Stochastic Processes MATH5835, P. Del Moral UNSW, School of - PowerPoint PPT Presentation

Stochastic Processes MATH5835, P. Del Moral UNSW, School of Mathematics & Statistics Lectures Notes 1 Consultations (RC 5112): Wednesday 3.30 pm � 4.30 pm & Thursday 3.30 pm � 4.30 pm 1/25

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Stochastic processes ◮ Random dynamical system. ◮ Sequential simulation of random variables. 3/25

Stochastic processes ◮ Random dynamical system. ◮ Sequential simulation of random variables. ◮ Randomness: 3/25

Stochastic processes ◮ Random dynamical system. ◮ Sequential simulation of random variables. ◮ Randomness: occurring with undefinite aim/pattern/regularity, odd and unpredictable, unknown, unidentified, out of place,. . . ◮ Simulation: 3/25

Stochastic processes ◮ Random dynamical system. ◮ Sequential simulation of random variables. ◮ Randomness: occurring with undefinite aim/pattern/regularity, odd and unpredictable, unknown, unidentified, out of place,. . . ◮ Simulation: imitation, mimicking, feigning, pretending, duplicate/replica/clone, counterfeit, fake,. . . 3/25

Stochastic processes ◮ Random dynamical system. ◮ Sequential simulation of random variables. ◮ Randomness: occurring with undefinite aim/pattern/regularity, odd and unpredictable, unknown, unidentified, out of place,. . . ◮ Simulation: imitation, mimicking, feigning, pretending, duplicate/replica/clone, counterfeit, fake,. . . ⇓ The generation of random numbers is too important to be left to chance. Robert Coveyou [Studies in Applied Mathematics, III (1970), 70-111.] 3/25

Lost in the Great Sloan Wall The Tau Zero Foundation � interstellar travels in any dimension. Tony Gonzales random travelling plans in the universe lattices at superluminal speeds ◮ dimension 1 and 2 from Reykjavik: 4/25

Lost in the Great Sloan Wall The Tau Zero Foundation � interstellar travels in any dimension. Tony Gonzales random travelling plans in the universe lattices at superluminal speeds ◮ dimension 1 and 2 from Reykjavik: Main drawbacks: infinite returns back home.... 4/25

Lost in the Great Sloan Wall ◮ Free trip travel voucher to the 3d- Great Sloan Wall : 5/25

Lost in the Great Sloan Wall ◮ Free trip travel voucher to the 3d- Great Sloan Wall : Main advantage: finite mean returns back home. 5/25

Lost in the Great Sloan Wall ◮ Free trip travel voucher to the 3d- Great Sloan Wall : Main advantage: finite mean returns back home. Main drawbacks: Wanders off in the infinite universe and never returns back home! 5/25

Lost in the Great Sloan Wall - Why?? ◮ Was it predictable? 6/25

Lost in the Great Sloan Wall - Why?? ◮ Was it predictable? ◮ What is the stochastic model? 6/25

Lost in the Great Sloan Wall - Why?? ◮ Was it predictable? ◮ What is the stochastic model? ◮ How to simulate it? 6/25

Lost in the Great Sloan Wall - Why?? ◮ Was it predictable? ◮ What is the stochastic model? ◮ How to simulate it? ◮ Is there some math. formulae? 6/25

Lost in the Great Sloan Wall - Why?? ◮ Was it predictable? ◮ What is the stochastic model? ◮ How to simulate it? ◮ Is there some math. formulae? 6/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model = 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? � blackboard. 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? � blackboard. ◮ simulation ? � 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? � blackboard. ◮ simulation ? � flip coins! 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? � blackboard. ◮ simulation ? � flip coins! ◮ Math analysis ? � 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? � blackboard. ◮ simulation ? � flip coins! ◮ Math analysis ? � intuition/blackboard. 7/25

Lost in the Great Sloan Wall - The stochastic model Stoch. model =Simple random walks on Z d : d = 1 ⇒ X n = X n − 1 + U n U n = +1 or − 1 proba 1 / 2 with ◮ dimension 2, 3, and any d ? � blackboard. ◮ simulation ? � flip coins! ◮ Math analysis ? � intuition/blackboard. More rigorous analysis for d = 2 , 3 = Project N o 1 ⊕ 7/25

Meeting Alice in Wonderland Alice + White rabbit ∈ Polygonal labyrinth (no communicating edges!): 1 17 2 16 3 15 4 14 5 13 6 12 7 11 8 10 9 ◮ Chances to meet after 5 moves : > 25% ◮ Chances to meet after 12 moves : > 51% ◮ Chances to meet after 40 moves : > 99% 8/25

