advertisement

Mathematics for Computer Science Applications of Random Walk MIT 6.042J/18.062J •Physics — Brownian motion •Finance — stocks, options •Algorithms — web search, clustering Random Walks Albert R Meyer, Lec 14W.1 May 12, 2010 Graph With Probable Transitions 1/4 1/4 1/2 G 1/4 Lec 14W.4 May 12, 2010 Distribution Over Nodes 1/4 1/4 1/4 1/4 1/4 1/2 1 1/2 1/2 1/2 1 1/2 2/3 1/4 1/4 1/4 1 00 What are p’B, p’O, p’G after 1 step? 1 Albert R Meyer, 1/4 1/3 0 (pB, pO, pG) Suppose you start at B: (1, 0, 0) 1/2 1/2 1/4 1/4 1/4 1/2 2/3 B Lec 14W.2 May 12, 2010 Distribution Over Nodes Outgoing-edge probabilities sum to 1 1/3 O Albert R Meyer, Albert R Meyer, Distribution Over Nodes 1/3 0 1/3 1/4 2/3 1/4 1/4 1/4 1 Lec 14W.6 May 12, 2010 00 1/2 1/2 1/4 1/4 1 2/3 1/4 1 1 1 , , 2 4 4 Dist after 1 step: (p’B, p’G, p’O) Dist after 1 step: only get places from B, so Dist after 2 steps: (p’’B, p’’O, p’’G) Albert R Meyer, May 12, 2010 1 1 1 , , 2 4 4 Lec 14W.7 Albert R Meyer, May 12, 2010 1 Distribution Over Nodes Distribution Over Nodes 1/3 1/3 1/4 1/2 1/2 1/4 1/4 1 1/4 2/3 1/4 1 1 1 , , 2 4 4 Dist after 1 step: p’’O = Pr{B to O|at B}•p’B + Pr{O to O|at O}•p’o + Pr{G to O|at G}•p’G Albert R Meyer, 1/2 1/2 1/2 1/4 Albert R Meyer, 1/4 2/3 7/24 1/2 1 1 5 7 , , 2 24 24 1/2 1/4 1/4 2/3 pG 1/2 1 May 12, 2010 1/4 1/3 2/3 ptG 1 May 12, 2010 Finding Stationary Dist. 1/3 distribution (pB, pO, pG) is stationary if next-step distribution is the same. What is a stationary dist. here? Albert R Meyer, ptB ptO Albert R Meyer, Stationary Distribution pB May 12, 2010 …and as t? May 12, 2010 pO = 5/24 distribution after t steps? (p’’B, p’’O, p’’G) 1/4 1 1 1 , , 2 4 4 Distribution Over Nodes distribution after 2 steps: Albert R Meyer, 1/4 1/4 p’’O = Pr{B to O|at B}•p’1/2 B 1/3 1/2 1 Dist after 1 step: May 12, 2010 5/24 1/4 2/3 1/4 + Pr{O to 1/3 O|at O}•p’ o 1/4 + Pr{G to0O|at G}•p’ G Distribution Over Nodes 1/4 1/4 Lec 14W.15 pB pO 1/4 1/3 2/3 pG 1 pB = pB’ = (1/2)pB + 1pG pO = pO’ = (1/4)pB + (1/3)pO pG = pG’ = (1/4)pB + (2/3)pO pB + pO + pG = 1 Albert R Meyer, May 12, 2010 Lec 14W.16 2 Finding Stationary Dist. pO 1/4 1/2 1/4 pB Google Page Rank 1/3 View the entire web as a graph • vertices are webpages • edge (u,v) exists if link from page u to page v • Pr{go to v from u} = 1/outdeg(u) Find stationary distribution {pu} Rank u above v if pu > pv. 2/3 pG 8 3 4 , , 15 15 15 solving for (pB, pO, pG): Albert R Meyer, May 12, 2010 Lec 14W.17 Questions on Stationary Dist Albert R Meyer, May 12, 2010 Further Questions 1/3 O • Does a stationary dist exist?(if graphYes finite) • Is it unique? Sometimes • Does a random walk Sometimes approach it from any starting distribution? – How quickly? Varies Albert R Meyer, May 12, 2010 Lec 14W.19 Lec 14W.18 1/4 B 1/2 2/3 1/4 G 1 • Pr{ever reach O | start at B} • Pr{reach G before O | start at B} • Average # steps for B to O Albert R Meyer, May 12, 2010 Lec 14W.20 Gambler’s Ruin • Decide to place $1 bets until either going broke or reaching some target amount of money. • What is Pr{reach target}? Gambler’s Ruin (let’s go to Vegas) May 12, 2010 Lec 14W.21 Albert R Meyer, May 12, 2010 Lec 14W.22 3 Gambling: Fair Case Gambling: Fair Case Suppose we’re playing a fair game: • Pr{win bet} = 1/2. What is Pr{reach $200} if we start with $100? 