassen Total Realizable monotonicity Stochastic Monotonicty order Proof(valuetools2007) Proof Partial Realizable monotonicity Stochastic Monotonicty order Counter Example Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Markovian Discrete Event Systems(MDES) MDES are dynamic systems evolving asynchronously and interacting at irregular instants called event epochs . They are defined by: a state space X a set of events E a set of probability measures P transition function Φ P ( e ) ∈ P denotes the occurrence probability Event An event e is an application defined on X , that associates to each state x ∈ X a new state y ∈ X . Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Markovian Discrete Event Systems(MDES) Transition function with events Let X i be the state of the system at the i th event occurrence time. The transition function Φ : X × E → X , X n +1 = Φ( X n , e n +1 ) Φ must to obey to the following property to generate P : � p ij = P ( φ ( x i , E ) = x j ) = P ( E = e ) e | Φ( x i , e )= x j Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Discrete Time Markov Chains (DTMC) DTMC { X 0 , X 1 , ..., X n +1 , ... } : stochastic process observed at points { 0 , 1 , ..., n + 1 } . ◮ It constitutes a DTMC if: ∀ n ∈ N and ∀ x i ∈ X : P ( X n +1 = x n +1 | X n = x n , X n − 1 = x n − 1 , ..., X 0 = x 0 ) = P ( X n +1 = x n +1 | X n = x n ) . The one-step transition probability p ij are given in a non-negative, stochastic transition matrix P : p 00 p 01 p 02 0 1 . . . p 10 p 11 p 12 P = P (1) = [ p ij ] . . . B C p 20 p 21 p 22 B C . . . B C . . . @ ... A . . . . . . Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Discrete Time Markov Chains (DTMC) A probability transition matrix P , can be described by a transition function Transition function in a DTMC Φ : X × U → X , is a transition function for P where : U is a random variable taking values in an arbitrary probability space U , such that: ∀ x , y ∈ X : P (Φ( x , U ) = y ) = p xy X n +1 = Φ( X n , U n +1 ) Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Stochastic ordering Stochastic ordering Stochastic ordering Let T and V be two discrete random variables and Γ an increasing set defined on X � � T � st V ⇔ P ( T = x ) ≤ P ( V = x ) , ∀ Γ x ∈ Γ x ∈ Γ Definition (Increasing set) Any subset Γ of X is called an increasing set if x � y and x ∈ Γ implies y ∈ Γ . Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Stochastic ordering Stochastic ordering Example Let ( X , � ) be a partial ordered state space, X = { a , b , c , d } . a � b � d , and a � c � d , Increasing sets:Γ 1 = { a,b,c,d } , Γ 2 = { b,c,d } , Γ 3 = { b,d } , Γ 4 = { c,d } , Γ 5 = { d } . V 1=(0.4,0.2,0.1,0.3) V 2=(0.2,0.1,0.3,0.4) V 1 � st V 2 On a : For Γ 1 = { a,b,c,d } : 0 . 4 + 0 . 2 + 0 . 1 + 0 . 3 ≤ 0 . 2 + 0 . 1 + 0 . 3 + 0 . 4 For Γ 2 = { b,c,d } : 0 . 2 + 0 . 1 + 0 . 3 ≤ 0 . 1 + 0 . 3 + 0 . 4 For Γ 3 = { b,d } : 0 . 2 + 0 . 3 ≤ 0 . 1 + 0 . 4 For Γ 4 = { c,d } : 0 . 1 + 0 . 3 ≤ 0 . 3 + 0 . 4 For Γ 5 = { d } :0 . 3 ≤ 0 . 4 Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Stochastic ordering Stochastic monotonicity Stochastic monotonicity P a transition probability matrix of a time-homogeneous Markov chain { X n , n ≥ 0 } taking values in X endowed with relation order � . { X n , n ≥ 0 } is st-monotone if and only if, ∀ ( x , y ) | x � y and ∀ increasing set Γ ∈ X � � p xz ≤ p yz z ∈ Γ z ∈ Γ Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Realizable monotonicity Realizable monotonicity P a stochastic matrix defined on X . P is realizable monotone, if there exists a transition function , such that Φ preserves the order relation. ∀ u ∈ U : if x � y then Φ( x , u ) � Φ( y , u ) Event monotonicity The model is event monotone, if the transition function by events preserves the order ie. ∀ e ∈ E ∀ ( x , y ) ∈ X x � y = ⇒ Φ( x , e ) � Φ( y , e ) A system is realizable monotone means that there exists a finite set of events E for which the system is event monotone Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Realizable monotonicity and perfect sampling Monotonicity and perfect sampling Principe Produce exact sampling of stationary distribution (Π) of a DTMC. One trajectory per state. The algorithm stops when all trajectories meet the same state coupling The evolution of the trajectories will be confused. If the model is event monotone Run only trajectories from minimal and maximal states. All other trajectories are always between these trajectories. If there is coupling at time t so all the other trajectories have also coalesced. ◮ The tool PSI 2 was developed to implement this method of simulation (JM.Vincent). Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Total order Relations between monotonicity concepts Total Order ( X , E ) : MDES P : Transition matrix Strassen P : Monotone ∃E : ( X , E ) Monotone Total order Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Total order Relations between monotonicity concepts Total Order ( X , E ) : MDES P : Transition matrix Strassen P : Monotone ∃E : ( X , E ) Monotone Total Valuetools2007 order ( X , E ) Monotone P ( E ) : Monotone Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Partial Order Relation between monotonicty concepts (Partial Order) Partial Order ( X , E ) : MDES P : Transition matrix Strassen P : Monotone ∃E : ( X , E ) Monotone Total Valuetools2007 order ( X , E ) Monotone P ( E ) : Monotone Proof P ( E ) : Monotone ( X , E ) Monotone Partial order Different Monotonicity Definitions in stochastic modelling
Introduction Models ?? St-Monotonicity Realizable monotonicity Relations between monotonicities Event monotonicity in � Conclusion Partial Order Relation between monotonicty concepts (Partial Order) The reciprocal is not true ( X , E ) : MDES P : Transition matrix Strassen P : Monotone ∃E : ( X , E ) Monotone Total Valuetools2007 order ( X , E ) Monotone P ( E ) : Monotone Proof P ( E ) : Monotone ( X , E ) Monotone Partial order ∃ ? E : ( X , E ) Monotone P : Monotone and P ( E ) = P Counter Example Different Monotonicity Definitions in stochastic modelling
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