Meeting Alice in Wonderland - Why?? ◮ Was it predictable? 9/25

Meeting Alice in Wonderland - Why?? ◮ Was it predictable? ◮ What is the stochastic model? 9/25

Meeting Alice in Wonderland - Why?? ◮ Was it predictable? ◮ What is the stochastic model? ◮ How to simulate it? 9/25

Meeting Alice in Wonderland - Why?? ◮ Was it predictable? ◮ What is the stochastic model? ◮ How to simulate it? ◮ Is there some math. formulae? 9/25

Meeting Alice in Wonderland - Why?? ◮ Was it predictable? ◮ What is the stochastic model? ◮ How to simulate it? ◮ Is there some math. formulae? 9/25

Meeting Alice in Wonderland- The stochastic model Stoch. model = 10/25

Meeting Alice in Wonderland- The stochastic model Stoch. model =Random walk on (finite & complete) graphs G := ( V , E ) = ( Vertices , Edges ) 10/25

Meeting Alice in Wonderland- The stochastic model Stoch. model =Random walk on (finite & complete) graphs G := ( V , E ) = ( Vertices , Edges ) Neighborhood systems: On the set of vertices x ∼ y = ⇒ ( x , y ) ∈ E = set of edges ⇓ Set of neighbors of x ∈ V := N ( x ) = { y ∈ V : y ∼ x } 10/25

Meeting Alice in Wonderland- The stochastic model Stoch. model =Random walk on (finite & complete) graphs G := ( V , E ) = ( Vertices , Edges ) Neighborhood systems: On the set of vertices x ∼ y = ⇒ ( x , y ) ∈ E = set of edges ⇓ Set of neighbors of x ∈ V := N ( x ) = { y ∈ V : y ∼ x } Stochastic model: X n uniform r.v. on the set of neighbors N ( X n − 1 ) � 1 P ( X n = y | X n − 1 = x ) = # N ( X n − 1 ) 1 N ( X n − 1 ) ( y ) 10/25

Meeting Alice in Wonderland - Simulation ⊕ Analysis ◮ Simulation ? � 11/25

Meeting Alice in Wonderland - Simulation ⊕ Analysis ◮ Simulation ? � flip coins! 11/25

Meeting Alice in Wonderland - Simulation ⊕ Analysis ◮ Simulation ? � flip coins! ◮ Math analysis ? � 11/25

Meeting Alice in Wonderland - Simulation ⊕ Analysis ◮ Simulation ? � flip coins! ◮ Math analysis ? � blackboard. 11/25

Meeting Alice in Wonderland - Simulation ⊕ Analysis ◮ Simulation ? � flip coins! ◮ Math analysis ? � blackboard. Other application domains? 11/25

Meeting Alice in Wonderland - Simulation ⊕ Analysis ◮ Simulation ? � flip coins! ◮ Math analysis ? � blackboard. Other application domains? Ex.: web-graph analysis (ranking ⇒ recommendations) 11/25

The MIT Blackjack team Mr M. card shuffle tracking and P. Diaconis magic number ≤ 5 shuffles = ⇒ possible predictions ≥ 90% , ⇒ possible predictions ≥ 40% ! 6 shuffles = Some questions? 12/25

The MIT Blackjack team Mr M. card shuffle tracking and P. Diaconis magic number ≤ 5 shuffles = ⇒ possible predictions ≥ 90% , ⇒ possible predictions ≥ 40% ! 6 shuffles = Some questions? ◮ Perfect shuffling ? 12/25

The MIT Blackjack team Mr M. card shuffle tracking and P. Diaconis magic number ≤ 5 shuffles = ⇒ possible predictions ≥ 90% , ⇒ possible predictions ≥ 40% ! 6 shuffles = Some questions? ◮ Perfect shuffling ? ⇔ fully unpredictable cards. 12/25

The MIT Blackjack team Mr M. card shuffle tracking and P. Diaconis magic number ≤ 5 shuffles = ⇒ possible predictions ≥ 90% , ⇒ possible predictions ≥ 40% ! 6 shuffles = Some questions? ◮ Perfect shuffling ? ⇔ fully unpredictable cards. ◮ A shuffle ? 12/25

The MIT Blackjack team Mr M. card shuffle tracking and P. Diaconis magic number ≤ 5 shuffles = ⇒ possible predictions ≥ 90% , ⇒ possible predictions ≥ 40% ! 6 shuffles = Some questions? ◮ Perfect shuffling ? ⇔ fully unpredictable cards. ◮ A shuffle ? ⇔ A permutation of 52 or 78 cards. 12/25