1/2 In general, if we start with $n Pr{reach $T} = n/T What about an unfair game? What about Pr{reach $600} if we start with $500? 5/6 Albert R Meyer, May 12, 2010 Lec 14W.23 Albert R Meyer, Gambling: Slightly Unfair May 12, 2010 Lec 14W.24 US Roulette Betting black in US roulette What is Pr{reach $500+100} starting with $500? (5/6 when fair) < 1 / 37,000 What is Pr{reach $1,000,100} starting with $1,000,000? ( 1 when fair) Image by MIT OpenCourseWare. < 1 / 37,000 no matter how many $ at start! Pr{win bet} = 18/38 = 9/19 < 1/2 Albert R Meyer, May 12, 2010 Lec 14W.25 Albert R Meyer, Gambler’s Ruin T May 12, 2010 Lec 14W.26 Gambler’s Ruin • Play $1 bets until going broke or make enough money. Parameters • p ::= Pr{win $1 bet} • n ::= initial capital • T ::= gambler’s target Question: What is Pr{reach target}? Albert R Meyer, May 12, 2010 Lec 14W.27 ”target" $$$ n "initial capital" # of bets Albert R Meyer, May 12, 2010 Lec 14W.28 4 Dow Jones Trend Gambler’s Ruin View as random walk on a line. $0 $1 … n-1 … n+1 n T p • p ::=Pr{win a bet} k • q ::= 1-p = Pr{lose a bet} random steps with “up” bias? T-1 k+1 k k-1 q What is Pr{reach T before 0}? Albert R Meyer, Lec 14W.29 May 12, 2010 Albert R Meyer, Lec 14W.30 May 12, 2010 © Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse General Approach p … n-1 n n+1 General Approach p … … q + Pr{target + wn-1 | n-1} • Pr{lose • q bet} May 12, 2010 n+1 … we have a linear recurrence! Albert R Meyer, Lec 14W.32 May 12, 2010 Linear Recurrence wn = rn 1 w1 r 1 for r ::= q/p 1. wn+1 = (1/p)wn - (q/p)wn-1 w0 = 0 (Gambler is broke) wT = 1 (Gambler is at target) Twist: we don’t know w1 But wT = 1, so can solve for w1. Then Solve using generating functions and get: May 12, 2010 wn = pwn+1 + qwn-1, so wn+1 = (1/p)wn - (q/p)wn-1 Lec 14W.31 Linear Recurrence Albert R Meyer, n q wn ::= Pr{hit target | start at n} == Pr{target wn+1 | n+1} • Pr{win p bet} Albert R Meyer, n-1 wn = Lec 14W.34 Albert R Meyer, rn 1 rT 1 May 12, 2010 Lec 14W.35 5 Winning when Biased Against wn = r 1 n < r 1 T r r Profit $100 in US Roulette n T p = 18/38 intended profit Tn 1 = r q = 20/38 1/r = 9/10 Pr{Profit $100} < (9/10)100 < 1/37,648 Suppose p < q, so r ::= q/p > 1. Albert R Meyer, Lec 14W.36 May 12, 2010 Profit $200 in US Roulette p = 18/38 q = 20/38 1/r = 9/10 Pr{Profit $200} < (9/10)200 < 1/70,000,000 Albert R Meyer, What About the Fair Case? wn = rn 1 rT 1 May 12, 2010 Lec 14W.39 (r ::= q/p = 1) •Uh oh, dividing by 0. •Use l’Hôpital’s Rule limr1 Albert R Meyer, Lec 14W.38 May 12, 2010 d(rn-1)/dr nrn-1 = = d(rT-1)/dr TrT-1 Albert R Meyer, May 12, 2010 n T Lec 14W.40 Team Problems Problems 1&2 Albert R Meyer, May 12, 2010 Lec 14W.41 6 MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.